Coupled-Cluster Theory - Theory...

Coupled-Cluster Theoryfor

Nuclear Structure

Vom Fachbereich Physikder Technischen Universität Darmstadt

zur Erlangung des Gradeseines Doktors der Naturwissenschaften

(Dr. rer. nat.)

genehmigte Dissertation vonM.Sc. Sven Binder

aus Mannheim

Darmstadt 2014D17

Referent: Prof. Dr. R. Roth

Korreferent: Prof. Dr. J. Wambach

Tag der Einreichung: 05.02.2014

Tag der Prüfung: 23.04.2014

Abstract

Nuclear Hamiltonians constructed within chiral effective field theory providean unprecedented opportunity to access nuclear phenomena based on low-energyquantum chromodynamics and, in combination with sophisticated many-bodymethods, allow for an ab initio description of nuclei without resorting to phe-nomenology.

This work focuses on the inclusion of chiral two-, and in particular three-bodyHamiltonians into many-body calculations, with emphasis on the formal and com-putational aspects related to the three-body interactions.

Through similarity renormalization group evolutions, the chiral Hamiltoniansare transformed into a form in which strong short-range correlations are tamed inorder to accelerate the convergence in the subsequent many-body calculations.

The many-body method mainly used is an angular-momentum coupled for-mulation of coupled-cluster theory with an iterative treatment of singly and dou-bly excited clusters, and two different approaches to non-iteratively include effectsof triply excited clusters. Excited nuclear states are accessed via the equation-of-motion coupled-cluster framework.

The extension of coupled-cluster theory to three-body Hamiltonians is con-sidered to verify the approximate treatment of three-nucleon interactions via thenormal-ordering two-body approximation as a highly efficient and accurate wayto include three-nucleon interactions into the many-body calculations, particu-larly for heavier nuclei.

Using a single chiral Hamiltonian whose low-energy constants are fitted tothree- and four-body systems, a qualitative reproduction of the experimental trendof nuclear binding energies, from 16O up to 132Sn, is achieved, which hints at thepredictive power of chiral Hamiltonians, even in the early state of developmentthey are at today.

Zusammenfassung

Nukleare Hamiltonoperatoren die aus chiraler effektiver Feldtheorie abgeleitetwerden bieten eine einzigartige Gelegenheit, nukleare Phänomene auf Grundlageniederenergetischer Quantenchromodynamik zu untersuchen. In Verbindung mitfortgeschrittenen Vielteilchenmethoden ermöglicht dies eine ab initio Beschrei-bung von Atomkernen ohne auf Phänomenologie zurückzugreifen.

Die vorliegende Arbeit beschäftigt sich mit der Inklusion chiraler Zwei-, undinsbesondere Dreinukleonen-Hamiltonoperatoren in Vielteilchenrechnungen, mitSchwerpunkt auf den formalen und rechnerischen Aspekten der Behandlung derDreinukleonenwechselwirkungen.

Durch Evolution mittels der Similarity Renormalization Group werden diechiralen Hamiltonoperatoren derart transformiert, dass die starken kurzreichwei-tigen Korrelationen gemildert werden um die Konvergenz in den anschließendenVielteilchenrechnungen zu beschleunigen.

Die hauptsächlich eingesetzte Vielteilchenmethode ist eine drehimpulsgekop-pelte Formulierung von Coupled-Cluster-Theorie mit einer iterativen Behandlungvon ein- und zweifach angeregten Clustern, sowie einer nicht-iterativen Berück-sichtigung dreifach angeregter Cluster. Angeregte Kernzustände werden über dieCoupled-Cluster Bewegungsgleichungsmethode bestimmt.

Es wird die Erweiterung von Coupled-Cluster-Theorie auf Dreiteilchen-Ha-miltonoperatoren betrachtet um die Behandlung von Dreinukleonen-Wechselwir-kungen in der Normalordnungsapproximation zu verifizieren als eine hochef-fiziente und akkurate Methode diese Wechselwirkungen näherungsweise in Viel-teilchenrechnungen einzubeziehen, insbesondere für schwere Kerne.

Ein einzelner Hamiltonoperator dessen Niederenergiekonstanten in Drei- undVierteilchensystemen bestimmt wurden genügt, um den experimentellen Trendnuklearer Bindungsenergien von 16O bis 132Sn qualitativ zu reproduzieren was,trotz ihres gegenwärtig frühen Entwicklungsstadiums, auf das Potential chiralerWechselwirkungen hinweist Vorhersagen zu ermöglichen.

Contents

1 Introduction 1

1.1 Ab Initio Nuclear Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Chiral Nuclear Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Similarity Renormalization Group . . . . . . . . . . . . . . . . . . . . . . . . . . 141.4 Hartree-Fock Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.5 Normal-Ordering Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 211.6 Configuration Interaction and No-Core Shell Model . . . . . . . . . . . . 25

1.6.1 Full Configuration Interaction . . . . . . . . . . . . . . . . . . . . . . . 261.6.2 Truncated Configuration Interaction . . . . . . . . . . . . . . . . . . . 281.6.3 No-Core Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.6.4 Importance-Truncated No-Core Shell Model . . . . . . . . . . . . 31

2 Coupled-Cluster Theory 33

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.2 The Exponential Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.3 Coupled-Cluster Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.4 Effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.5 The ΛCCSD Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.6 Expectation Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.7 The ΛCCSD(T) Energy Correction . . . . . . . . . . . . . . . . . . . . . . . . . . 552.8 The Completely-Renormalized Coupled-Cluster Method CR-CC(2,3) 572.9 Equation-of-Motion Coupled Cluster . . . . . . . . . . . . . . . . . . . . . . . . 62

2.9.1 Reduced Density Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Coupled-Cluster Theory for Nuclear Structure I

Contents

3 Coupled-Cluster Theory for Three-Body Hamiltonians 69

3.1 CCSD for Three-Body Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . 703.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.1.2 The CCSD Equations for Three-Body Hamiltonians . . . . . . . 71

3.2 Effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.3 The ΛCCSD Equations for Three-Body Hamiltonians . . . . . . . . . . . 873.4 The ΛCCSD(T) Energy Correction for Three-Body Hamiltonians . . . 91

4 Spherical Coupled-Cluster Theory 95

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.2 Spherical Tensor Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.3 Angular-Momentum Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.4 One-Body Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.5 Cross-Coupled Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.5.1 Scalar Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.5.2 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.6 Diagram Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.6.1 Antisymmetrized Diagram Coupling . . . . . . . . . . . . . . . . . . 1164.6.2 Cross-Coupled Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.7 Spherical CCSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.8 Convergence Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.9 Spherical CCSD for Three-Body Hamiltonians . . . . . . . . . . . . . . . . . 136

4.9.1 Three-Body Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . 1364.9.2 Conversion to Reduced Format . . . . . . . . . . . . . . . . . . . . . . . 1414.9.3 Spherical CCSD Equations for Three-Body Hamiltonians . . . 144

4.10 Spherical ΛCCSD(T) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1454.11 Spherical ΛCCSD(T) for Three-Body Hamiltonians . . . . . . . . . . . . . 1474.12 The CR-CC(2,3) Energy Correction . . . . . . . . . . . . . . . . . . . . . . . . . . 1484.13 Spherical EOM-CCSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1544.14 Spherical Reduced Density Matrices . . . . . . . . . . . . . . . . . . . . . . . . 156

5 Results 161

5.1 Comparison of the IT-NCSM with the Coupled-Cluster Method . . . 1645.2 CCSD with SRG-Transformed Chiral Two-Body Hamiltonians . . . . 1685.3 Reduced-Cutoff Chiral Three-Body Interaction . . . . . . . . . . . . . . . . 1725.4 Relevance of the E3max Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1785.5 The ΛCCSD(T) Energy Correction . . . . . . . . . . . . . . . . . . . . . . . . . . 1825.6 The CR-CC(2,3) Energy Correction . . . . . . . . . . . . . . . . . . . . . . . . . . 187

II Coupled-Cluster Theory for Nuclear Structure

Contents

5.7 CCSD with Explicit 3N Interactions . . . . . . . . . . . . . . . . . . . . . . . . . 1915.8 ΛCCSD(T) with Explicit 3N Interactions . . . . . . . . . . . . . . . . . . . . . 195

5.8.1 Benchmark of the NO2B Approximation . . . . . . . . . . . . . . . 1955.8.2 Approximation Schemes for the Amplitudes . . . . . . . . . . . . 199

5.9 Ab Initio Description of Heavy Nuclei . . . . . . . . . . . . . . . . . . . . . . . 2025.9.1 Self-Consistent Hartree-Fock Reference Normal-

Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2045.9.2 Role of the SRG Model Space . . . . . . . . . . . . . . . . . . . . . . . . 2065.9.3 Results for Heavy Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

6 Conclusion 221

A Excited Nuclear States 225

B Trapped Neutrons 235

C CCSD Diagrams and Spherical Expressions 247

C.1 Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248C.2 Spherical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250C.3 Diagrams for Three-Body Hamiltonians . . . . . . . . . . . . . . . . . . . . . 254C.4 Spherical Equations for Three-Body Hamiltonians . . . . . . . . . . . . . 257

D Effective Hamiltonian Diagrams and Spherical Expressions 269

D.1 Spherical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270D.2 Diagrams for Three-Body Hamiltonians . . . . . . . . . . . . . . . . . . . . . 274D.3 Spherical Equations for Three-Body Hamiltonians . . . . . . . . . . . . . 279

E ΛCCSD Diagrams and Spherical Expressions 291

E.1 Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292E.2 Spherical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293E.3 Spherical Equations for Three-Body Hamiltonians . . . . . . . . . . . . . 295

F Spherical Reduced Density Matrix 307

G ΛCCSD(T) Spherical Expressions 315

G.1 Spherical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316G.2 Spherical Equations for Three-Body Hamiltonians . . . . . . . . . . . . . 323

Coupled-Cluster Theory for Nuclear Structure III

Contents

H EOM-CCSD Diagrams and Spherical Expressions 327

H.1 Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328H.2 Spherical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329H.3 Spherical Equations (Scalar) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

I Publications 339

Bibliography 344

IV Coupled-Cluster Theory for Nuclear Structure

Chapter 1

Introduction

Chapter 1. Introduction

1.1 Ab Initio Nuclear Structure

Nuclear physics is surprisingly complex. Starting around the 1950s, early micro-scopic models of the atomic nucleus considered it as a compound of elementarybuilding blocks – the nucleons – where the large mean free path of these nucle-ons inside the nucleus suggested that they form some sort of weakly interactinggas. The nuclear interaction itself was quickly identified as being caused by me-son exchange which helped the understanding of two-nucleon properties. There-fore, this early picture of nuclear physics gave rise to the hope that the nuclearmany-body problem could be solved using meson-exchange interactions – suchas the one-boson exchange model [1], or the more recent CD-Bonn potential [2]– in many-nucleon systems, and that, because of the apparent weak interactionamong the nucleons, these many-body problems could eventually be solved us-ing perturbation-theory based methods.

The discovery of Quantum Chromodynamics (QCD) put the earlier insights intoperspective: Nucleons are not fundamental but are rather composed of quarksand gluons. However, in retrospect, color confinement at low energies justifies anapproximate treatment of nucleons as being fundamental. Of course, a funda-mental theory of the nuclear interaction necessarily has to be derived from QCD.However, the non-perturbative nature of QCD in the low-energy regime relevantfor nuclear physics has defied any attempts of a direct derivation so far. Thereare attempts to extract nuclear potentials from QCD calculations on the lattice,but these calculations do not yet operate at the physical quark masses. Today, themost promising path towards QCD-based nuclear interactions is via chiral effective

field theory [3–12], an effective theory in terms of low-energy degrees of freedomconstrained by the symmetries of the underlying theory. This leads to a set of con-

sistent QCD-based many-nucleon interactions which can be used in ab initio nu-clear structure calculations. However, the practical treatment of such many-bodyforces is far from trivial.

Once the Hamiltonian is known, the focus is on the solution of the many-body problem. Here, the early weakly-interacting-gas picture of the nucleus turnsout to be too simplistic. Instead, the nuclear many-body problem has a complexstructure, particularly due to strong short-range correlations induced by the nu-clear interaction (see [13] for a discussion based on the Argonne V18 [14] poten-tial). In practice, in order to obtain realistic solutions for the many-body problem,vast computational resources are required which have simply not been available

2 Coupled-Cluster Theory for Nuclear Structure

1.1. Ab Initio Nuclear Structure

in the past. The former sentiment towards the many-body problem is capturedin a quote by Igal Talmi from 1993 about the task of solving the non-relativisticSchrödinger equation for a many-body system with strong interactions [15]:

"Such a problem cannot be treated exactly by many-body theory.

Not even useful approximation procedures have been developed."

Nowadays, considering the impressive advances that many-body theory has made,this statement seems too pessimistic. By applying renormalization-group tech-niques to the initial nuclear Hamiltonian, for instance in the framework of theSimilarity Renormalization Group (SRG) [16, 17], the troubling short-range correla-tions can be weakened which then eases the burden on the many-body method.But also nuclear many-body methods themselves have seen much progress in re-cent years. For instance, using SRG-transformed interactions, the No-Core Shell

Model (NCSM) [18, 19] and its Importance-Truncated extension (IT-NCSM) [20, 21]provide quasi-exact solutions of the Schrödinger equation for nuclei in the p -shelland even beyond. For medium-mass and heavy closed-shell nuclei the Coupled-

Cluster Method [22–27] has been established as one of the most powerful approx-imate schemes and the recently introduced In-Medium Similarity Renormalization

Group [28–30] approach has also been successfully applied in this mass regioneven for open-shell nuclei.

In this work, non-relativistic configuration-space based ab initio approaches tothe nuclear many-body problem are considered in which all nucleonic degrees offreedom r i , ms i

, m t i, . . . , r A , msA

, m tA are taken into account explicitly. All infor-

mation about the system is therefore contained in the A-body state |Ψ⟩ that livesin the Hilbert space V ,

V =

A∧

i=1

L 2

R3⊗¦

± 1

2

©

⊗¦

± 1

2

©

. (1.1)

Since the nucleus is a purely Fermionic system, the antisymmetric product∧

en-forces the Pauli exclusion principle on the many-body state. According to thepostulates of quantum mechanics, the stationary nuclear state |Ψ⟩ is solution ofthe nonrelativistic stationary Schrödinger equation

H |Ψ⟩ = E |Ψ⟩ , (1.2)

i.e., it is an eigenfunction of the nuclear Hamiltonian operator H ,

H =1

A

A∑

i<j

(p i − p j )2

2m+

A∑

i<j

V NNi j+

A∑

i<j<k

V 3Ni j k+λCM HCM . (1.3)

Coupled-Cluster Theory for Nuclear Structure 3


In (1.3), for convenience a scalable (via the parameterλCM) center-of-mass harmonic-oscillator potential

HCM =1

2 A mP

2

CM+

1

2(A m ΩCM)

2 R CM−3

2ħhΩCM (1.4)

is added to suppress center-of-mass motion. Although the nuclear Hamiltonianderived from chiral effective field theory formally contains up to A-nucleon in-teractions, present-day nuclear applications consider only Hamiltonians of theform (1.3), where only up to three-nucleon interactions are included. One strategyfor the numerical solution of the partial differential equation H |Ψ⟩= E |Ψ⟩ is to putit into its weak form [31], in which |Ψ⟩ has to satisfy the equation

⟨Φ| (H − E ) |Ψ⟩ = 0 , ⟨Ψ|Ψ⟩ = 1 , ∀ |Φ⟩ ∈ V . (1.5)

The weak formulation is a common starting point for the application of Galerkinmethods for discretizing the original continuous operator eigenvalue problem.Once the problem is discretized, approximate solutions (ES ,ΨS) may be obtainedfrom finite-dimensional subspaces VS ⊂ V . For instance, the Rayleigh-Ritz pro-cedure for the ground-state energy and wavefunction – the foundation of manynuclear many-body methods such as Hartree-Fock or configuration interaction –may straightforwardly be applied to the finite-dimensional case,

ES = min|Ψ⟩∈VS , |Ψ⟩6=0

⟨Ψ|H |Ψ⟩⟨Ψ|Ψ⟩ , |ΨS⟩ = argmin

|Ψ⟩∈VS , |Ψ⟩6=0

⟨Ψ|H |Ψ⟩⟨Ψ|Ψ⟩ . (1.6)

In this work, the finite-dimensional model spacesVS are always spanned by a finiteset of A-body Slater determinant basis functions |Φµ,

VS = spann

|Φµ⟩ :µ= 1, . . . , dimVS

o

, (1.7)

with

|Φµ⟩ = A |φµ1. . .φµA

) (1.8)

=1p

A !

∑

π∈S(N )

sgn(π) |φπ(µ1) . . .φπ(µA )) , (1.9)

where |φµ1. . .φµA

) denotes the tensor-product state constructed from the single-particle states |φµ1

⟩, . . . , |φµA⟩,

|φµ1. . .φµA

) = |φµ1⟩⊗ · · ·⊗ |φµA

⟩ , (1.10)

which is not subjected to antisymmetrization. A common choice for the single-particle wavefunctions in nuclear structure calculations from which the Slater



determinants are build are l s -coupled harmonic-oscillator wavefunctions [32],which in coordinate representation read

⟨rστ|n (l s )j m j t m t ⟩ (1.11)

=∑

m l ms

l s j

m l ms m j

CG

Rnl (r ) Yl m l(θ ,ϕ) χ (s )

ms(σ) χ (t )

mt(τ) ,

where Yl m l(θ ,ϕ) are the spherical harmonics, χ (s )ms

(σ) are the spinor functions andRnl (r ) are the radial wavefunctions that satisfy the radial single-particle Schrödin-ger equation [32, 33]

− ħh2

2 m r 2

∂

∂ r

r 2 ∂

∂ r

+ħh2l (l +1)

2 m r 2+

1

2m Ω2 r 2

Rnl (r ) = εnl Rnl (r ) , (1.12)

for a harmonic-oscillator potential of frequency Ω. Working in this framework,what is left in order to solve the nuclear Schrödinger equation for an approximatewavefunction, is to specify an Ansatz for the wavefunction and to choose a con-venient Slater-determinant basis set.

Two ab initio many-body methods are considered in this work. The first one isthe (Importance-Truncated) No-Core Shell Model, employing a linear Ansatz

|Ψ⟩ =

1+

A∑

n=1

C (NCSM)n

|Φ⟩ (1.13)

for the many-body state, where C(NCSM)n generates all possible n-particle-n-hole

(npnh) excitations of a single-determinant reference state |Φ⟩. The second ab initio

many-body method is Coupled-Cluster theory, corresponding to an exponentialform of the wave operator,

|Ψ⟩ = e

A∑

n=1

Tn |Φ⟩ , (1.14)

where the Tn also are npnh excitation operators. The No-Core Shell Model iswidely used in nuclear structure calculations [34–54]. It is a universal tool to studythe nuclear system in which ground and excited states as well as properties are ac-cessible in the same framework. The numerical solution of the Schrödinger equa-tion is obtained from large-scale diagonalizations of the Hamiltonian projectedonto a finite space, which is a standard task in the calculation of quantum sys-tems and benefits greatly from parallel computing architectures available nowa-days. The curse of dimensionality of the No-Core Shell Model – which limits the



method to p -shell nuclei due to the exponential growth of the Slater-determinantbasis dimension – can be overcome by the Importance Truncation [21] which al-lows to select the many-body basis states according to their importance for thecalculation at hand, allowing to incorporate basis states well out of reach of thestandard No-Core Shell Model while keeping its variational character and wellpreserving the original translational invariance.

After the exponential Ansatz was introduced by Coester and Kümmel in thelate 1950s [22, 23], Čížek and Paldus laid the foundation for its application inmany-body Fermionic theory [24,55,56]. Nowadays, the Coupled-Cluster methodhas emerged as one of the most powerful methods in high-precision quantumchemistry calculations. In the quantum chemistry context, many variants of theCoupled-Cluster method have been developed over the years, starting from Coup-

led-Cluster with Singles and Doubles excitations (CCSD), going to the perturbativeinclusion of triples- and even quadruples-excitations contributions [57–59], andmany more. Although introduced in nuclear physics, the Coupled-Cluster methodhas not seen as much attention there as it has in the quantum chemistry commu-nity. In the 1990s, Mihaila and Heisenberg [60] brought the method back to the fo-cus of nuclear physicists, and more recently Dean, Hagen, Papenbrock, et al. madesignificant progress in establishing the method particularly in medium-mass andnuclear reaction calculations [27,61–75]. In this work, the Coupled-Cluster meth-ods considered for ground-state calculations are the CCSD approximation, andthe ΛCCSD(T) as well as the CR-CC(2,3) method for the non-iterative inclusion oftriples contributions to the energy. Excited states are accessed with the EOM-CCSD Ansatz [76]. The reason for employing these rather low-order approxi-mations of the full Coupled-Cluster method lies in the rather hard interactionencountered in the case of nuclear physics, which causes strong multi-nucleoncorrelations. This results in the necessity of large basis sets in order to obtain con-verged results with respect to the many-body model space size, which rendersthe application of higher-order Coupled-Cluster approximations practically im-possible. Even for the case of CCSD, the standard formulation of the method inm -scheme basis representation proves to be not practical anymore beyond 40Ca.Therefore, the spherical Coupled-Cluster scheme for closed-shell nuclei, originallyintroduced by Hagen, Papenbrock, et al. [27] is used throughout this work. Thisscheme achieves the required reduction of computational complexity which inprinciple makes the method applicable for closed-shell nuclei across the nuclearchart. Another substantial difference to quantum chemistry applications is theaforementioned need to incorporate three-body forces. This can be achieved by ei-



ther the approximate consideration through the use of effective two-body Hamil-tonians, or by extending Coupled-Cluster theory to explicitly treat three-bodyHamiltonians. Both approaches have first been considered again by Hagen, Pa-penbrock, et al. in 4He proof-of-principle calculations [63] and will be extended tothe medium-mass regime using the spherical formulation in this work.

This work is organized as follows: Basic aspects of ab initio nuclear structurephysics are reviewed in Chapter 1, such as the nuclear interaction and some auxil-iary methods used in the calculations. Chapter 2 presents the traditional Coupled-Cluster theory which is generalized for three-body Hamiltonians in Chapter 3.The following chapter discusses the spherical formulation of Coupled-Cluster the-ory and results are presented in Chapter 5. Finally, a conclusion is given in Chap-ter 6. The Appendix provides results of proof-of-principle calculations, and acompilation of diagrams and spherical equations that entered this work. Dia-grams for many standard Coupled-Cluster method can also be found in [25, 26].For documentational purposes, the spherical equations are presented exactly asthey are used in the computer implementation.



1.2 Chiral Nuclear Interactions

The determination of the nuclear interaction is a long-standing problem in theo-retical nuclear physics [10]. Although it is well-known that QCD is the underlyingtheory, a direct derivation of the nuclear interaction from QCD is not possible yet,due to the non-perturbative nature of QCD in the low-energy regime relevant fornuclear physics.

Phenomenological approaches, such as the Argonne V18 potential [14, 77],have been successful in describing two-nucleon (NN) properties. In the NN sector,the nuclear interaction already has a rather complicated form built from all opera-tor structures that can contribute [78], but the corresponding radial functions canbe determined from a large base of experimental data. However, the description offinite nuclei beyond the two-nucleon system requires the incorporation of many-nucleon forces, and these are difficult to deal with in such an approach. On theone hand, with the number of nucleons involved the number of operator struc-tures grows dramatically while, on the other hand, the experimental data baseshrinks. Furthermore, many-nucleon interactions need to be defined consistently

to the NN interaction [79].

Therefore, physical insight is needed to proceed. Such physical insight wasalready inherent in the first attempts of a field-theoretic description of the nuclearinteraction based on Yukawa’s idea of pion exchange [80], but these were onlypartly successful as well. While the one-pion exchange could be used to under-stand NN scattering data, the multi-pion exchange picture failed. The discoveryof heavy mesons then led to the one-boson exchange model [81], which could ac-curately describe the NN interaction. However, not for all of the bosons used inthis model experimental evidence exists. Finally, the discovery of QCD and theintroduction of the concept of effective field theories [3–12] allowed to formulate atheory of nuclear interactions rooted in QCD. In order for an effective theory towork, a separation of scales is required, each scale with its own set of relevant de-grees of freedom. In the case of QCD, these scales are identified as the asymptoticfree and the hadronic phase, which makes hadrons the more appropriate choiceas degrees of freedom for low-energy QCD than quarks and gluons. Furthermore,due to the large mass gap in the hadron spectrum between the pions and the heav-ier mesons, the most relevant degrees of freedom for low-energy nuclear physicsclearly are the nucleons and pions.

According to Weinberg [3], an effective field theory can be obtained by con-


1.2. Chiral Nuclear Interactions

structing the most general Lagrangian for these degrees of freedom which is con-sistent with the symmetries of the underlying theory. Since such a Lagrangianusually contains infinitely many terms and accompanying low-energy constants,for practical applications a scheme has to be devised that allows to group andselect these terms according to their individual importance. Such characteriza-tion is provided by a power counting scheme introduced by Weinberg for the La-grangian in the chiral expansion, in which it is expanded in a power series inQ/Λχ , where the soft scale Q is a momentum typical for the interaction and thehard scale Λχ is the limit where the theory is expected to break down. In order tomake real progress over the old pion-exchange theories, which already had nu-cleons and pions as degrees of freedom, chiral symmetry needs to be taken intoaccount as an important constraint on the theory. The resulting chiral effective field

theory (χEFT) then represents the solution to the problems mentioned above: Itis clearly connected to QCD via the effective field theory framework by retainingall relevant symmetries of QCD. Furthermore, it not only gives rise to consistenttwo- and many-body interactions, but through power-counting it also allows toidentify the most important of the many operator structures. Since nuclear inter-actions from chiral effective field theory will be employed throughout this work,a more detailed review in the spirit of Refs. [10–12] is given in the following.

Chiral symmetry is closely related to vanishing quark masses, and for theenergy scales relevant in the nuclear structure context, the up and down quarkmasses may be considered approximately zero, which motivates to focus on chi-ral symmetry in the up and down sector of QCD. The two-flavor QCD Lagrangianhas the form

LQCD = q (i γµDµ−M )q − 1

4Gµν ,a G µνa

, (1.15)

where q = (u , d )T are the quark fields, Dµ is the gauge-covariant derivative,M =

diag(mu , md ) denotes the quark mass matrix and Gµν ,a is the gluon field strengthtensor. Chiral symmetry is revealed when the Lagrangian is written in terms ofleft- and right-handed quark fields qL and qR ,

LQCD = qL i γµDµqL + qR i γµDµqR

− qLM qR − qRM qL − 1

4Gµν ,a G µνa

. (1.16)

From (1.16) follows that in the limit of vanishing quark masses – also referred toas the chiral limit – left- and right-handed quark fields are decoupled andLQCD be-comes invariant under separate flavor rotations among the left- and right-handed



quark fields,

qL −→ q ′L= e−iθ L ·τ/2 qL (1.17)

qR −→ q ′R= e−iθ R ·τ/2 qR , (1.18)

generated by Pauli matrices τ in flavor space, resulting in the SU(2) chiral symme-

try group SU(2)L × SU(2)R 1. The corresponding conserved currents may also beexpressed in terms of axial and vector currents belonging to an axial and vectorsubgroup of the chiral symmetry group, SU(2)A ,SU(2)V ⊂ SU(2)L×SU(2)R . In real-ity, quarks possess non-vanishing masses and, consequently, chiral symmetry isexplicitly broken. However, the quark masses of the order of a few MeV are muchsmaller than the typical hadronic mass scale of the order of 1 GeV and, there-fore, chiral symmetry is expected to be at least approximately preserved. Chiralsymmetry should be manifest in the hadron spectrum in form of multiplets cor-responding to SU(2)L×SU(2)R . However, only multiplets corresponding to SU(2)Vare observed in nature, hinting at a spontaneous breaking of SU(2)A . Since SU(2)Ahas 3 generators, its spontaneous breaking gives rise to the existence of 3 Nambu-Goldstone modes or, more precisely, pseudo-Nambu-Goldstone modes, due to thesmall explicit chiral symmetry breaking. Indeed, in this case, the experimentallyobserved hadron spectrum provides candidates for Nambu-Goldstone bosons inform of the pions π= (π±,π0). Then, also the remarkable mass gap between theseunnaturally light pions and the other hadrons may simply be explained by theGoldstone nature of the pions which are massless in the chiral limit but in real-ity acquire a small mass from the explicit breaking of chiral symmetry due to thesmall but non-vanishing quark masses.

The most general effective Lagrangian for pions and nucleons may be writtenin the form

LχEFT = Lππ+LπN +LN N + . . . . (1.19)

Considering the pion-only part Lππ for illustration, the Lagrangian is given by

Lππ = L (2)ππ+L (4)

ππ+O (π6) , (1.20)

where the superscript denotes the number of derivatives or pion-mass insertionswhich has to be even for the pion-only Lagrangian. For example, the lowest-order

1Additional symmetries of the QCD Lagrangian which a not relevant for the present discussionare U (1)V , corresponding to quark number conservation and U (1)A , broken on the quantum leveland therefore also referred to as the U (1) anomaly.



contribution L (2)ππ reads

L (2)ππ=

1

2∂µπ · ∂ µπ+

1

2F 2π

(∂µπ ·π)2+O (π6) , (1.21)

where Fπ is a low-energy constant related to the pion-decay constant which canbe determined experimentally to be Fπ = 92.4 MeV [6]. In order to apply a powercounting scheme, the effective Lagrangian has to be expanded in powers of a softscale over a hard scale Q/Λχ . Due to the Goldstone nature of the pions the softscale Q is associated with external momenta or the pion mass. The hard scale Λχis usually chosen around the mass of the ρmeson which is the lightest meson thatcannot be identified as a Goldstone boson associated with chiral symmetry break-ing. Beyond Λχ the theory is expected to break down because the ρ dynamicswould have to be taken into account explicitly. This is why Λχ is also referred to aschiral symmetry-breaking scale. Once the effective Langrangian has been expandedin Q/Λχ , the Weinberg power counting scheme

ν = −4+2N +2L+∑

i

∆i , ∆i = d i +n i

2−2 , (1.22)

is used to determine the power-counting order ν of a given Feynman diagram,where N denotes the number of nucleon lines, L the number of pion loops, andthe sum runs over all vertices for which∆i is the dimension of vertex i that is calcu-lated in terms of the number of derivatives d i and the number n i of nucleon linesat this vertex. This perturbative treatment of the effective Lagrangian in powers ofQ/Λχ motivated by chiral symmetry is referred to as chiral perturbation theory [6,7].

The inclusion of nucleons in the effective Lagrangian poses a problem due tothe mass mN of the nucleon which is not small compared to the hard scale and,therefore, does not allow for a perturbative treatment. This problem can be over-come in the heavy-baryon formalism [82, 83] in which heavy baryons are treatednon-relativistically by further expanding in terms of 1/mN , so the nucleons are re-garded as static sources of pions. A more detailed discussion is beyond the scopeof this section, however, it should be noted that for the chiral interactions used inthis work the power counting that is employed is the Weinberg power countingscheme (1.22), whose validity has been questioned (see Ref. [84] for a discussion).

As a consequence of the chiral expansion approach, consistent nuclear forcesemerge as a hierarchy in the power-counting order ν , as depicted in Figure 1.1. Atleading order (LO) corresponding to ν = 0, for instance, the NN interaction is givenby two NN contact terms, represented by the diagram



b (1.23)

and a one-pion-exchange contribution

b b (1.24)

which gives rise to a tensor force already at leading order. It should be noted thatfor ν = 1 all terms vanish due to parity and time-reversal constraints. Therefore,what is referred to as next-to-leading order (NLO) actually corresponds to ν = 2 andfrom there, for any given ν > 2, the next order corresponds to an increase of ν byone. Apparently, according to Figure 1.1, at LO and NLO only two-nucleon forcesexist. Three-nucleon forces arise at next-to-next-to-leading order (N2LO, etc.),= andfour-nucleon forces do not appear before N3LO. This way, chiral perturbation the-ory reproduces the observed hierarchy of nuclear forces in which the importanceof many-nucleon forces decrease with the number of active nucleons involved.The three-nucleon interaction at N2LO is represented by the three diagrams

b b b b b b (1.25)

in which the 5 low-energy constants c i , i = 1, 3, 4 and cD , cE enter. The c i , however,are already determined in the two-nucleon and pion-nucleon sector, which leavesonly cD and cE , assigned to the two-nucleon-contact with one-pion-exchange andthe three-nucleon-contact diagrams respectively, as new low-energy constants thathave to be experimentally determined from the three-body system. At present,the part of Figure 1.1 that is currently available for nuclear-structure calculationsis given by the NN interaction up to N3LO and the 3N interaction up to N2LO,which constitute the interactions mainly used in this work. When being evalu-ated, the diagrams have to be regularized by a cutoff momentum. For the NNinteraction, a regulator cutoff momentum of ΛNN = 500 MeV is used, while for the3N interaction, the regulator cutoff momenta of Λ3N = 500 MeV or 400 MeV willmainly be employed. 2

2The regularization cutoff momenta ΛNN and Λ3N should not be confused with the chiral break-down scale Λχ .



NN 3N 4N

LO(Q/Λχ )0

b b b

NLO(Q/Λχ )2

rs b b b

b

b

b

b

b

b

b

b

b

b

b

b

b

N2LO

(Q/Λχ )3b b b

b

b + . . .

b b b

b b b

N3LO

(Q/Λχ )4

rs b bb

b

b

b

b

rsb

b

b

b

b

b

b

b + . . .

b

b

b

b b

b

b

b

b

+ . . .

b

b

b

b

b

+ . . .

Figure 1.1: Nuclear forces from chiral perturbation theory based on the Weinberg power counting

[10]. Solid lines represent nucleon propagators, dashed lines pions. Different symbols

(small/large dots, etc.) denote different types of vertices [10].



1.3 Similarity Renormalization Group

Nuclear structure calculations are severely complicated by the existence of the re-pulsive core in the nucleon-nucleon potential. Nuclear structure theory aims atthe low-energy description of nuclei, but the short-range correlations from therepulsive core introduce an energy scale that is not easily resolved by the setof harmonic-oscillator Slater-determinant basis functions used in practical many-body calculations, resulting in a slow convergence of these calculations with re-spect to model-space size. These correlations manifest themselves in form of theinteraction operator coupling low- and high-momentum states. The approach ofthe Similarity Renormalization Group (SRG) [16, 17, 85] is to unitarily transform theinitial interaction to a more diagonal form that suppresses the coupling betweenlow- and high-momentum states. Due to the unitary nature of the transformation,in principle, no information is lost and low-energy nuclear structure may then bedescribed by low-energy degrees of freedom.

The initial Hamiltonian H0 is continuously unitarily transformed by the actionof a unitary operator Uα depending on a continuous real parameter α,

Hα = U †α

H0 Uα . (1.26)

The derivative of the transformed Hamiltonian with respect to α,

ddα

Hα =h

−U †α

dUα

dα, Hα

i

, (1.27)

motivates the definition of ηα,

ηα = −U †α

dUα

dα, (1.28)

as the generator of the transformation, such that the flow (1.27) of the Hamiltonianalong the unitary path reads

dHα

dα=

h

ηα, Hα

i

, η†α= −ηα . (1.29)

This equation is equivalent to the one-step unitary transformation (1.26) and, there-fore, the focus may be shifted away from finding the explicit transformation op-erators Uα to finding anti-Hermitean generators ηα that let the Hamiltonian flowalong an appropriate path for the problem at hand. The common choice for ηαin the context of nuclear structure calculations is given by the commutator of the


1.3. Similarity Renormalization Group

intrinsic kinetic energy operator, Tint = T − TCM, with the SRG-evolved Hamilto-nian [85],

ηα =

2µ

ħh2

2 h

Tint, Hα

i

, (1.30)

in terms of the reduced nucleon mass µ. This generator lets the momentum-spacerepresentation of the Hamiltonian flow towards band-diagonal form and thus, asdesired, leads to a decoupling of low- and high-momentum degrees of freedom.As shown in the example of Figure 1.3, this in turn accelerates the convergence ofthe many-body calculations with respect to model-space size.

In order to bring the operator flow equation (1.29) using generator (1.30) intomatrix-element representation in k -body space, resolutions of the k -body identityof the form

1(k ) =

∑

p

|φ(k )p⟩⟨φ(k )

p| (1.31)

have to be inserted between adjacent operators, leading to

ddα⟨φ(k )i |Hα|φ

(k )j ⟩ =

2µ

ħh2

2¨∑

pq

⟨φ(k )i |Tint|φ(k )p⟩⟨φ(k )

p|Hα|φ(k )q

⟩⟨φ(k )q|Hα|φ(k )j ⟩

−2∑

pq

⟨φ(k )i |Hα|φ(k )p⟩⟨φ(k )

p|Tint|φ(k )q

⟩⟨φ(k )q|Hα|φ(k )j ⟩

+∑

pq

⟨φ(k )i |Hα|φ(k )p⟩⟨φ(k )

p|Hα|φ(k )q

⟩⟨φ(k )q|Tint|φ(k )j ⟩

«

. (1.32)

Since the resolutions of the identities have to be truncated at some point in practi-cal calculations this introduces errors which have to be monitored (see Section 5.9.3and [86]).

At this point a crucial difference between the SRG transformation and otherrenormalization group approaches, such as Vlow-k [87] should be mentioned. Inthe latter, with increasing transformation parameter the UV cutoff of the interac-tion is lowered and, therefore, removing the high-energy scale from the interac-tion, as depicted in Figure 1.2. Therefore, for Vlow-k the transformation parametercorresponds to a cutoff in the momentum scale. For the SRG on the other hand, as



E→

E →

E→

E →

↑

←

տ

E→

E →

ր

ւ

initial Vlow-k SRG

Figure 1.2: Comparison of different renormalization approaches for the initial Hamiltonian (see

text and [88]).

the flow parameter α increases, the Hamiltonian is driven to band-diagonal formwhere α−1 is related to the width of the band. High-energy modes are still presentin the interaction, only their coupling to low-energy modes is suppressed. Conse-quently, in the case of the SRG, the flow-parameter does not correspond to a cutoffin the momentum scale – which is also clear from the unitary nature of the SRGtransformation.

A major drawback of the SRG transformation (and all other renormalizationtreatments) is the induction of many-body interactions. From evaluation of thecommutators

dHα

dα=

2µ

ħh2

2 hh

Tint, Hα

i

, Hα

i

(1.33)

it becomes apparent that each infinitesimal evolution step generates operatorswith particle ranks exceeding the original rank of the interaction. Therefore, atthe end of the evolution the transformed Hamiltonian will contain induced many-body interactions up to the number of nucleons in the system,

Hα = H (1)α+ H (2)

α+ H (3)

α+ . . . + H (A)

α. (1.34)

In order to preserve the unitarity of the transformation, if the evolved Hamilto-nian is to be used in an A-body calculation, all induced operators up to the A-bodylevel would have to be maintained during and after the SRG flow. For practicalreasons, typically only operators up to the three-body level can be kept while theothers have to be discarded, resulting in a formal violation of unitarity of the trans-formation. In many-body calculations this violation of unitarity will emerge as a


1.3. Similarity Renormalization Group

α= 0.00 fm4

E →

E ′

↓

0 5 10 15 20Nmax

-8-6-4-202

E[M

eV]

α= 0.04 fm4

E →

E ′

↓

0 5 10 15 20Nmax

-8-6-4-202

E[M

eV]

α= 0.08 fm4

E →

E ′

↓

0 5 10 15 20Nmax

-8-6-4-202

E[M

eV]

α= 0.16 fm4

E →

E ′

↓

0 5 10 15 20Nmax

-8-6-4-202

E[M

eV]

Figure 1.3: Effect of the SRG evolution on the matrix elements of the NN+3N-full interaction for

the triton channel (J π, T ) = (1/2+, 1/2) in an antisymmetrized three-body harmonic-

oscillator Jacobi basis with ħhΩ = 28 MeV. Plotted are the absolute values of the matrix

elements, where light colors represent large values, and dark colors represent values

near zero (also see [89]). Embedded in the matrix plots are convergence patterns of

triton ground-state energies obtained from the NCSM.



dependence of the observables on the flow parameter α. As long as the observ-ables are independent of the flow parameter it is assumed that unitarity has notbeen violated and, therefore, conclusions about the initial Hamiltonian may bedrawn from the transformed Hamiltonian. However, once unitarity is corruptedso much that flow-parameter dependence sets in, this will no longer be possible.In order to discuss the results obtained from the SRG framework described above,it is convenient to define and investigate the following three Hamiltonians:

(i) NN-only : An initial two-body Hamiltonian H NN0

is evolved and during theflow only induced operators up to the two-body level are kept.

(ii) NN+3N-induced : An initial two-body Hamiltonian H NN0

is evolved and dur-ing the flow only induced operators up to the three-body level are kept.

(iii) NN+3N-full : An initial two- plus three-body Hamiltonian H NN+3N0

is evolvedand during the flow only induced operators up to the three-body level arekept.

In this way contributions of induced three- and higher-body interactions out ofthe initial Hamiltonian can be quantified. The NN+3N-full Hamiltonian repre-sents the most complete Hamiltonian considered in this work since it contains thefull currently available set of chiral interactions. A flow-parameter dependence ofthe energy eigenvalues hints at significant four- and higher-body interaction con-tributions stemming from the initial two- or three-body interaction. These con-siderations are illustrated in Figure 1.4 (also see [17]): For 4He, the ground-stateenergies obtained for the NN-only Hamiltonian show a strong flow-parameter de-pendence, demonstrating the importance of SRG-induced many-body forces thatare not considered in this type of calculation. This flow-parameter dependencedoes not allow to make any prediction of where the result of the untransformedHamiltonian would come out. However, when induced three-body forces aretaken into account using the NN+3N-induced Hamiltonian, this flow-parameterdependence vanishes, indicating that unitarity of the transformation is already re-stored by the inclusion of induced three-body interactions. This way a predictionfor the bare chiral NN Hamiltonian can be made, but the agreement with experi-ment is rather poor. Including the initial chiral 3N interaction via the NN+3N-fullHamiltonian still produces results that show no dependence on the flow param-eter, and at the same time also significantly improve the agreement with experi-ment. Therefore, chiral Hamiltonians are capable to provide an accurate descrip-tion of the 4He system.


1.4. Hartree-Fock Method

For the heavier nucleus 12C, the situation is similar with the exception of anemerging flow-parameter dependence for the NN+3N-full Hamiltonian. Since for12C no flow-parameter dependence is observed for the NN+3N-induced Hamil-tonian, the NN+3N-full results suggest the existence of sizable four- (or higher-)body interactions out of the initial 3N interaction. At this point, this flow-para-meter dependence prevents any attempts of making robust predictions, similar tothe NN-only case. In order to restore predictive capabilities for the NN+3N-fullHamiltonian in calculations of medium-mass nuclei, a modified initial chiral 3Ninteraction will be employed that has a reduced regulator cutoff momentum, asdiscussed in more detail in Section 5.3.

1.4 Hartree-Fock Method

The fact that nucleons show properties of non-interacting Fermions, such as thelow density or the long mean free path of the nucleons within the nucleus [78],suggests the applicability of independent-particle methods such as the Hartree-Fock

method that is widely used in many-body theory, such as atomic and nuclearphysics, or quantum chemistry [78, 90].

In the Hartree-Fock method the nuclear interaction is not neglected but re-placed by a mean-field potential that is generated by the nucleons and in whichthey are assumed to move independently. The independent-particle picture al-lows to approximate the many-body wavefunction by a single Slater determinant,which is then determined by minimizing its energy expectation value accordingto the variational principle. This is achieved by optimizing the single-particle or-bitals from which the Slater determinant is built. These orbitals have to be deter-mined in a self-consistent manner because the orbitals determine the mean-fieldthe nucleons feel, which in turn determines the orbitals in which the nucleonsmove.

The Hartree-Fock method typically serves two purposes: On the one hand,it is used an approximate many-body method that provides the best approxi-mation to a many-body wavefunction from the set of single Slater determinants.On the other hand, it provides a set of optimized single-particle orbitals, whichcan subsequently be used as starting point for a more sophisticated many-bodymethod such as Coupled Cluster. The latter is the main purpose of the Hartree-Fock method in this work. Most of the time, the single-particle orbitals used in the



NN only NN+3N induced NN+3N full

4He

2 4 6 8 10 12 14 16 ∞Nmax

-29

-28

-27

-26

-25

-24

-23

.

E[M

eV]

2 4 6 8 10 12 14 ∞Nmax

Exp.

2 4 6 8 10 12 14 ∞Nmax

NN only NN+3N induced NN+3N full

12C

2 4 6 8 10 12 14 ∞Nmax

-110

-100

-90

-80

-70

-60

.

E[M

eV]

2 4 6 8 10 12 ∞Nmax

Exp.

2 4 6 8 10 12 ∞Nmax

Î

α= 0.04 fm4 α= 0.05 fm4 α= 0.0625 fm4 α= 0.08 fm4 α= 0.16 fm4

Figure 1.4: IT-NCSM ground-state energies for 4He and 12C for the NN-only, NN+3N-induced,

and NN+3N-full Hamiltonian for a sequence of SRG flow parameters [17]. Both nu-

clei the energies show no flow-parameter dependence for the NN+3N induced Hamilto-

nian, indicating no relevant induced four- or higher-body interactions out of the initial

NN interaction. The flow-parameter dependence for the NN+3N-full Hamiltonian in12C suggests the emergence of significant induced higher-body interactions out of the

initial 3N interaction.


1.5. Normal-Ordering Approximation

normal-ordering and Coupled-Cluster calculations have been determined from apreceding Hartree-Fock calculation.

The Hartree-Fock method is well covered in the literature, a standard treat-ment of this matter can be found in [33, 78, 90], and the extension to incorporatethree-body Hamiltonians is discussed in [91].

1.5 Normal-Ordering Approximation

As already discussed in the previous section, for accurate ab initio nuclear-structurecalculations the inclusion of three-body interactions is mandatory [17, 29, 30, 70,92, 93]. Along with it comes an array of formal and technical difficulties, suchas the increased complexity of the equations to be solved, or the treatment of in-tractably large Hamiltonian matrix representations. On the other hand, ab initio

calculations using only two-body Hamiltonians are a standard practice and do notsuffer from one of the aforementioned problems. Therefore, the construction ofeffective lower-rank interactions that approximate the original three-body interac-tion represents an economic way to include three-body effects in nuclear-structurecalculations using the standard two-body Hamiltonian framework.

The effective lower-rank interaction used in this work is obtained from thenormal-ordered two-body approximation [63, 92]. The idea behind this approx-imation is based on the observation that when a three-body operator h3 that isgiven in normal order with respect to the vacuum 3,

h3 =1

36

∑

pqr s t u

⟨pqr |h3|s t u ⟩ a †p

a †q

a †ra u a t a s , (1.35)

is represented in normal order with respect to some single-determinant A-bodyreference state |Φ⟩, the resulting operator has non-vanishing components also at

3The vacuum |0⟩ is understood as the state containing no nucleons such that a p |0⟩= 0.



lower particle ranks,

h3 =1

6

∑

i j k

⟨i j k |h3|i j k ⟩

+1

2

∑

pqi j

⟨i j p |h3|i j q ⟩ a †p

a q

+1

4

∑

pqr s i

⟨pqi |h3|r s i ⟩ a †p

a †q

a s a r

+1

36

∑

pqr s t u


a †q

a †ra u a t a s . (1.36)

Here, i , j , k denote orbitals occupied in |Φ⟩ and . . . indicates normal orderingwith respect to |Φ⟩. Since some contributions of the original three-body operatorhave been demoted to lower particle ranks, the residual three-body interactionoperator

1

36

∑

pqr s t u


a †q

a †ra u a t a s (1.37)

in (1.36) may be discarded, and yet still allowing to include three-body inter-action effects in a computational framework capable of handling at most two-body Hamiltonians. This particular scheme, in which the residual three-bodyoperator is discarded and the remaining zero-, one-, and two-body parts are in-cluded in the calculation, is referred to as normal-ordered two-body approxima-tion (NO2B) [63, 92], or normal-ordering approximation for short. Back in theparticle-vacuum representation, the three-body operator h3 in NO2B approxima-tion then reads [94] 4

hNO2B3

=1

6w0−

1

2

∑

pq

⟨p |w1|q ⟩ a †p

a q +1

4

∑

pqr s

⟨pq |w2|r s ⟩ a †p

a †q

a s a r , (1.38)

with definitions of the matrix elements of the normal-ordered interaction opera-tors

w0 =∑

i j k

⟨i j k |h3|i j k ⟩ , (1.39)

⟨p |w1|q ⟩ =∑

i j

⟨i j p |h3|i j q ⟩ , (1.40)

4The negative sign in front of the one-body part is intended, see [94].


1.5. Normal-Ordering Approximation

and

⟨pq |w2|r s ⟩ =∑

i

⟨pqi |h3|r s i ⟩ . (1.41)

A systematic study of the NO2B approximation using the IT-NCSM was per-formed in [92] from which Figure 1.5 is taken. Results obtained from the NO2Bprovide a good approximation to the ones obtained using explicit three-bodyHamiltonians, with deviations of about 2% for the case of 4He and about 1% for16O. Together with results for 40Ca, the conclusion can be drawn that the NO2Bapproximation works particularly well for heavier nuclei. This is also confirmedby Coupled-Cluster calculations for medium-mass nuclei [95, 96] in Section 5.7and 5.8.1.

Coupled-Cluster calculations usually employ a Hartree-Fock basis. The trans-formation from the spherical harmonic-oscillator single-particle basis into the Har-tree-Fock basis reads

|p m tp⟩(HF) =

∑

α

C αp mtp|αm tp

⟩ , (1.42)

where p = n p (l p sp )jp is a shorthand notation for the set of all quantum numbersexcept isospin, and the isospin tp = 1/2 will mostly be suppressed. Therefore, theHartree-Fock reference state |Φ⟩ is given by a superposition of Slater determinantsbuilt from harmonic-oscillator single-particle states,

|Φ⟩ = A | i m t i⟩(HF) ⊗ . . . ⊗ |k m tk

⟩(HF)

=∑

α

· · ·∑

γ

C αi mti

. . .Cγ

k mtk|αm t i

. . . γm tk⟩ . (1.43)

Using angular-momentum coupled three-body matrix elements, the zero-bodyresult of the normal-ordering with respect to the Hartree-Fock reference state isgiven by

w0 =∑

i mti

∑

j mt j

∑

k mtk

∑

α

∑

β

∑

γ

∑

δ

∑

ε

∑

κ

∑

JJ

ˆJ 2

× C αi mti

Cβ

j mt jCγ

k mtkC δ

i mtiC ε

j mt jC κ

k mtk

× ⟨αβγ, m t im t j

m tk|h3|δεκ, m t i

m t jm tk⟩

J

JM

J

JM

, (1.44)



4He 16O

-25.5

-25.0

-24.5

-24.0

.

E[M

eV]

NN+3N-ind.

-120

-115

-110NN+3N-ind.

2 4 6 8 10 12 14 16 18 20Nmax

-28.5

-28.0

-27.5

-27.0

.

E[M

eV]

NN+3N-full

2 4 6 8 10 12 14 16 18 20Nmax

-150

-145

-140

-135NN+3N-full

explicit Î

NO2B ◊ Í

α= 0.04 fm4 α= 0.05 fm4 α= 0.0625 fm4 α= 0.08 fm4

Figure 1.5: Comparison of IT-NCSM ground-state energies of 4He and 16O for the SRG-evolved

NN+3N induced and NN+3N full Hamiltonians using explicit 3N interactions and

the NO2B approximation [92]. The calculations employed a HO basis with oscillator

frequency ħhΩ = 20 MeV.


1.6. Configuration Interaction and No-Core Shell Model

where here and in the following the summations∑

i mti

,∑

j mt j

, and∑

k mtk

(1.45)

are understood to run only over orbitals |i m t i⟩, |j m t j

⟩, and |k m tk⟩ that are occupied

in the reference state |Φ⟩. The one-body contribution reads

⟨p mp m tp|w1|qmq m tq

⟩ = p−2∑

i mti

∑

j mt j

∑

α

∑

β

∑

γ

∑

δ

∑

JJ

ˆJ 2

× C αi mti

Cβ

j mt jCγ

i mtiC δ

j mt j

× ⟨αβp , m t im t j

m tp|h3|γδq , m t i

m t jm tq⟩

J

JM

J

JM

, (1.46)

and the normal-ordered two-body matrix elements are given by

⟨pq , tp tq |w2|r s , tr ts ⟩J M J MT M T T M T

= J −2∑

i mti

∑

α

∑

β

∑

J

ˆJ 2

× C αi mti

Cβ

i mti⟨pqα, tp tq m t i

|h3|r sβ , tr ts m t i⟩

J

JM

J

JM

T M T T M T

. (1.47)

The normal-ordered one- and two-body matrix elements – as given above – arestill given in the harmonic-oscillator basis and, therefore, need to be transformedto the HF basis after the normal ordering.

1.6 Configuration Interaction and No-Core ShellModel

The No-Core Shell Model (NCSM) [21, 51, 54] is an ab initio approach to the nuclearmany-body problem. All A nucleons are considered active, i.e., no degrees of free-dom of the system are combined into effective ones as it is for example done inthe traditional single-particle shell model. The nuclear wavefunction is expandedin a set of Slater determinant basis functions, where the expansion coefficients areobtained from a large-scale Hamiltonian matrix diagonalization. The method is



fully variational and yields the exact solution for the model space under consid-eration. Therefore, this method always provides an upper bound for the ground-state energy and, once convergence is reached with respect to model space size,constitutes an ideal tool against which other many-body methods may be bench-marked.

The NCSM belongs to the class of the Configuration Interaction (CI) methods.Since the CI methods and its common truncated variants CISD, CISDT etc. (seeSection 1.6.2) also share more formal analogies than the NCSM with the Coupled-Cluster method, it is worthwhile to briefly review these methods as well and tointroduce basic conventions and notations.

1.6.1 Full Configuration Interaction

In the CI scheme, the continuous Hamiltonian eigenproblem is Galerkin-discre-tized and converted into a finite-dimensional approximation scheme by introduc-ing a finite set ΛSP of discrete single-particle basis functions |φk ⟩ 5

ΛSP =n

|φk ⟩ : k = 1, . . . , dimΛSP

o

. (1.48)

As discussed in Section 1.1, a common choice for these single-particle basis func-tions are the harmonic-oscillator wavefunctions |n k (l k sk )jk m jk

tk m tk⟩ (or linear com-

binations of these, if canonical orbitals obtained from the Hartree-Fock methodare used). For the purpose of deriving appropriate truncation schemes for themany-body basis it is important to note that to each single-particle basis state |φk ⟩a quantum number ek may be assigned, which characterizes the energy of the single-

particle state |φk ⟩, and is defined by

ek = 2n k + l k . (1.49)

For the Full Configuration Interaction (FCI) method, the constraint in the total num-ber of single-particle states, dimΛSP <∞, is the only truncation for the model spacethat is introduced. Consequently, the FCI many-body basis ΛFCI consists of allSlater determinants constructed from the single-particle basis set ΛSP, which arenot equivalent in the sense that they only differ by single-particle index permuta-tions. So the many-body basis may be constructed as

Λ(FCI) =n

|Φk ⟩= A |φk1. . .φkA

) : k1 < · · ·< kA = 1, . . . , dimΛSP

o

, (1.50)

5The single-particle basis is assumed to be orthonormal throughout this work.



with A being the antisymmetrization operator (1.8), and the FCI model spaceV (FCI) is then given by

V (FCI) = spann

Λ(FCI)o

. (1.51)

For this model space the matrix representation (Hk l ) of the Rayleigh-Ritz projec-tion of the Hamiltonian is calculated,

Hk l = ⟨Φk |H |Φl ⟩ , (1.52)

from which, via diagonalization, the expansion of the Hamiltonian eigenstates inthe basis Λ(FCI) may be obtained,

∑

l

Hk l C l = E Ck ⇒ |Φ(FCI)⟩ =∑

k

Ck |Φk ⟩ . (1.53)

Since the Hamilton matrix (Hk l ) is obtained from orthogonal projection techniquesit is guaranteed that the FCI wavefunction |Ψ(FCI)⟩ is the best approximation to theexact wavefunction that, in the sense of residual minimization, can be computedfrom the model space V (FCI). This is also the reason why the FCI method is fullyvariational. The main limitation of the method is due to the rapid growth of themany-body basis which goes as

dimV (FCI) =(dimΛSP)!

A ! (dimΛSP−A)!, (1.54)

and, thus, is scaling factorially with particle number A and single-particle basissize dimΛSP.

Regarding truncations of the FCI scheme, it is common practice to introduce aparametrization of the FCI wavefunction using the concept of the reference stateand corresponding excitation operators. The reference state

|Φ⟩ = A |φi 1. . .φi A

) (1.55)

is a single Slater determinant build from the set of single-particle orbitals φi 1

that minimize the energy functional

Erefφi 1

, . . . ,φi A= min

i 1< ...< i A

e i 1+ · · ·+ e i A

, (1.56)

with the constraint that the index set i 1, . . . , i A has of course the correct numberof proton and neutron states for the nucleus under consideration. The referencestate may, therefore, serve as zero-order approximation to the wavefunction with



corresponding zero-order energy Eref. Once the reference state is determined, thesingle-particle basis index set is divided into the orbitals occupied by the refer-ence state (referred to as hole states) and the unoccupied (particle) states, with thefollowing notational convention :

hole states : i , j , k , . . . ∈ occupied in |Φ⟩particle states : a ,b , c , . . . ∈ unoccupied in |Φ⟩any state : p ,q , r, . . .

(1.57)

The n-particle-n-hole (npnh) excitation |Φa 1...a n

i 1...i n⟩ of the reference determinant is de-

fined as the Slater determinant in which, relative to the reference state, n holestates have been replaced by n particle states, i.e.,

|Φa 1...a n

i 1...i n⟩ = (a †

a 1a i 1)(a †

a 2a i 2) . . . (a †

a na i n) |Φ⟩

= a †a 1

. . . a †a n

a i n. . . a i 1

|Φ⟩ , (1.58)

and the corresponding npnh excitation operator generating all possible npnh ex-citations reads

Cn =1

(n !)2

∑

i 1,...,i na 1,...,a n

ca 1...a n

i 1...i na †

a 1. . . a †

a na i n

. . . a i 1. (1.59)

In terms of excitation operators (1.59), the FCI wavefunction can be parametrizedby the linear Ansatz

|Ψ(FCI)⟩ = (1+ C (FCI)) |Φ⟩ , C (FCI) =

A∑

n=1

C (FCI)n

. (1.60)

Since C(FCI)m |Φ⟩ vanishes for excitation ranks m > A, the FCI wave operator C natu-

rally terminates at the ApAh level.

1.6.2 Truncated Configuration Interaction

A natural way to truncate the full CI Ansatz (1.60) is to truncate the excitationlevel X (C (FCI)) of the FCI wave operator at some value M , i.e., X (C (CIM)) = M < A.The corresponding truncated CI variant is then referred to as CIM, or as CI with

Singles and Doubles excitations (CISD) for M = 2, as CI with Singles, Doubles and

Triples excitations (CISDT) for M = 3, and so on. This truncation is justified by theexpectation that in the expansion (1.60) higher excitation ranks are less relevant



than lower ones, provided that the reference state is sufficiently close to |Ψ(FCI)⟩.Therefore, using the CIM Ansatz the Schrödinger equation reads

H

1+

M∑

n=1

C (CIM)n

|Φ⟩ = E (CIM)

1+

M∑

n=1

C (CIM)n

|Φ⟩ . (1.61)

In analogy to the Coupled-Cluster method, a set of coupled equations for theenergy E (CIM) and the amplitudes c

a 1...a k

i 1...1kcan be obtained by left-projecting the

CIM Schrödinger equation (1.61) onto the reference |Φ⟩ and excited determinants|Φa 1...a k

i 1...i k⟩,

⟨Φ| H

1+M∑

n=1

C(CIM)n

|Φ⟩ = E (CIM)

⟨Φa 1

i 1| H

1+M∑

n=1

C(CIM)n

|Φ⟩ = E (CIM) ca 1

i 1, ∀ a 1, i 1

......

⟨Φa 1...a M

i 1...i M| H

1+M∑

n=1

C(CIM)n

|Φ⟩ = E (CIM) ca 1...a M

i 1...i M, ∀ a 1, . . . , i M .

(1.62)

It is noteworthy that in a specific line of (1.62) not all excitation ranks of C (CIM)

contribute, because

⟨Φa 1...a k

i 1...i k|H

1+

M∑

n=1

C (CIM)n

|Φ⟩ = ⟨Φa 1...a k

i 1...i k|H

1+

nmax∑

n=nmin

C (CIM)n

|Φ⟩, (1.63)

with

nmin = max¦

1 , k −X (H )©

(1.64)

nmax = min¦

M , k +X (H )©

. (1.65)

Particularly, the energy has an explicit dependence on the C (CIM) coefficients of theform

E (CIM) = E (CIM)n

ca 1

i 1

o

, . . . ,n

ca 1...a X (H )

i 1...i X (H )

o

, (1.66)

while, due to the coupled nature of (1.62), there is of course an implicit depen-dence on all coefficients. From a physical point of view, the presence of the CIenergy in the truncated CI amplitude equations is troubling. This is because, forthe truncated CI case, an unequal scaling of both sides of the amplitude equationsis introduced as the number of particles in the system is increased. The energy and



the CIM coefficients scale differently, leading to violation of size extensivity [25]. Inquantum-chemistry applications, this violation of size extensivity of the truncatedCI method poses a serious problem which makes the size-extensivity preservingCoupled-Cluster method more favorable there.

1.6.3 No-Core Shell Model

An alternative truncation of the FCI parametrization which aims not at excita-tion rank but rather on excitation energy is employed by the No-Core Shell Model

(NCSM). If the excitation energy of a Slater determinant relative to the unperturbedreference state is defined by

ea 1...a n

i 1...i n≡

n∑

k=1

ea k− e i k

, (1.67)

then the NCSM again uses a linear parametrization of the wavefunction similarto the FCI parametrization,

|Ψ(NCSM)⟩ =

1+

A∑

n=1

C (NCSM)n

|Φ⟩ , (1.68)

with excitation operators C(NCSM)n

C (NCSM)n

=1

(n !)2

∑′

i 1,...,i na 1,...,a n

ca 1...a n

i 1...i na †

a 1. . . a †

a na i n

. . . a i 1(1.69)

where the summations∑′

i 1,...,i na 1,...,a n

(1.70)

are constrained to maximum excitation energies, generated by the operator stringa †

a 1. . . a †

a na i n

. . . a i 1acting on the reference state, according to

ea 1...a n

i 1...i n≤ Nmax . (1.71)

This Nmax truncation is of particular significance in NCSM calculations using aharmonic-oscillator basis, since despite of the use of single-particle coordinatesthis truncation allows for any choice of Nmax an exact factorization of the NCSMwavefunction into a center-of-mass and a relative part [97],

|Ψ(NCSM)⟩ = |Ψint⟩ ⊗ |ΨCM⟩ , (1.72)

and, therefore, avoiding mixing center-of-mass and intrinsic excitations. This isthe reason why the harmonic-oscillator basis, although exhibiting an undesiredasymptotic behavior [98], is commonly preferred over other basis sets.



1.6.4 Importance-Truncated No-Core Shell Model

Despite its heavy use and long history of success in nuclear structure calculations,the NCSM becomes intractable for nuclei beyond 16O, even on massively parallelcomputational architectures. This is because the NCSM requires unmanageablylarge Nmax-parametrized model spaces in order to obtain model-space convergedresults. A more sophisticated truncation scheme is introduced by the Importance-

Truncated No-Core Shell Model (IT-NCSM) [20, 21].

The IT-NCSM is motivated by the observation that many of the basis states ofa NCSM model space do not significantly contribute to the expansion of a specificwavefunction. Therefore, an a priori criterion is introduced that estimates the im-portance of specific basis states for the expansion of some eigenstate |Ψ⟩. This wayonly the most relevant basis states can be selectively incorporated in the many-body basis, allowing to go to larger values of Nmax than would be accessible in thestandard NCSM.

In order to do so, a small reference space Vref (typically a full NCSM model spacewith small Nmax) is specified from which a first approximation, the reference state

|Ψref⟩ (not to be confused with the reference state |Φ⟩, which does not enter theconsiderations in this section ), is calculated,

H |Vref|Ψref⟩ = εref |Ψref⟩ . (1.73)

With this information at hand, the importance κµ of a basis state |Φµ⟩ 6∈Λref outsidethe reference space can be estimated from perturbation theory in first order as

κµ = −⟨Φµ|H |Ψref⟩εµ−εref

(1.74)

(a detailed review can be found in [21]). Obviously, regarding preserving transla-tionally invariance, |Φµ⟩ are taken from full NCSM model spaces of large enoughNmax to guarantee Vref ⊂ V (NCSM). The IT-NCSM model space is consequently de-fined as the reference space and its extension spanned by all determinants whichhave an importance measure larger than some fixed κmin,

V (IT-NCSM) = spann

|Φµ⟩ ∈ V (NCSM) : κµ ≥ κmin

o

. (1.75)

The IT-NCSM is clearly a variational approach, converging to the exact result asκmin goes to zero,

|Ψ(NCSM)⟩ = limκmin→0

|Ψ(IT-NCSM)⟩ , (1.76)



as well as all observables computed from the wavefunction which typically showa smooth κmin-dependence and usually can successfully be extrapolated to theκmin = 0 limit.


Chapter 2

Coupled-Cluster Theory

Chapter 2. Coupled-Cluster Theory

2.1 Introduction

Violation of size extensivity and exponential scaling of the model space have pro-ven to be the main limitations of CI-like methods in quantum-chemistry appli-cations. In the nuclear context the importance of size extensivity has not beenestablished yet but the unfortunate scaling behavior has limited ab initio methodsto the regime of p -shell nuclei anyway. By abandoning the variational princi-ple, however, the Coupled-Cluster method [22–27, 99–103] overcomes these ob-stacles, being size extensive in all truncation orders and exhibiting a polynomial,rather than exponential, scaling of the model space size. In this and the follow-ing sections two-body Hamiltonians are considered exclusively. The generalizationCoupled-Cluster theory to three-body Hamiltonians is postponed to Chapter 3.

2.2 The Exponential Ansatz

Seeking for alternative ways to solve the many-body Schrödinger equation, in-stead of focusing on truncations of the linear CI-like scheme, more general, andthis nonlinear, parametrizations may be considered. Among these, the exponen-

tial Ansatz for the Coupled-Cluster wavefunction is probably the most powerfulknown to date,

|Ψ⟩ = e T |Φ⟩ , T =

A∑

n=1

Tn , (2.1)

where the cluster operator T is defined in close analogy to the CI case, with com-ponents

T1 = =1

(1!)2

∑

a i

t aia †

aa i (2.2)

T2 = =1

(2!)2

∑

ab i j

t abi ja †

aa †

b a j a i (2.3)

...

Tn = =1

(n !)2

∑

a 1...a ni 1...i n

ta 1...a n

i 1...i na †

a 1. . . a †

a na i n

. . . a i 1 . (2.4)

As in Eqs. (2.2)-(2.4), regarding the diagrammatic treatment of CC equations, it iscustomary to work with operators that are in normal order relative to the reference


2.2. The Exponential Ansatz

state, indicated by . . ., where it is understood that all particle creation or hole an-nihilation operators a †

a, a i are to the left of all particle annihilation or hole creation

operators a a , a †i . Then, as for the standard normal-ordering prescription, expec-

tation values of normal-ordered operator products in the reference state, whichserves as new vacuum, vanish, for example

⟨Φ|a †a

a †b a j a i |Φ⟩ = ⟨Φ|Φab

i j⟩ = 0 . (2.5)

In the case of pure excitation operators, such as Tn , or pure de-excitation operators,the operator string is automatically in normal order with respect to the referencestate and the brackets . . . may be dropped. The Hamiltonian operator is put innormal-ordered form as well. A general two-body Hamiltonian in standard formreads

H = h0+ h1+ h2 (2.6)

= h0+∑

pq

⟨p |h1|q ⟩ a †p

a q +1

4

∑

pqr s

⟨pq |h2|r s ⟩ a †p

a †q

a s a r , (2.7)

where antisymmetrized two-body matrix elements ⟨pq |h2|r s ⟩ are introduced,

⟨pq |h2|r s ⟩ = (pq |h2|r s )− (pq |h2|s r ) . (2.8)

In normal-ordered form, the operator is given by

H = h0+∑

i

⟨i |h1|i ⟩+∑

pq

⟨p |h1|q ⟩ a †p

a q+1

2

∑

i j

⟨i j |h2|i j ⟩

+∑

pqi

⟨p i |h2|qi ⟩ a †p

a q+1

4

∑

pqr s

⟨pq |h2|r s ⟩ a †p

a †q

a s a r , (2.9)

where it is conventional to introduce the reference expectation value ⟨Φ|H |Φ⟩, theone-body Fock operator FN , and the two-body interaction operator VN , leading to

H = ⟨Φ|H |Φ⟩+∑

pq

f pqa †

pa q+

1

4

∑

pqr s

v pqr sa †

pa †

qa s a r (2.10)

≡ ⟨Φ|H |Φ⟩+ FN + VN , (2.11)

for which the diagrammatic representations used in this work are shown in Fig-ure 2.1, and whose matrix elements are given by

⟨Φ|H |Φ⟩ = h0+∑

i

⟨i |h1|i ⟩+1

2

∑

i j

⟨i j |h2|i j ⟩ , (2.12)

f pq≡ ⟨p | f |q ⟩ = ⟨p |h1|q ⟩+

∑

i

⟨p i |h2|qi ⟩ , (2.13)

v pqr s≡ ⟨pq |v |r s ⟩ = ⟨pq |h2|r s ⟩ . (2.14)



FN : b l b l b l b l

VN : b b b b b b b b

b b b b b b

Figure 2.1: Goldstone diagrams representing the Fock operator FN and antisymmetrized Goldstone

diagrams for VN that come from distinct corresponding Hugenholtz diagrams.

Anticipating the treatment of three-body Hamiltonians later on, the matrix ele-ments v

pqr s are understood as the antisymmetrized matrix elements of the two-

body part VN of the normal-ordered Hamiltonian, which for two-body Hamiltoni-ans coincide with the ordinary antisymmetrized matrix elements of the two-bodyHamiltonian h2 in standard form.

Using the normal-ordered Hamiltonian,

HN = H −⟨Φ|H |Φ⟩ , (2.15)

and after subtracting the zero-body contribution, the Schrödinger equation canbe written in the form

HN e T |Φ⟩ = ∆E e T |Φ⟩ , (2.16)

in which the quantity

∆E ≡ E −⟨Φ|H |Φ⟩ (2.17)

is called the correlation energy. Since ⟨Φ|H |Φ⟩ is the expectation value of the Hamil-tonian in the reference state, it is also referred to as reference energy Eref,

Eref ≡ ⟨Φ|H |Φ⟩ . (2.18)


2.2. The Exponential Ansatz

Clearly, the total energy is then given by the sum of reference and correlation en-ergy. An alternative formulation of (2.16) is obtained by left-multiplication withe−T , arriving at

H |Φ⟩ = ∆E |Φ⟩ , (2.19)

where the (normal-ordered) Coupled-Cluster effective Hamiltonian H is defined as

H ≡ e−T HN e T . (2.20)

The form (2.19) of the Schrödinger equation is of particular importance and will bethe starting point for the derivation of the Coupled-Cluster equations. It is con-venient since all the complicated exponential structure of the Coupled-ClusterAnsatz is absorbed in H . By left-multiplication with e−T in (2.16), the ampli-tudes of the cluster operator have been separated from the energy so that the en-ergy will not appear in the amplitude equations. Since it was the presence of theenergy in the CI amplitude equations which caused violation of size extensivitythere, Coupled-Cluster theory circumvents this problem by introducing the effec-tive Hamiltonian [25].

Since T † 6= −T , the transformation (2.20) is not unitary, resulting in a non-Hermitean effective Hamiltonian. The transformation is, however, a similaritytransformation (which is why (2.19) is also referred to as similarity transformed

Schrödinger equation) and, therefore, the spectrum of the original Hamiltonian isnot altered. But the non-Hermitecity leads to an asymmetric expression for theenergy,

⟨Φ| e−T HN e T |Φ⟩ = ∆E , (2.21)

which is not subject to variational conditions, causing truncated Coupled-Clustermethods not to give an upper bound for the exact energy, contrary to methods forwhich the variational principle is fulfilled.

The exponential form (2.1) of the wavefunction is exact in the sense that for theuntruncated case it reproduces the Full CI wavefunction. In fact, the CI excitationoperators may be expressed by the Coupled-Cluster cluster operators by the one-to-one correspondence

C1 = T1 (2.22)

C2 =1

2!T 2

1+ T2 (2.23)

C3 =1

3!T 3

1+ T1T2+ T3 , (2.24)



or, for general excitation operators,

Cn =

n∑

k=1

1

k !

∑

m1...mk

δm1+...+mk , n

k∏

j=1

Tm j, (2.25)

which elucidates the relationship between Coupled Cluster and CI. However, theCoupled-Cluster Ansatz performs differently to CI when the cluster operator istruncated to some excitation rank M ,

T (M ) =

M∑

n=1

Tn . (2.26)

As in the CI case, the corresponding Coupled-Cluster method is called CCM, orfor M = 2 CCSD, and so on. Due to its nonlinear nature, the Coupled-ClusterAnsatz allows to generate higher-order excitations from products of lower-orderexcitation operators (Figure 2.2).

The individual importance of the terms in (2.22)-(2.25) may be roughly esti-mated using simple considerations. For Hartree-Fock bases, the T1 operator issmall 1 and, therefore, (2.22) is expected to contribute little. Since T 2

1is even

smaller than T1, the first term in (2.23) will, consequently, also contribute very lit-tle, leaving T2 in (2.23) as the dominant term. Analogously, the only term in (2.24)that does not involve T1 is the triples excitations cluster operator T3, which is ex-pected to be dominant. However, going to higher excitation types C4,5,..., the con-nected cluster operators T4,5,... are expected to become less relevant, since they rep-resent a simultaneous correlation of the corresponding number of nucleons. Forthat reason, T2 is typically already significantly more important than T3. In sum-mary, for a general excitation operator (2.25), the contributions from T2 and T3

clusters are expected to have most relevance.

2.3 Coupled-Cluster Equations

Analogously to the exact case (2.21), for a given (truncated) CCM methods with

T ≈ T (M) = T1 + T2 + . . . + TM , (2.27)

the expression for the correlation energy∆E (M) =∆E (t (M)) as function of the clusteramplitudes

t (M ) ≡nn

t ai

o

,n

t abi j

o

, . . . ,n

ta 1...a M

i 1,...,i M

oo

, (2.28)

1Here, an operator is called small if its matrix elements have small absolute values.


2.3. Coupled-Cluster Equations

b bb b b bbC bC bC bC bC

bC bC bC bC bC bCbC bC bC bC bC bC bCbC bC bC bC bC bC bC

|Φ〉

b bb bC b bbC bC bC bC bC

bC bC bC bC bC bCbC bC b bC bC bC bCbC bC bC bC bC bC bC

T1 |Φ〉

b bCbC b b bbC bC bC bC bC

bC bC bC bC bC bCbC bC b bC bC bC bCbC bC bC bC bC bC b

T2 |Φ〉

bC bCbC bC bC bbC bC bC bC bC

bC bC b bC bC bCb b bC b bC bC bCbC bC bC bC bC bC b

T1 T2 T2 |Φ〉

Figure 2.2: Illustration of excitation types in the CC ansatz.

can be derived by left-projecting the similarity-transformed Schrödinger equation

H (M) |Φ⟩ = ∆E (M) |Φ⟩ (2.29)

with

H (M) ≡ e−T (M) HN e T (M) , (2.30)

onto the reference state. In analogy to the CI case, a coupled set of algebraic equa-tions for the determination of the amplitudes t (M) is obtained by left-projectingthe similarity-transformed Schrödinger equation onto the excited determinants|Φa 1...a n

i 1...i n⟩ with n ≤M , i.e.,

⟨Φ| H (M) |Φ⟩ = ∆E (M) (2.31)

⟨Φai| H (M) |Φ⟩ = 0 , ∀ a , i (2.32)

⟨Φabi j| H (M) |Φ⟩ = 0 , ∀ a ,b , i , j (2.33)

...

⟨Φa 1...a M

i 1...i M| H (M) |Φ⟩ = 0 , ∀ a 1, . . . , a M , i 1, . . . , i M . (2.34)

In the case of CCSD, for example, the T1 and T2 amplitudes can be determined bysolving the system given by (2.32) and (2.33). Above expressions can be signifi-cantly simplified once it is recognized that the expansion of the effective Hamilto-nian H (M), containing two exponentials of the cluster operator, actually terminatesat finite expansion order. This is due to T (M) being an excitation operator, and the



finite expression for H (M) in case of two-body Hamiltonians HN reads 2

H (M) = HN +1

1!

HN , T (M)

+1

2!

HN , T (M)

, T (M)

+1

3!

HN , T (M)

, T (M)

, T (M)

+1

4!

HN , T (M)

, T (M)

, T (M)

, T (M)

. (2.35)

This commutator expansion of the effective Hamiltonian can be further simplifiedto the form

H (M) =

HN e T (M)

C, (2.36)

where the subscript C restricts the expressions inside the brackets to connected

terms, where the Hamiltonian has at least one contraction with each cluster oper-ator. Consequently, since a two-body Hamiltonian can be contracted with a max-imum number of 4 operators, the expansion of the exponential can be restrictedto terms containing not more than 4 cluster operators. Therefore, the most conve-nient form of the general Coupled-Cluster amplitude equations reads

⟨Φa 1...a n

i 1...i n|

HN e T (M)

C|Φ⟩ = 0 , n = 1, . . . , M . (2.37)

For CCSD [61, 65, 66, 70, 105–114], where the cluster operator is truncated at thesingles and doubles excitations,

T (CCSD) = T1 + T2 , (2.38)

the energy and amplitude equations, after expansion of the exponential, are thengiven by

∆E (CCSD) = ⟨Φ|h

HN

T1+ T2+1

2!T 2

1

i

C|Φ⟩ (2.39)

0 = ⟨Φai|h

HN

1+ T1+ T2+1

2!T 2

1+ T1T2+

1

3!T 3

1

i

C|Φ⟩ (2.40)

0 = ⟨Φabi j|h

HN

1+ T1+ T2+1

2!T 2

1+ T1T2

+ 1

2!T 2

2+ 1

3!T 3

1+ 1

2!T 2

1T2+

1

4!T 4

1

i

C|Φ⟩ . (2.41)

2Proving the non-terminating form of the commutator expansion is an easy exercise [104]:Setting H (M) = e−αT (M)HN e αT (M) , for the derivate holds dH (M)/dα = e−αT (M) [HN , T (M)]e αT (M) andconsequently d2H (M)/dα2 = e−αT (M) [[HN , T (M)], T (M)]e αT (M) and so on. Plugging into H (M) =∑

n

1

n !αn

dn H (M)

dαn

α=0

and setting α to 1 gives the commutator expansion.


2.3. Coupled-Cluster Equations

(SA)

+ f ai

(SB a )

+∑

c k

f kc t a c

i k

(SBb )

+ 12

∑

c d k

v a kc d t c d

i k

(SBc )

− 12

∑

c k l

v k li c t a c

k l

(SC a )

+∑

c

f ac t c

i

(SC b )

−∑

i

f ki t a

k

(SC c )

+∑

c k

v a ki c t c

k

(SDa )

− 12

∑

c d k l

v k lc d t a d

k l t ci

(SDb )

− 12

∑

c d k l

v k lc d t c d

i l t ak

(SDc )

+∑

c d k l

v k lc d t d a

l i t ck

(SE a )

−∑

c k

f kc t c

i t ak

(SEb )

+∑

c d k

v a kc d t c

i t dk

(SE c )

−∑

c k l

v k li c t a

k t cl

(SF )

−∑

c d k l

v k lc d t a

k t ci t d

l

= 0 , ∀ a , i

Figure 2.3: Algebraic expressions for the CCSD T1 amplitude equations.

Cluster operator products, such as 1

4!T 4

2, have already been left out in (2.39)–(2.41)

since their excitation rank is too high for the Hamiltonian to de-excite the resultingdeterminants to the state the equation is projected on. Therefore, for a Hamilto-nian with excitation rank X (H ), in the Tn amplitude equation only cluster operatorproducts P (T (M)) may appear with excitation ranks X (P (T (M))) ≤ X (HN ) + X (Tn ).Evaluating Eqs. (2.39)-(2.41) in terms of matrix elements of the operators involvedis a standard task using diagrammatic techniques [25,26]. The diagrams are listedin Appendix C.1, and the corresponding algebraic expressions are given in Fig-ures 2.3 and 2.4. In front of each term the assigned diagram is indicated wherethe naming convention has been taken from [26] 3. This facilitates the identifica-tion of the corresponding spherical expression presented later in this work. Theexpression for the correlation energy reads

∆E (CCSD) =(E A)

+1

4

∑

ab i j

vi j

ab t abi j

(E B )

+∑

a i

f ia

t ai

(EC )

+1

2

∑

ab i j

vi j

ab t ai

t bj

, (2.42)

and it is noteworthy that the expression above is also valid for all higher-orderCoupled-Cluster methods such as CCSDT, CCSDTQ, etc., provided that two-bodyHamiltonians are used. This stems from the obvious fact that it is not possible toform closed diagrams using a two-body interaction and cluster operators beyondT2. The correlation energy depends on all cluster operator amplitudes, of course,but the T3, T4, . . . amplitudes enter implicitly through the solution of the Coupled-Cluster amplitude equations.

3In Ref. [26], the algebraic expression for (DHa) has an incorrect sign.



(DA)

+ v abi j

(D B a )

+ Pab

∑

c

f bc t a c

i j

(D Bb )

− Pi j

∑

k

f kj t ab

i k

(D Bc )

+ 12

∑

c d

v abc d t c d

i j

(D Bd )

+ 12

∑

k

v k li j t ab

k l

(D B e )

+ Pab Pi j

∑

c k

v k bc j t a c

i k

(DC a )

+ 14

∑

c d k l

v k lc d t c d

i j t abk l

(DC b )

+ Pi j

∑

c d k l

v k lc d t a c

i k t b dj l

(DC c )

− 12

Pi j

∑

c d k l

v k lc d t d c

i k t abl j

(DC d )

− 12

Pab

∑

c d k l

v k lc d t a c

l k t d bi j

(DDa )

+ Pi j

∑

c

v abc j t c

i

(DDb )

− Pab

∑

k

v k bi j t a

k

(DE a )

− Pi j

∑

c k

f kc t ab

k j t ci

(DEb )

− Pab

∑

c k

f kc t cb

i j t ak

(DE c )

+ Pab Pi j

∑

c d k

v a kc d t d b

k j t ci

(DE d )

− Pab Pi j

∑

c k l

v k li c t cb

l j t ak

(DE e )

− 12

Pab

∑

c d k

v k bc d t c d

i j t ak

(DE f )

+ 12

Pi j

∑

c k l

v k lc j t ab

k l t ci

(DE g )

+ Pab

∑

c d k

v k ac d t d b

i j t ck

(DE h)

− Pi j

∑

c k l

v k lc i t ab

l j t ck

(DFa )

+∑

c d

v abc d t c

i t dj

(DF b )

+∑

k l

v k li j t a

k t bl

(DF c )

− Pab Pi j

∑

c k

v k bc j t a

k t ci

(DG a )

+ 12

∑

c d k l

v k lc d t ab

k l t ci t d

j

(DG b )

+ 12

∑

c d k l

v k lc d t c d

i j t ak t b

l

(DG c )

− Pab Pi j

∑

c d k l

v k lc d t d b

l j t ak t c

i

(DG d )

− Pi j

∑

c d k l

v k lc d t ab

l j t ck t d

i

(DG e )

− Pab

∑

c d k l

v k lc d t d b

i j t al t c

k

(DHa )

− Pab

∑

c d k

v k bc d t a

k t ci t d

j

(DHb )

+ Pi j

∑

c k l

v k lc j t a

k t bl t c

i

(DI )

+∑

c d k l

v k lc d t a

k t bl t c

i t dj = 0 , ∀ a ,b , i , j

Figure 2.4: Algebraic expressions for the CCSD T2 amplitude equations.


2.4. Effective Hamiltonian

The Coupled-Cluster amplitude equations, which are of the form

G (t ) = 0 (2.43)

are usually converted into a fixed-point problem

I (t (n )) = t (n+1) , I (t ∗) = t ∗ , (2.44)

and solved iteratively. A common choice for the iteration scheme is

(0)ta 1...a n

i 1...i n= 0

(n+1)ta 1...a n

i 1...i n=

⟨Φa 1...a ni 1...i n

| (HN e T )C(t(n )) |Φ⟩

fi 1i 1+ ...+ f

i ni n− f

a 1a 1− ...− f

a na n

, FN → F oN ,

(2.45)

where in the amplitude equations the Fock operator FN is replaced by its off-diagonal F o

N part [26]. An alternative iteration scheme that leads to more stableiterations is considered in Section 4.8.

2.4 Effective Hamiltonian

Once the CCSD amplitude equations have been solved, the effective Hamiltonian

H (CCSD) = e−T (CCSD)

HN e T (CCSD)

(2.46)

may be constructed explicitly from the cluster amplitudes t (CCSD). Recalling thecommutator expansion of the effective Hamiltonian (2.20), it is apparent that theCCSD effective Hamiltonian H (CCSD) will contain up to six-body operator terms,

H (CCSD) = H0+ H1+ H2+ H3+ H4+ H5+ H6 , (2.47)

which are generated by the four-fold commutator (2.35). Since each Hk is assumedto be in normal-ordered form, H (CCSD) can directly be written as

H (CCSD) = H0+∑

pq

H pqa †

pa q+

1

4

∑

pqr s

H pqr sa †

pa †

qa s a r

+1

36

∑

pqr s t u

H pqrs t u a †

pa †

qa †

ra u a t a s + . . . ,

≡ H0+ Hopen , (2.48)

again employing the short-hand notation

H p1...pn

q1...qn≡ ⟨p1 . . . pn |Hn |q1 . . .qn ⟩ (2.49)



H1 : b r b r b r b r

H2 : b b b b b b b b

b b b b b b

H3 : b b b b b b b b b . . .

H4 : b b b b . . .

Figure 2.5: Antisymmetrized Goldstone diagrams for H1 and H2 as well as selected H3 and H4

diagrams that arise from distinct corresponding Hugenholtz diagrams.

for antisymmetrized matrix elements, and where Hopen denotes the part of theeffective Hamiltonian with open Fermion lines, as opposed to the closed effectiveHamiltonian diagrams H0. It should be noted that the non-Hermitecity of theeffective Hamiltonian implies that

H pq ...r s ...6=

H r s ...pq ...

∗. (2.50)

For the Coupled-Cluster methods considered in this work that only involve two-body Hamiltonians, only the one- and two-body parts of H (CCSD) will be needed.The expressions for the matrix elements depend on the particle-hole characterof the orbitals and, therefore, the Hamiltonian is split into different particle-hole



H0 = ∆E (CCSD)

H ia = f i

a +∑

c k

v i ka c t c

k

H ab = f a

b +∑

c k

v a kb c t c

k −∑

k

t ak H

kb −

12

∑

c k l

v k lb c t a c

k l

H ij = f i

j +∑

c k

v i kj c t c

k +∑

c

t cj H

ic +

12

∑

c d k

v i kc d t c d

j k

H ai = 0

H i j

ab= v

i j

ab

H a j

ib=

χ ′′′a j

ib+ 1

2

∑

c

t ci H

a j

cb+∑

c k

vk j

cbt a c

i k

H a ib c = v a i

b c −∑

k

v k ib c t a

k

H i kj a = v i k

j a +∑

c

v i kc a t c

j

H abc d =

χ ′ab

c d +12

∑

k l

v k lc d t ab

k l

H i j

k l= v

i j

k l+ Pk l

∑

c

t cl

χ ′i j

k c+ 1

2

∑

c d

vi j

c dt c d

k l

H abc i = v ab

c i +∑

d

v abc d t d

i + Pab

∑

k

t bk

χ ′a k

i c −∑

k

t abk i H

kc

+Pab

∑

d k

t d bk i H

a kc d +

12

∑

k l

t abk l H

k lc i

H i aj k =

χ ′′i a

j k +∑

c

t c aj k H

ic

H abi j = 0

Figure 2.6: Algebraic expressions for the effective Hamiltonian matrix elements. For the defini-

tions of the intermediates χ see Figure 2.7.



χ ′a i

b c = v a ib c −

12

∑

k

v k ib c t a

k

χ ′a j

ib= v

a j

ib− 1

2

∑

k

vk j

ibt a

k +∑

c

t ci

χ ′a j

cb

χ ′′a j

ib= v

a j

ib− 1

2

∑

k

vk j

ibt a

k +12

∑

c

t ci

χ ′a j

cb

χ ′′′a j

ib= v

a j

ib−∑

k

vk j

ibt a

k +12

∑

c

t ci H

a j

cb

χ ′i k

j a = v i kj a +

12

∑

c

v i kc a t c

j

χ ′ab

c d = v abc d − Pab

∑

k

t bk

χ ′a k

c d

χ ′ab

c i = v abc i +

12

∑

d

v abc d t d

i − Pab

∑

k

t bk

χ ′′a k

c i

χ ′i a

j k = v i aj k −

12

∑

l

v i lj k t a

l

χ ′′i a

j k = v i aj k −

∑

l

v i lj k t a

l − Pj k

∑

c

t ck

χ ′′′a i

j c

+ Pj k

∑

c l

t c al k H

i lj c +

12

∑

c d

t c dj k H

i ac d

Figure 2.7: Intermediates used in Figure 2.6.


2.5. The ΛCCSD Equations

topologies, for the one-body part H1

H1 =∑

i j

H ija †

ia j +

∑

i a

H iaa †

ia a +

∑

a i

H aia †

aa i +

∑

ab

H aba †

aa b

(2.51)

and analogous for H2, where the algebraic expressions of the corresponding ma-trix elements are listed in Figures 2.6 and 2.7. In effective Hamiltonian terms theCCSD T1 and T2 amplitude equations may be represented as

H ai= 0 , ∀ a , i (2.52)

H abi j

= 0 , ∀ a ,b , i , j (2.53)

emphasizing that theH ph andH pp

hh matrix elements of H (CCSD) vanish if the CCSDequations are satisfied by the cluster amplitudes.

Having the effective Hamiltonian in explicit form allows to formulate furtherapplications like ΛCCSD(T) or EOM-CCSD – basically being a diagonalization ofH (CCSD) – in terms of effective Hamiltonian matrix elements which results in morecompact expressions.

2.5 The ΛCCSD Equations

The Coupled-Cluster Λ operator appears on several occasions in Coupled-Clustertheory, as in the expression for the the energy derivative in the context of responsetreatment of properties,

ddλ∆E (λ) = ⟨Φ|

1+Λ dH (λ)

dλ|Φ⟩ , (2.54)

or in the fundamental energy functional,

E (CC)(Λ, T ) = ⟨Φ|

1+Λ

H |Φ⟩ , (2.55)

which, when stationary, gives the Coupled-Cluster correlation energy. From there,it attains importance in the calculation of higher-order contributions to the en-ergy. Furthermore, because the Λ operator is determined from solving the left-eigenfunction equation for H , the Λ equations are to a large extent equivalent toequations encountered in Equation-of-Motion Coupled-Cluster theory. Since themain motivation for the Λ operator is from the properties treatment which is ofminor interest in this work, only a very brief review is given in the following.



The Λ operator parametrizes the bra counterpart ⟨Ψ| of the Coupled-Clusterground state |Ψ⟩ according to

⟨Ψ| = ⟨Φ| ( 1+Λ) e−T , (2.56)

which satisfies the biorthonormality condition

⟨Ψ|Ψ⟩ = 1 (2.57)

and where

Λ =

A∑

n=1

Λn (2.58)

is a sum of npnh de-excitation operators Λn ,

Λ1 = =1

(1!)2

∑

a i

λiaa †

ia a (2.59)

Λ2 = =1

(2!)2

∑

ab i j

λi j

ab a †ia †

ja b a a (2.60)

...

Λn = =1

(n !)2

∑

a 1...a ni 1...i n

λi 1...i n

a 1...a na †

i 1. . . a †

i na a n

. . . a a 1 . (2.61)

The corresponding amplitude equations can be obtained by considering theSchrödinger equation

⟨Ψ|HN = ∆E ⟨Ψ| (2.62)

and using Ansatz (2.56)

⟨Φ| ( 1+Λ) e−T HN = ∆E ⟨Φ| ( 1+Λ) e−T . (2.63)

Right-multiplication with e T leads to a formulation in terms of the effective Hamil-tonian,

⟨Φ| ( 1+Λ) H = ∆E ⟨Φ| ( 1+Λ) , (2.64)

which, using

H = H01+ Hopen , (2.65)


2.5. The ΛCCSD Equations

can be cast in the energy-independent form

⟨Φ| ( 1+Λ) Hopen = 0 . (2.66)

In the case of ΛCCSD, where the Λ operator is approximated as

Λ ≈ Λ(CCSD) = Λ1+Λ2 , (2.67)

and the effective Hamiltonian Hopen is replaced by H (CCSD)open , the Λ1 and Λ2 ampli-

tudes can be obtained from solving the system of linear equations [76, 115–121]

⟨Φ| ( 1+Λ1+Λ2 ) H (CCSD)open

|Φai⟩ = 0 (2.68)


|Φabi j⟩ = 0 . (2.69)

Since these systems of equations are typically very large, they are also solved it-eratively, and a similar iteration scheme as for CCSD, Eq. (2.45), can be set up.From excitation rank considerations follows that only one-, two- and three-bodycomponents of H (CCSD)

open enter the above system which also can be written as [96]

0 = ⟨Φ|n

(1+Λ1) H1

C+

(Λ1+Λ2) H2

C+

Λ2 H3

C

o

|Φai⟩ (2.70)

0 = ⟨Φ|n

(1+Λ1+Λ2) H2

C+

Λ2 H1

C

+

Λ1 H1

DC+

Λ2 H3

C

o

|Φabi j⟩ . (2.71)

where the label DC represents disconnected operator products. As for the T (CCSD)

amplitude equations, these expressions can be evaluated using standard diagram-matic techniques (for diagrams see Appendix E.1). The resulting equations for theΛ1 and Λ2 amplitudes in terms of effective Hamiltonian matrix elements are listedin Figure 2.8 and the corresponding spherical expressions are appended in Ap-pendix E.2. In Figure 2.8, the matrix elements of the three-body operator H3 havebeen expressed in terms of matrix elements of lower-rank operators accordingto [26, 96]

H c d ik l a

= −Pk l

∑

m

H i ma l

t c dk m+ Pc d

∑

e

H i da e

t c ek l

(2.72)

H i j c

k b l = −∑

d

t c dk l

vi j

d b (2.73)

H c j d

ab k = −∑

l

t c dk l

vl j

ab (2.74)



(Λ1A)

+ H ia

(Λ1 B)

+∑

c

λic H

ca

(Λ1C )

−∑

k

λka H

ik

(Λ1D)

+∑

c k

λkc H

c ik a

(Λ1 E )

+ 12

∑

c d k

λi kc d H

c da k

(Λ1 F )

− 12

∑

c k l

λk la c H

i ck l

(Λ1G )

+ 12

∑

c d e k l

λk lc d H

d ie a t c e

k l

(Λ1H)

− 12

∑

c d k l m

λk lc d H

m il a t c d

k m = 0 , ∀ a , i

(Λ2A)

+ H i j

ab

(Λ2 B)

+ Pab Pi j λj

bH i

a

(Λ2C )

+ Pi j

∑

c

λic H

c j

ab

(Λ2D)

− Pab

∑

k

λka H

i j

k b

(Λ2 E )

+ Pab

∑

c

λi ja c H

cb

(Λ2 F )

− Pi j

∑

k

λi kab H

j

k

(Λ2G )

+ Pab Pi j

∑

c k

λi ka c H

c j

k b

(Λ2H)

+ 12

∑

c d

λi j

c dH c d

ab

(Λ2 I )

+ 12

∑

k l

λk lab H

i j

k l

(Λ2 J )

− 12

Pab

∑

c d k l

λk lc a t c d

k l vi j

d b

(Λ2 K )

− 12

Pi j

∑

c d k l

λk ic d t c d

k l vl j

ab= 0 , ∀ a ,b , i , j

Figure 2.8: Algebraic expressions for the ΛCCSD Λ1 and Λ2 amplitude equations.

in order to avoid the use of six-index quantities. Therefore, it should be noted thatin effective Hamiltonian terms it can be written

(Λ1G)+ (Λ1H) =b b b

=1

4

∑

c d k l

λk lc dH c d i

k l a, (2.75)

(Λ2J) =b b b

=1

4Pab Pi j

∑

c k l

λk lc aH i j c

k b l , (2.76)


2.6. Expectation Values

and

(Λ2K) =b b b

=1

4Pab Pi j

∑

c d k

λk ic dH c j d

ab k . (2.77)

2.6 Expectation Values

The evaluation of expectation values

¬

G¶

≡ ⟨Ψ| G |Ψ⟩⟨Ψ|Ψ⟩ =

⟨Φ| e T †G e T |Φ⟩

⟨Φ| e T †e T |Φ⟩

(2.78)

is inherently more complicated in Coupled-Cluster theory than in methods thatemploy a linear parametrization of the wavefunction. The reason is that the ex-pression ⟨Φ|e T †

GN e T |Φ⟩, in which both excitation and de-excitation type operatorsappear, does not terminate as for instance the expression ⟨Φ|e−T GN e T |Φ⟩ does, inwhich only excitation operators appear.

As for the Coupled-Cluster energy, which in the untruncated case must of coursebe the same as the Hamiltonian expectation value, any expectation value may beseparated into its reference and correlation part,

¬

G¶

= ⟨Φ|G |Φ⟩+¬

GN

¶

, (2.79)

with the correlation part being the non-trivial task in the expectation value cal-culation. However, the denominator cancels against the disconnected parts fromthe nominator [100],

¬

GN

¶

=⟨Φ| e T †

GN e T |Φ⟩⟨Φ| e T †

e T |Φ⟩= ⟨Φ|

e T †

GN e T

C|Φ⟩ , (2.80)

leaving a connected form for the correlation part. In order to detach the generalproblem of the expectation value evaluation from the specific operator G , the n-

body reduced density operator

γ(n )N =

∑

p1...pnq1...qn

|q1 . . .qn ⟩γN

q1...qn

p1...pn⟨p1 . . .pn | (2.81)

with matrix elementsγN

q1...qn

p1...pn= ⟨Φ|

h

e T † a †p1

. . . a †pn

a qn. . . a q1

e Ti

C|Φ⟩ (2.82)



γN

ia =

(γhpA)+ λi

a

γN

ai =

(γphA)+ t a

i

(γphB)+∑

c k

λkc t a c

i k

(γphC)−∑

c k

λck t a

k t ci

(γphD)− 1

2

∑

c d k l

λk lc d t c d

k i t al

(γphE)− 1

2

∑

c d k l

λk lc d t c a

k l t di

γN

ba =

(γppA)+∑

k

λka t b

k

(γppB)+ 1

2

∑

c k l

λk lc a t cb

k l

γN

j

i =(γhhA)−∑

c

λjc t c

i

(γhhB)− 1

2

∑

c d k

λk j

c dt c d

k i

Figure 2.9: Algebraic expressions for the one-body response density matrix elements.

is introduced. It is clearly independent from the operator under consideration,and the expectation value of any n-body operator in normal order may be ex-pressed as the contraction of the operator matrix elements with the density matrixelements,

¬

GN

¶

=

n∑

k=1

1

(k !)2

∑

p1...pkq1...qk

⟨p1 . . . pk |g N |q1 . . .qk ⟩γN

q1...qk

p1...pk. (2.83)

For the case of one-body operators O, the reference part of the expectation valuemay directly be incorporated in the contraction by the use of a modified densitymatrix γp

q ,

¬

O¶

=∑

pq

⟨p |o|q ⟩ γqp

, γqp≡¨

(γN )qp +δpq : p ,q ∈ holes

(γN )qp : else .

(2.84)

For a two-body operator V in vacuum normal order, a formulation of the expecta-tion value involving only the two-body matrix elements ⟨pq |v |r s ⟩ is easily foundas [26]

¬

V¶

=1

4

∑

pqr s

⟨pq |v |r s ⟩γN

r s

pq+∑

pq

∑

i

⟨p i |v |qi ⟩

γN

q

p+

1

2

∑

i j

⟨i j |v |i j ⟩ .

(2.85)


2.6. Expectation Values

γN

abc i =

γppphA

+∑

k

t abk i λ

kc

γppphB

+ Pab

∑

k

t ak t b

i λkc

γppphC

+ 12

Pab

∑

d k l

t d ak l t b

i λk ld c

γppphD

+ Pab

∑

d k l

t b di l t a

k λk lc d

γppphE

− 12

∑

d k l

t abk l t d

i λk lc d

γppphF

− Pab

∑

d k l

t ak t b

l t di λ

k lc d

γppphG

− Pab

∑

d k l

t a dk i t b

l λk lc d

γN

abi j = Pab Pi j

¨(γ

pphhA)+ 1

4t ab

i j

(γpphhB)+ t a

i t bj

(γpphhC)+

∑

c k

t c ak i t b

j λkc

(γpphhD)− 1

2

∑

c k

t abk j t c

i λkc

(γpphhE)− 1

2

∑

c k

t cbi j t a

k λkc

(γpphhF)−

∑

c k

t ak t c

i t bj λ

kc

(γpphhG)+

∑

c d k l

t c ak i t b d

j l λk lc d

(γpphhH)− 1

4

∑

c d k l

t c dk i t ab

l j λk lc d

(γpphhI)− 1

4

∑

c d k l

t c ak l t d b

i j λk lc d

(γpphhJ)+ 1

16

∑

c d k l

t abk l t c d

i j λk lc d

(γpphhK)− 1

2

∑

c d k l

t c dk i t a

l t bj λ

k lc d

(γpphhL)− 1

2

∑

c d k l

t c ak l t d

i t bj λ

k lc d

(γpphhM)−

∑

c d k l

t b dj l t a

k t ci λ

k lc d

(γpphhN)+ 1

4

∑

c d k l

t abk l t c

i t dj λ

k lc d

(γpphhO)+ 1

4

∑

c d k l

t c di j t a

k t bl λ

k lc d

(γpphhP)+

∑

c d k l

t ak t c

i t bl t d

j λk lc d

«

Figure 2.10: Algebraic expressions for the two-body response density matrix elements.



γN

i ab j =

γhpphA

+ t aj λ

ib

γhpphB

+∑

c k

t c ak j λ

k icb

γhpphC

−∑

c k

t ak t c

j λi kb c

γN

i j

ab=(γhh

ppA)+ λ

i j

ab

γN

abc d =

(γppppA)+ 1

2

∑

k l

t abk l λ

k lc d

(γppppB)+ Pab

∑

k l

t ak t b

l λk lc d

γN

i j

k l=(γhh

hhA)+ 1

2

∑

c d

t c dk l λ

i j

c d

(γhhhhB)+ Pk l

∑

c d

t ck t d

l λi j

c d

γN

i aj k =

γhphhA

−∑

c

t c aj k λ

ic

γhphhB

− Pj k

∑

c

t cj t a

k λic

γhphhC

− 12

Pj k

∑

c d l

t c dl j t a

k λl ic d

γhphhD

− Pj k

∑

c d l

t a dk l t c

j λi lc d

γhphhE

+ 12

∑

c d l

t c dj k t a

l λi lc d

γhphhF

+ Pj k

∑

c d l

t cj t d

k t al λ

i lc d

γhphhG

+ Pj k

∑

c d l

t c aj l t d

k λi lc d

γN

a ib c =

γphppA

+∑

k

t ak λ

k ib c

γN

i j

k a=

γhhhpA

−∑

c

t ck λ

i jc a

Figure 2.11: Algebraic expressions for the two-body response density matrix elements, continued.


2.7. The ΛCCSD(T) Energy Correction

By solving the Λ equations an alternative form of the density matrix elements

(γN )q1...qn

p1...pn= ⟨Φ|

h

1+Λ

a †p1

. . . a †pn

a qn. . . a q1

e T

C

i

C|Φ⟩ (2.86)

may be used, which, unlike (2.82) leads to terminating expressions. The CCSDone- and two-body matrix elements of the reduced density matrix derived from(2.86) are listed in Figures 2.9-2.11, and the spherical expressions can be found inAppendix F.

2.7 The ΛCCSD(T) Energy Correction

While the CCSD equations are rather easy to solve, the solution of the full CCSDTequations is far out of reach for all but the lightest nuclei. Nonetheless, higher-order excitations may be included in the calculations via a combination of the iter-ative, infinite-summation techniques obtained from solving the Coupled-Clusterequations for low-rank clusters, with an a posteriori non-iterative correction dueto higher-rank clusters, typically based on perturbation-theory considerations.There is an abundance of different methods that have emerged in the field ofquantum chemistry, such as CCSD[T] [122,123], CCSD(T) [124], CCSD(TQf) [125],ΛCCSD(T) [57, 126], ΛCCSD(TQf) [127], CCSD(2)T [128–131], CCSD(2) [128–131],CR-CCSD(T) [132–136], CR-CCSD(TQ) [132–136], CR-CC(m ,m ′) [117,119,121,137,138], CR-CC(2,3)+Q [139], LR-CCSD(T) [140], or LR-CCSD(TQ) [140]. In this work,besides the CR-CC(2,3) correction, the main focus will be on the ΛCCSD(T) non-iterative energy correction [57,126] due to its rather simple structure but yet accu-rate results. However, in this section only a brief overview is given, because themethod is discussed in more detail in the context of three-body Hamiltonians, seeSection 3.4.

Starting point for the derivation of the ΛCCSD(T) correction is an expansionof the Coupled-Cluster energy functional

E (CC) = ⟨Φ| (1+Λ) H |Φ⟩ (2.87)

for CCSDT up to fourth order perturbative contributions relative to the CCSDground-state wavefunction, which can be formulated in converged CCSD T1, T2

and ΛCCSD Λ1, Λ2 amplitudes, and by determining the corresponding energy cor-rection from this functional [57, 58]. This results in an expression for the energy



δE (ΛCCSD(T)) =1

(3!)2

∑

ab ci j k

λi j k

ab c

1

εab ci j k

t ab ci j k

λi j k

ab c= Pa/b c Pk/i j

∑

d

v d kb c λ

i j

a d− Pc/ab Pi/j k

∑

l

vj k

l cλi l

ab

+ Pi/j k Pa/b c λia v b c

j k + Pi/j k Pa/b c f ia λ

j k

b c

t ab ci j k = Pa/b c Pk/i j

∑

d

v b cd k t a d

i j − Pc/ab Pi/j k

∑

l

v l cj k t ab

i l

Pp/qr = 1− Tpq − Tp r

Figure 2.12: Algebraic expressions for the computation of δE (ΛCCSD(T)).

correction of the form

δE (ΛCCSD(T)) (2.88)

=1

(3!)2

∑

ab ci j k

⟨Φ| Λ

F odN+ VN

|Φab ci j k⟩ 1

εab ci j k

⟨Φab ci j k|

VN T2

C|Φ⟩ ,

where F odN is the off-diagonal part of the normal-ordered Fock operator FN and the

energy denominator εab ci j k is defined as

εab ci j k

≡ f ii+ f

j

j + f kk− f a

a− f b

b− f c

c. (2.89)

The corrected total energy is therefore given by

E (ΛCCSD(T)) = Eref+∆E (CCSD)+δE (ΛCCSD(T)) . (2.90)

Explicit expressions for the evaluation of (2.88) can be found in Figure 2.12. Be-cause the enormous number 4, namely h3p3, of six-index amplitudes cannot bestored at once, it is common practice to organize the calculation according to

δE (ΛCCSD(T)) =∑

i<j<k

1

3!

∑

ab c

λi j k

ab c

1

εab ci j k

t ab ci j k

(2.91)

where the bracket is evaluated for each i , j , k index combination separately, re-quiring only the storage of p3 tensors. The particle index sum is not restricted toa < b < c in order to use optimized BLAS [141] routines for the calculation of theλ

i j k

ab c and t ab ci j k tensors.

4The number of hole orbitals is denoted by h and p is the number of particle orbitals.


2.8. The Completely-Renormalized Coupled-Cluster Method CR-CC(2,3)

2.8 The Completely-Renormalized Coupled-ClusterMethod CR-CC(2,3)

The completely renormalized Coupled-Cluster methods (CR-CC) and particularthe CR-CC(2,3) version [114, 117, 119, 120, 137, 138] are promising alternatives tothe ΛCCSD(T) approach, since they are more complete, and, therefore, expectedto be more accurate. The CR-CC(m , m ′) methods are based on asymmetric energyexpressions and the moment expansion of the full CI energy, defining the methodof moments of the Coupled-Cluster equations [132–136, 142, 143]. This frame-work encompasses all sorts of energy corrections and its comprehensive structuregreatly facilitates the theoretical discussion. In this work, for instance, this featureis exploited by using CR-CC(2,3) as the base from which theΛCCSD(T) method forthree-body Hamiltonians is derived in Section 3.4. Apart from the theoretical con-text, the CR-CC(2,3) method will also be used in practical applications because itis worthwhile to have multiple triples correction methods at hand. This is becauseas long as full CCSDT calculations remain too expensive, the quality of triples cor-rection approaches has to be estimated from the comparison of different methods.Furthermore, it is encouraging to note that the final CR-CC(2,3) equations are ac-tually not significantly more complex than theirΛCCSD(T) counterparts, resultingin a similar implementational effort.

Using the definitions of the left and right Coupled-Cluster eigenstates,

⟨Ψ| = ⟨Φ| (1+Λ)e−T and |Ψ⟩ = e T |Φ⟩ , (2.92)

the exact correlation energy

∆E = ⟨Ψ|HN |Ψ⟩ (2.93)

may be expressed in terms of the Λ operator and the effective Hamiltonian,

∆E = ⟨Φ|

1+Λ

H |Φ⟩ . (2.94)

Of course, in practice neither Λ nor H are known. However, the exact correlationenergy can still be obtained if in (2.94) the CCSD effective Hamiltonian is used,provided that an appropriate redefinition of the operator acting on the left refer-ence state is employed,

∆E = ⟨Φ| L H (CCSD) |Φ⟩ , (2.95)



with L being a de-excitation operator of the form

L =

A∑

n=1

Ln , Ln =1

(n !)2

∑

i 1...i na 1...a n

l i 1...i n

a 1...a na †

ia †

ja †

k a c a b a a . (2.96)

Indeed, if L is determined such that ⟨Φ|L represents the lowest-energy left eigen-state 5 of H (CCSD),

⟨Φ| L H (CCSD) = ∆E ⟨Φ| L , (2.97)

and assuming the normalization

⟨Φ|L |Φ⟩ = 1 , (2.98)

it is easy to check that (2.95) holds. The ultimate goal is to split (2.95) into theinformation provided by CCSD and the information for the contributions beyondCCSD, which can be completely absorbed in the operator L . In order to do so, itis convenient to insert a resolution of the identity of the form

1 = |Φ⟩⟨Φ|+ P +Q , (2.99)

with projection operators

P = P1+ P2 (2.100)

Q = P3+ · · ·+ PA (2.101)

where

Pn =∑

i 1<···<i na 1<···<a n

|Φa 1...a n

i 1...i n⟩⟨Φa 1...a n

i 1...i n| , (2.102)

between the L and H (CCSD) operators in (2.95). This allows to make use of theproperties of H (CCSD)

⟨Φ| H (CCSD) |Φ⟩ = ∆E (CCSD) , (2.103)

⟨Φai| H (CCSD) |Φ⟩ = 0 , (2.104)

⟨Φabi j| H (CCSD) |Φ⟩ = 0 , (2.105)

which immediately allows to write (2.95) in the form

∆E = ∆E (CCSD) + ⟨Φ| L Q H (CCSD) |Φ⟩ (2.106)

= ∆E (CCSD) + δE . (2.107)

5Since H (CCSD) is given by a similarity transformation of HN , both operators exhibit the samespectrum and, consequently, the lowest eigenvalue of H (CCSD) is ∆E .



Thus, the exact energy correction δE for the contributions beyond CCSD is givenby

δE = ⟨Φ| L Q H (CCSD) |Φ⟩ (2.108)

and the only unknown in this expression is L . Consequently, in the followingit is L for which approximations are introduced in order to derive manageableexpressions.

Since the goal is to derive an energy correction δE (T) due to triply excited clus-ters, the projector Q is approximated by the space spanned by the triply exciteddeterminants,

Q ≈ P3 , (2.109)

which projects the L3 component out of L in the expression 6

LQ = L3 P3 . (2.110)

Thus, the triples energy correction δE (T) reads

δE (T) = ⟨Φ| L3 P3 H (CCSD) |Φ⟩ . (2.111)

Introducing the matrix elements of L3,

li j k

ab c = ⟨Φ| L3 |Φab ci j k⟩ , (2.112)

and using the definition of the so-called generalized moments of the CCSD equa-tions [132,133, 135, 136, 144]

Mab ci j k

= ⟨Φab ci j k| H (CCSD) |Φ⟩ , (2.113)

the triples energy correction can be cast into a form given as contraction of the L3

amplitudes with the moments,

δE (T) =1

(3!)2

∑

ab ci j k

li j k

ab c Mab ci j k

. (2.114)

Still, the moments Mab ci j k only carry CCSD information, while all information be-

yond CCSD is contained in the yet unknown operator L3. In the CR-CC(2,3)

6An alternative point of view is of course to approximate L3 by its three-body part L3.



method [114, 117, 119, 120, 137, 138], the L3 operator is determined in a quasi-perturbative manner, using the expression [117,119]

⟨Φ| L3 = ⟨Φ|

1+Λ(CCSD)

H (CCSD) R (CCSD)3 , (2.115)

which exploits the formal similarity between the L and the Λ operators [117,119],and where

R (CCSD)3 =

P3

∆E (CCSD)−H (CCSD)(2.116)

is the reduced resolvent of H (CCSD) in the subspace spanned by the triply exciteddeterminants, which has the property

P3

∆E (CCSD)−H (CCSD)P3

∆E (CCSD)−H (CCSD)

P3 = P3 . (2.117)

This allows to write the triples correction in the form

δE (T) = ⟨Φ|

1+Λ(CCSD)

H (CCSD) R (CCSD)3 H (CCSD) |Φ⟩ . (2.118)

In order to avoid the explicit construction of the reduced resolvent R (CCSD)3 in the

above expression, by right-multiplication with

P3


P3 (2.119)

making use of (2.117) and projecting onto |Φab ci j k ⟩, the CR-CC(2,3) Ansatz (2.115) for

L3 may be written as∑

l<m<nd<e< f

⟨Φd e f

l m n |


|Φab ci j k⟩ l l m n

d e f

= ⟨Φ|

1+Λ(CCSD)

H (CCSD) |Φab ci j k⟩ , (2.120)

which can be cast into the energy-independent form

−∑

l<m<nd<e< f

⟨Φd e f

l m n | H (CCSD)open

|Φab ci j k⟩ l l m n

d e f= ⟨Φ|

1+Λ(CCSD)

H (CCSD)open

|Φab ci j k⟩ . (2.121)

This formulation of CR-CC(2,3) is invariant under arbitrary rotations of occupiedand unoccupied orbitals. This requirement can be lifted due to the fact that thecalculations in this work use Hartree-Fock, and thus fixed, orbitals. Then, the sys-tem of equations (2.120) or (2.121) can be replaced by a non-iterative expression,such as [117,119, 137, 138]

li j k

ab c = ⟨Φ|

1+Λ(CCSD)

H (CCSD)open

|Φab ci j k l⟩

Dab ci j k

−1

, (2.122)



δE (CR−CC(2,3)) =1

(3!)2

∑

ab ci j k

li j k

ab c Mab ci j k

Mab ci j k

= Pab c Tab ci j k

Tab ci j k

= Pi j /k

1

2

∑

e

H abk e

t e ci j− 1

2

∑

m

J m ci j

t abk m

li j k

ab c = Ni j k

ab c /Dab ci j k

Ni j k

ab c = Pab c Γi j k

ab c

Γi j k

ab c = Pi j /k

h1

2λk

cH i j

ab +1

2λ

i j

ab H kc

+ 1

2

∑

e

λi je cH k e

ab− 1

2

∑

m

λk mabH i j

m c

i

Dab ci j k

= H ii+H j

j +H kk−H a

a−H b

b−H c

c

−H a ia i−H b i

b i−H c i

c i−H a j

a j −Hb j

b j −Hc j

c j

−H a ka k−H b k

b k−H c k

c k+H i j

i j +H i ki k+H j k

j k

+H abab+H a c

a c+H b c

b c

+H i j a

i j a +H i k ai k a+H j k a

j k a +Hi j b

i j b +H i k bi k b+H j k b

j k b

+H i j c

i j c +H i k ci k c+H j k c

j k c −H ab iab i−H a c i

a c i−H b c i

b c i

−H ab j

ab j −Ha c j

a c j −Hb c j

b c j −H ab kab k−H a c k

a c k−H b c k

b c k

H ab iab i

=∑

m

v i mab

t abi m

H i j a

i j a = −∑

e

v i ja e

t a ei j

J m ci j

= H m ci j−∑

e

H me

t e ci j

Figure 2.13: Algebraic expressions for the calculation of the CR-CC(2,3) energy correction for

ground states [138].



employing the denominator

Dab ci j k

= ∆E (CCSD)−⟨Φab ci j k| H (CCSD) |Φab c

i j k⟩ (2.123)

= −3∑

n=1

⟨Φab ci j k| H (CCSD)

n|Φab c

i j k⟩ . (2.124)

The working equations for the CR-CC(2,3) method can be found in [138], wherethey are presented in a form that also includes excited-state corrections. In Fig-ure 2.13, the simplified version of these equations is given that only considersthe ground-state triples correction. Some minor modifications have already beenmade for convenience later on regarding the angular-momentum coupled formu-lation.

2.9 Equation-of-Motion Coupled Cluster

In addition to ground-state wavefunctions and properties, excited states and theirproperties can be accessed within the Coupled-Cluster framework. In this work,the Equation-of-Motion CCSD (EOM-CCSD) [26,76] approach is employed, wherefor excited states |Ψ(CCSD)

µ ⟩ a linear and thus CI-like excitation operator, truncated atthe 2p2h excitation level,

R(CCSD)µ

= Rµ,0+ Rµ,1+ Rµ,2 (2.125)

= Rµ,0+ + (2.126)

=

rµ

0+∑

a i

rµa

ia †

aa i +

∑

ab i j

rµab

i ja †

aa †

b a j a i (2.127)

is used to generate the excited state from the CCSD ground state,

|Ψ(CCSD)µ

⟩ = R(CCSD)µ

e T (CCSD) |Φ⟩ . (2.128)

For the excited state |Ψ(CCSD)µ ⟩ the Schrödinger equation reads

HN R(CCSD)µ

e T (CCSD) |Φ⟩ = ∆E (CCSD)µ

R(CCSD)µ

e T (CCSD) |Φ⟩ , (2.129)

which, due to the commutation relation [R(CCSD)µ , T (CCSD)] = 0 can, analogously to

the ground-state case, be formulated as an eigenvalue problem of the effectiveHamiltonian,

H (CCSD)R(CCSD)µ

|Φ⟩ = ∆E (CCSD)µ

R(CCSD)µ

|Φ⟩ . (2.130)


2.9. Equation-of-Motion Coupled Cluster

Disconnected terms can further be removed from this equation by subtracting theground-state equation H (CCSD) |Φ⟩=∆E (CCSD) |Φ⟩,

h

H (CCSD),R(CCSD)µ

i

|Φ⟩ = ω(CCSD)µ

R(CCSD)µ

|Φ⟩ , (2.131)

where

ω(CCSD)µ

≡ ∆E (CCSD)µ

−∆E (CCSD) (2.132)

is the excitation energy relative to the ground state. Since the effective Hamil-tonian H (CCSD) = H0 + H (CCSD)

open , with H0 = ∆E (CCSD), consists of two parts – with

and without external lines – and the commutator [H0,R(CCSD)µ ] clearly vanishes,

the EOM-CCSD eigenvalue equation can be put into the convenient form

H (CCSD)open R

(CCSD)µ

C|Φ⟩ = ω(CCSD)

µR(CCSD)µ

|Φ⟩ , (2.133)

serving as starting point for the derivation of algebraic expressions for the R(CCSD)µ

amplitudes. By left-projecting (2.133) onto singly and doubly excited determi-nants, an eigenvalue problem for the amplitudes (rµ)ai and (rµ)ab

i j is obtained

⟨Φai|

H (CCSD)open R

(CCSD)µ

C|Φ⟩ = ω(CCSD)

µ

rµa

i(2.134)

⟨Φabi j|

H (CCSD)open R

(CCSD)µ

C|Φ⟩ = ω(CCSD)

µ

rµab

i j. (2.135)

The constant amplitude

rµ

0may afterwards be calculated separately from the

solution of (2.134) and (2.135) according to

⟨Φ|

H (CCSD)open R

(CCSD)µ

C|Φ⟩ = ω(CCSD)

µ

rµ

0. (2.136)

The corresponding diagrams can be found in Appendix H.1, and the algebraic ex-pressions are listed in Figure 2.14. The last 4 terms in the R2 amplitude equationsstem from contributions of the three-body part H3 of the effective Hamiltonian tothe R2 equations which have been explicitly expressed in terms of cluster ampli-tudes and in order to avoid storing the three-body matrix elements of H3 [76] (fora similar discussion, see Section 3.3).

The excited bra state ⟨Ψ(CCSD)µ |may be parametrized as [76]

⟨Ψ(CCSD)µ

| = ⟨Φ| L(CCSD)µ

e−T (CCSD)

, (2.137)



ω r0 =(R0A)+∑

a i

r ai H

ia

(R0B)+ 1

4

∑

ab i j

r abi j H

i j

ab

⟨Φai |

H (CCSD)open R

(CCSD)µ

C|Φ⟩ =

(R1A)+∑

c

r ci H

ac

(R1B)−∑

k

r ak H

ki

(R1C)+∑

c k

r ck H

a ki c

(R1D)+∑

c k

r a ci k H

kc

(R1E)+ 1

2

∑

c d k

r c di k H

a kc d

(R1F)+ − 1

2

∑

c k l

r a ck l H

k li c

⟨Φabi j |

H (CCSD)open R

(CCSD)µ

C|Φ⟩ =

(R2A)+ Pi j

∑

c

r ci H

abc j

(R2B)− Pab

∑

k

r ak H

k bi j

(R2C)+ Pab

∑

c

r a ci j H

bc

(R2D)− Pi j

∑

k

r abi k H

kj

(R2E)+ 1

2

∑

c d

r c di j H

abc d

(R2F)+ 1

2

∑

k l

r abk l H

k li j

(R2G)+ Pab Pi j

∑

c k

r a ci k H

k bc j

(R2H)+ Pab

∑

c d k

r dk t a c

i j Hb k

c d

(R2I)+ 1

2Pab

∑

c d k l

r d ak l v k l

c d t cbi j

(R2J)− Pi j

∑

c k l

r cl t ab

i k Hk l

j c

(R2K)+ 1

2Pi j

∑

c d k l

r c di l t ab

j k v k lc d

Figure 2.14: Algebraic expressions for the R(CCSD) amplitude equations. The index µ has been

dropped.



where L(CCSD)µ is a de-excitation operator

L(CCSD)µ

= Lµ,0+ Lµ,1+ Lµ,2 (2.138)

= Lµ,0+ + (2.139)

=

lµ

0+∑

a i

lµi

aa †

ia a +

∑

ab i j

lµi j

aba †

ia †

ja b a a . (2.140)

Since ⟨Ψ(CCSD)µ | satisfies the Schrödinger equation

⟨Φ| L(CCSD)µ

e−T (CCSD)

HN = ∆E (CCSD)µ

⟨Φ| L(CCSD)µ

e−T (CCSD)

(2.141)

it follows that ⟨Φ| L(CCSD)µ is also eigenfunction of the effective Hamiltonian,

⟨Φ| L(CCSD)µ

H (CCSD) = ∆E (CCSD)µ

⟨Φ| L(CCSD)µ

. (2.142)

As for the right eigenproblem, the left one can be formulated in a way that directlyprovides the excitation energy ω(CCSD)

µ ,

⟨Φ| L(CCSD)µ

H (CCSD)open =

∆E (CCSD)µ

−H0

⟨Φ| L(CCSD)µ

, (2.143)

and thus, recalling thatH0 =∆E (CCSD),

⟨Φ| L(CCSD)µ

H (CCSD)open = ω(CCSD)

µ⟨Φ| L(CCSD)

µ. (2.144)

Unlike the right eigenvalue equation, the left one has no restriction to connecteddiagrams. Furthermore, it has the same structure as the ΛCCSD equations, and,therefore, the corresponding diagrams are identical. The only difference is in di-agrams (Λ1A) and (Λ2A), which for the EOM-CCSD case translate into

b

=

lµ

0χ i

a,

b b

=

lµ

0v

i j

ab .(2.145)

However, it can be shown that

lµ

0=δµ0 [26] and, therefore, these diagrams van-

ish for excited states and are consequently left out in the equations. In conclusion,the effective Hamiltonian matrix, being of non-Hermitean nature, possesses two

sets of normalized eigenvectorsn

L(CCSD)µ

o

andn

R(CCSD)µ

o

,

L(CCSD)µ

≡

lµ

0,n

lµi

a

o

,n

lµi j

ab

o

(2.146)

R(CCSD)µ

≡

rµ

0,n

rµa

i

o

,n

rµab

i j

oT

, (2.147)



with

L(CCSD)µ

· L(CCSD)µ

= 1 (2.148)

R(CCSD)µ

· R(CCSD)µ

= 1 , (2.149)

which share the same eigenvalues but are otherwise distinct. As is well-knownfrom the theory of non-Hermitean eigenvalue problems [145], the eigenvectorsn

L(CCSD)µ

o

andn

R(CCSD)µ

o

are not orthogonal among themselves but satisfy a biorthog-

onality relation

L(CCSD)µ

· R(CCSD)ν

= δµν (2.150)

where the originally normalized left eigenvector L(CCSD)µ

has to be rescaled accord-ing to

L(CCSD)µ

→ 1

L(CCSD)µ

· R(CCSD)µ

L(CCSD)µ

(2.151)

to achieve unit overlap with the corresponding right eigenvector

⟨Ψ(CCSD)µ

|Ψ(CCSD)ν

⟩ = ⟨Φ| L(CCSD)µ

e−T (CCSD)

e T (CCSD)

R(CCSD)ν

|Φ⟩ = δµν . (2.152)

In order to achieve unit overlap, the choice of L(CCSD)µ

as the vector to be scaled isof course arbitrary. However, to be consistent with the ground-state solution theright vector is normalized to unity and the left vector is rescaled [76].

2.9.1 Reduced Density Matrices

Properties of excited states as well as transition properties can be calculated withinthe EOM-CCSD approach from the left and right solutions of the effective Hamil-tonian eigenproblem by means of a reduced density matrix (ρµνN )

q1...qnp1...pn

defined sim-ilarly to its ground-state counterpart (2.82),

(ρµνN )

q1...qn

p1...pn= ⟨Φ|

h

L(CCSD)µ

a †p1

. . . a †pn

a qn. . . a q1

e T (CCSD)

CR(CCSD)ν

i

C|Φ⟩ . (2.153)

The quantity of interest is a generalized expectation value of a n-body operator G ,¬

G¶

µν= ⟨Ψ(CCSD)

µ| G |Ψ(CCSD)

ν⟩

= ⟨Ψ(CCSD)µ

|Ψ(CCSD)ν

⟩ ⟨Φ| G |Φ⟩+¬

GN

¶

µν, (2.154)



which is evaluated analogously to (2.83) in terms of the reduced density matrix as

¬

GN

¶

µν=

n∑

k=1

1

(k !)2

∑

p1...pkq1...qk

⟨p1 . . .pk |g N |q1 . . .qk ⟩ (ρµνN )q1...qk

p1...pk. (2.155)

Again, the one-body operator expectation value may be written in the compactform

¬

O¶

µν=

∑

pq

⟨p |o|q ⟩ (ρµν )qp

, (2.156)

(ρµν )qp≡

¨

(ρµνN )

qp +δµν δpq : p ,q ∈ holes

(ρµνN )

qp : else ,

(2.157)

where the orthogonality of the left and right eigenstates of the effective Hamilto-nian has been taken into account. This approach works for excited states as well asfor ground states, provided that the operators L

(CCSD)0 ,R

(CCSD)0 and vectors L

(CCSD)0 ,

R(CCSD)0 , defined as

L(CCSD)0 ≡ 1+Λ , L

(CCSD)0 =

1,λia,λi j

ab

(2.158)

R(CCSD)0 ≡ 1 , R

(CCSD)0 =

1,0,0

, (2.159)

are assigned to the ground state solutions. As is discussed in Ref. [76], transitionmoments ⟨Ψ(CCSD)

µ |Ψ(CCSD)ν ⟩ are not well defined due to the non-Hermitecity of the

effective Hamiltonian. Therefore, products of left and right transition moments¬

O¶

µν

¬

O¶

νµ= ⟨Ψ(CCSD)

µ| O |Ψ(CCSD)

ν⟩ ⟨Ψ(CCSD)

ν| O |Ψ(CCSD)

µ⟩ (2.160)

are computed instead since these products correspond to the squares of the tran-sition moments which are the only observables in the first place.

Reduced density matrices are not used in actual calculations in this work, con-sequently no equations are presented, but in later sections useful remarks aboutthe spherical treatment of reduced density matrixes as well as the appropriatenormalization (2.152) of the left and right eigenvectors are given.


Chapter 3

Coupled-Cluster Theory forThree-Body Hamiltonians

Chapter 3. Coupled-Cluster Theory for Three-Body Hamiltonians

3.1 CCSD for Three-Body Hamiltonians

3.1.1 Introduction

The normal-ordering approximation discussed in Section 1.5 represents an effec-tive way for the approximate incorporation of three-, or even higher-nucleon in-teraction effects in many-body calculations that are able to handle effective in-teractions up to the two-body level. Nevertheless, the desire to include the fullthree-body interaction still persists, at least for the purpose to benchmark possi-ble approximation schemes.

The inclusion of higher-body interactions into the NCSM framework is – con-ceptually – relatively simple. A highly efficient implementation of the three-bodyinteraction matrix element handling in the IT-NCSM allows for computations ofnuclei even beyond the p shell. The treatment of the full three-body force in theCoupled-Cluster framework comes along with a significant increase of diagramsto be evaluated, resulting in a larger implementational effort. But once imple-mented, the CCSD method for three-body Hamiltonians benefits greatly from itsgentle scaling behaviour as well as from efficient matrix element handling [86] togo beyond the s d shell.

A first derivation of the corresponding CCSD equations in a factorized formwas published in 2007 [63]. Here, in this work, the unfactorized derivation is pre-sented, resulting in less compact but structurally simpler expressions. Diagramfactorization often comes along with increased memory requirements since in-termediates have to be stored. On the other hand, the computational runtime isusually strongly dominated by only a few diagrams that need to be singled out toreceive special implementational care.


3.1. CCSD for Three-Body Hamiltonians

3.1.2 The CCSD Equations for Three-Body Hamiltonians

The derivation of the three-body CCSD equations requires the same formal stepsas for the two-body case. The Hamiltonian in second-quantized standard formreads

H = h0+ h1+ h2+ h3 (3.1)

= h0+∑

pq

⟨p |h1|q ⟩ a †p

a q +1

4

∑

pqr s

⟨pq |h2|r s ⟩ a †p

a †q

a s a r

+1

36

∑

pqr s t u


a †q

a †ra u a t a s . (3.2)

In terms of normal-ordered operator strings the Hamiltonian is represented by

H = h0+∑

i

⟨i |h1|i ⟩+∑

pq

⟨p |h1|q ⟩ a †p

a q+1

2

∑

i j

⟨i j |h2|i j ⟩

+∑

pqi

⟨p i |h2|qi ⟩ a †p

a q+1

4

∑

pqr s

⟨pq |h2|r s ⟩ a †p

a †q

a s a r

+1

6

∑

i j k

⟨i j k |h3|i j k ⟩+ 1

2

∑

pqi j

⟨i j p |h3|i j q ⟩ a †p

a q

+1

4

∑

pqr s i

⟨pqi |h3|r s i ⟩ a †p

a †q

a s a r

+1

36

∑

pqr s t u


a †q

a †ra u a t a s , (3.3)

which again can be cast into the compact form

H = ⟨Φ|H |Φ⟩+∑

pq

f pqa †

pa q+

1

4

∑

pqr s

v pqr sa †

pa †

qa s a r

+1

36

∑

pqr s t u

wpqrs t u a †

pa †

qa †

ra u a t a s , (3.4)

or ,

H = ⟨Φ|H |Φ⟩+ FN + VN + WN . (3.5)

In (3.5), ⟨Φ|H |Φ⟩ is the reference state expectation value

⟨Φ|H |Φ⟩ = h0+∑

i

⟨i |h1|i ⟩+1

2

∑

i j

⟨i j |h2|i j ⟩+ 1

6

∑

i j k

⟨i j k |h3|i j k ⟩ , (3.6)



and the following definitions are analogous to the two-body CCSD case,

f pq≡ ⟨p | f |q ⟩ = ⟨p |h1|q ⟩+

∑

i

⟨p i |h2|qi ⟩+ 1

2

∑

i j

⟨i j p |h3|i j q ⟩ , (3.7)

v pqr s≡ ⟨pq |v |r s ⟩ = ⟨pq |h2|r s ⟩+

∑

i

⟨pqi |h3|r s i ⟩ , (3.8)

wpqrs t u ≡ ⟨pqr |w |s t u ⟩ = ⟨pqr |h3|s t u ⟩ . (3.9)

Apart from the normal-ordered three-body part WN , the Hamiltonian (3.5)

H = ⟨Φ|H |Φ⟩+ HN (3.10)

has the same topology as for the two-body case, leading to the exact same equa-tions with replaced definitions for the matrix elements of the normal-ordered op-erators FN , VN and reference expectation value ⟨Φ|H |Φ⟩. Therefore, all new expres-sions are generated from WN and will consequently always involve the matrix el-ements ⟨pqr |w |s t u ⟩. This observation also allows to write any quantity, such asthe correlation energy or amplitude expressions, for instance, in the form

∆E (CCSD) = ∆E(CCSD)NO2B + ∆E

(CCSD)3B (3.11)

0 = T(CCSD)1,NO2B + T

(CCSD)1,3B (3.12)

0 = T(CCSD)2,NO2B + T

(CCSD)2,3B , (3.13)

where the quantity with label "NO2B" denotes the usual algebraic expressions al-ready known from the Coupled-Cluster theory for two-body Hamiltonians, butwith implied usage of the re-definitions of the normal-ordered Hamiltonians ma-trix elements Eref, f

pq , and v

pqr s , Eqs. (3.6), (3.7) and (3.8), whereas the quantity with

label "3B" denotes all the new terms due to the presence of the residual normal-ordered three-body interaction operator WN .

The expansion for the effective Hamiltonian for an arbitrary CC method trun-cated at the MpMh excitation level,

H (M) = e−T (M)HN e T (M) , (3.14)

again terminates, this time, due to the six external lines of WN , after the six-foldcommutator at the lastest,

H (M) = HN +

6∑

n=1

1

n !

h

. . .h

︸︷︷︸

n times

HN , T (M)i

, . . . , T (M)i

︸︷︷︸

n times

. (3.15)



From this clearly follows that no more than six T (M) operators will appear in indi-vidual diagrams for the effective Hamiltonian, and consequently in any Coupled-Cluster equations. For CCSD, as can be seen below, the maximum number ofT (CCSD) operators appearing in the equations is actually only five instead of six.

The Coupled-Cluster equations then follow from left-projection of the similar-ity transformed Schrödinger equations onto the reference state and correspondingexcited determinants,

⟨Φa 1...a n

i 1...i n| H (M) |Φ⟩ = ⟨Φa 1...a n

i 1...i n|

HN e T (M)

C|Φ⟩ , n = 0, . . . , M . (3.16)

As mentioned before, the operator parts FN and VN produce the exact same ex-pressions as for the two-body case, which contributions to the CCSD energy andamplitude equations are abbreviated as ∆E

(CCSD)NO2B resp. T(CCSD)

1,NO2B and T(CCSD)2,NO2B and are

not written explicitly. Excitation rank considerations 1

X

⟨Φa 1...a n

i 1...i n|

+X

WN

+X

e T (CCSD) != 0 (3.17)

yield the following operator expression for the CCSD energy equation

∆E (CCSD) = ∆E(CCSD)NO2B + ⟨Φ|

h

WN

T1+ T2+1

2!T 2

1+ T1T2+

1

3!T 3

1

i

C|Φ⟩ (3.18)

and for the T1 and T2 amplitude equations

0 = T(CCSD)1,NO2B

+ ⟨Φai|h

WN

1+ T1+ T2+1

2!T 2

1+ T1T2

+ 1

2!T 2

2+ 1

3!T 3

1+ 1

2!T 2

1T2+

1

4!T 4

1

i

C|Φ⟩ (3.19)

0 = T(CCSD)2,NO2B

+ ⟨Φabi j|h

WN

1+ T1+ T2+1

2!T 2

1+ T1T2+

1

2!T 2

2+ 1

3!T 3

1+ 1

2!T 2

1T2

+ 1

3!T 3

1T2+

1

2!T1T 2

2+ 1

4!T 4

1+ 1

5!T 5

1

i

C|Φ⟩ (3.20)

The energy equation may be simplified right away due to the requirement ofproducing closed diagrams which is clearly only possible for the cluster operatorproducts 2 T1T2 and T 3

1 , so that

∆E (CCSD) =∆E(CCSD)NO2B + ⟨Φ|

h

WN

T1T2+1

3!T 3

1

i

C|Φ⟩ . (3.21)

1The excitation ranks of |Φa 1...a n

i 1...i n⟩ and ⟨Φa 1...a n

i 1...i n| are understood to be n and −n , respectively.

2This is because their number of external lines match the number of external lines of the resid-ual normal-ordered three-body interaction operator WN .



b b b b b b b b b b b b

X = 0 X = 0 X = 0 X = 0


X = 1 X = 1 X = 1 X =−1


X =−1 X =−1 X = 2 X = 2


X =−2 X =−2 X = 3 X =−3

excitation level X (WN ) 3 2 1 0 -1 -2 -3naming convention T1 — — — — T1A T1B T1Cnaming convention T2 — — T2A T2B T2C T2D T2E

Figure 3.1: Topology and excitation level of the three-body part WN of the normal-ordered Hamilto-

nian. The table introduces the diagram naming convention that characterizes diagrams

by the excitation level of the WN operator part involved. There are no X (WN )≥ 2 con-

tributions to the CCSD T1 and T2 equations since this would require at least 5 external

lines from the bra determinant.



Analogous considerations lead to minor simplifications of the T1 and T2 amplitudeequations,

0 = T(CCSD)1,NO2B

+ ⟨Φai|h

WN

T2+1

2!T 2

1+ T1T2+

1

2!T 2

2+ 1

3!T 3

1+ 1

2!T 2

1T2+

1

4!T 4

1

i

C|Φ⟩ (3.22)

and

0 = T(CCSD)2,NO2B

+ ⟨Φabi j|h

WN

T1+ T2+1

2!T 2

1+ T1T2+

1

2!T 2

2+ 1

3!T 3

1+ 1

2!T 2

1T2

+ 1

3!T 3

1T2+

1

2!T1T 2

2+ 1

4!T 4

1+ 1

5!T 5

1

i

C|Φ⟩ . (3.23)

The evaluation of (3.21)-(3.23) in terms of matrix elements is straightforward usingstandard diagrammatic techniques. In order to catch all topologically distinct dia-grams it is recommended to do a Hugenholtz analysis first before translating eachHugenholtz diagram in one equivalent antisymmetrized Goldstone diagram [26].These diagrams are listed in Section C.3 and the corresponding algebraic expres-sions are listed in Figures 3.2-3.4. The naming convention has been chosen ac-cording to the excitation level of the WN operator part involved in the diagram.The topology of WN along with the corresponding excitation level is given in Fig-ure 3.1. Unlike for two-body Hamiltonians, where the algebraic expression for∆E (CCSD) is also valid for all higher-order Coupled-Cluster method, in the case ofthree-body Hamiltonians this expression only holds for the CCSD approximation.This is because a three-body interaction also allows to form a closed diagram viacontraction with a T3 operator,

⟨Φ|

WN T3

C|Φ⟩ =

b b b

=1

(3!)2

∑

ab ci j k

wi j k

ab c t ab ci j k

. (3.24)

Also, it should be noted that the total CCSD ground-state energy reads

E (CCSD) = Eref + ∆E(CCSD)NO2B + ∆E

(CCSD)3B , (3.25)

where from the definition of Eref as a reference-state expectation value it is clearthat it is not affected, compared to the NO2B treatment, by including the residualnormal-ordered three-body interaction in the calculations.



∆E (CCSD) = ∆E(CCSD)NO2B

(ED)+ 1

4

∑

c d e k l m

w k l mc d e t c

k t d el m

(EE)+ 1

6

∑

c d e k l m

w k l mc d e t c

k t dl t e

m

(T1Aa)+ 1

2

∑

c d k l

w k l ac d i t c

k t dl

(T1Ab)+ 1

4

∑

c d k l

w k l ac d i t c d

k l

(T1Ba)+ 1

2

∑

c d e k l

w k l ac d e t c

k t dl t e

i

(T1Bb)+ 1

4

∑

c d e k l

w k l ac d e t c d

k l t ei

(T1Bc)+ 1

2

∑

c d e k l

w k l ac d e t c

k t d el i

(T1Bd)− 1

2

∑

c d k l m

w k l mc d i t c

k t dl t a

m

(T1Be)− 1

4

∑

c d k l m

w k l mc d i t c d

k l t am

(T1Bf)− 1

2

∑

c d k l m

w k l mc d i t c

k t d al m

(T1Ca)− 1

2

∑

c d e k l m

w k l mc d e t c

k t dl t a

m t ei

(T1Cb)− 1

4

∑

c d e k l m

w k l mc d e t c d

k l t am t e

i

(T1Cc)− 1

2

∑

c d e k l m

w k l mc d e t c

k t d el i t a

m

(T1Cd)− 1

2

∑

c d e k l m

w k l mc d e t c

k t d al m t e

i

(T1Ce)+ 1

2

∑

c d e k l m

w k l mc d e t c

k t dl t e a

m i

(T1Cf)+ 1

4

∑

c d e k l m

w k l mc d e t c d

k l t e am i

(T1Cg)− 1

4

∑

c d e k l m

w k l mc d e t c a

k l t d ei m + t a

i (NO2B) = 0 , ∀ a , i

Figure 3.2: Algebraic expressions for ∆E (CCSD) and the CCSD T1 amplitude equations for three-

body Hamiltonians.



Pab Pi j

n

(T2Aa)+ 1

4

∑

c k

w k abc i j t c

k

(T2Ba)+ 1

2

∑

c d k

w ab kc j d t c

i t dk

(T2Bb)− 1

2

∑

c k l

w k b li j c t a

k t cl

(T2Bc)+ 1

4

∑

c d k

w ab kc j d t c d

i k

(T2Bd)− 1

4

∑

c k l

w k b li j c t a c

k l

(T2Ca)+ 1

4

∑

c d e k

w a k bc d e t c

i t dk t e

j

(T2Cb)−

∑

c d k l

w k l bi c d t a

k t cl t d

j

(T2Cc)+ 1

4

∑

c k l m

w k l mi c j t a

k t cl t b

m

(T2Cd)+ 1

8

∑

c d e k

w k abc d e t c

k t d ei j

(T2Ce)+ 1

4

∑

c d e k

w a k bc d e t c

i t d ek j

(T2Cf)− 1

2

∑

c d k l

w a k lc d j t c

i t d bk l

(T2Cg)− 1

2

∑

c d k l

w k l bi c d t a

k t c dl j

(T2Ch)+

∑

c d k l

w k l bc d j t d

l t a ci k

(T2Ci)+ 1

8

∑

c k l m

w k l mc i j t c

k t abl m

(T2Cj)+ 1

4

∑

c k l m

w k l mi c j t a

k t cbl m

(T2Da)− 1

2

∑

c d e k l

w k l bc d e t c

k t al t d

i t ej

(T2Db)+ 1

2

∑

c d k l m

w k l mc d j t c

k t di t a

l t bm

(T2Dc)− 1

4

∑

c d e k l

w a k lc d e t c

i t dj t b e

k l

(T2Dd)− 1

2

∑

c d e k l

w a k lc d e t c

i t bk t d e

j l

(T2De)+

∑

c d e k l

w a k lc d e t c

i t dk t e b

l j

(T2Df)− 1

4

∑

c d e k l

w k l bc d e t c

k t al t d e

i j

(T2Dg)+ 1

4

∑

c d e k l

w k l ac d e t c

k t dl t e b

i j

(T2Dh)+ 1

4

∑

c d k l m

w k l mi c d t a

k t bl t c d

j m

(T2Di)+ 1

2

∑

c d k l m

w k l mi c d t a

k t cj t b d

l m

(T2Dj)−

∑

c d k l m

w k l mi c d t a

k t cl t d b

m j

(T2Dk)+ 1

4

∑

c d k l m

w k l mc i d t c

k t dj t ab

l m

(T2Dl)+ 1

8

∑

c d e k l

w b k lc d e t a c

i j t d ek l

(T2Dm)− 1

8

∑

c d e k l

w a k lc d e t c d

i j t b ek l

(T2Dn)+ 1

2

∑

c d e k l

w k l bc d e t a c

i k t d el j

o

+ . . .

Figure 3.3: Algebraic expressions for the CCSD T2 amplitude equations for three-body Hamilto-

nians.



+ Pab Pi j

n

(T2Do)− 1

8

∑

c d k l m

w k l mj c d t ab

i k t c dl m

(T2Dp)+ 1

8

∑

c d k l m

w k l mi c d t ab

k l t c dj m

(T2Dq)− 1

2

∑

c d k l m

w k l mc d j t a c

i k t d bl m

(T2Dr)− 1

4

∑

c d k l m

w k l mc d i t c

k t dl t ab

m j

(T2Ea)+ 1

4

∑

c d e k l m

w k l mc d e t c

k t al t d

i t bm t e

j

(T2Eb)− 1

4

∑

c d e k l m

w k l mc d e t c

k t dl t a

m t e bi j

(T2Ec)− 1

4

∑

c d e k l m

w k l mc d e t c

k t dl t e

i t abm j

(T2Ed)+ 1

8

∑

c d e k l m

w k l mc d e t c

k t al t b

m t d ei j

(T2Ee)−

∑

c d e k l m

w k l mc d e t c

k t di t a

l t e bm j

(T2Ef)+ 1

8

∑

c d e k l m

w k l mc d e t c

k t di t e

j t abl m

(T2Eg)+ 1

4

∑

c d e k l m

w k l mc d e t a

k t ci t d

j t b el m

(T2Eh)+ 1

4

∑

c d e k l m

w k l mc d e t a

k t ci t b

l t d ej m

(T2Ei)− 1

2

∑

c d e k l m

w k l mc d e t e

j t a ci k t d b

l m

(T2Ej)+ 1

8

∑

c d e k l m

w k l mc d e t e

j t c dk i t ab

l m

(T2Ek)− 1

8

∑

c d e k l m

w k l mc d e t c

j t abi k t d e

l m

(T2El)− 1

8

∑

c d e k l m

w k l mc d e t b

k t d el m t a c

i j

(T2Em)+ 1

8

∑

c d e k l m

w k l mc d e t b

m t c ak l t d e

i j

(T2En)− 1

2

∑

c d e k l m

w k l mc d e t b

m t a ci k t d e

l j

(T2Eo)− 1

4

∑

c d e k l m

w k l mc d e t e

m t a ci j t b d

k l

(T2Ep)+ 1

16

∑

c d e k l m

w k l mc d e t e

m t c di j t ab

k l

(T2Eq)+ 1

2

∑

c d e k l m

w k l mc d e t d

l t a ci k t e b

m j

(T2Er)− 1

4

∑

c d e k l m

w k l mc d e t e

m t abi k t c d

j l

o

+ t abi j (NO2B) = 0 , ∀ a ,b , i , j

Figure 3.4: Algebraic expressions for the CCSD T2 amplitude equations for three-body Hamilto-

nians, continued.



3.2 Effective Hamiltonian

As already stated in Section 3.1.2, the expression for the effective Hamiltonian foran arbitrary CC method truncated at the MpMh excitation level,

H (M) = e−T (M) HN e T (M) , (3.26)

terminates after the six-fold commutator with the cluster operator

H (M) = FN +

2∑

n=1

1

n !

h

FN , T (M)i(n )

+ VN +

4∑

n=1

1

n !

h

VN , T (M)i(n )

+ WN +

6∑

n=1

1

n !

h

WN , T (M)i(n )

, (3.27)

where [·, ·](n ) denotes the n-fold commutator. From (3.27) it is evident that all newcontributions are given by the last line of (3.27),

WN +

6∑

n=1

1

n !

h

WN , T (M)i(n )

=

WN e T (M)

C, (3.28)

again stemming from WN alone, which may be emphasized by the expression

H (M) = H (M)NO2B +

WN e T (M)

C. (3.29)

In the case of CCSD, H (CCSD) now contains up to nine-body operators, as is appar-ent from the example

b b b

. (3.30)

If the cluster operator amplitudes have been determined from the CCSD equationsincluding the residual normal-ordered three-body interaction WN , then the zero-body matrix element H0 of H (CCSD) is again given by the corresponding CCSDcorrelation energy (3.21),

H0 = ∆E (CCSD) , (3.31)

and the H ph and H pp

hh matrix elements vanish because they correspond to theCCSD T1 and T2 equations. Expressions for the complete one- and two-body part



b b b b b b

b b b

b b b b

Figure 3.5: Selected topologies of H3 and H4, generated by the residual normal-ordered three-body

interaction WN , that enter the ΛCCSD equations.

of H (CCSD) are listed in Figures 3.6-3.9, and the corresponding diagrams and spher-ical expressions can be found in Appendices D.2 and D.3. For the three- and four-body part of H (CCSD), expressions have been evaluated only for the selected topolo-gies shown in Figure 3.5, which are the only diagrams required for ΛCCSD usingthree-body Hamiltonians, as can be seen in Figures 3.10-3.12.



H ia = H i

a (NO2B)(H h

p A)+ 1

4

∑

c d k l

w i k la c d t c d

k l

(H hp B)+ 1

2

∑

c d k l

w i k la c d t c

k t dl

H ab = H a

b (NO2B)(H

pp A)+ 1

4

∑

c d k l

w a k lb c d t c d

k l

(Hp

p B)− 1

4

∑

c d k l m

w k l mb c d t c d

k l t am

(Hp

p C)+ 1

2

∑

c d k l m

w k l mb c d t c

k t a dl m

(Hp

p D)+ 1

2

∑

c d k l

w a k lb c d t c

k t dl

(Hp

p E)− 1

2

∑

c d k l m

w k l mb c d t c

k t dl t a

m

H ij = H i

j (NO2B)(H h

hA)+ 1

4

∑

c d k l

w i k lc d j t c d

k l

(H hh

B)+ 1

4

∑

c d e k l

w i k lc d e t c d

k l t ej

(H hh

C)− 1

2

∑

c d e k l

w i k lc d e t c d

k j t el

(H hh

D)+ 1

2

∑

c d k l

w i k lc d j t c

k t dl

(H hh

E)+ 1

2

∑

c d e k l

w i k lc d e t c

k t dl t e

j

H i j

ab= H i j

ab(NO2B)

(H hhpp A)+

∑

c k

wi j k

ab ct c

k

H a ib c = H a i

b c (NO2B)

H phpp A

+ 12

∑

d k l

w i k lb c d t a d

k l

H phpp B

+∑

d l

w a i lb c d t d

l

H phpp C

+∑

d k l

w i k lb c d t a

k t dl

H i kj a = H i k

j a (NO2B)

H hhhp

A

− 12

∑

c d l

w i k la c d t c d

j l

H hhhp

B

+∑

c l

w i k lj a c t c

l

H hhhp

C

−∑

c d l

w i k la c d t c

j t dl

Figure 3.6: Algebraic expressions for the effective Hamiltonian one- and two-body matrix elements

for three-body Hamiltonians.



H abc d = H ab

c d (NO2B)(H

pppp A)− 1

2Pab

∑

e k l

w a k lc d e t b e

k l

(Hpp

pp B)+

∑

e k

w ab kc d e t e

k

(Hpp

pp C)+ 1

2

∑

e k l m

w k l mc d e t ab

l m t ek

(Hpp

pp D)+ 1

2Pab

∑

e k l m

w k l mc d e t a

k t b el m

(Hpp

pp E)−

∑

e k l m

w b k lc d e t a

k t el

(Hpp

pp F)+ 1

2Pab

∑

e k l m

w k l mc d e t a

l t bm t e

k

H i j

k l= H i j

k l(NO2B)

(H hhhh

A)− 1

2Pk l

∑

c d m

wi j m

k c dt c d

m l

(H hhhh

B)+

∑

c m

wi j m

k l ct c

m

(H hhhh

C)+ 1

2

∑

c d e m

wi j m

c d et d e

k l t cm

(H hhhh

D)+ 1

2Pk l

∑

c d e m

wi j m

c d et c d

l m t ek

(H hhhh

E)+

∑

c d m

wi j m

c d lt c

k t dm

(H hhhh

F)+ 1

2Pk l

∑

c d e m

wi j m

c d et d

k t el t c

m

H a j

ib= H a j

ib(NO2B)

H phph

A

− 12

∑

c d k

wa j k

b c dt c d

i k

H phph

B

+ 12

∑

c k l

wj k l

b c it a c

k l

H phph

C

+∑

c k

wa j k

b c it c

k

H phph

D

− 12

∑

c d k l

wj k l

b c dt c d

i l t ak

H phph

E

− 12

∑

c d k l

wj k l

b c dt a d

k l t ci

H phph

F

+∑

c d k l

wj k l

b c dt a c

i k t dl

H phph

G

+∑

c d k

wa j k

b c dt c

k t di

H phph

H

−∑

c k l

wj k l

b c it c

k t al

H phph

I

−∑

c d k l

wj k l

b c dt a

l t ck t d

i

Figure 3.7: Algebraic expressions for the effective Hamiltonian two-body matrix elements for three-

body Hamiltonians, continued.



H abc i = H ab

c i (NO2B)

H ppph

A

−∑

d k

w ab kc d i t d

k

H ppph

B

+ 12

Pab

∑

d e k l

w a k lc d e t d e

i k t bl

H ppph

C

+ 12

Pab

∑

d e k l

w a k lc d e t b d

k l t ei

H ppph

D

+ Pab

∑

d e k l

w a k lc d e t b e

i l t dk

H ppph

E

− 12

∑

d k l m

w k l mc d i t ab

l m t dk

H ppph

F

− 12

Pab

∑

d e k l m

w k l mc d i t b d

k l t am

H ppph

G

−∑

d e k

w ab kc d e t d

k t ei

H ppph

H

+ Pab

∑

d k l

w b k lc d i t d

k t al

H ppph

I

+ 12

∑

d e k l m

w k l mc d e t ab

i m t dk t e

l

H ppph

J

− Pab

∑

d e k l m

w k l mc d e t b e

i l t dk t a

m

H ppph

K

+ 14

Pab

∑

d e k l m

w k l mc d e t d e

i l t bk t a

m

H ppph

L

− 12

Pab

∑

d e k l m

w k l mc d e t a e

l m t bk t d

i

H ppph

M

− 12

∑

d e k l m

w k l mc d e t ab

l m t dk t e

i

H ppph

N

− Pab

∑

d e k l m

w a k lc d e t d

k t bl t e

i

H ppph

O

+ 12

Pab

∑

d k l m

w k l mc d i t a

m t bl t d

k

H ppph

P

+ 12

Pab

∑

c d e k l m

w k l mc d e t a

m t bl t d

k t ei

H ppph

Q

− 14

∑

d e k l m

w k l mc d e t ab

k i t d el m

H ppph

R

+ 14

∑

d e k l m

w k l mc d e t ab

k l t d ei m

H ppph

S

− 12

Pab

∑

d e k l m

w k l mc d e t a d

k l t e bm i





H i aj k = H i a

j k (NO2B)

H hphh

A

+∑

c l

w i a lj k c t c

l

H hphh

B

+ 12

Pj k

∑

c d l m

w i l mj c d t a c

l m t dk

H hphh

C

+ 12

Pj k

∑

c d l m

w i l mj c d t c d

k l t am

H hphh

D

+ Pj k

∑

c d l m

w i l mj c d t a d

k m t cl

H hphh

E

− 12

∑

c d e l

w a i lc d e t d e

j k t cl

H hphh

F

+ 12

Pj k

∑

c d e l

w a i lc d e t c d

l k t ej

H hphh

G

−∑

c l m

w i l mj c k t c

l t am

H hphh

H

− Pj k

∑

c d l

w a i lc d k t c

l t dj

H hphh

I

+ 12

∑

c d e l m

w i l mc d e t a e

k j t cl t d

m

H hphh

J

+ Pj k

∑

c d e l m

w i l mc d e t a d

k m t cl t e

j

H hphh

K

− 12

Pj k

∑

c d e l m

w i l mc d e t a d

l m t ck t e

j

H hphh

L

− 12

Pj k

∑

c d e l m

w i l mc d e t d e

m j t ck t a

l

H hphh

M

+ Pj k

∑

c d e l m

w i l mc d e t a e

j m t cl t d

k

H hphh

N

− Pj k

∑

c d l m

w i l mj c d t a

m t cl t d

k

H hphh

O

+ 12

Pj k

∑

c d e l

w a i lc d e t c

l t dk t e

j

H hphh

P

− 12

Pj k

∑

c d e l m

w i l mc d e t a

m t cl t d

k t ej

H hphh

Q

+ 14

∑

c d e l m

w i l mc d e t c a

j k t d el m

H hphh

R

− 14

∑

c d e l m

w i l mc d e t c d

j k t a el m

H hphh

S

+ 12

Pj k

∑

c d e l m

w i l mc d e t c d

j l t e am k





H i j a

k b l= H i j a

k b l(NO2B)

H hhphph

A

+ wi j a

k b l

H hhphph

B

+ Pk l

∑

c m

wi j m

k b ct a c

l m

H hhphph

C

+ 12

∑

c d

wi j a

cb dt c d

k l

H hhphph

D

− Pk l

∑

c

wi j a

b c lt c

k

H hhphph

E

+∑

m

wi j m

b k lt a

m

H hhphph

F

−∑

c d m

wi j m

b c dt a d

k l t cm

H hhphph

G

− Pk l

∑

c d m

wi j m

b c dt a d

l m t ck

H hhphph

H

+ 12

∑

c d m

wi j m

b c dt c d

k l t am

H hhphph

I

+ Pk l

∑

c m

wi j m

b c lt c

k t am

H hhphph

J

+∑

c d

wi j a

cb dt c

k t dl

H hhphph

K

+∑

c d m

wi j m

b c dt c

k t dl t a

m

H a j b

c d i= H a j b

c d i(NO2B)

H phppph

A

+ wa j b

c d i

H phppph

B

+ Pab

∑

e k

wa j k

c d et b e

i k

H phppph

C

+ 12

∑

k l

wk j l

c d it ab

k l

H phppph

D

− Pab

∑

k

wb k j

c d it a

k

H phppph

E

−∑

e

wab j

c d et e

i

H phppph

F

+∑

e k l

wj k l

c d et ab

i l t ek

H phppph

G

+ Pab

∑

e k l

wj k l

c d et b e

i l t ak

H phppph

H

− 12

∑

e k l

wj k l

c d et ab

k l t ei

H phppph

I

− Pab

∑

e k

wa j k

c d et b

k t ei

H phppph

J

+∑

k l

wk j l

c d it a

k t bl

H phppph

K

−∑

e k l

wj k l

c d et a

k t bl t e

i

Figure 3.10: Algebraic expressions for selected three-body effective Hamiltonian matrix elements

for three-body Hamiltonians.



H ab ki j c = H ab k

i j c (NO2B)

H pphhhp

A

+ w ab ki j c

H pphhhp

B

+ 12

∑

d e

w ab kc d e t d e

i j

H pphhhp

C

+ Pab Pi j

∑

d l

w b k lj c d t a d

i l

H pphhhp

D

+ 12

∑

l m

w k l mi j c t ab

l m

H pphhhp

E

− 12

Pab

∑

d e l m

w k l mc d e t a d

i j t b el m

H pphhhp

F

+ 12

Pab Pi j

∑

d e l m

w k l mc d e t a d

i l t b ej m

H pphhhp

G

+ 14

∑

d e l m

w k l mc d e t ab

l m t d ei j

H pphhhp

H

− 12

Pi j

∑

d e l m

w k l mc d e t ab

i l t d ej m

H pphhhp

I

+ Pi j

∑

d

w ab kc d j t d

i

H pphhhp

J

− Pab

∑

l

w k l bi j c t a

l

H pphhhp

K

− 12

Pab

∑

d e l

w k l bc d e t d e

i j t al

H pphhhp

L

− Pab Pi j

∑

d e l

w a k lc d e t b e

j l t di

H pphhhp

M

+ 12

Pi j

∑

d l m

w k l mc d j t ab

l m t di

H pphhhp

N

+ Pab Pi j

∑

d l m

w k l mc d i t b d

j m t al

H pphhhp

O

− Pab

∑

d e l

w a k lc d e t b e

i j t dl

H pphhhp

P

+ Pi j

∑

d l m

w k l mc d i t ab

j m t dl

H pphhhp

Q

+∑

d e

w ab kd e c t d

i t ej

H pphhhp

R

− Pab Pi j

∑

d l

w l b kd j c t a

l t di

H pphhhp

S

+∑

l m

w l m ki j c t a

l t bm

H pphhhp

T

+ 12

∑

d e l m

w k l mc d e t d e

i j t al t b

m

H pphhhp

U

− Pab Pi j

∑

d e l m

w k l mc d e t b e

j m t al t d

i

H pphhhp

V

+ 12

∑

d e l m

w k l mc d e t ab

l m t di t e

j

H pphhhp

W

− Pab

∑

d e l m

w k l mc d e t e b

i j t am t d

l

H pphhhp

X

− Pi j

∑

d e l m

w k l mc d e t ab

m j t ei t d

l

H pphhhp

Y

− Pab

∑

d e l

w k l bc d e t a

l t di t e

j

H pphhhp

Z

+ Pi j

∑

d l m

w k l mc d j t a

l t bm t d

i

H pphhhp

AA

+∑

d e l m

w k l mc d e t a

l t bm t d

i t ej

Figure 3.11: Algebraic expressions for selected three-body effective Hamiltonian matrix elements

for three-body Hamiltonians, continued.


3.3. The ΛCCSD Equations for Three-Body Hamiltonians

H i j k l

ab c d=(H hhhh

pppp A)+ Pc d

∑

e

wi j c

ab et e d

k l

(H hhhhpppp B)− Pk l

∑

m

wi j m

ab kt c d

m l

(H hhhhpppp C)− Pc d

∑

e m

wi j m

ab et e d

k l t cm

(H hhhhpppp D)− Pk l

∑

e m

wi j m

ab et c d

m l t ek

Figure 3.12: Algebraic expressions for selected four-body effective Hamiltonian matrix elements

for three-body Hamiltonians, continued.

3.3 The ΛCCSD Equations for Three-BodyHamiltonians

The ΛCCSD equations for three-body Hamiltonians can straightforwardly be de-rived in analogy to the two-body case. In terms of the Λ(CCSD) and H (CCSD) opera-tors, the ΛCCSD equations may again be cast in the form


|Φai⟩ = 0 (3.32)


|Φabi j⟩ = 0 , (3.33)

now using Eq. (3.27) as the underlying definition for the effective Hamiltonian.When evaluated in terms of the Λn and Hn operators, such as in Eq. (2.70)-(2.71),the projection onto the singly excited determinants is identical to (3.34), but theprojection onto the doubly excited determinants obtains two new terms, resultingin [96]

0 = ⟨Φ|n

(1+Λ1) H1

C+

(Λ1+Λ2) H2

C+

Λ2 H3

C

o

|Φai⟩ (3.34)

0 = ⟨Φ|n

(1+Λ1+Λ2) H2

C+

Λ2 H1

C

+

Λ1 H1

DC+

(Λ1+Λ2) H3

C+

Λ2 H4

C

o

|Φabi j⟩ . (3.35)

The two new terms are

⟨Φ|

Λ1 H3

C|Φab

i j⟩ =

b b b

(3.36)

and



⟨Φ|

Λ2 H4

C|Φab

i j⟩ =

b b b b

(3.37)

which require a three-body interaction vertex in order to comply with the con-nectedness condition, for instance

⟨Φ|

Λ2 H4

C|Φab

i j⟩ =

b b b+

b b b

+b b b

+b b b

. (3.38)

It is important to realize that only the WN -contributions to H3 and H4 enter (3.36)and (3.37), i.e.,

⟨Φ|

Λ1 (H3,NO2B+ H3,3B)

C|Φab

i j⟩ = ⟨Φ|

Λ1 H3,3B

C|Φab

i j⟩ (3.39)

⟨Φ|

Λ2 (H4,NO2B+ H4,3B)

C|Φab

i j⟩ = ⟨Φ|

Λ2 H4,3B

C|Φab

i j⟩ . (3.40)

As before, in order to circumvent storage of the three- and four-body effectiveHamiltonian matrix elements, the WN contributions to H pph

hhp , H hhphph , H php

pph andH pppp

pppp in terms of interaction matrix elements and cluster amplitudes are directlyinserted into the ΛCCSD equations. The resulting ΛCCSD equations for three-body Hamiltonians are listed in Figures 3.13-3.14 and the spherical expressionscan be found in Appendix E.3. It should be noted that once the expressions forthe effective Hamiltonian are inserted into the contractions, the permutation op-erators Pk l and Pc d may each be replaced by a factor of 2 because, for example,orbitals k and l are always summed over and additionally always appear as anindex pair in antisymmetrized matrix elements.


3.3. The ΛCCSD Equations for Three-Body Hamiltonians

(Λ3B1

A)+ 1

4

∑

c d k l

λk lc d w c d i

k l a

(Λ3B1

B)+ 1

8

∑

c d e f k l

λk lc d w c d i

a e f te f

k l

(Λ3B1

C)+

∑

c d e k l m

λk lc d w d i m

l a e t c ek m

(Λ3B1

D)+ 1

8

∑

c d k l m n

λk lc d w i m n

k l a t c dm n

(Λ3B1

E)− 1

4

∑

c d e f k l m n

λk lc d w i m n

a e f t c ek l t d f

m n

(Λ3B1

F)+ 1

2

∑

c d e f k l m n

λk lc d w i m n

a e f t c ek m t

d f

l n

(Λ3B1

G)+ 1

16

∑

c d e f k l m n

λk lc d w i m n

a e f t c dm n t

e f

k l

(Λ3B1

H)− 1

4

∑

c d e f k l m n

λk lc d w i m n

a e f t c dk m t

e f

l n

(Λ3B1

I)+ 1

2

∑

c d e k l

λk lc d w c d i

a e l t ek

(Λ3B1

J)− 1

2

∑

c d k l m

λk lc d w i m d

k l a t cm

(Λ3B1

K)− 1

4

∑

c d e f k l m

λk lc d w i m d

a e f te f

k lt c

m

(Λ3B1

L)−

∑

c d e f k l m

λk lc d w c i m

a e f td f

l mt e

k

(Λ3B1

M)+ 1

4

∑

c d e k l m n

λk lc d w i m n

a e l t c dm n t e

k

(Λ3B1

N)+

∑

c d e k l m n

λk lc d w i m n

a e k t d el n t c

m

(Λ3B1

O)− 1

4

∑

c d e f k l m n

λk lc d w c i m

a e f td f

k lt e

m

(Λ3B1

P)+ 1

2

∑

c d e k l m n

λk lc d w i m n

a e k t c dl n t e

m

(Λ3B1

Q)+ 1

4

∑

c d e f k l

λk lc d w c d i

e f a t ek t

f

l

(Λ3B1

R)−

∑

c d e k l m

λk lc d w m d i

e l a t cm t e

k

(Λ3B1

S)+ 1

4

∑

c d k l m n

λk lc d w m ni

k l a t cm t d

n

(Λ3B1

T)+ 1

8

∑

c d e f k l m n

λk lc d w m ni

e f a te f

k lt c

m t dn

(Λ3B1

U)−

∑

c d e f k l m n

λk lc d w m ni

w f a td f

l nt c

m t ek

(Λ3B1

V)+ 1

8

∑

c d e f k l m n

λk lc d w m ni

e f a t c dm n t e

k tf

l

(Λ3B1

W)− 1

2

∑

c d e f k l m n

λk lc d w m ni

e f a tf d

k lt c

n t en

(Λ3B1

X)− 1

2

∑

c d e f k l m n

λk lc d w m ni

e f a t c dnl t

f

kt e

m

(Λ3B1

Y)− 1

2

∑

c d e f k l m n

λk lc d w m d i

e f a t cm t e

k tf

l

(Λ3B1

Z)+ 1

2

∑

c d e k l m n

λk lc d w m ni

e l a t cm t e

k t dn

(Λ3B1

AA)+ 1

4

∑

c d e f k l m n

λk lc d w m ni

e f a t cm t e

k t dn t

f

l+ λi

a [NO2B] = 0 , ∀ a , i

Figure 3.13: Algebraic expressions for the ΛCCSD Λ1 amplitude equations for three-body Hamil-

tonians.



Pab Pi j

Λ3B2,J

A

+ 14

∑

c k l

λk lc a w

i j c

k b l

Λ3B2,J

B

+ 14

∑

c d k l m

λk lc a Pk l w

i j m

k b dt d c

m l

Λ3B2,J

C

+ 18

∑

c d e k l

λk lc a w

i j c

d b et d e

k l

Λ3B2,J

D

+ 14

∑

c d k l

λk lc a Pk l w

i j c

d b lt d

k

Λ3B2,J

E

− 14

∑

c k l m

λk lc a w

i j m

k b lt c

m

Λ3B2,J

F

+ 14

∑

c d e k l m

λk lc a w

i j m

d e bt e c

k l t dm

Λ3B2,J

G

+ 14

∑

c d e k l m

λk lc a Pk l w

i j m

d b et e c

m l t dk

Λ3B2,J

H

− 18

∑

c d e k l m

λk lc a w

i j m

d b et d e

k l t cm

Λ3B2,J

I

− 14

∑

c d k l m

λk lc a Pk l w

i j m

d b lt d

k t cm

Λ3B2,J

J

+ 14

∑

c d e k l

λk lc a w

i j c

d b et d

k t el

Λ3B2,J

K

− 14

∑

c d e k l m

λk lc a w

i j m

b e dt d

k t el t c

m

Λ3B2,K

A

+ 14

∑

c d k

λk ic d w

c j d

ab k

Λ3B2,K

B

+ 14

∑

c d e k l

λk ic d Pc d w

c j l

ab et e d

l k

Λ3B2,K

C

+ 18

∑

c d k l m

λk ic d w

l j m

ab kt c d

l m

Λ3B2,K

D

− 14

∑

c d k l

λk ic d Pc d w

l j d

ab kt c

l

Λ3B2,K

E

+ 14

∑

c d e k

λk ic d w

c j d

ab et e

k

Λ3B2,K

F

− 14

∑

c d e k l m

λk ic d w

l m j

e abt c d

m k t el

Λ3B2,K

G

− 14

∑

c d e k l m

λk ic d Pc d w

l j m

ab et e d

m k t cl

Λ3B2,K

H

+ 18

∑

c d e k l m

λk ic d w

l j m

ab et c d

l m t ek

Λ3B2,K

I

− 14

∑

c d e k l

λk ic d Pc d w

c j l

ab et d

l t ek

Λ3B2,K

J

+ 14

∑

c d k l m

λk ic d w

l j m

ab kt c

l t dm

Λ3B2,K

K

+ 14

∑

c d e k l m

λk ic d w

l j m

ab et c

l t dm t e

k

Λ3B2,L

A

+∑

c k

λkc w

i j c

ab k

Λ3B2,L

B

+∑

c d k

λkc w

i j c

ab dt d

k

Λ3B2,L

C

−∑

c k l

λkc w

i j l

ab kt c

l

Λ3B2,L

D

−∑

c d k l

λkc w

i j l

ab dt c d

k l

Λ3B2,L

E

+∑

c d k l

λkc w

i j l

ab dt c d

k l

Λ3B2,M

A

+ 12

∑

c d e k l

λk lc d w

i j c

ab et e d

k l

Λ3B2,M

B

− 12

∑

c d k l m

λk lc d w

i j m

ab kt c d

m l

Λ3B2,M

C

+ 12

∑

c d e k l m

λk lc d w

i j m

ab et d e

k l t cm

Λ3B2,M

D

− 12

∑

c d e k l m

λk lc d w

i j m

ab et c d

m l t ek +λ

i j

ab[NO2B] = 0 , ∀ a ,b , i , j

Figure 3.14: Algebraic expressions for the ΛCCSD Λ2 amplitude equations for three-body Hamil-

tonians.


3.4. The ΛCCSD(T) Energy Correction for Three-Body Hamiltonians

3.4 The ΛCCSD(T) Energy Correction for Three-BodyHamiltonians

The derivation of the ΛCCSD(T) method given in [57] is not quite transparent,making it difficult to extend it to three-body Hamiltonians in an analogous way asit is derived for two-body Hamiltonians. Furthermore, the way the method is pre-sented in [57] also makes it difficult to realize the types of approximations that leadto the final result. Therefore, in this section the ΛCCSD(T) method is rederived asan approximation to the superior CR-CC(2,3) method [114,117,119,120,137,138],this way facilitating the identification of new terms corresponding to the inclusionof three-body interactions, and helping to understand the approximate nature ofΛCCSD(T).

Starting from the CR-CC(2,3) method, ΛCCSD(T) is easily derived as a series ofapproximations to CR-CC(2,3). The generalized moments of the CCSD equations

Mab ci j k

= ⟨Φab ci j k|

HN e T (CCSD)

C|Φ⟩ (3.41)

are approximated by restricting to terms at most linear in the cluster operator,

Mab ci j k

≈ ⟨Φab ci j k|

HN

1+ T1+ T2

C|Φ⟩ . (3.42)

Since the main focus of this section is the extension of ΛCCSD(T) to three-bodyHamiltonians and to identify new terms arising from the presence of the resid-ual normal-ordered three-body interaction operator WN in the normal-orderedHamiltonian HN , the moments are split into the contributions from the normal-ordered two-body approximation, in the following denoted as M

ab ci j k (NO2B), and

the contributions due to WN , denoted as Mab ci j k (3B),

Mab ci j k

= Mab ci j k(NO2B) + M

ab ci j k(3B) . (3.43)

The expressions for Mab ci j k (NO2B) and M

ab ci j k (3B) in terms of interaction and cluster

operators are given by

Mab ci j k(NO2B) = ⟨Φab c

i j k|

FN + VN

1+ T1+ T2

C|Φ⟩ (3.44)

= ⟨Φab ci j k|

VN

1+ T1+ T2

C|Φ⟩ (3.45)

and

Mab ci j k(3B) = ⟨Φab c

i j k|

WN

1+ T1+ T2

C|Φ⟩ , (3.46)



and programmable expressions in terms of matrix elements of the operators in-volved can be found in Figure 3.15.

In order to simplify the CR-CC(2,3) expression for L3 to the form used inΛCCSD(T), the reduced resolvent R3 is replaced by its simplified Møller-Plessetform [57,58],

R (CCSD)3 = − P3

H (CCSD)open

≈ − P3

FN

(3.47)

=∑

i<j<ka<b<c

εab ci j k

−1

|Φab ci j k⟩⟨Φab c

i j k| (3.48)

where εab ci j k is defined for the two-body Hamiltonian case,

εab ci j k= f i

i+ f

j

j + f kk− f a

a− f b

b− f c

c. (3.49)

The latter approximation is equivalent to replacing the H (CCSD)open on the left-hand

side of system (2.121) corresponding to CR-CC(2,3), by the FN operator. Further-more, in order to arrive at ΛCCSD(T), the effective Hamiltonian H (CCSD)

open on theright-hand side of system (2.121) is approximated by its leading contribution,which is HN . These approximations allow to replace system (2.121) by the sim-plified form

−∑

l<m<nd<e< f

⟨Φd e f

l m n | FN |Φab ci j k⟩ = ⟨Φ|

1+Λ(CCSD)

HN |Φab ci j k⟩ , (3.50)

which immediately leads to a convenient expression for the L3 amplitudes,

li j k

ab c =

εab ci j k

−1

⟨Φ|

1+Λ(CCSD)

HN |Φab ci j k⟩ . (3.51)

Again, in order to identify new terms arising from the presence of WN in thenormal-ordered Hamiltonian, the L3 amplitudes are split into their NO2B partand their part due to WN ,

li j k

ab c = li j k

ab c (NO2B) + li j k

ab c (3B) , (3.52)

which are given by

li j k

ab c (NO2B) =

⟨Φ|

Λ1 VN

DC|Φab c

i j k⟩+ ⟨Φ|

Λ2 FN

DC|Φab c

i j k⟩

+ ⟨Φ|

Λ2 VN

C|Φab c

i j k⟩

εab ci j k

−1

(3.53)


3.4. The ΛCCSD(T) Energy Correction for Three-Body Hamiltonians

and

li j k

ab c (3B) =

⟨Φ|WN |Φab ci j k⟩+ ⟨Φ|

Λ1 WN

C|Φab c

i j k⟩ (3.54)

+⟨Φ|

Λ2 WN

C|Φab c

i j k⟩

εab ci j k

−1

, (3.55)

and a similar splitting, analogously to (3.25), is done for the energy correctionδE (ΛCCSD(T)),

δE (ΛCCSD(T)) = δE(ΛCCSD(T))NO2B + δE

(ΛCCSD(T))3B , (3.56)

where the individual parts are given by

δE(ΛCCSD(T))NO2B =

1

(3!)2

∑

ab ci j k

li j k

ab c (NO2B)Mab ci j k(NO2B) (3.57)

and

δE(ΛCCSD(T))3B =

1

(3!)2

∑

ab ci j k

li j k

ab c (NO2B)Mab ci j k(3B)

+ li j k

ab c (3B)Mab ci j k(NO2B) + l

i j k

ab c (3B)Mab ci j k(3B)

. (3.58)

The final programmable expressions for the L3 amplitudes and the energy cor-rection are listed in Figure 3.15 and the corresponding spherical expression canbe found in Appendix G.2. In summary, the total ΛCCSD(T) ground-state energyconsists of several parts

E (ΛCCSD(T)) = Eref + ∆E(CCSD)NO2B + δE

(ΛCCSD(T))NO2B

+ ∆E(CCSD)3B + δE

(ΛCCSD(T))3B , (3.59)

where the 3B contributions are only present if the residual normal-ordered three-body interaction is included in the calculations. Furthermore, as it is discussedin [96], it is interesting to note that from the many-body perturbation (MBPT)point of view, the significance of the contributions due to the T3 clusters changes byincluding the three-body interaction WN . In the NO2B approximation, T3 contri-butions show up at second MBPT order for the wavefunction and at fourth orderin the energy δE (ΛCCSD(T)), but the inclusion of WN moves these contributions to firstorder for the wavefunction and second order for the energy.



One appealing aspect of the ΛCCSD(T) correction the simple structure thatmakes the method efficient and easy to implement. However, the degree of ap-proximations that enter the method makes it necessary to examine the perfor-mance of the method compared to more accurate approaches such as CR-CC(2,3).

δE (ΛCCSD(T)) =1

(3!)2

∑

ab ci j k

li j k

ab cM

ab ci j k

li j k

ab c=

¨

li j k

ab c(NO2B)

(LA)− Pab/c

∑

l

wi j k

ab lλl

c

(LB)+ Pi j /k

∑

d

wi j d

ab cλk

d

(LC)+ 1

2Pi j /k

∑

d e

w d e kab c λ

i j

d e

(LD)+ 1

2Pab/c

∑

l m

wi j k

l m cλl m

ab

(LE)+ Pab/c Pi j /k

∑

d l

wi j d

ab lλk l

c d

(LF)+ w

i j k

ab c

«

εab ci j k

−1

li j k

ab c(NO2B) = λ

i j k

ab c

Mab ci j k = M

ab ci j k (NO2B)

(MA)− Pab/c

∑

l

w ab li j k t c

l

(MB)+ Pi j /k

∑

d

w ab ci j d t d

k

(MC)+ 1

2Pi j /k

∑

d e

w ab cd e k t d e

i j

(MD)+ 1

2Pab/c

∑

l m

w l m ci j k t ab

l m

(ME)+ Pab/c Pi j /k

∑

d l

w ab li j d t c d

k l

(MF)+ w ab c

i j k

Mab ci j k (NO2B) = t ab c

i j k

Figure 3.15: Algebraic expressions for the calculation of theΛCCSD(T) energy correction for three-

body Hamiltonians. See Figure 2.12 for the expressions for t ab ci j k

and λi j k

ab c.


Chapter 4

Spherical Coupled-Cluster Theory

Chapter 4. Spherical Coupled-Cluster Theory

4.1 Introduction

The m -scheme formulation of Coupled-Cluster theory used in previous chaptersallows for CCSD computations for 40Ca up to about 8 oscillator shells using mod-erate computational resources, whereas the ΛCCSD(T) correction is already out ofreach even for smaller numbers of oscillator shells. Manageable three-body calcu-lations in the m -scheme framework are restricted to 4He and even there to smallmodel spaces [63].

Of course, the best way to reduce the computational expense is by exploitationof symmetries. In this work, only closed-shell nuclei are considered for which thenucleons fill complete (sub-) shells shown in Figure 4.1. For such nuclei the clus-ter operator is a rank-zero spherical tensor operator and spherical symmetry maybe exploited using angular-momentum algebra. Spherical Coupled-Cluster the-ory was first discussed in 2010 [27]. For two-body Hamiltonians, the sphericalformulation significantly extends the region of the nuclear chart accessible withCoupled-Cluster theory up to the heavy nuclei regime. Calculations involvingthree-body Hamiltonians can be performed up to medium-mass nuclei, wherethey benefit particularly from the efficient matrix element handling using a J T -coupled scheme [86]. The price to be paid for this decrease of computational de-mand by exploiting symmetries is an increased complexity of the initially rathersimple Coupled-Cluster equations and the corresponding computer implementa-tion.

For the following discussion of the spherical formulation, a convenient changein notation is introduced: Single-particle m -scheme states are denoted with a baror with separated angular-momentum projection quantum number as

|p ⟩ ≡ |p mp ⟩ ≡ |n p (l p sp ) jp mp tp m tp⟩ (4.1)

while a complete angular-momentum shell is represented by

|p ⟩ ≡ |n p (l p sp ) jp tp m tp⟩ . (4.2)

The approach followed in this work to obtain spherical Coupled-Cluster equationsis by angular momentum coupling of the corresponding m -scheme diagrams.Therefore, a brief summary of the relevant aspects of angular-momentum alge-bra and definitions used in this work is given in the following sections.


4.1. Introduction

1s 1s1/2 (2) [2] (M )

1p1p3/2 (4) [6]1p1/2 (2) [8] (M )

1d1d5/2 (6) [14]

2s2s1/2 (2) [16]1d3/2 (4) [20] (M )

1f1f7/2 (8) [28] (M )

1f5/2 (6) [38]2p2p3/2 (4) [32]

2p1/2 (2) [40]

1g1g9/2 (10) [50] (M )1g7/2 (8) [58]

2d2d5/2 (6) [64]2d3/2 (4) [68]

3s 3s1/2 (2) [70]

1h1h11/2 (12) [82] (M )1h9/2 (10) [92]

2f2f7/2 (8) [100]2f5/2 (6) [106]

3p3p3/2 (4) [110]3p1/2 (2) [112]

1i1i13/2 (14) [126] (M )

1i11/2 (12) [154]

2g2g9/2 (10) [136]

2g7/2 (8) [162]

3d3d5/2 (6) [142]

3d3/2 (4) [168]4s 4s1/2 (2) [164]

1j15/2 (16) [184] (M )

0ħhΩ

1ħhΩ

2ħhΩ

n

3ħhΩ

4ħhΩ

5ħhΩ

6ħhΩ

Figure 4.1: Schematic odering of single-particle states [146] used to generate the spherical reference

state. Magic nucleon numbers are marked as (M ).



4.2 Spherical Tensor Operators

An irreducible spherical tensor operator M(J ) of rank J is defined as the set of 2J+1 op-erators M(J )

M=−J ,...,J that transform under rotations R the same way as the sphericalharmonics do, i.e.,

D(R) M(J )M D†(R) =

J∑

M ′=−J

D(J )

M ′M M(J )

M ′ , (4.3)

where D(R) is the rotation operator corresponding to the rotation R acting on theHilbert space and D

(J )

M ′M are the Wigner D functions. However, the Hermiteanadjoint operator (M(J ))† does not transform as an irreducible tensor operator ac-cording to (4.3), its components rather transform as

(M(J )M )

† ∼ (−1)J−MM(J )−M . (4.4)

It is, therefore, customary to define the generalized Hermitean adjoint of an irre-ducible spherical tensor operator according to,

( ˆM†)(J ) =n

( ˆM†)(J )M=−J ,...,J

o

(4.5)

with

( ˆM†)(J )M ≡ (−1)J+M (M

(J )−M )

† , (4.6)

which transforms the same way as the original spherical tensor operator does,

( ˆM†)(J )M ∼ M

(J )M . (4.7)

This difference in transformation property of spherical tensor operators and itsHermitean adjoints has to be taken into account when they are subjected to angular-momentum coupling. When coupling angular momenta, the M component K(J )M

of an irreducible tensor operator K(J ) may be constructed from the direct productof two irreducible tensor operators M(JM) and N(JN) using Clebsch-Gordan coeffi-cients,

K(J )M =

∑

MMMN

JM JN J

MM MN M

CG

M(JM)MM

N(JN)MN

, (4.8)

which may be written in a short-hand tensor product notation as

K(J )M =

¦

M(JM) ⊗ N

(JN)©(J )

M. (4.9)


4.2. Spherical Tensor Operators

Since generalized Hermitean adjoint operators also transform as irreducible spher-ical tensor operators, mixed couplings between these and standard spherical tensoroperators are allowed. Coupling Hermitean adjoint operators by the prescrip-tion (4.8) is also meaningful because the operators involved have the same trans-formation properties. It yields an operator that transforms as the Hermitean ad-joint of an irreducible spherical tensor operator of rank J ,

∑

MMMN

JM JN J

MM MN M

CG

M(JM)MM

†

N(JN)MN

†(4.10)

= (−1)J−M∑

MMMN

JM JN J

MM MN −M

CG

ˆM

†(JM)

MM

ˆN

†(JN)

MN

(4.11)

= (−1)J−M

ˆM

†(JM)

⊗ˆN

†(JN)

(J )

−M

. (4.12)

When working with matrix representations of spherical tensor operators theWigner-Eckart theorem is particularly valuable. It allows for a factorization of ma-trix elements of spherical tensor operators into a geometric part that depends onthe projection quantum numbers but does not depend on the actual operator un-der consideration, and a reduced matrix element that is operator-specific and inde-pendent of the projections. Therefore, if K(J ) is an irreducible spherical two-bodytensor operator of rank J , for matrix elements of its components K

(J )M holds 1

⟨pq |K(J )M |r s ⟩

Jpq M pq Jr s M r s

= (−1)Jpq−M pq

Jpq J Jr s

−M pq M M r s

3j

⟨pq ||K(J )||r s ⟩

Jpq Jr s

, (4.14)

where

⟨pq ||K(J )||r s ⟩

Jpq Jr s

(4.15)

is the reduced matrix element of K(J ). Since the geometric part is already known forall operators, all operator-specific information is encoded in the reduced matrixelements. Furthermore, many problems may be formulated in reduced matrixelements exclusively, thus avoiding the geometric part completely. Two important

1For one-body operators the Wigner-Eckart theorem analogously reads

⟨p mp |K(J )|q mq ⟩ = (−1)jp−mp

jp J jq

−mp M mq

3j⟨p ||K(J )||q ⟩ . (4.13)



properties of spherical tensor operators immediately follows from (4.14) due to thepresence of the Wigner 3j symbol: The angular momenta involved have to satisfythe triangular condition

|Jpq − Jr s | ≤ J ≤ Jpq + Jr s , (4.16)

and the projections have to fulfill the condition

−M pq +M r s +M = 0 . (4.17)

4.3 Angular-Momentum Coupling

The concept of irreducible spherical tensor operators may be applied to the cre-ation and annihilation operators of second quantization. The 2jp +1 operators

a †p≡

n

a †p ,mp=−jp ,...,jp

o

(4.18)

transform as components of an irreducible spherical tensor operator of rank jp .They may, therefore, be subjected to angular-momentum coupling, e.g.,

n

a †p⊗ a †

q

o(J )

M=

∑

mp mq

jp jq J

mp mq M

CG

a †p ,mp

a †q ,mq

. (4.19)

When (4.19) is applied to the vacuum, it gives an antisymmetrized two-particlestate coupled to good angular momentum J and projection M ,

∑

mp mq

jp jq J

mp mq M

CG

|p mp q mq ⟩ ≡ |p q

J M

⟩ . (4.20)

The coupled bra state is defined as the Hermitean adjoint of the coupled ket state2, i.e.,

⟨p q

J M

| =∑

mp mq

jp jq J

mp mq M

CG

⟨p mp q mq | =

|p q

J M

⟩†

. (4.21)

It is important to note that the coupling line always denotes the coupling of thestates and not the coupling of the operators that create them, which makes a differ-ence because of the inverted order of the positions of the operators and the Slaterdeterminant entries for bra determinants, as can be seen from

⟨p mp q mq | = ⟨0| a q ,mqa p ,mp

. (4.22)

2If the Clebsch-Gordan coefficients are chosen to be real.


4.3. Angular-Momentum Coupling

The generalized Hermitean adjoint of a †p ,mp

is given by

ˆa p ,mp≡ (−1)jp−mp

a †p ,−mp

†

(4.23)

= (−1)jp−mp a p ,−mp(4.24)

and states created by the action of this operator will be denoted as

⟨p mp | ≡ ⟨0| ˆa p ,mp= (−1)jp−mp ⟨p −mp | . (4.25)

These bra states may be coupled among themselves or also to ket states, e.g.,

⟨p | . . . |q ⟩

J M

=∑

mp mq

jp jq J

mp mq M

CG

⟨p mp | . . . |q mq ⟩ (4.26)

=∑

mp mq

(−1)jp−mp

jp jq J

−mp mq M

CG

⟨p mp | . . . |q mq ⟩ (4.27)

where the rules of angular-momentum algebra hold and no distinction betweenbra and ket states has to be made any more. Equation (4.27) may also be reversed,i.e.,

⟨p mp | . . . |q mq ⟩ =∑

J M

(−1)jp−mp

jp jq J

−mp mq M

CG

⟨p | . . . |q ⟩

J M

. (4.28)

A coupled standard bra state can easily be expressed in terms of states generatedby the generalized Hermitian adjoints by

⟨p q

J M

| = (−1)J−M ⟨p q

J−M

| . (4.29)

Angular-momentum algebra is well-known and exhaustively documented [147],so no details such as orthogonality relations for coupling coefficients and recou-pling transformations etc. are given here. However, one particularly useful rela-tion that is easily confirmed is listed for future reference

∑

mp mq

jp jq J

−mp mq M

CG

jp J ′ jq

−mp M ′ mq

3j

= (−1)2jp (−1)J+M J −1 δJ J ′ δM−M ′ , (4.30)

where hat factors are defined by

=p

2j +1 . (4.31)



It should be noted that in the definition of coupled antisymmetric states (4.20)no normalization factor has been introduced, as it often is in the literature [33,78]. Therefore, these states are unnormalized, and the normalized states wouldbe given by

|p q

J M

⟩norm ≡ Npq |p q

J M

⟩ , (4.32)

with normalization factor

Npq ≡p

1+(−1)J δn p nqδl p lq

δjp jqδmtp mtq

1+δn p nqδl p lq

δjp jqδmtp mtq

. (4.33)

Consequently, if p and q represent two like nucleons in the same shell, this factorprevents coupling to odd angular momenta which is a feature also shared by theunnormalized states.

In most parts of this work the X coefficients [148]

jp jq Jpq

jr js Jr s

Jp r Jqs J

X

= Jpq Jr s Jp r Jqs

jp jq Jpq

jr js Jr s

Jp r Jqs J

9j

(4.34)

are used in favor of Wigner 9j symbols for angular momentum recouplings offour angular momenta in order to keep the expressions short. It is convenient tointroduce a similar definition also for Wigner 6j symbols,

¨

jp jq jr

js j t ju

«

X

= p q r

¨

jp jq jr

js j t ju

«

6j

, (4.35)

in order to shorten the equations encountered in the spherical formulation ofΛCCSD(T) (see. Appendix G.1).

4.4 One-Body Operators

The purpose of this section is to define coupled one-body matrix elements whichare inevitably encountered when Coupled-Cluster diagrams are evaluated in thespherical scheme. For a one-body spherical tensor operator component K(J )M thedefinition of coupled matrix elements used in this work is

⟨p |K(J )M |q ⟩

J ′M ′

=∑

mp mq

(−1)p−mp

jp jq J ′

−mp mq M ′

CG

⟨p mp |K(J )M |q mq ⟩ .

(4.36)


4.4. One-Body Operators

The decoupling employs the same phase factor, i.e.,

⟨p mp |K(J )M |q mq ⟩ =∑

J ′M ′

(−1)p−mp

jp jq J ′

−mp mq M ′

CG

⟨p |K(J )M |q ⟩

J ′M ′

.

(4.37)

The coupled one-body matrix elements may be expressed in terms of reducedone-body matrix elements as

⟨p |K(J )M |q ⟩

J−M

= (−1)2jp (−1)J−M J −1 ⟨p ||K(J )||q ⟩ , (4.38)

and, furthermore,

⟨p |K(J )M |q ⟩

J ′M ′

= δJ J ′ δM−M ′ ⟨p |K(J )M |q ⟩

J−M

.

(4.39)

So the only non-vanishing coupling corresponds to the rank J and the negativeprojection −M of the spherical tensor operator. This can be seen as follows. Re-placing the m -scheme matrix elements in

⟨p |K(J )M |q ⟩

J ′M ′

=∑

mp mq

(−1)p−mp

jp jq J ′

−mp mq M ′

CG

⟨p mp |K(J )M |q mq ⟩

(4.40)

by their reduced matrix elements according to the Wigner-Eckart theorem leadsto

⟨p |K(J )M |q ⟩

J ′M ′

=∑

mp mq

(−1)p−mp

jp jq J ′

−mp mq M ′

CG

× (−1)p−mp

jp J jq

−mp M mq

3j

⟨p ||K(J )||q ⟩ , (4.41)

where the phases cancel and the useful relation (4.30) can be applied to give

⟨p |K(J )M |q ⟩

J ′M ′

= (−1)2jp (−1)J′+M ′

( J ′)−1 δJ J ′ δM−M ′ ⟨p ||K(J )||q ⟩ . (4.42)

This is equivalent to (4.39) and (4.38) follows immediately.



4.5 Cross-Coupled Matrix Elements

4.5.1 Scalar Case

Matrix elements of scalar two-body spherical tensor operatorsK(0)0 such as the two-body Hamiltonian are routinely used in the J -coupled representation

⟨pq |K(0)0 |r s ⟩J M J M

=∑

mp mqmr ms

jp jq J

mp mq M

CG

jr js J

mr ms M

CG

× ⟨p mp q mq | K(0)0 |r mr s ms ⟩ (4.43)

which will be referred to as standard coupling. For the Hamiltonian, being a rank-zero spherical tensor operator, the matrix elements are diagonal in J and M .

An alternative coupling scheme that appears naturally in the derivation ofspherical Coupled-Cluster equations is the cross-coupled scheme. Following [148]3, two types of cross-coupled matrix elements may be defined, referred to as cross-

coupled scheme A (CCA)

⟨p q |K(0)0 |r s ⟩

J M

J M

≡ (−1)J−M ⟨p q |K(0)0 |r s ⟩

J−M

J M

(4.44)

= (−1)J−M∑

mp mqmr ms

(−1)jp−mp

jp jr J

−mp mr −M

CG

× (−1)jq−mq

jq js J

−mq ms M

CG

⟨p mp q mq | K(0)0 |r mr s ms ⟩ (4.45)

and cross-coupled scheme B (CCB)

⟨p q |K(0)0 |r s ⟩

J M

J M

≡ (−1)J−M ⟨p q |K(0)0 |r s ⟩

J−M

J M

(4.46)

= (−1)J−M∑

mp mqmr ms

(−1)jp−mp

jp js J

−mp ms −M

CG

× (−1)jq−mq

jq jr J

−mq mr M

CG

⟨p mp q mq | K(0)0 |r mr s ms ⟩ (4.47)

3However, a different style for the cross-coupling lines is used in order to avoid confusion withstandard coupling lines.


4.5. Cross-Coupled Matrix Elements

where the coupling line runs across the operator. According to the original defi-nition in [148], in both cases the phase (−1)J−M and the negative projection are as-sociated with the coupling involving the state p , although the choice is arbitraryfor the scalar case. However, this assignment is convenient for the general caseas well, which is why the original choice from [148] is kept in this work. As thestandard coupled matrix elements, cross-coupled matrix elements are diagonal inthe angular momenta and projections, i.e.,

⟨p q |K(0)0 |r s ⟩

J M

J ′M ′

= δJ J ′ δM M ′ ⟨p q |K(0)0 |r s ⟩

J M

J M

(4.48)

⟨p q |K(0)0 |r s ⟩

J M

J ′M ′

= δJ J ′ δM M ′ ⟨p q |K(0)0 |r s ⟩

J M

J M

, (4.49)

as is proven for the general case in the next section. The transformation equationsbetween the standard coupling and the cross-coupled schemes are listed in Fig-ure 4.2. The transformations for the reduced matrix elements follow directly fromthe simple relation between reduced and standard matrix elements for the case ofscalar spherical tensor operators,

⟨pq |K(0)|r s ⟩J M J M

= J −1 ⟨pq ||K(0)0 ||r s ⟩J J

(4.50)

and analogously for the cross-coupled matrix elements. Of course, these transfor-mations are only a special case of the more general transformations in Figure 4.3,but they are more efficient because for the Wigner 9j symbols appearing in thegeneral transformations may be replaced by 6j symbols as in Figure 4.2.

4.5.2 General Case

For applications involving spherical tensor operators of rank different from zerothe previous definition of cross-coupled matrix elements has to be generalized.Cross-coupled matrix elements of general spherical tensor operators are straight-forwardly defined in analogy to the scalar case by

⟨p q |K(J )M |r s ⟩

Jp r M p r

Jqs Mqs

≡ (−1)Jp r−M p r ⟨p q |K(J )M |r s ⟩

Jp r−M p r

Jqs Mqs

(4.51)



⟨p q |K(0)0 |r s ⟩

J M

J M

=∑

J ′( J ′)2 (−1)jq+jr+J+J ′

¨

jp jr J

js jq J ′

«

6j

⟨pq |K(0)0 |r s ⟩

J ′M ′ J ′M ′

⟨p q |K(0)0 |r s ⟩

J M

J M

= −∑

J ′( J ′)2 (−1)jq+jr−J

¨

jp js J

jr jq J ′

«

6j

⟨pq |K(0)0 |r s ⟩

J ′M ′ J ′M ′

⟨pq |K(0)0 |r s ⟩

J M J M

=∑

J ′( J ′)2 (−1)jq+jr+J+J ′

¨

jp jr J ′

js jq J

«

6j

⟨p q |K(0)0 |r s ⟩

J ′M ′

J ′M ′

≡ CCAtoStd

pq

r sJ ; J ′

⟨p q |K(0)0 |r s ⟩

J ′M ′

J ′M ′

⟨pq |K(0)0 |r s ⟩

J M J M

= −∑

J ′( J ′)2 (−1)jq+jr−J ′

¨

jp js J ′

jr jq J

«

6j

⟨p q |K(0)0 |r s ⟩

J ′M ′

J ′M ′

≡ CCBtoStd

pq

r sJ ; J ′

⟨p q |K(0)0 |r s ⟩

J ′M ′

J ′M ′

Figure 4.2: Transformations between the standard and cross-coupled schemes for scalar spherical

tensor operators.



and

⟨p q |K(J )M |r s ⟩

Jp s M p s

Jqr Mqr

≡ (−1)Jp s−M p s ⟨p q |K(J )M |r s ⟩

Jp s−M p s

Jqr Mqr

. (4.52)

The cross-coupled matrix elements as defined in (4.51) and (4.52) also allowa factorization into a geometric part and a reduced matrix element in the usualsense

⟨p q |K(J )M |r s ⟩

Jp r M p r

Jqs Mqs

= (−1)Jp r−M p r

Jp r J Jqs

−M p r M Mqs

3j

⟨p q ||K(J )||r s ⟩

Jp r

Jqs

(4.53)

and

⟨p q |K(J )M |r s ⟩

Jp s M p s

Jqr Mqr

= (−1)Jp s−M p s

Jp s J Jqr

−M p s M Mqr

3j

⟨p q ||K(J )||r s ⟩

Jp s

Jqr

. (4.54)

Only the cross-coupled A case (4.53) is derived in the following. By means ofangular-momentum recouplings the cross-coupled matrix element may be writ-ten in terms of standard coupled matrix elements as

⟨p q |K(J )M |r s ⟩

Jp r M p r

Jqs Mqs

= (−1)Jp r−M p r

∑

JM

Jp r Jqs J−M p r Mqs M

CG

×∑

Jpq Jr s

jp jr Jp r

jq js Jqs

Jpq Jr s J

X

∑

M pq M r s

(−1)Jpq−M pq

Jpq Jr s JM pq M r s M

CG

× ⟨pq |K(J )M |r s ⟩

Jpq−M pq Jr s M r s

(4.55)



Replacing the standard coupled matrix element by its reduced matrix element,

⟨p q |K(J )M |r s ⟩

Jp r M p r

Jqs Mqs

= (−1)Jp r−M p r

∑

JM

Jp r Jqs J−M p r Mqs M

CG

×∑

Jpq Jr s

jp jr Jp r

jq js Jqs

Jpq Jr s J

X

∑

M pq M r s


CG

Jpq J Jr s

M pq M M r s

3j

× ⟨pq ||K(J )||r s ⟩

Jpq Jr s

, (4.56)

allows application of the useful relation (4.30) to give

∑

M pq M r s


CG

Jpq J Jr s

M pq M M r s

3j

= (−1)J+M J −1 δJ J δM−M

(4.57)

which in turn eliminates the JM summations, leading to

⟨p q |K(J )M |r s ⟩

Jp r M p r

Jqs Mqs

= (−1)Jp r−M p r (−1)J+M J −1

Jp r Jqs J

−M p r Mqs −M

CG

×∑

Jpq Jr s

jp jr Jp r

jq js Jqs

Jpq Jr s J

X

⟨pq ||K(J )||r s ⟩

Jpq Jr s

. (4.58)

By introducing a Wigner 3j symbol and introducing the short-hand notation forthe reduced cross-coupled matrix element,

⟨p q ||K(J )||r s ⟩

Jp r

Jqs

≡∑

Jpq Jr s

jp jr Jp r

jq js Jqs

Jpq Jr s J

X

⟨pq ||K(J )||r s ⟩

Jpq Jr s

(4.59)

Eq. (4.58) can be written as

⟨p q |K(J )M |r s ⟩

Jp r M p r

Jqs Mqs

= (−1)Jp r−M p r

Jp r J Jqs

−M p r M Mqs

3j

⟨p q ||K(J )||r s ⟩

Jp r

Jqs

, (4.60)



arriving at (4.53).

The non-vanishing cross-coupled A matrix elements

⟨p q |K(J )M |r s ⟩

Jp r M p r

Jqs Mqs

(4.61)

of a spherical tensor operator component K(J )M fulfill the triangular and projectionconditions

|Jp r − Jqs | ≤ J ≤ Jp r + Jqs , −M p r +Mqs +M = 0 , (4.62)

and an analogous statement holds for the cross-coupled scheme B. This followsimmediately from the presence of the Wigner 3j symbol in (4.60).

Figure 4.3 summarizes the transformation formulas from standard to crosscoupling and vice versa for the case of reduced matrix elements, which are morecompact than for the standard matrix elements because the trivial geometricalfactor does not appear any more.



⟨p q ||K(J )||r s ⟩

Jp r

Jqs

=∑

Jp r Jqs

jp jr Jp r

jq js Jqs

Jpq Jr s J

X

⟨pq ||K(J )||r s ⟩

Jpq Jr s

⟨p q ||K(J )||r s ⟩

Jp r

Jqr

= (−1)jr+js−Jr s

∑

Jp s Jqr

jp js Jp s

jq jr Jqr

Jpq Jr s J

X

⟨pq ||K(J )||r s ⟩

Jpq Jr s

⟨pq ||K(J )||r s ⟩

Jpq Jr s

=∑

Jp r Jqs

jp jq Jpq

jr js Jr s

Jp r Jqs J

X

⟨p q ||K(J )||r s ⟩

Jp r

Jqs

≡ CCAtoStd

pq Jpq ; Jp r

r s Jr s ; Jqs

⟨p q ||K(J )||r s ⟩

Jp r

Jqs

⟨pq ||K(J )||r s ⟩

Jpq Jr s

= (−1)jr+js−Jr s

∑

Jp s Jqr

jp jq Jpq

js jr Jr s

Jp s Jqr J

X

⟨p q ||K(J )||r s ⟩

Jp s

Jqr

≡ CCBtoStd

pq Jpq ; Jp s

r s Jr s ; Jqr

⟨p q ||K(J )||r s ⟩

Jp s

Jqr

Figure 4.3: Transformations between the standard and cross-coupled schemes for reduced matrix

elements of general spherical tensor operators K(J ).


4.6. Diagram Coupling

4.6 Diagram Coupling

The aim of this section is the evaluation of angular-momentum coupled diagrams. Anoutstanding review of this matter is given in [148] and thus only a brief review isgiven here. A coupled diagram is obtained by coupling the external lines of thecorresponding m -scheme diagrams to good angular momentum, i.e., as schemat-ically indicated below,

(4.63)∑

ma mb

ja jb J

ma mb M

CG

a b

≡a b

J M

.

If these m -scheme diagrams represent matrix elements of some operator, the cou-pled diagrams clearly represent coupled matrix elements of this operator. For exam-ple, the (DBc) contribution (see Figure 2.4) to the coupled T2 matrix elements maybe written as

a i b j

J M

J M

= ⟨ a b | t2 | i j ⟩

J M J M

(D Bc )←−∑

ma mb

∑

m i m j

ja jb J

ma mb M

CG

j i j j J

m i m j M

CG

× 1

2

∑

c d

⟨a b |v |c d ⟩ ⟨c d |t2|i j ⟩ . (4.64)

This expression is not quite satisfactory yet since for its evaluation still m -schemematrix elements of the operators involved are required. The purpose of the cou-pling techniques to be presented below is to replace m -scheme matrix elementsand the corresponding summations over angular momentum projection quantumnumbers by products of coupled matrix elements. For instance, the above contri-bution may in the end be written in a fully angular-momentum-coupled formula-tion as

⟨ a b | t2 | i j ⟩

J M J M

(D Bc )←− 1

2

∑

c d

∑

J M

⟨ a b | v | c d ⟩

J M J M

⟨ c d | t2 | i j ⟩

J M J M

. (4.65)



Since the external lines are already explicitly coupled, the focus is on couplingof the internal lines. An internal line consists of a bra and a ket part which aresummed over, i.e.,

∑

p mp

⟨p mp | . . . |p mp ⟩ or∑

p mp

|p mp ⟩⟨p mp | . (4.66)

It is the projection quantum number summation that is of main interest in the fol-lowing, so the angular momentum summation will be ignored. Since an internalline typically corresponds to the outgoing part of some operator and the ingo-ing part of another operator, the bra and ket states will belong to different matrixelements. Internal lines may be coupled right away to zero angular momentumusing

∑

mp

⟨p mp | . . . |p mp ⟩ = (−1)2jp p ⟨p | . . . |p ⟩00

, (4.67)

∑

mp

|p mp ⟩⟨p mp | = p |p ⟩⟨p |00

. (4.68)

The simple proof is given in [148] but it is instructive to repeat it here. Usingthe identity

(−1)jp−mp p

jp jp 0

mp −mp 0

CG

= 1 , (4.69)

(4.67) is easily verified∑

mp

⟨p mp | . . . |p mp ⟩

= p

∑

mp

(−1)jp−mp

jp jp 0

mp −mp 0

CG

⟨p mp | . . . |p mp ⟩ (4.70)

= (−1)2jp p

∑

mp

(−1)jp−mp

jp jp 0

−mp mp 0

CG

⟨p mp | . . . |p mp ⟩ (4.71)

= (−1)2jp p ⟨p | . . . |p ⟩00

(4.72)

and (4.68) follows similarly.



By coupling the internal lines, example (4.64) becomes

⟨ a b | t2 | i j ⟩

J M J M

(D Bc )←− 1

2

∑

c d

c d ⟨ a b | v | c d ⟩⟨ c d | t2 | i j ⟩

J M J M00

00

. (4.73)

The projection summation has been eliminated in favor of coupling lines, but theabove form still has the disadvantageous property of coupling lines connectingdifferent matrix elements. However, these matrix elements can be disentangledby recoupling transformations of the internal lines for which an example is givenin the below:

Matrix elements connected by scalar coupled internal lines of the form

|pq ⟩⟨p q |00

00

(4.74)

may be disentangled by the transformation

p p |pq ⟩⟨p q |00

00

=∑

J M

|pq ⟩⟨pq |J M J M

. (4.75)

In order to prove this, the idea is to couple the two internal lines to good totalangular momentum. Since both angular momenta involved are zero the result istrivial,

p q |pq ⟩⟨p q |00

00

= p q

n

|pq ⟩⟨p q |0

0

o(0)

0. (4.76)

Having arrived at a fully coupled expression, the disentanglement of the matrixelement is only a matter of one standard recoupling transformation of the angularmomenta,

p q

n

|pq ⟩⟨p q |0

0

o(0)

0= p q

∑

J J ′

jp jp 0

jq jq 0

J J ′ 0

X︸︷︷︸

−1p

−1q J δJ J ′

n

|pq ⟩⟨p q |J J ′

o(0)

0(4.77)

=∑

J

Jn

|pq ⟩⟨p q |J J

o(0)

0. (4.78)



p q |pq ⟩⟨p q |

00

00

=∑

J M

|pq ⟩⟨pq |

J M J M

p q ⟨p q | . . . |pq ⟩

00

00

=∑

J M

⟨pq | . . . |pq ⟩

J M J M

p q ⟨p | . . . |q ⟩⟨q | . . . |p ⟩

00

00

=∑

J M

(−1)J−M (−1)jp+jq−J ⟨p | . . . |q ⟩⟨q | . . . |p ⟩

J M J−M

p q ⟨p | . . . |q ⟩ . . . |p ⟩ . . . ⟨q |

00

00

= (−1)2jq

∑

J M

(−1)J−M (−1)jr+js−J ⟨p | . . . |q ⟩ . . . |p ⟩ . . . ⟨q |

J M J−M

Figure 4.4: Internal line recoupling transformations involving two scalar coupled lines.

Breaking up the scalar coupling leads to

∑

J

Jn

|pq ⟩⟨p q |J J

o(0)

0=

∑

J

J∑

M M ′

J J 0

M M ′ 0

CG

|pq ⟩⟨p q |J M J M ′

(4.79)

=∑

J M

(−1)J−M |pq ⟩⟨p q |J M J−M

(4.80)

since

J J 0M M ′ 0

CG= (−1)J−M J −1δM−M ′ . By means of (4.29) the final result is obtained,

∑

J M

(−1)J−M |pq ⟩⟨p q |J M J−M

=∑

J M

|pq ⟩⟨pq |J M J M

. (4.81)

Internal line recouplings for other situations may be derived analogously. Themost frequent cases are summarized in Figure 4.4. The example (4.64) can stillbe further simplified since the matrix elements involved do not depend on the



total projection quantum numbers and thus the corresponding summation maybe replaced by a factor,

⟨ a b | t2 | i j ⟩

J M J M

(D Bc )←− 1

2

∑

c d

∑

J

J 2 ⟨ a b | v | c d ⟩

J M 0 J M 0

⟨ c d | t2 | i j ⟩

J M 0 J M 0

, (4.82)

where M 0 is an arbitrary, physically allowed value for the projection. More compli-cated diagrams than example (4.64) require other transformations to disentangleindividual matrix elements because non-scalar coupled internal lines may appearor the external lines need to be recoupled. The most convenient way to evaluatesuch diagrams is to chose recouplings of lines that involve at least one scalar cou-pled line since the transformations clearly become more simple with the numberof scalar lines.

For a simple demonstration of such transformations, let [p ] denote either |p ⟩ or⟨p |. Matrix elements connected by one scalar and one non-scalar coupled internallines may be disentangled by the transformation

[p ][q ][r ][s ]

00

J M

= (−1)jp+js −1p

∑

J ′M ′

∑

J ′′M ′′

(−1)J+J ′ J ′ J ′′

×¨

J J ′ J ′′

jr js jq

«

6j

J ′ J ′′ J

M ′ M ′′ M

CG

[p ][q ][r ][s ]

J ′M ′ J ′′M ′′

(4.83)

To see this, coupling the expression to total angular momentum JM is againtrivial due to the zero angular momentum involved,

[p ][q ][r ][s ]

00

J M

=∑

JM

0 J J0 M M

CG

n

[p ][q ][r ][s ]

0

J

o(J )

M(4.84)

=n

[p ][q ][r ][s ]

0

J

o(J )

M. (4.85)

This can then be recoupled using X coefficients and the total coupling may again



be broken up, arriving at

[p ][q ][r ][s ]

00

J M

=∑

J ′ J ′′

jp jq J ′

jr js J ′′

0 J J

X

n

[p ][q ][r ][s ]

J ′ J ′′

o(J )

M(4.86)

=∑

J ′M ′

∑

J ′′M ′′

J ′ J ′′ J

M ′ M ′′ M

CG

jp jq J ′

jr js J ′′

0 J J

X

[p ][q ][r ][s ]

J ′M ′ J ′′M ′′

. (4.87)

Finally, from the simplification of X coefficients involving one zero angular mo-mentum,

jp jq J ′

jr js J ′′

0 J J

X

= (−1)jp+js −1p(−1)J+J ′ J ′ J ′′

¨

J J ′ J ′′

jr js jq

«

6j

, (4.88)

Eq. (4.83) follows immediately.

Again, other transformations of this kind are equally easy to derive and themost commonly encountered ones are summarized in Figure 4.5 for reference.These rules simplify significantly if for the final coupling of two states only thescalar coupling is allowed, for instance if these states belong to the bra and ketstates of a scalar one-body operator, as discussed in Section 4.4. The transforma-tion rules for the case in which states [r ] and [s ] are allowed to scalar couplingonly is given in the bracketed expressions in Figure 4.5.

4.6.1 Antisymmetrized Diagram Coupling

As for the CCSD T2 m -scheme equations, some expressions are antisymmetrizedby the action of permutation operators, such as

⟨p q |g |r s ⟩ = Pp q Pr s (p q |g |r s )

= (1− Tp q ) (1− Tr s ) (p q |g |r s ) , (4.89)

where ⟨p q |g |r s ⟩denotes the antisymmetrized expression obtained from the actionof the permutation operators on some non-antisymmetric expression (p q |g |r s ).For instance, for the (DBe) contribution to the CCSD T2 amplitudes,

⟨a b |t2|ı ⟩(D B e )← Pa b Pı

∑

c k

⟨k b |v |c ⟩ t a c

ı k, (4.90)



[p ][q ][r ][s ]

00

J M

= (−1)jp+js −1p

∑

J ′M ′

∑

J ′′M ′′(−1)J+J ′ J ′ J ′′

×¨

J J ′ J ′′

jr js jq

«

6j

J ′ J ′′ J

M ′ M ′′ M

CG

[p ][q ][r ][s ]

J ′M ′ J ′′M ′′

= − −1p −1

s (−1)jp+jq−J [p ][q ][r ][s ]

J M 00

[p ][q ][r ][s ]

00

J M

= −1p

∑

J ′M ′

∑

J ′′M ′′(−1)J+J ′+J ′′ J ′ J ′′

×¨

J J ′ J ′′

js jr jq

«

6j

J ′ J ′′ J

M ′ M ′′ M

CG

[p ][q ][r ][s ]

J ′M ′ J ′′M ′′

= −1p −1

r (−1)jq+jr−J [p ][q ][r ][s ]

J M 00

[p ][q ][r ][s ]

00

J M

= −1q (−1)jp+jq+jr+js

∑

J ′M ′

∑

J ′′M ′′(−1)J J ′ J ′′

×¨

J J ′ J ′′

jr js jp

«

6j

J ′ J ′′ J

M ′ M ′′ M

CG

[p ][q ][r ][s ]

J ′M ′ J ′′M ′′

= − −1q −1

s [p ][q ][r ][s ]

J M 00

[p ][q ][r ][s ]

00

J M

= −1q (−1)jp+jq

∑

J ′M ′

∑

J ′′M ′′(−1)J+J ′′ J ′ J ′′

×¨

J J ′ J ′′

js jr jp

«

6j

J ′ J ′′ J

M ′ M ′′ M

CG

[p ][q ][r ][s ]

J ′M ′ J ′′M ′′

= −1q −1

r [p ][q ][r ][s ]

J M 00

Figure 4.5: Internal line recoupling transformations for one scalar and one non-scalar coupling

lines. The expressions in brackets correspond to the case in which only a scalar coupling

of [r ] and [s ] is allowed.



the non-antisymmetric part would be

(a b |t2|ı ) =∑

c k

⟨k b |v |c ⟩ t a c

ı k. (4.91)

In practice, the expression (p q |g |r s ) is calculated for the whole index range, whichallows to access elements with swapped indices, (q p |g |r s ), for instance. The anti-symmetrized expression is then simply obtained by

⟨p q |g |r s ⟩ = (p q |g |r s ) − (q p |g |r s ) − (p q |g |s r ) + (q p |g |s r ) . (4.92)

For the spherical case, an antisymmetrized coupled diagram is obtained by cou-pling the antisymmetrized m -scheme expressions,

⟨ p q | g | r s ⟩

Jpq M pq Jr s M r s

=∑

mp mqmr ms

jp jq Jpq

mp mq M pq

CG

jr js Jr s

mr ms M r s

CG

⟨p q |g |r s ⟩ (4.93)

Formally, the evaluation requires the coupling of each individual index permuta-tion of the m -scheme expression,

⟨ p q | g | r s ⟩

Jpq M pq Jr s M r s

=∑

mp mqmr ms

jp jq Jpq

mp mq M pq

CG

jr js Jr s

mr ms M r s

CG

×

(p q |g |r s )− (q p |g |r s )− (p q |g |s r )+ (q p |g |s r )

(4.94)

= ( p q | g | r s )

Jpq M pq Jr s M r s

− ( q p | g | r s )

Jpq M pq Jr s M r s

− ( p q | g | s r )

Jpq M pq Jr s M r s

− ( q p | g | s r )

Jpq M pq Jr s M r s

. (4.95)



Reversing the couplings that go from the right to the left introduces phases,

⟨ p q | g | r s ⟩

Jpq M pq Jr s M r s

= ( p q | g | r s )

Jpq M pq Jr s M r s

− (−1)jp+jq−Jpq ( q p | g | r s )

Jpq M pq Jr s M r s

− (−1)jr+js−Jr s ( p q | g | s r )

Jpq M pq Jr s M r s

+ (−1)jp+jq−Jpq (−1)jr+js−Jr s ( q p | g | s r )

Jpq M pq Jr s M r s

. (4.96)

This result suggests the definition of a permutation operator

Ppq (Jpq ) ≡ 1− (−1)jp+jq−Jpq Tpq (4.97)

so that the antisymmetrized coupled expression can be written as

⟨ p q | g | r s ⟩

Jpq M pq Jr s M r s

= Ppq (Jpq ) Pr s (Jr s ) ( p q | g | r s )

Jpq M pq Jr s M r s

. (4.98)

In the m -scheme, if an expression ⟨p q |g |r s ) is already antisymmetric in the orbitalpair p q , an additional antisymmetrizer Pp q corresponds to a factor of 2, e.g.,

⟨p q |g |r s ) =1

2Pp q ⟨p q |g |r s ) , (4.99)

which is sometimes used to write the equations in a more symmetric form. Thesame holds for the spherical case for the operators Ppq (Jpq ), i.e., for an expressionthat is already antisymmetric in the sense that

Tpq ⟨ p q | g | r s )

Jpq M pq Jr s M r s

= − (−1)jp+jq−Jpq ⟨ p q | g | r s )

Jpq M pq Jr s M r s

, (4.100)

an additional antisymmetrization operator may introduced by

⟨ p q | g | r s )

Jpq M pq Jr s M r s

=1

2Ppq (Jpq ) ⟨ p q | g | r s )

Jpq M pq Jr s M r s

. (4.101)

The above results, that for the spherical case an original m -scheme permuta-tion operator Pp q can simply be replaced by the operator Ppq (Jpq ), only holds ifthe states that are permuted also are the states that are coupled together to angu-lar momentum Jpq . This is typically the case for diagrams that are evaluated in



the standard coupling scheme, but not if a diagram is first evaluated in the cross-coupled scheme and then transformed to standard coupling. In the latter case, fora component K(J )M of a spherical tensor operator, the transformation that generatesan antisymmetrized standard coupled reduced matrix element from a non-antisymmetric

cross-coupled A expression is given by

⟨pq ||K(J )||r s ⟩

Jpq Jr s

=∑

Jp r Jqs

¨

jp jq Jpq

jr js Jr s

Jp r Jqs J

X

− (−1)jr+js−Jr s

jp jq Jpq

js jr Jr s

Jp s Jqr J

X

Tr s

+ (−1)jp+jq−Jpq (−1)jr+js−Jr s

jq jp Jpq

js jr Jr s

Jqs Jp r J

X

Tpq Tr s

− (−1)jp+jq−Jpq

jq jp Jpq

jr js Jr s

Jqr Jp s J

X

Tpq

«

(p q ||K(J )||r s )

Jp r

Jqs

≡ CCAtoStd(A)

pq Jpq ; Jp r

r s Jr s ; Jqs

(p q ||K(J )||r s )

Jp r

Jqs

. (4.102)

Since this transformation depends on the rank of the spherical tensor operator,simplifications are possible for the case of scalar tensor operators. For these, thetransformation for the standard matrix elements reads 4

⟨pq |K(0)0 |r s ⟩J M J M

=

¨∑

J ′

( J ′)2 (−1)jq+jr+J+J ′

¨

jp jr J ′

js jq J

«

6j

1+ Tpq Tr s

+∑

J ′

( J ′)2 (−1)jq+jr+J ′

¨

jp js J ′

jr jq J

«

6j

Tpq + Tr s

«

(p q |K(0)0 |r s )

J ′M ′0

J ′M ′0

(4.103)

4For half-integer values of jp , jq , jr , js .



for which the shorthand notation

⟨pq |K(0)0 |r s ⟩J M J M

= CCAtoStd(A)

pq

r sJ ; J ′

(p q |K(0)0 |r s )

J ′M ′0

J ′M ′0

(4.104)

is used. For the scalar case, this transformation may easily be written in terms ofthe non-antisymmetric cross-coupling transformations as

CCAtoStd(A)

pq

r sJ ; J ′

= CCAtoStd

pq

r sJ ; J ′

1+ Tpq Tr s

−CCBtoStd

pq

r sJ ; J ′

Tpq + Tr s

. (4.105)

For partial antisymmetrizations in which, e.g., only orbitals p and q are to be an-tisymmetrized, the corresponding transformation may be obtained from (4.102)or (4.103) simply by setting the other permutation operator Tr s to 0.

4.6.2 Cross-Coupled Evaluation

As mentioned in the previous chapter, it is sometimes advantageous to evaluate adiagram in the cross-coupled scheme first, before transforming it to the standardcoupling. This is because in order to obtain a cross-coupled matrix element lessinternal and external recouplings may be required than for the standard couplingand, thus, less summations over intermediate angular momenta and coupling co-efficients may be required.

For instance, the (R2G) contribution

b b (4.106)

to the EOM-CCSD R2 amplitudes, when directly evaluated in standard coupledevaluation, is given by



⟨ab ||R(J )2 ||i j ⟩

Jab Ji j

(R2G)←− (−1)Jab−M ab

Jab J Ji j

−M ab M M i j

−1

3j

Pab (Jab ) Pi j (Ji j )

× (−1)j i+j j−Ji j

∑

c k

∑

J J ′ J ′′

¨

J ′ J ′′ Jab

jb ja jc

«

6j

¨

J ′′ J ′′′ Ji j

j i j j jk

«

6j

J ′ ( J ′′)2 J ′′′

×∑

M ′M ′′

(−1)J′+M ′

J ′ J ′′ Jab

M ′ −M ′′ −M ab

CG

J ′′′ J ′′ Ji j

−M ′ M ′′ M ab

CG

×

J ′ J J ′′′

M ′ M −M ′

3j

⟨c a ||R(J )2 ||i k ⟩J ′ J ′′′

⟨b k |H2|c j ⟩

J ′′M ′′

J ′′M ′′

, (4.107)

where the triple sum over angular momenta and the product of two coupling coef-ficients cannot be factorized in any way. This is the result of the many recouplingsthat are required in order to disentangle the two matrix elements.

On the other hand, if the m -scheme expression is first transferred into cross-coupled form A, that requires the couplings

a i b j

Ja i Jb j

= ⟨a b ||R(J )2 ||i j ⟩

Ja i

Jb j

,

then it becomes apparent that in this case the (a i ) and (b j ) coupling lines alreadybelong to distinct matrix elements, requiring recoupling transformations for thec and k internal lines only. Therefore, the resulting expression for the standardcoupled reduced matrix element is much simpler,

⟨ab ||R(J )2 ||i j ⟩

Jab Ji j

(R2G)←− CCAtoStd(A)

ab Jab ; Ja i

i j Ji j ; Jb j

×∑

c k

(−1)jc+jk−Jb j ⟨a c ||R(J )2 ||i k ⟩

Ja i

Jb j

⟨b k |H2|c j ⟩

Jb j Mb j

Jb j Mb j

, (4.108)


4.7. Spherical CCSD

and it is more efficient because the only sums over total angular momenta arethe ones in the transformation from the cross-coupled to the standard couplingscheme. Furthermore, the problem factorizes into first evaluating the expressionin cross-coupled form, which cost is basically determined by

∑

c kand the transfor-

mation to standard coupling afterwards. Another inconvenience of the form (4.107)is that it can not longer solely be evaluated using optimized matrix multiplicationroutines, which is due to the appearance of orbital indices in both matrix elementsand coupling coefficients.

4.7 Spherical CCSD

Using the techniques described in previous sections, the spherical Coupled-Clusterequations are easily obtained. For CCSD, the m -scheme amplitude equations areof the form

0 = + . . . , ∀ a , ıb l

ıa(4.109)

0 = + . . . , ∀ a , b , ı , .b b

ıa b(4.110)

Therefore, coupling both sides of these equations by taking linear combinationsof m -scheme expressions, the coupled formulation follows immediately,

0 = + . . . , ∀ a , ib l

ia

00

(4.111)

0 = + . . . , ∀ a ,b , i , j , J , M ,b b

ia jb

J M J M

(4.112)

where the right hand side diagrams have to be evaluated according to diagramcoupling techniques. In (4.111) and (4.112), the scalar character of the cluster op-erator T has already been taken into account by constraining the coupling of exter-nal lines to total angular momenta and projection to 00 for the T1 matrix elementsand J M for bra and ket coupling in the T2 matrix elements. The energy expression

E =

b b

+

b l

+

b b



consists of closed diagrams only, and is, therefore, not subjected to external linecoupling, but the internal lines will have to be coupled in order to get an expres-sion in terms of coupled matrix elements only. The result for the spherical energyexpression is

∆E (CCSD) =(E A)

+1

4

∑

c d k l

∑

J

J 2 ⟨c d |t2|k l ⟩

J M J M

⟨k l |v |c d ⟩

J M J M

(E B )

+∑

c k

⟨c |t1|k ⟩

00

⟨k | f |c ⟩

00

(EC )

+1

2

∑

c d k l

−1c−1

d

∑

J

J 2 ⟨c |t1|k ⟩

00

⟨d |t1|l ⟩

00

⟨k l |v |c d ⟩

J M J M

, (4.113)

and the spherical CCSD T1 and T2 amplitude equations are listed in Appendix C.2.

The matrix elements of the normal-ordered Hamiltonian (2.10) should also beexpressed in terms of coupled matrix elements. Using the identity

p q ⟨p q |v |pq ⟩00

00

= p q ⟨p q |v |pq ⟩00

00

=∑

J

J 2 ⟨pq |v |pq ⟩J M J M

, (4.114)

the corresponding expressions are straightforwardly found to be

⟨Φ|H |Φ⟩ = h0−∑

i

i ⟨ı |h1|i ⟩00

+1

2

∑

i j

∑

J

J 2 ⟨i j |v |i j ⟩

J M J M

(4.115)

⟨p | f |q ⟩00

= ⟨p |h1|q ⟩00

− −1p

∑

i

∑

J

J 2 ⟨p i |v |qi ⟩

J M J M

. (4.116)

The spherical CCSD equations be written as

⟨Φ|H (CCSD)|Φ⟩ = ∆E (CCSD) (4.117)

⟨a |H (CCSD) |i ⟩00

= 0 , ∀ a , i (4.118)

⟨ab |H (CCSD)|i j ⟩

J M J M

= 0 , ∀ a ,b , i , J , M . (4.119)


4.8. Convergence Acceleration

Using this notation, the spherical analogue to the standard iterative scheme (2.45)for the self-consistent solution of the CCSD equations reads

(0)⟨a |t1|i ⟩00

= (0)⟨ab |t2|i j ⟩

J M J M

= 0

(n+1)⟨a |t1|i ⟩00

= ⟨a |H (CCSD)(t (n ))|i ⟩00

⟨a | f |a ⟩00

−⟨ı | f |i ⟩00

−1

(n+1)⟨ab |t2|i j ⟩

J M J M

= ⟨ab |H (CCSD)(t (n ))|i j ⟩

J M J M

×

−1a⟨a | f |a ⟩

00

+ −1b ⟨b | f |b ⟩

00

− −1i ⟨ı | f |i ⟩

00

− −1j ⟨ | f |j ⟩

00−1

(4.120)

where, as in the m -scheme case, FN → F oN is set in the amplitude equations. The

n = 1 amplitudes are then easily obtained from the relations

⟨a |H (CCSD)(t (0))|i ⟩00

= a ⟨a | f |i ⟩00

(4.121)

⟨ab |H (CCSD)(t (0))|i j ⟩

J M J M

= ⟨ab |v |i j ⟩

J M J M

. (4.122)

4.8 Convergence Acceleration

In practical applications the iteration schemes (2.45) or (4.120) for solving theCCSD amplitude equations or the analogous scheme for solving the ΛCCSD am-plitude equations is not sufficient due to slow convergence, or even divergence, ofthe iterations. Consequently, the iterations have to be accelerated and stabilized.There are mainly two possibilities to improve the situation. On the one hand,the iteration scheme (2.45) may be used and the resulting sequence of vectorst (k ) may be used to construct an improved, faster converging vector sequence.Methods like the simple mixing [149], the Anderson mixing [150], or the Broyden

mixing [149] achieve such a stabilized and accelerated convergence. On the otherhand, the standard iteration scheme (2.45) itself may be modified, which may thenagain be combined with convergence accelerators such as the Broyden mixing.Both approaches are considered in the following.

According to (2.44), the Coupled-Cluster amplitude equations are convertedinto a fixed-point problem

t (n+1) = I (t (n )) , t ∗ = I (t ∗) . (4.123)



If the iterations are divergent, the simple mixing method, in which the new vectoris given by

t (n+1) = α I (t (n )) + (1−α) t (n ) , (4.124)

may help to bring the sequence to convergence. By mixing the new vector I (t (n ))

with the old one, regulated by the mixing parameter α, this mixing guaranteesthat the iteration scheme does not depart too far from the initial guess, whichslows the overall convergence down but at the same time makes it more robustagainst poor choices of the start vector.

The Anderson and Broyden mixing presented Figures 4.6 and 4.7 are moresophisticated convergence acceleration methods based on the multidimensionalNewton method and they are widely used in the quantum chemistry context. De-tailed reviews can be found in [149–151] and only the practical application is dis-cussed in the following. As for the simple mixing, the Anderson and Broydenmixings also have a mixing parameter α but both methods have an additionalbackward range parameter M which determines the number of previous vectorst (n ) that should be taken into consideration for constructing the new vector t (n+1).It can be easily verified that the simple mixing is contained in the Anderson mix-ing if for the latter M = 0 is chosen.

Figure 4.8 shows an illustration of the convergence rates of the Anderson, Broy-den and simple mixing and for the case of no mixing at all. Convergence is mon-itored in terms of the norm of the residual vector, i.e.,

residual(k) = I (t (k ))− t (k )

2. (4.125)

The no-mixing case exhibits a slow divergence, making the use of convergence-enhancing methods mandatory. The simple mixing, with an for all methods uni-versally chosen mixing parameter α= 0.6 leads to convergence, however, at a veryslow rate. Therefore, as mentioned above, mixing methods do not only improvethe convergence, they may also be vital for the iterations to converge at all. TheAnderson and Broyden show a very similar performance which is far superior tothe previous scenarios, leading to residuals below 10−7 within 20–30 iterations.Of course, the convergence is improved as the backward range parameter M isincreased, i.e., more information about previous iterations is used in order to de-termine the new vector.

Figure 4.9 shows the influence of the mixing parameter α for the Anderson andBroyden mixing for fixed range M . In the cases presented in Figure 4.9, where the



t (1) = I (t (0))

For k = 1, 2, . . .

mk =mink , M

F (k ) = I (t (k ))− t (k )

Determine β (k ) =

β(k )0 , . . . ,β

(k )mk

Tthat solves

minβ=(β0,...,βmk )

T

F (k−mk ), . . . , F (k )

β

2,

mk∑

i=0

βi = 1

t (k+1) = (1−α)mk∑

i=0

β(k )i t (k−mk+i )+α

mk∑

i=0

β(k )i I (t (k−mk+i ))

Figure 4.6: The Anderson convergence acceleration method [150].

vector series is nicely converging, αmay be chosen large since there is no need toslow the convergence down. As can be seen in in Figure 4.9, fastest convergenceis achieved at α= 0.6. Therefore, the combination of parameters M ≈ 8 and α≈ 0.6

typically is a good choice for obtaining fast and robust convergence.

As already mentioned above, another possibility – which does not seem tohave been paid much attention in the past – to enhance the convergence of theCoupled-Cluster amplitude equations is by introducing a new iteration scheme,alternative to (2.45), which is used to generate the vector sequence used in themixing methods. In the following, an improved iteration scheme is proposed. Inorder to do so, it is instructive to review how the standard iteration scheme (2.45)is constructed. Using the T1 amplitude equations

0 = . . .(SC a )

+∑

c

f ac

t ci

(SC b )

−∑

k

f ki

t ak

(SC c )

+∑

c k

v a ki c

t ck+ . . . (4.126)

as an example, the amplitude t ai may be introduced on the left-hand side by adding

t ai Da

i on both sides and diving by Dai , arriving at

t ai=

1

Dai

n

. . .(SC a )

+∑

c

f ac

t ci

(SC b )

−∑

k

f ki

t ak

(SC c )

+∑

c k

v a ki c

t ck+ · · ·+ t a

iDa

i

o

. (4.127)



t (1) = I (t (0))

For m = 1, 2, . . .

t (m+1) = t (m )+α F (m )−m−1∑

n=m

wn γm n u (n )

where

m =max1, m −M

γm n =m−1∑

k=m

c mk βk n

βk n =

w 201+a

−1

k n

c mk =wk

∆F (k )†

F (m )

a k n =wk wn

∆F (n )†∆F (k )

u (n ) =α∆F (n )+∆t (n )

∆t (n ) =

t (n+1)− t (n )

/F (n+1)−F (n )

2

∆F (n ) =

F (n+1)−F (n )

/F (n+1)−F (n )

2

F (k ) = I (t (k ))− t (k )

Figure 4.7: The (modified) Broyden algorithm [149].



Anderson

0 5 10 15 20 25 30

10-6

10-4

0.01

1

iteration

resi

dual

Broyden

0 5 10 15 20 25 30

10-6

10-4

0.01

1

iteration

resi

dual

α = 0.6, M = 2 (), 4 (), 6 (), 8 (), 10 ()

simple mixing:

no mixing:

68NiNN+3N-fullħhΩ= 24 MeV

E3max = 18

αSRG = 0.04 fm4

Figure 4.8: Comparison of the Anderson and Broyden method to the simple mixing or with no

mixing at all, for fixed mixing parameter α and varying backward range M , for the

CCSD amplitude equations.



Anderson

0 5 10 15 20 25 30

10-6

10-4

0.01

1

iteration

resi

dual

Broyden

0 5 10 15 20 25 30

10-6

10-4

0.01

1

iteration

resi

dual

M = 8, α = 0.2 (), 0.4 (), 0.6 (), 0.8 (), 1.0 ()

68NiNN+3N-fullħhΩ= 24 MeV

E3max = 18

αSRG = 0.04 fm4

Figure 4.9: Comparison of the Anderson and Broyden mixing for fixed backward range M and

varying mixing parameter α, for the CCSD amplitude equations.



The denominator Dai is chosen to be

Dai= f i

i− f a

a. (4.128)

As a result of this choice of Dai , the t a

i amplitudes are removed from the expressioninvolving the contractions with the Fock operator,

(SC a )

+∑

c

f ac

t ci

(SC b )

−∑

k

f ki

t ak+ t a

iDa

i=

∑

c 6=a

f ac

t ci−∑

k 6=i

f ki

t ak

. (4.129)

Thus, another way to describe this procedure is to move the terms involving asingle t a

i and a single Fock matrix element to the left side and to divide by theprefactor Da

i in order to isolate t ai . For the T2 equations, there is an analogous

procedure. Equation (4.127) is then used as starting point to set up the standarditerative scheme,

(n+1)t ai=

1

Dai

n

. . .(SC a )

+∑

c

f ac(n )t c

i

(SC b )

−∑

k

f ki(n )t a

k

(SC c )

+∑

c k

v a ki c

(n )t ck+ · · ·+ t a

iDa

i

o

. (4.130)

The definition of the denominator Dai eliminates the specific amplitude t a

i on theright-hand side from the terms (SCa) and (SCb), which involve a contraction of theT1 operator with the Fock operator, but in other terms such as (SCc) the t a

i remain.It would be more natural, as it is done in the Jacobi scheme for linear systems, toeliminate t a

i completely from the right-hand side. For the example contributionsconsidered in (4.127), this is achieved simply by using the modified denominator

Dai= f i

i− f a

a− v a i

i a. (4.131)

Unlike this example, it is not possible to remove all occurrences of t ai from the

right-hand side, as can be seen, for instance, for the diagram

(SEb )

+∑

c d k

v a kc d

t ci

t dk

. (4.132)

In such cases, the strategy followed here is to simply pick one of these t ai and move

it to the left-hand side. One possible realization of the improved iteration scheme,in terms of spherical denominators Da

i and Dabi j (J ), is given in Figures 4.10 and 4.11.

Diagrams of the T2 amplitude equations that are evaluated in cross-coupled formare not easy to deal with and have been excluded from the considerations 5. Sincethis iteration scheme is analogous to the Jacobi scheme for linear systems, it willbe referred to as Jacobi iteration scheme in the following.



Dai=

(SC a )

− −1a⟨a | f |a ⟩

00

(SC b )

+ −1a⟨ı | f |i ⟩

00

(SC c )

+ −2a

∑

J

J 2 ⟨a i |v |i a ⟩

J M J M

(SDa )

− 1

2−2

a

∑

d k l

∑

J

J 2 ⟨a d |t2|k l ⟩

J M J M

⟨k l |v |a d ⟩

J M J M

(SDb )

− 1

2−2

a

∑

c k l

∑

J

J 2 ⟨i l |v |c d ⟩

J M J M

⟨c d |t2|i l ⟩

J M J M

(SDc )

+ −2a

∑

d l

−2d

∑

J J ′

J 2 ( J ′)2 ⟨i l |v |a d ⟩

J M J M

⟨d a |t2|l i ⟩

J ′M ′ J ′M ′

(SE a )

− −2a⟨ı | f |a ⟩

00

⟨a |t1|i ⟩00

(SEb )

− −2a

∑

d k

−1k

∑

J

J 2 ⟨a k |v |a d ⟩

J M J M

⟨d |t1|k ⟩

00

(SE c )

+ −2a

∑

c l

−1c

∑

J

J 2 ⟨i l |v |i c ⟩

J M J M

⟨c |t1|l ⟩

00

(SF )

− −3a

∑

d k l

−1d

∑

J

J 2 ⟨k l |v |a d ⟩

J M J M

⟨a |t1|k ⟩

00

⟨d |t1|l ⟩

00

Figure 4.10: Denominator for the Jacobi iteration for the CCSD T1 amplitude equations.



Dabi j(J ) = (1+ Tab ) (1+ Ti j )

¨(D B a )

− 1

2−1

b⟨b | f |b ⟩

00

(D Bb )

+ 1

2−1

j⟨ | f |j ⟩

00

(D Bc )

+ 1

8⟨ab |v |ab ⟩

J M J M

(D Bd )

+ 1

8⟨i j |v |i j ⟩

J M J M

(DC a )

+ 1

16

∑

c d

⟨i j |v |c d ⟩

J M J M

⟨c d |t2|i j ⟩

J M J M

(DC c )

− 1

4−2

i

∑

c d k

∑

J ′

( J ′)2 ⟨k i |v |c d ⟩

J ′M ′ J ′M ′

⟨c d |t2|k i ⟩

J ′M ′ J ′M ′

δj i j l

(DC d )

− 1

4−2

a

∑

c k l

∑

J ′

( J ′)2 ⟨c a |t2|k l ⟩J ′M ′ J ′M ′

⟨k l |v |c a ⟩

J ′M ′J ′M ′

δja jd

(DE a )

− 1

2−2

i

∑

c

⟨c |t1|i ⟩00

⟨ı | f |c ⟩00

(DEb )

− 1

2−2

a

∑

k

⟨a |t1|k ⟩

00

⟨k | f |a ⟩

00

(DE e )

+ 1

4−1

a

∑

k

⟨a |t1|k ⟩

00

⟨kb |v |ab ⟩

J M J M

(DE f )

− 1

4−1

i

∑

c

⟨c |t1|i ⟩00

⟨i j |v |c j ⟩

J M J M

(DE g )

− 1

2−2

a

∑

c k

−1c

∑

J ′

( J ′)2 ⟨c |t1|k ⟩

00

⟨k a |v |c a ⟩

J ′M ′J ′M ′

(DE h)

+ 1

2−2

i

∑

c k

−1c

∑

J ′

( J ′)2 ⟨c |t1|k ⟩

00

⟨k i |v |c i ⟩

J ′M ′ J ′M ′

(DG a )

+ 1

8−1

b−1

c

∑

k l

⟨a |t1|k ⟩

00

⟨b |t1|l ⟩

00

⟨k l |v |ab ⟩

J M J M

(DG b )

+ 1

8−1

i−1

j

∑

c d k l

⟨c |t1|i ⟩00

⟨d |t1|j ⟩

00

⟨i j |v |c d ⟩

J M J M

(DG e )

− 1

2−3

a

∑

c k l

−1c

∑

J ′

( J ′)2 ⟨a |t1|l ⟩

00

⟨c |t1|k ⟩

00

⟨k l |v |c a ⟩

J ′M ′J ′M ′

«

Figure 4.11: Denominator for the Jacobi iteration for the CCSD T2 amplitude equations.



Figure 4.12 shows a comparison of the standard and Jacobi iteration schemecombined with the Broyden mixing for 68Ni CCSD calculations at the HO frequen-cies ħhΩ= 24 MeV and 40 MeV, where the former frequency corresponds to the op-timal frequency at which the convergence of the CCSD equations is typically alsothe best. At the optimal frequency (Figure 4.12 top), both iteration schemes per-form similarly, where the Jacobi scheme converges a little faster. The real use of theJacobi scheme, however, is when the equations do not converge quickly or do evendiverge. This is illustrated in Figure 4.12 at the bottom, where for the not optimalfrequency the standard iteration scheme converges slower, but the convergencerate of the Jacobi scheme remains unchanged. Even in cases where the equationsare highly divergent in the standard iteration scheme, a combination of the Ja-cobi scheme with a low-α Broyden mixing often leads to convergence. Of course,evaluating the denominators in Figures 4.10 and 4.11 is significantly more expen-sive than for the standard scheme. However, particularly in CCSD calculationsfor three-body Hamiltonians where the iteration steps become quite costly, usingthese improved denominators saves a significant amount of computing time.

5These are the diagrams (DBe), (DCb), (DEc), (DEd), (DGc), and (DGd).



ħhΩ= 24 MeV

0 5 10 15 20 25 30

10-6

10-4

0.01

1

iteration

resi

dual

ħhΩ= 40 MeV

0 5 10 15 20 25 30

10-6

10-4

0.01

1

iteration

resi

dual

Broyden : α = 0.6, M = 8

Jacobi :

standard :

68NiNN+3N-full

E3max = 14

αSRG = 0.04 fm4

Figure 4.12: Comparison of the standard and the Jacobi iteration scheme for the CCSD amplitude

equations.



4.9 Spherical CCSD for Three-Body Hamiltonians

4.9.1 Three-Body Matrix Elements

The J T -coupled matrix element scheme [86] provides fast access to matrix ele-ments of the form 6

⟨ a b c | w | d e f ⟩

Jab Tab Jd e Td e

JMTMT JMTMT

, (4.133)

whereas for Coupled-Cluster applications matrix elements of the form

⟨ a m tab m tb

c m tc| w | d m td

e m tef m t f

⟩

Jab M ab Jd e M d e

Jc f M c f

(4.134)

are required 7. In (4.134), the isospin projections are written explicitly in orderto stress that the isospin is not coupled there. However, the matrix elements thatare stored will be reduced matrix elements that correspond to the more compactisospin-coupled form,

⟨ a b c | w | d e f ⟩

Jab MabTab MTab

Jd e Md eTd e MTd e

Jc f Mc fTc f MTc f

. (4.135)

Ignoring isospin for the moment, the matrix elements (4.135) can be expressed

6Total angular momentum and isospin projectionsM andMT have a fixed value of 1/2 becausethe interaction is independent ofM due to rotational invariance, and through the an isospin av-eraging the interaction also becomes independent ofMT .

7It is actually the reduced matrix elements that are required.


4.9. Spherical CCSD for Three-Body Hamiltonians

in terms of the standard coupled matrix elements (4.134) as

⟨ a b c | w | d e f ⟩

Jab M ab Jd e M d e

Jc f M c f

= (−1)Jab−M ab

Jab Jd e Jc f

−M ab M d e M c f

3j

×∑

J(−1)J−jc−Jab Jc f

ˆJ 2

¨

Jab Jd e Jc f

j f jc J

«

6j

⟨ a b c | w | d e f ⟩

Jab Jd e

JM JM

.

(4.136)

To proof this, straightforward recouplings lead to

⟨ a b c | w | d e f ⟩

Jab M ab Jd e M d e

Jc f M c f

=∑

mc m f

(−1)jc−mc

jc j f Jc f

−mc m f M c f

CG

×∑

JM

Jab jc JM ab mc M

CG

∑

J ′M ′

Jd e j f J ′M d e m f M ′

CG

⟨ a b c | w | d e f ⟩

Jab Jd e

JM J ′M ′

,

(4.137)

where the diagonality of the three-body matrix elements in J andM can be ex-ploited to arrive at

⟨ a b c | w | d e f ⟩

Jab M ab Jd e M d e

Jc f M c f

=∑

J⟨ a b c | w | d e f ⟩

Jab Jd e

JM JM

(4.138)

×∑

mc m fM(−1)jc−mc

jc j f Jc f

−mc m f M c f

CG

Jab jc J

M ab mc M

CG

Jd e j f J

M d e m f M

CG

︸︷︷︸

≡ Ω

.

Then, the relation [147]

∑

αβδ

(−1)b+β

a b c

α β γ

CG

b e d

−β ε δ

CG

a f d

α φ δ

CG

= (−1)b+c+d+ f c d 2 e−1

c f e

γ φ ε

CG

¨

a b c

e f d

«

6j

(4.139)

can be written in the form



∑

αβδ

(−1)b−β

b a c

−β α γ

CG

e b d

ε β δ

CG

f a d

φ α δ

CG

= (−1)−2a−b+2c+3d−e c d 2 e−1

c f e

γ φ ε

CG

¨

a b c

e f d

«

6j

(4.140)

which allows to simplify Ω to give

Ω = (−1)J−jc+Jab J −1ab

Jc fˆJ 2

×

Jc f Jd e Jab

M c f M d e M ab

CG

¨

Jab Jd e Jc f

j f jc J

«

6j

(4.141)

= (−1)Jab−M ab

Jab Jd e Jc f

−M ab M d e M c f

3j

× (−1)J−jc−Jab Jc fˆJ 2

¨

Jab Jd e Jc f

j f jc J

«

6j

. (4.142)

Plugging this into (4.138),

⟨ a b c | w | d e f ⟩

Jab M ab Jd e M d e

Jc f M c f

= (−1)Jab−M ab

Jab Jd e Jc f

−M ab M d e M c f

3j

×∑


ˆJ 2

¨

Jab Jd e Jc f

j f jc J

«

6j

⟨ a b c | w | d e f ⟩

Jab Jd e

JM JM

,

(4.143)

gives the desired expression (4.136).

It is clear that the above result holds for the isospin as well. Furthermore, sinceall projection-quantum-number dependence is in the prefactor

(−1)Jab−M ab

Jab Jd e Jc f

−M ab M d e M c f

3j

, (4.144)



the corresponding reduced matrix elements may immediately be defined as

⟨ab c ||w ||d e f ⟩

Jab Jd e

Jc f

≡∑


ˆJ 2

¨

Jab Jd e Jc f

j f jc J

«

6j

⟨ a b c | w | d e f ⟩

Jab Jd e

JM JM

,

(4.145)

or, including isospin,

⟨ab c ||w ||d e f ⟩

Jab Tab Jd e Td e

Jc f Tc f

≡∑


ˆJ 2

¨

Jab Jd e Jc f

j f jc J

«

6j

×∑

T(−1)T −tc−Tab Tc f T 2

¨

Tab Td e Tc f

t f tc T

«

6j

⟨ a b c | w | d e f ⟩

Jab Tab Jd e Td e

JMTMT JMTMT

.

(4.146)

The prefactor (4.144) looks similar to the geometric part of the traditional defini-tion of reduced matrix elements (4.14), however, the rank of the tensor, which iszero in this case, does not appear in the Wigner 3j symbol any more. Instead, thethird total angular momentum appears in the 3j symbol, resulting in the require-ments

|Jab − Jd e | ≤ Jc f ≤ Jab + Jd e , (4.147)

and

|Tab −Td e | ≤ Tc f ≤ Tab +Td e (4.148)

for non-vanishing matrix elements.

When needed, the isospin m -scheme matrix elements are calculated from the



isospin-coupled ones by straightforward decoupling,

⟨ a m tab m tb

c m tc| w | d m td

e m tef m t f

⟩

Jab M ab Jd e M d e

Jc f M c f

=∑

Tab M Tab

∑

Td e M Td e

∑

Tc f M Tc f

(−1)Tab−M Tab

Tab Td e Tc f

−M TabM Td e

M Tc f

3j

×

ta tb Tab

m tam tb

M Tab

CG

td te Td e

m tdm te

M Td e

CG

× (−1)tc−mtc

tc t f Tc f

m tcm t f

M Tc f

CG

⟨ab c ||w ||d e f ⟩

Jab Tab Jd e Td e

Jc f Tc f

. (4.149)

Since the total projections M Tab, M Td e

, M Tc fare fixed by the m -scheme isospin pro-

jections, the corresponding summations may be removed. The 29 transformationcoefficients T [. . . ]may easily be precomputed and stored, so that the transforma-tion simply reads

⟨ a m tab m tb

c m tc| w | d m td

e m tef m t f

⟩

Jab M ab Jd e M d e

Jc f M c f

=∑

Tab

∑

Td e

∑

Tc f

T

m tam tb

m tc

m tdm te

m t f

Tab Td e Tc f

⟨ab c ||w ||d e f ⟩

Jab Tab Jd e Td e

Jc f Tc f

. (4.150)

Due to the coupling running across the operator, these matrix elements are notHermitean. Nevertheless, they fulfill the symmetry relation

⟨d e f ||w ||ab c ⟩

Jd e Td e Jab Tab

Jc f Tc f

= (−1) (−1)Jab+Jd e+Jc f (−1)jc+j f −Jc f

× (−1) (−1)Tab+Td e+Tc f (−1)tc+t f −Tc f ⟨ab c ||w ||d e f ⟩

Jab Tab Jd e Td e

Jc f Tc f

(4.151)

that still allows to exploit the original Hermitecity of the Hamiltonian in standardcoupling.



4.9.2 Conversion to Reduced Format

In theJ T -coupled scheme, the matrix elements (4.133) are stored for orbital indexcombinations of the form

a ≥b ≥ c ,

d ≤ a , e ≤¨

b : for a = d

d : else, f ≤

¨

c : for a = d ∧b = d

e : else. (4.152)

This does not permit to directly access matrix elements with arbitrary orbital in-dices. Since the purpose of the J T -coupled scheme is the calculation of m -scheme matrix elements which have simple index permutation relations, this isnot a drawback there. On the other hand, for the reduced matrix elements (4.146)that enter the spherical Coupled-Cluster equations, it is important to have fast ac-cess to all possible index combinations. Therefore, only the trivial index swapsin the first two orbitals in bra and ket as well as Hermitecity are exploited in thestorage scheme for the reduced matrix elements, i.e.,

a ≥b , d ≤ a , e ≤¨

b : for a = d

d : else. (4.153)

In order to calculate reduced matrix elements of arbitrary orbital index combina-tions, standardJ T -coupled matrix elements with the same orbital indices are re-quired. As mentioned before, these are not directly accessible in the J T -coupledscheme, so they have to be expressed in terms of index combinations that are avail-able. Two examples are given by

| p q r ⟩

Jpq

JM

= −∑

Jp r

(−1)jq+jr+Jpq+Jp r Jpq Jp r

¨

jq jp Jpq

jr J Jp r

«

6j

| p r q ⟩Jp r

JM

(4.154)

and

| p q r ⟩

Jpq

JM

= −∑

Jqr

(−1)jq+jr−Jqr Jpq Jqr

¨

jp jq Jpq

jr J Jqr

«

6j

| q r p ⟩

Jqr

JM

(4.155)



which can be shown as follows: For (4.154), straightforward angular-momentumrecoupling leads to

| p q r ⟩

Jpq

JM

=∑

Jp r


¨

jq jp Jpq

jr J Jp r

«

6j

n

| p r q ⟩

Jpq

o(Jpq r )J.

(4.156)

Exchanging the positions of q and r results in a sign from antisymmetry. Thereare no further phases because by exchanging q and r no angular momentum cou-plings are affected. Therefore, (4.154) follows immediately.

For (4.155), again, straightforward angular-momentum recoupling leads to

| p q r ⟩

Jpq

JM

=∑

Jqr

(−1)jp+jq+jr+J Jpq Jqr

¨

jp jq Jpq

jr J Jqr

«

6j

| p q r ⟩

Jqr

JM

. (4.157)

Using the relation

| p q r ⟩

Jqr

JM

= (−1)2 (−1)jp+Jqr−J | q r p ⟩

Jqr

JM

, (4.158)

Eq. (4.155) is reproduced.

From each of these two relations another trivial relation can be derived regard-ing permutations of the first two orbitals on the right-hand side, giving a phasefrom antisymmetry and reversion of the coupling direction.

It should be noted that the sign in (4.154) stems from antisymmetry when or-bitals q and r have been exchanged. It does not stem from angular-momentumconsiderations. Therefore, this sign arises only once, even if isospin is consideredas well. On the other hand, the sign in (4.155) does origin from angular-momentumconsiderations and will, therefore, cancel with the one from arising in the isospintransformation. The complete set of relevant transformation expressions, includ-ing isospin, is given in Figure 4.13.



| p q r ⟩

Jpq Tpq

JMTMT

= − (−1)jp+jq−Jpq (−1)1−Tpq | q p r ⟩

Jpq Tpq

JMTMT

| p q r ⟩

Jpq Tpq

JMTMT

= −∑

Jp r


¨

jq jp Jpq

jr J Jp r

«

6j

×∑

Tp r

(−1)1+Tpq+Tp r Tpq Tp r

¨1/2 1/2 Tpq

1/2 T Tp r

«

6j

| p r q ⟩

Jp r Tp r

JMTMT

| p q r ⟩

Jpq Tpq

JMTMT

=∑

Jp r

(−1)jp+jq+Jpq Jpq Jp r

¨

jq jp Jpq

jr J Jp r

«

6j

×∑

Tp r

(−1)1+Tpq Tpq Tp r

¨1/2 1/2 Tpq

1/2 T Tp r

«

6j

| r p q ⟩

Jp r Tp r

JMTMT

| p q r ⟩

Jpq Tpq

JMTMT

=∑

Jqr

(−1)jq+jr−Jqr Jpq Jqr

¨

jp jq Jpq

jr J Jqr

«

6j

×∑

Tqr

(−1)1−Tqr Tpq Tqr

¨1/2 1/2 Tpq

1/2 T Tqr

«

6j

| q r p ⟩

Jqr Tqr

JMTMT

| p q r ⟩

Jpq Tpq

JMTMT

= −∑

Jqr

Jpq Jqr

¨

jp jq Jpq

jr J Jqr

«

6j

×∑

Tqr

Tpq Tqr

¨1/2 1/2 Tpq

1/2 T Tqr

«

6j

| q r p ⟩

Jqr Tqr

JMTMT

Figure 4.13: Transformations between different index permutations for the angular momentum

coupling used in the J T -coupled scheme.



4.9.3 Spherical CCSD Equations for Three-Body Hamiltonians

Apart from the three-body matrix element handling, the spherical formulationof CCSD for three-body Hamiltonians does not require any new techniques. Thethree-body contributions to the normal-ordered Hamiltonian in terms of angular-momentum-coupled matrix elements read

⟨Φ|H |Φ⟩ = ⟨Φ|H |Φ⟩2B−1

6

∑

i j k

k

∑

J

J ⟨i j k ||w ||i j k ⟩

J J

0

, (4.159)

⟨p | f |q ⟩00

= ⟨p | f |q ⟩00

2B+1

2

∑

i j

∑

J

J ⟨i j p ||w ||i j q ⟩

J J

0

, (4.160)

and

⟨pq |v |r s ⟩J M J M

= ⟨pq |v |r s ⟩J M J M

2B− J −1∑

i

i ⟨pq ı ||w ||r s i ⟩J J

0

, (4.161)

where ⟨. . . ⟩2B denote the already known contributions from the two-body Hamil-tonian. The algebraic expressions for the spherical ∆E CCSD, T1 and T2 amplitudeequations are listed in Appendix C.4. The computational runtime is dominatedby the two diagrams

(T2Dc)

− 1

4Pab (J ) Pi j (J ) J −1 −1

i−1

j(−1)ja+jb−J

∑

c d e k l

∑

J ′ J ′′

(−1)J+J ′+J ′′ J ′ J ′′

×¨

J ′ J ′′ J

ja jb je

«

6j

⟨k l a ||w ||c d e ⟩

J ′ J

J ′′

⟨c |t1|i ⟩00

⟨d |t1|j ⟩

00

⟨e b |t2|k l ⟩

J ′M ′ J ′M ′

(4.162)

and(T2Eo)

− 1

4Pab (J ) Pi j (J )

∑

c d e k l m

∑

J ′ J ′′

J ′ ( J ′′)2¨

J ′ J ′′ J

ja jb jd

«

6j

¨

J ′ J ′′ J

ja jc jd

«

6j

× ⟨k l m ||w ||c d e ⟩

J ′ J ′

0

⟨a c |t2|i j ⟩J M J M

⟨b d |t2|k l ⟩

J ′M ′ J ′M ′

⟨e |t1|m ⟩00

, (4.163)

which, therefore, require special attention in the implementation.


4.10. Spherical ΛCCSD(T)

4.10 Spherical ΛCCSD(T)

When attempting to derive the spherical expression for δE (ΛCCSD(T)) from the m -scheme expression

δE (ΛCCSD(T)) =1

(3!)2

∑

a b cı k

λı k

a b c

1

εa b c

ı k

t a b c

ı k, (4.164)

the presence of the denominator εa b c

ı kis troubling at first glance. Each angular-

momentum projection appears in three instead of two matrix elements, and one ofthem even appears in a denominator. It is clear that in general such an expressioncannot be disentangled into separate spherical matrix elements. However, εa b c

ı kis

given by a sum of matrix elements of the Fock operator, which is a scalar one-body spherical tensor operator. Since its matrix elements are independent of theprojection

⟨p mp | f |q mq ⟩ = ⟨p 0| f |q0⟩ = − −1p⟨p | f |q ⟩

00

, (4.165)

the denominator may be drawn in front of the projection summations, thus al-lowing the angular-momentum coupling techniques from the previous sectionsto obtain the corresponding spherical expression 8. Setting

εa b c

ı k= εab c

i j k= − −1

i⟨ı | f |i ⟩

00

− −1j⟨ | f |j ⟩

00

− −1k⟨k | f |k ⟩

00

+ −1a⟨a | f |a ⟩

00

+ −1b⟨b | f |b ⟩

00

+ −1c⟨c | f |c ⟩

00

(4.166)

the δE (ΛCCSD(T)) correction reads

δE (ΛCCSD(T)) = − 1

(3!)2

∑

ab ci j k

1

εab ci j k

∑

J J ′ J ′′

(−1)jc+jk−J ′′

× (−1)J+J ′+J ′′ ⟨i j k ||ˆλ||ab c ⟩

J J ′

J ′′

⟨ab c ||ˆt ||i j k ⟩

J ′ J

J ′′

. (4.167)

8Other non-iterative energy corrections such as CR-CC(2,3) [138] also have a denominatorwhich, however, is not as simple as the one encountered for ΛCCSD(T) which then cannot betreated exactly in a spherical formulation, see Sections 2.8 and 5.6.



When evaluating the coupled expressions for the ˆλ, ˆt amplitudes, the permu-tation operators pose a problem, as, e.g., for the contribution

λi j k

ab c ← Pa/b c Pk/i j

∑

d

v d kb cλ

i j

a d (4.168)

to the λ amplitudes. Similar to Section 4.9.2, it is possible to couple the m -schemeexpression for a specific index combination and to work out the transformationthat leads to the antisymmetrized coupled expression. However, this transfor-mation is quite memory-consuming in practical applications. The most straight-forward approach for dealing with the permutations is to apply them before theangular-momentum coupling, and simply couple each resulting term, i.e.,

λi j k

ab c ←

1− Tab − Ta c

1− Ti k − Tj k

∑

d

v d kb cλ

i j

a d (4.169)

=

1− Tab − Ta c − Ti k − Tj k + Tab Ti k + Tab Tj k

+ Ta c Ti k + Ta c Tj k

∑

d

v d kb cλ

i j

a d

=(λA1)

+∑

d

v d kb cλ

i j

a d

(λATab )−∑

d

v d ka cλ

i j

b d

(λATa c )− . . . . (4.170)

The corresponding spherical expressions are listed in Appendix G.1 9. Theseexpression make heavy use of X coefficients. Since there are too many of them topre-store, only X coefficients for fixed values of J , J ′, J ′′ are calculated and cachedat a time 10. In order to accelerate the computation of these X coefficients, therelation for Wigner 9j symbols

a b c

d e f

g h j

9j

=∑

x

(−1)2x (2x +1)

¨

a b c

f j x

«

6j

¨

d e f

b x h

«

6j

¨

g h j

x a d

«

6j

(4.171)

is used in order to compute the X coefficients from pre-cached Wigner 6j sym-bols. Because of the presence of the coupling coefficients it is no longer possible

9The naming convention is such that, for example, (λATab) is the first term contributing to λ

after the permutation operator Tab has been applied.10This is of course because the energy correction contributions for fixed i , j , k , J , J ′, J ′′ are cal-

culated at a time.


4.11. Spherical ΛCCSD(T) for Three-Body Hamiltonians

to use optimized matrix-multiplication routines to compute the ˆt , ˆλ amplitudesfor given hole orbitals i , j , k . Therefore, there exists no longer a reason not to con-strain the particle index summation in the energy formula. A spherical expressionfor the energy correction with partially exploited antisymmetry that is used in theimplementation, is given by

δE (ΛCCSD(T)) = − 1

(3!)2

∑

a≥b ;ci≥j ;k

(2−δab ) (2−δi j )1

εab ci j k

×∑

J J ′ J ′′

(−1)jc+jk−J ′′ (−1)J+J ′+J ′′ ⟨i j k ||ˆλ||ab c ⟩

J J ′

J ′′


J ′ J

J ′′

, (4.172)

which may be confirmed by recognizing that εab ci j k is invariant under permutations

of orbitals and, for instance,

∑

ab

|ab

Jab

⟩ ⟨ab

Jab

| = ∑

a<b

+∑

a=b

+∑

a>b

!

|ab

Jab

⟩ ⟨ab

Jab

| (4.173)

=

∑

a<b

+∑

a=b

+∑

a<b

(−1) (−1)ja+jb−Jab

2

!

|ab

Jab

⟩ ⟨ab

Jab

| (4.174)

=∑

a≤b

(2−δab ) |ab

Jab

⟩ ⟨ab

Jab

| . (4.175)

4.11 Spherical ΛCCSD(T) for Three-Body Hamiltonians

The ΛCCSD(T) expressions for three-body Hamiltonians are translated into thespherical formulation analogous to the two-body Hamiltonian case. As before,the treatment of permutation operators Pab/c etc. is not trivial. For the three-body ΛCCSD(T) a more economic implementation has been chosen than straight-forwardly expanding the permutation operator Pab/c in terms of transpositions,Pab/c = 1− Ta c − Tb c . As an illustrative example, the expression

(LA)

− Pab/c

∑

l

wi j k

ab l λlc= −

1− Ta c − Tb c

∑

l

wi j k

ab l λlc

(4.176)

may also be written as

(LA1)−∑

l

wi j k

ab l λlc

LATa cTb c

− Pab

∑

l

wi j k

b c l λla

. (4.177)



The permutation operator Pab is easy to deal with because in the spherical schemeit may simply be replaced by Pab (J ) . As a second example, (LE)may be rewrittenas

(LE)

+ Pab/c Pi j /k

∑

d l

wi j d

ab l λk lc d

=(LE1)

+∑

d l

wi j d

ab l λk lc d

LETa cTb c

+ Pab

∑

d l

wi j d

b c l λk la d

(4.178)

LETi kTj k

+ Pi j

∑

d l

wj k d

ab l λi lc d

LETi k ,Tj kTa c ,Tb c

+ Pab Pi j

∑

d l

wj k d

b c l λi la d

,

again involving permutation operators that are most convenient for the transla-tion into the spherical scheme. The spherical expressions corresponding to theserepresentations of l

i j k

ab c and Mab ci j k can be found in Appendix G.2.

4.12 The CR-CC(2,3) Energy Correction

As reviewed in more detail in Section 2.8, the CR-CC(2,3) energy correction is ofthe form

δE (CR−CC(2,3)) =1

(3!)2

∑

a b cı k

lı k

a b cM

a b c

ı k, (4.179)

where lı k

a b care the amplitudes of the approximated left-eigenstate operator and

Ma b c

ı kare the generalized moments of the CCSD equations. As can be seen from

(2.122), the denominator

D a b c

ı k≡ D a b c

ı k(3) , (4.180)

where

D a b c

ı k(k ) = −

k∑

n=1

⟨Φa b c

ı k| H (CCSD)

n|Φa b c

ı k⟩ , (4.181)

(or D(k ) for short) involving one-, two-, and three-body effective Hamiltonian ma-trix elements, enters the definition of the l

ı k

a b camplitudes, such that the energy

expression can be stated as

δE (CR−CC(2,3)) =1

(3!)2

∑

ab ci j k

∑

ma mb mcmi m j mk

Nı k

a b c

1

D a b c

ı k

Ma b c

ı k. (4.182)


4.12. The CR-CC(2,3) Energy Correction

Such a denominator containing higher-than-one-body matrix elements is trou-bling when the energy expression (4.179) is translated into the spherical formula-tion, because the three-fold appearance of individual orbitals prevents the use oftraditional angular-momentum-coupling techniques. A projection-independentdenominator, on the other hand, may be pulled in front of the projection sumin (4.182), and the contraction of the N

ı k

a b cand M

a b c

ı kelements can be formulated

in angular-momentum-coupled form. One possibility is to truncate the denomi-nator at the k = 1 level,

D a b c

ı k≈ D a b c

ı k(1) = − ⟨Φa b c

ı k| H (CCSD)

1 |Φa b c

ı k⟩ , (4.183)

which has the same structure as the denominator encountered in ΛCCSD(T), withthe one-body effective Hamiltonian being replaced by the Fock operator, and isindependent from the projections. However, while D a b c

ı k(1) is the only denomi-

nator that can be treated exactly in spherical Coupled-Cluster theory, it shouldalso be attempted to include the higher-order denominators D a b c

ı k(2) and D a b c

ı k(3)

into the calculations, at least in an approximate form. The approach pursued inthis work to incorporate denominators beyond D a b c

ı k(1) is to replace the matrix

elements H p q

p q and H p q r

p q r entering the definition of D a b c

ı k(2) and D a b c

ı k(3) by their

projection-averaged counterpartsH pq

pqandH pqr

pqr, according to

H pq

pq= −2

p−2

q

∑

mp mq

⟨p q |H2|p q ⟩ =∑

J

J 2 ⟨pq |H2|pq ⟩J M J M

(4.184)

H pqr

pqr= −2

p−2

q−2

r

∑

mp mqmr

⟨p q r |H3|p q r ⟩ (4.185)

=∑

Jpq

∑

J

J 2 ⟨pqr |H3|pqr ⟩

Jpq

J M

Jpq

J M

. (4.186)

The resulting denominators will correspondingly be referred to as Dab c

i j k(2) and

Dab c

i j k(3), or D(2) and D(3) for short. The coupled matrix elements entering (4.186)

can be obtained from

⟨ab i |H3|ab i ⟩

Jab

J M

Jab

J M

= − J 2ab

∑

m

¨

jm j i Jab

J j i Jab

«

6j

⟨ab | t2 |i m ⟩

Jab M ab Jab M ab

⟨i m |v |ab ⟩

Jab M ab Jab M ab

(4.187)



and

⟨i j a |H3|i j a ⟩

Ji j

J M

Ji j

J M

= J 2i j

∑

e

¨

je ja Ji j

J ja Ji j

«

6j

⟨i j |v |a e ⟩

Ji j M i j Ji j M i j

⟨a e |t2|i j ⟩Ji j M i j

Ji j M i j

. (4.188)

This approximative treatment of the higher-order denominators will be justifiedby practical calculations in Section 5.6, where it is compared to results correspond-ing to an exact treatment of the denominators obtained from an m -scheme imple-mentation.

For the spherical derivation of the CR-CC(2,3) working equations, the couplingstrategy used for theΛCCSD(T) method in Section 4.10 is too cumbersome becauseof the expansion of the permutation operators involved, resulting in many andrather complex terms to consider in the final expressions. The advantage of thisapproach is little memory consumption in actual calculations. For CR-CC(2,3),an alternative route is followed, where the multi-index permutation operators areapplied to the coupled expressions. In order to do so, a different coupled form ofthe energy-correction expression is more convenient, 11

δE (CR−CC(2,3))

=1

(3!)2

∑

a≥b ;ci≥j ;k

∑

Jab Ji jJ

(2−δab ) (2−δi j ) ⟨i j k ||l ||ab c ⟩ ⟨ab c ||M||i j k ⟩

JabJabJi j Ji j

J JJJ

, (4.189)

in which the orbitals connected via angular-momentum coupling that are alsothe ones that are subject to permutation among each other. The application ofangular-momentum coupling and the permutation operators

Pab c = 1− Tab − Ta c − Tb c + Tab Ta c + Ta c Tab (4.190)

and

Pi j /k = 1− Ti k − Tj k (4.191)

at orbitals as they occur in (4.189) may then be evaluated using the transforma-tions Pab c (Jab , Jab c ) and Pi j /k (Ji j , Ji j k ) listed in Figure 4.14. These transformations

11It should be noted that in (4.189) reduced matrix elements are used while the expressionspresented in the following result in the non-reduced matrix elements.



are quite simple, requiring only 6j coupling coefficients and orbital permutations.However, all intermediate angular momenta and orbitals that are permuted haveto be held in memory in order to avoid to compute quantities more than once,making this approach more memory consuming, as mentioned earlier, but on theother hand more efficient.

Using the transformations from Figure 4.14 the generalized moments may thenbe calculated as

⟨ab c |M|i j k ⟩

Jab

J M

Ji j

J M

= Pab c (Jab , J ) ⟨ab c |T|i j k ⟩

Jab

J M

Ji j

J M

, (4.192)

with

⟨ab c |T|i j k ⟩

Jab

J M

Ji j

J M

= Pi j /k (Ji j , J )

¨

1

2Ji j Jab

∑

m

(−1)jc+jm+Ji j

¨

jc jm Ji j

jk J Jab

«

6j

⟨m c | J |i j ⟩Ji j M i j

Ji j M i j

⟨ab |t2|k m ⟩

Jab M ab Jab M ab

− 1

2Ji j Jab

∑

e

(−1)je+jc+Ji j

¨

jc je Ji j

jk J Jab

«

6j

⟨ab |H2|k e ⟩

Jab M ab Jab M ab

⟨e c |t2|i j ⟩Ji j M i j

Ji j M i j«

(4.193)

and where the angular-momentum-coupled form of the J intermediate reads

⟨m c | J |i j ⟩J M J M

= ⟨m c |H2|i j ⟩J M J M

+ −1m

∑

e

⟨e c |t2|i j ⟩J M J M

⟨m |H1|e ⟩00

. (4.194)

For the derivation of (4.193), the non-trivial identity∑

mc mkmm

∑

Mab Mi jM

Jab jc J

M ab mc M

CG

Ji j jk J

M i j mk M

CG

×

jm jc Ji j

mm mc M i j

CG

jk jm Jab

mk mm M ab

CG

= − (−1)jc+jm+Ji j J 2 Ji j Jab

¨

jc jm Ji j

jk J Jab

«

6j

(4.195)



∑

ma mb

ja jb Jab

ma mb M ab

CG

∑

M ab mc

Jab jc Jab c

M ab mc M ab c

CG

× Pab c |a ma b mb c mc ⟩

=

¨

1− (−1)ja+jb−Jab Tab +∑

J

ˆJ Jab

¨

jc jb Jja Jab c Jab

«

6j

Ta c TJabJ

−∑

J(−1)jb+jc+J +Jab ˆJ Jab

¨

jc ja Jjb Jab c Jab

«

6j

Tb c TJabJ

−(−1)ja+jb+Jab

∑

J

ˆJ Jab

¨

jc ja Jjb Jab c Jab

«

6j

Tab Ta c TJabJ

−∑

J(−1)jb+jc+J ˆJ Jab

¨

jc jb Jja Jab c Jab

«

6j

Ta c Tab TJabJ

«

| a b c ⟩

Jab

Jab c M ab c

≡ Pab c (Jab , Jab c ) | a b c ⟩

Jab

Jab c M ab c

∑

m i m j

j i j j Ji j

m i m j M i j

CG

∑

M i j mk

Ji j jk Ji j k

M i j mk M i j k

CG

× Pi/j k | i m i j m j k mk ⟩

=

¨

1 +∑

J

ˆJ Ji j

¨

jk j j Jj i Ji j k Ji j

«

6j

Ti k TJi jJ

−∑

J(−1)jk+j j+J +Ji j ˆJ Ji j

¨

jk j i Jj j Ji j k Ji j

«

6j

Tj k TJi jJ

«

| i j k ⟩

Ji j

Ji j k M i j k

≡ Pi j /k (Ji j , Ji j k ) | i j k ⟩

Ji j

Ji j k M i j k

Figure 4.14: Transformations required in to obtain angular-momentum coupled and antisym-

metrized expressions for Pab c |a b c ⟩ and Pi j /k |ı k ⟩, as needed in the CR-CC(2,3)

implementation. The action of, e.g., TJabJ is understood as replacing all occurences

of Jab on the right by J .



is helpful. Similarly, the N amplitudes may be calculated according to

⟨i j k |N |ab c ⟩

Jab

J M

Ji j

J M

= Pab c (Jab , J ) ⟨i j k |Γ|ab c ⟩

Jab

J M

Ji j

J M

(4.196)

where

⟨i j k |Γ|ab c ⟩

Jab

J M

Ji j

J M

= Pi j /k (Ji j , J )

¨

1

2δJab Ji j

−1k(−1)jc+jk+Jab+Ji j

×

⟨ i j |H2| a b ⟩

Jab M ab Jab M ab

⟨k |λ1|c ⟩

00

+ ⟨ i j |λ2| a b ⟩

Jab M ab Jab M ab

⟨k |H1|c ⟩

00

− 1

2(−1)jc+je+Ji j

∑

e

Jab Ji j

¨

jc je Ji j

jk J Jab

«

6j

⟨i j |λ2|e c ⟩


⟨k e |H2|ab ⟩

Jab M ab Jab M ab

+ 1

2(−1)jc+jm+Ji j

∑

m

Jab Ji j

¨

jc jm Ji j

jk J Jab

«

6j

⟨i j |H2|m c ⟩


⟨k m |λ2|ab ⟩

Jab M ab Jab M ab«

.

(4.197)



4.13 Spherical EOM-CCSD

Excited eigenstates of the Hamiltonian have good angular momentum and pro-jection, so these quantum numbers may be used to label right eigenstates as

|Ψ(CCSD)µ,J ,M ⟩ (4.198)

and the corresponding left eigenstates as

⟨Ψ(CCSD)µ,J ,M | . (4.199)

As in the m -scheme case, these eigenstates will be generated by the action of cor-responding excitation operators on the CCSD ground state. If T(J ) is a sphericaltensor operator, then the right eigenstate transforms as

|Ψ(CCSD)µ,J ,M ⟩ ∼ T

(J )M , (4.200)

but the left eigenstate transforms differently, according to

⟨Ψ(CCSD)µ,J ,M | ∼ (−1)J−M

T(J )−M . (4.201)

Therefore, by introducing excitation operators R(CCSD)µ,J ,M and de-excitation operators

L(CCSD)µ,J ,M that transform as spherical tensor operators,

R(CCSD)µ,J ,M , L

(CCSD)µ,J ,M ∼ T

(J )M , (4.202)

the spherical EOM-CCSD ansatz for the excited states then reads

|Ψ(CCSD)µ,J ,M ⟩ = R

(CCSD)µ,J ,M e T (CCSD) |Φ⟩ (4.203)

⟨Ψ(CCSD)µ,J ,M | = ⟨Φ| (−1)J−M

L(CCSD)µ,J ,−M e−T (CCSD)

, (4.204)

where the excitation index µ will be dropped in this and the following sections.As before, for EOM-CCSD, the excitation operator R(CCSD)

J ,M consists of a zero-, one-,and two-body part,

R(CCSD)J ,M = R

(J )0,M + R

(J )1,M + R

(J )2,M . (4.205)

It is clear that the zero-body part cannot generate any angular momentum and istherefore of the form

R(J )0,M = δJ 0 δM 0 R0 1 (4.206)


4.13. Spherical EOM-CCSD

with R0 being a number. This is consistent with the statement that the zero-bodypart of the excitation operator is only non-zero for excited states that have the samesymmetries as the ground state which is always a 0+ state in our case. Again, theexcitation operator satisfies an eigenvalue equation for the effective Hamiltonianof the form

H (CCSD)open R

(CCSD)J ,M

C|Φ⟩ = ω R

(J )M |Φ⟩ . (4.207)

Projecting this equation onto the singly and doubly excited m -scheme Slater de-terminants and coupling the resulting equations leads to the coupled form

⟨a |

H (CCSD)open R

(CCSD)J ,M

C|i ⟩

J−M

= ω ⟨a |R(CCSD)J ,M |i ⟩

J−M

(4.208)

⟨ab |

H (CCSD)open R

(J )M

C|i j ⟩

Jab M ab Ji j M i j

= ω ⟨ab |R(CCSD)J ,M |i j ⟩

Jab M ab Ji j M i j

(4.209)

which more conveniently can be formulated in terms of reduced matrix elementsas 12

⟨a ||

H (CCSD)open R

(CCSD)J

C||i ⟩ = ω ⟨a ||R(J )1 ||i ⟩ (4.210)

⟨ab ||

H (CCSD)open R

(CCSD)J

C||i j ⟩

Jab Ji j

= ω ⟨ab ||R(J )2 ||i j ⟩

Jab Ji j

. (4.211)

The corresponding equations are listed in Appendix H.2.

Analogous considerations for the left eigenvalue problem

⟨Φ| (−1)J−ML(CCSD)J ,−M H (CCSD)

open = ⟨Φ| (−1)J−ML(CCSD)J ,−M

lead to the spherical form

⟨i ||L(CCSD)J H (CCSD)

open ||a ⟩ = ω ⟨i ||L(J )1 ||a ⟩ (4.212)

⟨i j ||L(CCSD)J H (CCSD)

open ||ab ⟩

Ji j Jab

= ω ⟨i j ||L(J )2 ||ab ⟩

Ji j Jab

, (4.213)

for which the corresponding equations can be found in Appendix H.2.

12Since H (CCSD)open is a scalar under rotation,

H (CCSD)open R

(CCSD)J

Cis a spherical tensor operator of

rank J .



The eigenstates of H (CCSD) are also parity eigenstates. In order to target a spe-cific parity, the constraints

(−1)l a!= (−1)l i , for matrix elements of L(J )1 , R

(J )1 (4.214)

(−1)l a+lb!= (−1)l i+l j , for matrix elements of L(J )2 , R

(J )2 , (4.215)

may be enforced to obtain positive parity states 13, or

(−1)l a

!

6= (−1)l i , for matrix elements of L(J )1 , R(J )1 (4.216)

(−1)l a+lb

!

6= (−1)l i+l j , for matrix elements of L(J )2 , R(J )2 , (4.217)

in order to obtain access the negative parity spectrum.

If one is interested in the J = 0 spectrum only, R(CCSD)0 and L

(CCSD)0 are scalar ten-

sor operators and the spherical EOM-CCSD equations may significantly be sim-plified. As for CCSD, these equations may be formulated without the use of cou-pling coefficients except for cross-coupling transformations. Therefore, simplercode structures and optimized matrix-multiplication routines may again be em-ployed in order to accelerate the calculations. The scalar EOM-CCSD equationshave also been worked out and can be found in Appendix H.3.

4.14 Spherical Reduced Density Matrices

The m -scheme expression for the EOM-CCSD reduced density matrices in termsof the spherical tensor operators R

(CCSD)ν ,JR

and L(CCSD)µ,JL

reads

(ρµνN )

q1...qn

p1...pn= ⟨Ψ(CCSD)

µ,JL ,M L| a †

p1. . . a †

pna qn

. . . a q1 |Ψ(CCSD)

ν ,JR ,M R⟩

= ⟨Φ| (−1)JL−M L L(CCSD)µ,JL ,−M L

e−T (CCSD)

a †p1

. . . a †pn

a qn. . . a q1

e T (CCSD)

R(CCSD)ν ,JR ,M R

|Φ⟩

= ⟨Φ| (−1)JL−M L L(CCSD)µ,JL ,−M L

a †p1

. . . a †pn

a qn. . . a q1

e T (CCSD)

CR(CCSD)ν ,JR ,M R

|Φ⟩ . (4.218)

13This is because the CCSD ground state is a 0+ state.


4.14. Spherical Reduced Density Matrices

This translates into the usual m -scheme expressions in terms of amplitudes ra 1...a n

ı 1...ı n

and lı 1...ı n

a 1...a nprovided that the following identifications

r a 1...a n

ı 1...ı n= ⟨a 1 . . . a n |R(CCSD)

µ,JR ,M R|ı 1 . . . ı n ⟩ (4.219)

and

l ı 1...ı n

a 1...a n= ⟨ı 1 . . . ı n |(−1)JL−M L L

(CCSD)µ,JL ,−M L

|a 1 . . . a n ⟩ (4.220)

are implied.

The eigenvectors of the effective Hamiltonian need proper normalization inorder to compute the density. All vectors belong to the same excitation indexso this index will be dropped again. The vectors R

(CCSD)JR

and L(CCSD)JL

are normal-ized such they lead to the correct normalizations in the corresponding m -schemeexpressions for the operators R

(CCSD)JR ,M R

and (−1)JL−M L L(CCSD)JL ,−M L

. Therefore, the righteigenvector is rescaled according to

R(CCSD)JR

→ 1p

NRR

R(CCSD)JR

(4.221)

with

NRR = (R0)2+∑

a ı

⟨a |R(JR )

1,M R|ı ⟩ ⟨a |R(JR )

1,M R|ı ⟩

+1

4

∑

a b ı

⟨a b |R(JR )

2,M R|ı ⟩ ⟨a b |R(JR )

2,M R|ı ⟩

= (R0)2+∑

a i

⟨a ||R(JR )

1 ||i ⟩2+

1

4J −2

R

∑

ab i j

∑

Jab Ji j

h

⟨ab ||R(JR )

2 ||i j ⟩

Jab Ji j

i2

(4.222)

which follows from∑

a ı

⟨a |R(JR )

1,M R|ı ⟩ ⟨a |R(JR )

1,M R|ı ⟩ (4.223)

=∑

a i

(−1)ja−ma

ja J i

−ma M m i

3j

2

⟨a ||R(JR )

1 ||i ⟩2

(4.224)

=∑

a i

⟨a ||R(JR )

1 ||i ⟩2

(4.225)



and

1

4

∑

a b ı

⟨a b |R(JR )

2,M R|ı ⟩ ⟨a b |R(JR )

2,M R|ı ⟩ (4.226)

=1

4

∑

ab i j

∑

Jab MabJi j Mi j

∑

J ′ab

M ′ab

J ′i j

M ′i j

ja jb Jab

ma mb M ab

CG

j i j j Ji j

m i m j M i j

CG

×

ja jb J ′ab

ma mb M ′ab

CG

j i j j J ′i j

m i m j M ′i j

CG

× (−1)Jab−M ab

Jab JR Ji j

−M ab M M i j

3j

(−1)J′ab−M ′

ab

J ′ab JR J ′i j

−M ′ab M M ′

i j

3j

× ⟨ab ||R(JR )

2 ||i j ⟩

Jab Ji j

⟨ab ||R(JR )

2 ||i j ⟩

J ′ab

J ′i j

(4.227)

=1

4J −2

R

∑

ab i j

∑

Jab Ji j

h

⟨ab ||R(JR )

2 ||i j ⟩

Jab Ji j

i2

. (4.228)

In a next step, L(CCSD)JL

is rescaled in order to ensure unity of the state overlap

1 = NLR = ⟨Ψ(J )M |Ψ(J )M ⟩ , (4.229)

implying JL = JR ≡ J for the moment, which m -scheme expression is given interms of reduced matrix elements as

NLR = (−1)J−ML0 R0+ (−1)J−M

∑

a ı

⟨ı |L(J )1,−M |a ⟩ ⟨a |R(J )1,M |ı ⟩ (4.230)

+1

4(−1)J−M

∑

a b ı

⟨ı |L(J )1,−M |a b ⟩ ⟨a b |R(J )1,M |ı ⟩ , (4.231)

NLR = − J −2∑

a i

(−1)ja+j i−J ⟨i ||L(J )||a ⟩ ⟨a ||R(J )||i ⟩

+1

4J −2

∑

ab i j

∑

Ji j Jab

(−1)J+Ji j+Jab ⟨i j ||L(J )2 ||ab ⟩

Ji j Jab

⟨ab ||R(J )2 ||i j ⟩

Jab Ji j

. (4.232)

Since excited states are considered, L0 = 0 is implied, so the product L0R0 van-ishes. Eq. (4.232) may also be used to check biorthonormality of the left and righteigenvectors. A further check of the implementation, at least for JR = 0 states,


4.14. Spherical Reduced Density Matrices

may be done by computing the overlapNΛR of the Λ state with the right eigenvec-tors [76],

NΛR = R0+δJR 0

∑

a i

⟨i ||Λ1||a ⟩ ⟨a ||R(JR )

1 ||i ⟩

+1

4δJR 0

∑

ab i j

∑

J

⟨i j ||Λ2||ab ⟩

J J

⟨ab ||R(JR )

2 ||i j ⟩

J J

, (4.233)

which should vanish 14.

Using the spherical expression of the reduced density matrix, the expectationvalue of a one-body spherical tensor operator O

(J )M may be expressed as

¬

O(J )M

¶

µν= − (−1)J−M J −2

∑

pq

(−1)jp+jq−J ⟨p ||o (J )||q ⟩ ⟨q ||ρµν (J )N ||p ⟩ .

(4.234)

Since it depends on the projection simply by the phase, the calculations can besimplified by setting

M L =M R =M = 0 . (4.235)

14It is of course always possible to compute the corresponding m -scheme vector from the spher-ical solution and then check overlaps etc., which is a recommended way to verify things anyway.


Chapter 5

Results

Chapter 5. Results

The overarching goal of this work is to extend the range of ab initio nuclear struc-ture calculations to medium-mass and heavy nuclei. However, the computationalframework considered in this work has arrived at a level of complexity at whichthe interpretation of results is not entirely straightforward. The initial Hamilto-nian undergoes a renormalization treatment before it enters the many-body calcu-lation, which itself is a multi-step procedure. Truncations are involved in each ofthese steps and it is important to understand in which ways they affect the resultsand to carefully monitor their impact in order to be able to estimate the overallaccuracy of the calculations.

Although the accuracy of the Coupled-Cluster method can be estimated froman analysis of the contributions at different orders of the cluster expansion, amore direct approach through a comparison of Coupled-Cluster results with ex-act solutions is favorable, where possible. In Section 5.1, the quasi-exact resultsfor 16O from the IT-NCSM are compared to the various Coupled-Cluster ground-state methods considered in this work, and it is concluded that CCSD approachin combination with triples corrections can compete with the quasi-exact diago-nalizations for this nucleus.

Using NN-only Hamiltonians eliminates all problems and difficulties relatedto the treatment of 3N interactions and allows to demonstrate the capabilities ofthe many-body methods detached from technical limitations imposed by the 3Ninteractions. In Section 5.2 it is shown that at the two-body level Coupled-Clustercalculations can be performed across the nuclear chart, and that the input inter-action is the more limiting factor in such calculations.

In ground-state calculations beyond light nuclei, the standard chiral NN+3N-full Hamliltonian exhibits strong contributions from SRG-induced beyond-3N in-teractions which prevent any attempt to estimate the ground-state energies of thebare NN+3N-full Hamiltonian. In Section 5.3 a reduced cutoff-momentum vari-ant of the chiral 3N interaction is considered that exhibits a much reduced flow-parameter dependence and which will then be used for all of the following calcu-lations beyond light nuclei.

One of the most important truncations related to the inclusion of 3N interac-tions is an energy-truncation E3max in the 3N matrix elements, which is consideredin Section 5.4.


Triples corrections are indispensable in the Coupled-Cluster framework byproviding crucial contributions beyond CCSD, in particular for harder interac-tions. But even for soft interactions, where the contributions are rather small,triples corrections give important information about the convergence of the clus-ter expansion. General aspects of triples corrections in nuclear structure calcula-tions are discussed in Section 5.5 in the context of ΛCCSD(T), and the results forthe CR-CC(2,3) method, including a comparison to ΛCCSD(T), are presented inSection 5.6.

In practical calculations, the normal-ordered two-body approximation is crit-ical since, in particular for calculations involving heavy nuclei or triples correc-tions, the NO2B approximation reduces the computing time by orders of magni-tude. Since the calculations rely so heavily on this approximation it is worthwhileto verify its validity for medium-mass nuclei using CCSD and ΛCCSD(T) calcula-tions for three-body Hamiltonians in Sections 5.7 and 5.8.1.

Heavy nuclei are considered in Section 5.9. Advancing to larger mass num-bers requires to revisit truncations of the SRG model space, and the generationof large-Emax matrix element sets. Nevertheless, despite all technical difficulties,it is shown that reliable ab initio calculations can be performed even for nuclei asheavy as 132Sn.

Proof-of-principle calculations of excited states using the spherical EOM-CCSDformalism are attached in Appendix A. In these calculations the EOM-CCSDframework proves to be capable of describing selected low-lying states, makingit a more favorable approach to such states than the Random Phase Approxima-tion approach, for instance.

Finally, to extend the calculations beyond common nuclei, in Appendix B neu-trons trapped in an external potential are considered. These neutron systems al-low to study the extreme-isospin component of the nuclear interactions and serveas simple models for neutron-rich nuclei.


Chapter 5. Results

5.1 Comparison of the IT-NCSM with theCoupled-Cluster Method

Comparing CCSD ground-state energies with ones obtained from the IT-NCSMallows to estimate the quality of the CCSD approximation to the exact wavefunc-tion, because converged IT-NCSM results may for present purposes be regardedas the quasi-exact solutions of the Schrödinger equation. This is not the case forCCSD, since, even in the limit emax→∞, contributions from triple and higher-rankexcitations are missing in the wavefunction.

A direct comparison of the Nmax and emax model spaces for the IT-NCSM andCCSD is not immediately possible. For CCSD, a rough estimate of the maximumunperturbed excitation energy of the basis Slater determinants that is generatedby a operator product (T1)

n (T2)m is given by (n+2m )emaxħhΩ, while for the IT-NCSM

the maximum excitation energy is NmaxħhΩ. Therefore, for emax = Nmax, the maxi-mum excitation energy of CCSD exceeds the one of the IT-NCSM. This is why inFigure 5.1 the CCSD results seem to converge more quickly. On the other hand,only some determinants (those that are generated by certain products of excitationoperators) with such high energies are included in the CCSD model space whilethe IT-NCSM includes all determinants 1 up to the maximum excitation energyNmaxħhΩ.

Unlike the IT-NCSM, CCSD is not strictly variational but in practice non-variational behavior is practically never encountered. Therefore, the CCSD re-sults converge from above and the converged results are expected to lie somewhatabove the converged IT-NCSM because of the missing beyond-single-and-doubleexcitations in CCSD. These expectations are confirmed in Figure 5.1, where thethree types of Hamiltonians NN-only, NN+3N-induced, and NN+3N-full withregular cutoff momentum Λ3N = 500 MeV are considered in a harmonic-oscillatorbasis for a sequence of SRG flow parameters. For calculations involving three-body Hamiltonians, the normal-ordered two-body approximation is used. Thenucleus 16O has been chosen because it is at the upper end of IT-NCSM capa-bilities and at the same time marks the beginning of the medium-mass regime,which is the primary interest of this work. Postponing the detailed discussion ofCoupled-Cluster results to later sections, good agreement of CCSD and the IT-NCSM is apparent. In the model spaces that are considered, CCSD is converged

1More strictly speaking, all relevant determinants.


5.1. Comparison of the IT-NCSM with the Coupled-Cluster Method

beyond any doubt and for the IT-NCSM the extrapolations are trustworthy. Forthe IT-NCSM, the largest sources of uncertainties come from the importance trun-cation for non-vanishing κmin and the extrapolation to infinite model space sizesNmax, which are both of the order of 1 MeV. For the Coupled-Cluster calculations,the truncation of the cluster operator at the T2 excitation level represents the majorsource of uncertainty of the order of a few MeV, and, consequently, the observeddeviations from CCSD to the IT-NCSM are to a large extend attributable to thisapproximative nature of CCSD. While the NCSM is translational invariant fromthe outset and the IT-NCSM practically preserves this translational invariance,spurious center-of-mass contaminations may occur in truncated Coupled-Clustercalculations. However, these center-of-mass effects are expected to be small [152].

Conclusions about the bare Hamiltonians at SRG flow-parameter α = 0.0 fm4

can often be drawn by analyzing the flow-parameter dependence of ground-stateenergies of SRG-evolved Hamiltonians. Here, the different quality of the IT-NCSMand Coupled-Cluster results have to be considered. Since the IT-NCSM is a quasi-

exact method, the flow-parameter dependence may completely be attributed toomitted SRG-induced many-body interactions. On the other hand, the truncationsinherent in the CCSD or ΛCCSD(T) approach may cause flow-parameter depen-dence on their own. A simple example is given by truncation of the cluster opera-tor: Since softer interactions are expected to induce less correlations, it is expectedthat an approximate method such as CCSD performs better for softer interactionsthan for harder ones. A second source of flow-parameter dependence is identifiedas the E3max cut for three-body matrix elements, as discussed in later sections. Inprinciple, an E3max truncation should also cause a flow-parameter dependence inthe IT-NCSM, however, for all IT-NCSM model spaces Nmax ≤ 14 used in this work,the full set of required matrix elements was included.

A more detailed comparison of IT-NCSM and Coupled-Cluster ground-stateenergies for 16O – now also including triples corrections to the energy – can befound in Figure 5.2. In all cases the IT-NCSM energies lie halfway between theCCSD andΛCCSD(T) results. For the NN+3N-induced Hamiltonian atα= 0.04 fm4,for instance, CCSD yields -120.2 MeV, the IT-NCSM gives -121.8 MeV andΛCCSD(T)gives -123.6 MeV. From this follows that CCSD is even a little closer to the IT-NCSM than is ΛCCSD(T). Thus, a naive look at Figure 5.2 suggests CCSD to bethe more accurate approximation than is ΛCCSD(T). However, due to its varia-tional character, the true exact result is expected to lie below the IT-NCSM and inquantum-chemistry applications ΛCCSD(T) tends to overshoot the exact result alittle [153], so the exact result is actually expected to lie in between the IT-NCSM


Chapter 5. Results

NN only

2 4 6 8 10 12 14 16 18Nmax

-170

-160

-150

-140

.

E[M

eV]

IT-NCSM

16 14 12 10 8 6 4 2emax

CCSD

NN + 3N induced (NO2B)

2 4 6 8 10 12 14 16 18Nmax

-130

-120

-110

-100

-90

.

E[M

eV]

IT-NCSM

16 14 12 10 8 6 4 2emax

CCSD

NN + 3N full (NO2B)

2 4 6 8 10 12 14 16 18

Nmax

-150

-140

-130

-120

.

E[M

eV]

IT-NCSM

16 14 12 10 8 6 4 2emax

CCSD

Î

α= 0.04 fm4 α= 0.05 fm4 α= 0.0625 fm4 α= 0.08 fm4

Figure 5.1: Comparison of IT-NCSM and CCSD results for 16O and the three Hamiltonians

NN-only, NN+3N-induced, and NN+3N-full for a sequence of SRG flow parame-

ters. The single-particle basis is the harmonic-oscillator basis with oscillator frequency

ħhΩ= 20MeV. For the NN+3N Hamiltonians the NO2B approximation to the 3N in-

teraction was used with E3max = 12.


5.1. Comparison of the IT-NCSM with the Coupled-Cluster Method

NN-only

0.04 0.05 0.0625 0.08

α [fm4]

-175

-170

-165

-160

-155

.

E[M

eV]

16O

NN+3N-induced

0.04 0.05 0.0625 0.08

α [fm4]

-124

-122

-120

-118

.

E[M

eV

]

NN+3N-full

0.04 0.05 0.0625 0.08

α [fm4]

-150

-145

-140

.

E[M

eV

]

IT-NCSM CCSD ΛCCSD(T) CR-CC(2,3)

Figure 5.2: Comparison of extrapolated IT-NCSM ground-state energies with converged CCSD,

ΛCCSD(T), and CR-CC(2,3) results for 16O. Parameters of the calculations as in

Figure 5.1.

and ΛCCSD(T) results. This expectation is further confirmed by the CR-CC(2,3)results, which are the more accurate approximation to the exact triples correction,and which lie halfway between the IT-NCSM and ΛCCSD(T) results, such thatin many cases the IT-NCSM and CR-CC(2,3) ground-state energies agree withinthe remaining uncertainties. Furthermore, the spread between the results for thedifferent many-body methods for individual flow parameters becomes smallerwith increasing flow parameter. This is also expected, because, as already men-tioned above, for softer interactions approximate many-body methods should per-form better. In conclusion, for 16O, Coupled-Cluster theory with singles and dou-bles excitations combined with a corrective treatment of triples is able to provideground-state energies that can compete with the quasi-exact diagonalizations thatcan be performed within the IT-NCSM.


Chapter 5. Results

5.2 CCSD with SRG-Transformed Chiral Two-BodyHamiltonians

Calculations using SRG-transformed NN-only Hamiltonians do not provide muchuseful physical information due to the strong violation of the unitarity of the SRGtransformation caused by the omission of SRG-induced three- and higher-bodyinteractions in the NN-only approach. However, by ignoring three-body inter-actions for the moment, all additional complications related to the handling of3N interactions are avoided, which for example will later restrict the calculationsusing 3N interactions to the medium-mass regime. Furthermore, without 3N in-teractions, CCSD calculations become inexpensive and, therefore, the NN-onlyframework may be used to demonstrate basic capabilities and limitations of theCCSD implementation itself, without being constrained by limitations of the inputHamiltonian.

Figures 5.3 and 5.4 summarize NN-only results for the reference- and CCSDenergy for medium-mass and heavy nuclei ranging from 16O to 208Pb, using both,the harmonic-oscillator (HO) and Hartree-Fock (HF) basis. Considering the refer-ence energies in panel (a), it is apparent that for the HO basis it increases rapidlyas one departs from the optimal oscillator frequency, while for the HF case thereference energy is absolutely stable for the whole frequency range considered. Itis, therefore, striking to find in panel (b) only small deviations in the final CCSDresults in the HO and HF case, for frequencies up to ħhΩ = 32MeV. An extremeexample for 208Pb is given in Table 5.1: The CCSD energies are all very similarfor the HO and HF basis and for both oscillator frequencies ħhΩ= 20 and 36 MeV.The reference energy, however, differs by up to more than 7 GeV (!). So, at fre-quency ħhΩ = 36 MeV, CCSD contributes about 8 GeV to the energy for the HObasis, while for the HF basis it is only less than 900 MeV, both arriving at the sameresult within about 200 MeV. This may be regarded as a demonstration of Thou-less’ theorem [154].

The plots in panel (c) of Figures 5.3 and 5.4 show the convergence of CCSDenergies in HF basis with respect to the model space size emax at the optimal oscil-lator frequency for the individual nuclei. All nuclei show a similar convergencepattern, where at least in emax = 14 model spaces the results are finally fully con-verged. Therefore, in the NN-only framework, it is possible to obtain convergedCCSD results all over the nuclear chart. This is made possible by the sphericalformulation of CCSD. For the 208Pb calculations at emax = 14, for example, the m -


5.2. CCSD with SRG-Transformed Chiral Two-Body Hamiltonians

16O 40Ca 56Ni

20 25 30 35~Ω [MeV]

-18

-14

-10

-6

-2

.

Ere

f/A

[MeV

]

(a)

20 25 30 35~Ω [MeV]

20 25 30 35~Ω [MeV]

exp

20 25 30 35~Ω [MeV]

-18

-14

-10

-6

-2

.

E/A

[MeV

]

(b)

20 25 30 35~Ω [MeV]

20 25 30 35~Ω [MeV]

exp

4 6 8 10 12 14emax

-18

-14

-10

-6

-2

.

E/A

[MeV

]

(c)

4 6 8 10 12 14emax

4 6 8 10 12 14emax

exp

Î

α= 0.02 fm4 α= 0.04 fm4 α= 0.08 fm4

Figure 5.3: Reference energy and CCSD ground-state energy per nucleon for the nuclei 16O, 40Ca

and 56Ni, using SRG-transformed chiral N3LO two-nucleon interactions for a range

of flow parameters. Plots (a) and (b) are obtained from an emax = 14 model space, while

for (c) the optimal HO frequencies ħhΩ= 20 MeV (for 16O) and ħhΩ= 24 MeV (for 40Ca,56Ni) are used, which correspond to minima in both, the HO and the HF basis. Open

symbols represent the results for the results harmonic-oscillator basis, full symbols the

Hartree-Fock basis.


Chapter 5. Results

114Sn 132Sn 208Pb

20 25 30 35~Ω [MeV]

-30

-25

-20

-15

-10

-5

0

.

Ere

f/A

[MeV

]

(a)

20 25 30 35~Ω [MeV]

20 25 30 35~Ω [MeV]

exp

20 25 30 35~Ω [MeV]

-30

-25

-20

-15

-10

-5

0

.

E/A

[MeV

]

(b)

20 25 30 35~Ω [MeV]

20 25 30 35~Ω [MeV]

exp

4 6 8 10 12 14emax

-30

-25

-20

-15

-10

-5

0

.

E/A

[MeV

]

(c)

4 6 8 10 12 14emax

4 6 8 10 12 14emax

exp

Î

α= 0.02 fm4 α= 0.04 fm4 α= 0.08 fm4

Figure 5.4: As in Figure 5.3 for nuclei 114Sn, 132Sn and 208Pb. In each case the optimal oscillator

frequency is ħhΩ= 24 MeV.


5.2. CCSD with SRG-Transformed Chiral Two-Body Hamiltonians

208Pb :ħhΩ(HO) [MeV] Eref [MeV] E (CCSD) [MeV]

20 -1819 -313036 +5025 -2950

ħhΩ(HF) [MeV] Eref [MeV] E (CCSD) [MeV]20 -2300 -315736 -2171 -3035

Table 5.1: Reference energies and CCSD ground-state energies for 208Pb for oscillator frequencies

ħhΩ = 20 MeV and 36 MeV, highlighting Thouless’ theorem.

scheme formulation would require to solve the CCSD equations for about 300 bil-lion T amplitudes – which is way beyond present capabilities. For the sphericalscheme, on the other hand, the number of amplitudes totals to about 600 million,which can be dealt with using modern computers. As mentioned above, compar-ison with the experimental values at this stage is pointless because of the massiveflow-parameter dependence of the energies that does not allow to draw any con-clusion about the bare Hamiltonian.

In summary, spherical CCSD with SRG-transformed NN-only Hamiltonianshas been demonstrated to be practically applicable to nuclei over a large massrange. This is due to the fact that the interactions considered here are quite softand well-behaved from a computational point of view. Obviously, (at least) SRG-induced 3N interactions have to be included in the calculation in order to possiblyget rid of the flow-parameter dependence and it is expected that at least the chiral3N interaction needs to be included in any quantitative calculation. However, once3N interactions are included, the resulting interactions get more difficult to dealwith compared to the NN-only case. For example, the CCSD equations usuallydo not converge for the HO basis if the oscillator frequency does not coincide withthe optimal frequency, making using the HF basis mandatory.


Chapter 5. Results

5.3 Reduced-Cutoff Chiral Three-Body Interaction

The chiral 3N interaction [155] with regulator cutoff-momentum Λ3N = 500 MeVcan successfully be employed to describe light nuclei [17]. For example, IT-NCSMresults for 4He ground-state energies are presented in Figure 5.5 (top). For theNN-only Hamiltonian, the converged energies show a strong dependence on theflow parameter. Therefore, it can be concluded that the unitarity of the SRG trans-formation is strongly violated by omitting all induced three-and higher-body in-teractions during the SRG flow. The systematics of the flow-parameter depen-dence is such that the energy moves downwards for increasing flow parameter.So, from the NN-only plot in Figure 5.5, the bare (untransformed) result for thechiral NN interaction is expected to lie well above the experimental value but thestrong flow-parameter dependence prohibits any more detailed prediction. Oncethe SRG-induced 3N interactions are included in the calculations by using theNN+3N-induced Hamiltonian, the flow-parameter dependence of the convergedenergies practically vanishes. This implies that – for the range of flow param-eters considered here – induced four-body interactions are not relevant for thedescription of the 4He ground state. Consequently, the NN+3N-induced calcula-tions can be seen as unitarily equivalent to calculations with the chiral NN inter-action, which, however, misses the experimental value considerably. The chiralN3LO two-nucleon interaction is, therefore, not sufficient for reproducing the 4Heground-state energy. This is not too surprising since chiral 3N interactions al-ready appear at N2LO in the Weinberg power counting, which have not yet beentaken into account. Once these chiral 3N interactions are included via the NN+3N-

full Hamiltonian, the ground-state energies still show no flow-parameter depen-dence which allows to make a prediction for the bare NN+3N-full Hamiltonianand which also shows good agreement with the experimental value. The flow-parameter independence of the NN+3N-full results provide the important infor-mation that SRG-induced four-body interactions out of the initial 3N interactionare not relevant in this case.

For the heavier nucleus 12C shown in Figure 5.5 (middle), the situation is sim-ilar. The results for the NN-only Hamiltonian show a strong flow-parameter de-pendence and thus do not allow for any predictions for the bare chiral NN in-teraction. Inclusion of the induced 3N interaction eliminates the flow-parameterdependence but as in the case of 4He, the prediction for the chiral NN interac-tion is clearly underbound with respect to experiment. The effect of the initialchiral 3N interaction moves the results towards the experimental value, with a


5.3. Reduced-Cutoff Chiral Three-Body Interaction

NN-only

2 4 6 8 10 12 14 16 ∞Nmax

-29

-28

-27

-26

-25

-24

-23

.

E[M

eV]

4He

NN+3N-induced

2 4 6 8 10 12 14 ∞Nmax

Exp.

NN+3N-full

2 4 6 8 10 12 14 ∞Nmax

2 4 6 8 10 12 14 ∞Nmax

-110

-100

-90

-80

-70

-60

.

E[M

eV]

12C

2 4 6 8 10 12 ∞Nmax

Exp.

2 4 6 8 10 12 ∞Nmax

2 4 6 8 10 12 14 ∞Nmax

-180

-160

-140

-120

-100

-80

.

E[M

eV]

16O

2 4 6 8 10 12 ∞Nmax

Exp.

2 4 6 8 10 12 ∞Nmax

Î

α= 0.04 fm4 α= 0.05 fm4 α= 0.0625 fm4 α= 0.08 fm4 α= 0.16 fm4

Figure 5.5: IT-NCSM ground-state energies for the nuclei 4He, 12C and 16O for the three Hamil-

tonians NN-only, NN+3N-induced, and NN+3N-full. For 16O, also the results for

the 400 MeV cutoff-momentum NN+3N-full Hamiltonian are shown (open symbols),

which exhibit a much reduced flow-parameter dependence. The calculations were per-

formed using a HO basis with ħhΩ = 20 MeV and with full inclusion of 3N interactions.


Chapter 5. Results

slight overbinding for the flow-parameter values considered here. But since for12C there is an emerging flow-parameter dependence for the NN+3N-full results,no safe statement can be made for the bare NN+3N-full Hamiltonian. It is clearthat the violation of unitarity of the SRG transformation is caused by the omissionof induced 4N interactions out of the initial 3N – the initial NN interaction can beruled out as a source of sizeable induced 4N because of the absent flow-parameterdependence of the energies for the NN+3N-induced Hamiltonian.

This flow-parameter dependence of the NN+3N-full results is enhanced forincreasing mass number. Considering 16O, in Figure 5.5 (bottom), the results forthe standard NN+3N-full Hamiltonian (full symbols in the NN+3N-full plot), withthe Λ3N = 500 MeV cutoff momentum in the regularization function [155], showsa flow-parameter dependence of about 10 MeV for the particular range of flowparameters considered here. As for 12C, the flow-parameter dependence is eas-ily found to be caused by at least 4N interactions induced out of the initial 3N.At this point all predictive capabilities for the NN+3N-full Hamiltonian are lostwithin the framework used so far. The inclusion of induced 4N interactions inorder to reduce the flow-parameter dependence is computationally too demand-ing and, therefore, a modified interaction is used in the following: The NN+3N-fullplot for 16O also shows results for a reduced-cutoff interaction (open symbols), inwhich the regulator cutoff-momentum of the chiral 3N has been lowered fromits standard value Λ3N = 500MeV to Λ3N = 400MeV and the low-energy constantcE has been refitted to reproduce the 4He binding energy, while the other LECthat is exclusively related to chiral 3N interactions at N2LO, cD , which is fitted tothe triton half-life, remains unchanged. The reason why cD has not to be adaptedto the new parameters of the interaction is that altering the 3N cutoff or cE doesnot affect the results for the triton half-life. The values of the LECs for variouschoices of Λ3N can be found in Table 5.2. The main motivation for lowering thecutoff momentum is the observation that this way off-diagonal matrix elementsin the 3N interaction, which are a major source of induced many-body interac-tions, get suppressed (a detailed discussion can be found in [156,157]). Using this400 MeV cutoff interaction, the 16O ground-state energies show a much reducedflow-parameter dependence, which is the purpose of reducing the 3N cutoff inthe first place, as discussed below. Furthermore, the ground-state energies nowlie on top of the experimental value. Later on, for calculations of heavy nuclei inSection 5.9.3, also theΛ3N = 350MeV interaction will be used to study SRG-inducedmany-body interactions.



Λ3N c1 c3 c4 cD cE

[MeV] [GeV−1] [GeV−1] [GeV−1]500 –0.81 –3.2 5.4 –0.2 –0.205450 –0.81 –3.2 5.4 –0.2 –0.016400 –0.81 –3.2 5.4 –0.2 0.098350 –0.81 –3.2 5.4 –0.2 0.205

Table 5.2: Low-energy constants that parametrize the chiral 3N interaction, for various choices of

the 3N regular momentum cutoffΛ3N [86]. The constants c1, c3, and c4 are fixed through

the chiral NN interaction where they enter as well, and cD also remains unchanged

because it is fitted to the triton halflife which is not affected by altering Λ3N. The final

LEC cE is fitted to reproduce the 4He ground-state energy.

Since for the nucleus 16O and beyond only the reduced-cutoff interaction al-lows to obtain more or less flow-parameter independent ground-state energies, itwill be the customary choice for medium-mass nuclei. In Figures 5.6 and 5.7 stillboth, the standard and reduced-cutoff interaction are used for comparison, but inthe following sections only the reduced-cutoff interaction will be considered. Fig-ures 5.6 and 5.7 show CCSD ground-state energies for the medium-mass closed-shell nuclei 16,24O and 40,48Ca, using a harmonic-oscillator basis at fixed oscillatorfrequency ħhΩ= 20MeV. The 3N interactions are included via NO2B, but the largemodel spaces considered here require an additional cutoff parameter E3max in thethree-body matrix elements, as discussed in more detail in Section 5.4. For thepresent results E3max = 14 was used. Although this cut is in principle expected toaffect the results – particularly for the heavier nuclei 40,48Ca – on the large scalesused for the plots such E3max-effects do not play a significant role. Figures 5.6and 5.7 show essentially the same qualitative behavior as the IT-NCSM results for16O, only on a larger scale. In all cases, the NN+3N-induced Hamiltonian pro-vides results that are practically flow-parameter invariant and tend to underbind.The NN+3N-full Hamiltonian with standard 500 MeV regularization of the chiral3N interaction (full symbols in the NN+3N-full plot) shows a very strong flow-parameter dependence and for the flow parameters considered here, the evolvedNN+3N-full Hamiltonians show massive overbinding compared to experiment,which is even comparable to the NN-only case. On the other hand, the NN+3N-full Hamiltonian with the reduced-cutoff 3N interaction (open symbols in theNN+3N-full plot) provides results with a much reduced flow-parameter depen-dence at the level of the NN+3N-induced results, even for the heaviest nucleus


Chapter 5. Results

-170

-160

-150

-140

-130

-120

-110

-100

.

E[M

eV]

NN-only

exp.

NN+3N-ind.

16O~Ω = 20 MeV

NN+3N-full

2 4 6 8 10 12 14emax

-240

-220

-200

-180

-160

-140

-120

.

E[M

eV] exp.

2 4 6 8 10 12 14emax

24O~Ω = 20 MeV

2 4 6 8 10 12 14emax

Î

α= 0.04 fm4 α= 0.05 fm4 α= 0.0625 fm4 α= 0.08 fm4

Figure 5.6: Ground-state energies from CCSD for 16O and 24O for the NN-only, NN+3N-induced,

and NN+3N-full Hamiltonians. For the NN+3N-full Hamiltonian, the standard reg-

ularization (Λ3N = 500 MeV) and the low-cutoff variant (Λ3N = 400 MeV, open sym-

bols) are shown. The underlying single-particle basis is the harmonic-oscillator basis

and for all 3N Hamiltonians the NO2B approximation with E3max = 14 is used. Figure

taken from [92].



-600

-550

-500

-450

-400

-350

-300

-250

.

E[M

eV]

NN-only

exp.

NN+3N-ind.

40Ca~Ω = 20 MeV

NN+3N-full

2 4 6 8 10 12 14emax

-800

-700

-600

-500

-400

-300

.

E[M

eV]

exp.

2 4 6 8 10 12 14emax

48Ca~Ω = 20 MeV

2 4 6 8 10 12 14emax

Î

α= 0.04 fm4 α= 0.05 fm4 α= 0.0625 fm4 α= 0.08 fm4

Figure 5.7: As in Figure 5.6 for the nuclei 40Ca and 48Ca.


Chapter 5. Results

under consideration. Therefore, it is possible to use these results to make predic-tions for the reduced-cutoff NN+3N-full Hamiltonian. Furthermore, the agree-ment with the experimental values is impressive: Except for 40Ca, for which theNN+3N-induced energies already lie on top of experiment, all theoretical valuesare very close to the experimental ones, highlighting the predictive power of chi-ral Hamiltonians, even in the medium-mass regime. The good performance ofthe reduced-cutoff interaction is remarkable, considering the fact that no infor-mation beyond 4He went into its construction. In the following sections variousaspects of this interaction are investigated, also using more advanced many-bodytechniques.

5.4 Relevance of the E3max Cut

Calculations using three-body Hamiltonians are challenging because of the enor-mous number of matrix elements involved. In most cases, a full representation ofthe three-body Hamiltonian in an emax-truncated single-particle basis can neitherbe handled in the many-body calculation nor can it be generated in the first place.Therefore, an additional truncation parameter is required in order to reduce therepresentation of the Hamiltonian to a manageable size. As already mentionedin Section 5.3, the representation of the three-body Hamiltonian is constrained tomatrix elements ⟨pqr |w |s t u ⟩ satisfying an energy truncation of the form

max

ep + eq + er , es + e t + er

≤ E3max (5.1)

where E3max is the truncation parameter and e i = 2n i+l i denotes the single-particleharmonic-oscillator energy quantum number. Current typical values of E3max areat the order of 14. Since for an emax = 12 calculation the maximum allowed valueof ep + eq + er would be 36, E3max = 14 represents a potentially serious cut whoseimpact on the results of many-body calculations requires careful inspection.

It should be noted that the following discussion is based on 3N matrix elementsevolved in the SRG model space corresponding to the so-called 40C parametriza-tion (see Section 5.9.2) used for most of the medium-mass calculations presentedin this work or in calculations of other research groups [17, 29, 30, 86, 92, 96, 158],which makes it worthwhile to investigate the properties of these matrix elements.However, as discussed in Section 5.9.2, the 40C model space is not sufficient anymore for heavier nuclei. As a consequence, for the 40C SRG model space parame-trization, the E3max effect is artificially enhanced for the heavier nuclei considered


5.4. Relevance of the E3max Cut

NN+3N-induced

12 14

E3max

-130

-128

-126

-124

-122

.

E[M

eV

]

NN+3N-full

12 14

E3max

16O

NN+3N-induced

12 14

E3max

-370

-365

-360

-355

-350

.

E[M

eV

]

NN+3N-full

12 14

E3max

40Ca

NN+3N-induced

12 14

E3max

-450

-440

-430

-420

-410

.

E[M

eV

]

NN+3N-full

12 14

E3max

48Ca

NN+3N-induced

12 14

E3max

-520

-500

-480

-460

-440

.

E[M

eV

]

NN+3N-full

12 14

E3max

56Ni

α= 0.02 fm4 α= 0.04 fm4 α= 0.08 fm4

Figure 5.8: Comparison of CCSD ground-state energies for the nuclei 16O, 40Ca, 48Ca, and 56Ni for

the NN+3N-induced and NN+3N-full Hamiltonians in NO2B approximation with

E3max = 12 and E3max = 14. The calculations were performed in a HF basis with

ħhΩ = 24 MeV and emax = 12.


Chapter 5. Results

in this section. This issue is again addressed in Section 5.9.2, where results forconsiderably larger SRG model spaces are presented.

Figure 5.8 shows a compilation of CCSD ground-state energies for medium-mass nuclei ranging from 16O to 56Ni, for the NN+3N-induced and NN+3N-fullHamiltonians in NO2B approximation and the usual range of SRG flow parame-ters. The calculations have been performed for E3max = 12 as well as E3max = 14, andthe effect of the E3max cut is estimated by the deviation of the two sets of results.The effect of E3max clearly depends on how far the Hamiltonian has been evolved:Considering the softest (α= 0.08 fm4) Hamiltonians, for all but the heaviest nuclei56Ni the impact of E3max is practically negligible. For 56Ni, the E3max cut has an ef-fect of about of 3 MeV, so it is still small compared to the total energy scale of about500 MeV. For the harder interactions, the E3max cut shows sizeable effects alreadyfor the lighter nuclei, where the hardest interactions show the largest change inthe energies. For 56Ni and α= 0.02 fm4, for instance, the ground-state energies forthe NN+3N-induced Hamiltonian change about 12 MeV, which corresponds toabout 2.5 % of the total binding energy.

For the CCSD ground-state energies, there is also an interesting systematic ef-fect of E3max on the flow-parameter dependence. As is apparent from Figure 5.8,for the NN+3N-induced Hamiltonian the flow-parameter dependence decreaseswhen E3max is increased and it is an interesting question how much the flow-parameter dependence will be eventually reduced in the E3max→∞ limit. For theNN+3N-full Hamiltonians, on the other hand, the flow-parameter dependenceeven gets increased. This behavior is a consequence of the ordering of the CCSDground-state energies with respect to the SRG flow parameter. According to Fig-ure 5.7, for instance, the three-body parts of the NN+3N-induced and NN+3N-fullHamiltonians are repulsive. Consequently, the ground-state energies are expectedto move upwards as more of the 3N interactions is included by increasing E3max.Furthermore, since for the NN+3N-induced Hamiltonian the CCSD energies forsmaller flow parameters lie below the energies for larger flow parameters andharder Hamiltonians imply larger E3max effects, the flow-parameter dependence isreduced. For the NN+3N-full Hamiltonian, on the other hand, the CCSD energyordering is such that the energies for the harder Hamiltonians already lie abovethe energies for the softer Hamiltonians and, therefore, increasing E3max only en-larges the flow-parameter dependence. As can exemplarily be seen in Figure 5.7,the contribution of the induced 3N is much larger than it is for the low-cutoff ini-tial 3N. Therefore, it might be assumed that relative changes in the induced 3Nmay be much more visible than relative changes in the initial 3N. Consequently,


5.4. Relevance of the E3max Cut

∆(E3max) ∆(E3max)

α [fm4] NN+3N-induced NN+3N-full

[%] [%]0.02 0.5 0.7

16O 0.04 0.2 0.40.08 0.0 0.10.02 1.2 1.1

48Ca 0.04 0.5 0.50.08 0.2 0.0

Table 5.3: Comparison of the difference in the CCSD ground-state energies for E3max = 12 and

E3max = 14 for the NN+3N-induced and NN+3N-full Hamiltonian in NO2B approxi-

mation, for the nuclei 16O and 48Ca. The calculations were performed using a HF basis

with ħhΩ = 24 MeV and at emax = 12.

ground-state energies for the NN+3N-induced and the NN+3N-full Hamiltonianshould show a similar behavior, since the induced 3N interaction is of course alsoincluded in the latter. And in fact, the changes in the ground-state energies withincreasing E3max for NN+3N-induced and NN+3N-full are almost identical, as isevident from Table 5.3, hinting at the induced 3N as the driving cause.

The absolute values of the difference of the ground-state energies correspond-ing to E3max = 12 and E3max = 14 grow with increasing mass number, but so dothe overall energy scales. Thus, it cannot immediately be determined from Fig-ure 5.8 how the relative effect of E3max to the ground-state energies evolves withmass number. To this end, Figure 5.9 shows the relative size ∆(E3max) of the E3max

effect normalized to the E3max = 14 ground-state energy, given by

∆(E3max) =|E (E3max = 14)− E (E3max = 12)|

E (E3max = 14)/100% . (5.2)

The relative E3max effect ∆(E3max) shows a systematic increase with mass numberso that relative accuracy is increasingly lost as one goes to larger masses. Forthe hardest (α = 0.02 fm4) NN+3N-induced Hamiltonian, the relative E3max effectranges from 0.5 % for 16O to 2.5 % for 56Ni, which is already twice as large as for48Ca (1.2 %). For the NN+3N-full Hamiltonian, the relative size grows from 0.5 %for 16O to about 2 % for 56Ni. For the NN+3N-full Hamiltonian at α = 0.02 fm4,there is no such drastic increase compared to the NN+3N-induced case, but thereis for α= 0.04 fm4, which triples its size going from 48Ca to 56Ni. Therefore, usinghard interactions beyond the mass region A ≈ 60, the E3max = 14 cut is expected to


Chapter 5. Results

NN+3N-induced

0

0.5

1

1.5

2

2.5

3

.

∆(E

3m

ax)

[%]

16O 24O 40Ca 48Ca 56Ni

NN+3N-full

0

0.5

1

1.5

2

2.5

3

.

∆(E

3m

ax)

[%]

16O 24O 40Ca 48Ca 56Ni

α= 0.02 fm4 α= 0.04 fm4 α= 0.08 fm4

Figure 5.9: Change of CCSD ground-state energies for the NN+3N-induced and the NN+3N-

full Hamiltonian in NO2B approximation when increasing E3max = 12 to E3max = 14,

normalized to the E3max = 14 ground-state energy. The calculations were performed

using a HF basis with ħhΩ = 24 MeV and at emax = 12.

become a major source of uncertainty 2. The soft α = 0.08 fm4 Hamiltonians alsoshow a systematic increase but the effect itself is considerably smaller so that evenfor 56Ni the relative E3max effect does not exceed 0.5 %.

5.5 The ΛCCSD(T) Energy Correction

At the level of CCSD, the cluster expansion is not yet sufficiently converged in or-der for higher-excitation rank effects to be negligible. In this section, ΛCCSD(T) isused to assess the size of triples corrections. Since ΛCCSD(T) tends to overshootthe actual triples correction [153], it gives a more conservative estimate of triples-and higher-excitation contributions than, e.g., CR-CC(2,3) would do. Neverthe-less, ΛCCSD(T) and CR-CC(2,3) give sufficiently similar results so that the conclu-sions drawn in this section do not depend on the actual triples correction methodused.

Figure 5.10 shows the convergence of ground-state energies from CCSD (opensymbols) and ΛCCSD(T) (full symbols), for medium-mass nuclei for the NN+3N-induced and low-cutoff NN+3N-full Hamiltonian with respect to the model-spacesize. As in the case of CCSD, the ΛCCSD(T) energies are sufficiently converged in

2In Section 5.5 it is shown that the relative importance of ΛCCSD(T) grows with a slower ratethan the relative E3max effect, so E3max is presumably the more limiting factor for A > 60 nucleiregarding accuracy.



order to allow extrapolations to the emax→∞ limit. In Figure 5.11, the CCSD andΛCCSD(T) results from the largest model spaces are compared. For the NN+3N-full Hamiltonian at α = 0.02 fm4, the ΛCCSD(T) correction provides 6 MeV morebinding energy for 16O, and 25 MeV for 48Ca. For the soft Hamiltonians at α =0.08 fm4, the binding energy is increased by 1.5 MeV for 16O and 10 MeV for 48Ca.Therefore, in all cases – even for the softest Hamiltonians – the ΛCCSD(T) energycorrection gives significant contributions. Since for the NN+3N-induced Hamil-tonian the CCSD energies corresponding to smaller values of α already lie belowenergies for larger α, and smaller α cause even larger ΛCCSD(T) energy correc-tions, the flow-parameter dependence is increased after including the ΛCCSD(T)correction. On the other hand, for the NN+3N-full Hamiltonian the ordering ofthe CCSD results regarding flow parameter is reversed and the flow-parameterdependence is decreased by ΛCCSD(T). So the triples excitations correction hasthe opposite effect on the flow-parameter dependence than increasing the E3max

cut has.

Assuming fast convergence of the cluster expansion, which is justified by, e.g.,Figure 5.19 in Section 5.8.1, δE (ΛCCSD(T)) dominates over higher-order corrections,i.e.,

δE (ΛCCSD(T))

≫

Eexact− E (ΛCCSD(T))

. (5.3)

Therefore, the size of δE (ΛCCSD(T)) may be used to estimate the size of the contribu-tion of the neglected higher excitation ranks of the cluster operator. Figure 5.12shows the relative importance of the ΛCCSD(T) correction normalized to the totalenergy according to

∆(ΛCCSD(T)) =

δE (ΛCCSD(T))

E (ΛCCSD(T))/100% . (5.4)

For all nuclei considered, δE (ΛCCSD(T))makes up 3-6 % of the total binding energy forthe α = 0.02 fm4 Hamiltonians while its contributions for the α = 0.08 fm4 Hamil-tonians are only about 1-2 %. Thus, using soft Hamiltonians, quite accurate cal-culations can be performed, even for the heavier nuclei. Unlike the E3max cut, therelative uncertainties due to the cluster truncation seem not to increase stronglywith mass number. So the method should be applicable with similar relative ac-curacy even in the A > 60 mass region.

The ΛCCSD(T) correction may also be used to study the feasibility of CCSDT.Unlike ΛCCSD(T), including the full triple excitations via CCSDT leads to the non-linear CCSDT equations which have to be solved iteratively, which requires to store


Chapter 5. Results

NN+3N-induced

-130

-125

-120

-115

-110

.

E[M

eV]

exp

NN+3N-full

16O~Ω = 20 MeV

-180

-170

-160

-150

-140

.

E[M

eV]

exp

24O~Ω = 20 MeV

-380

-360

-340

-320

-300

.

E[M

eV]

exp

40Ca~Ω = 24 MeV

-440

-400

-360

-320

.

E[M

eV]

exp

48Ca~Ω = 28 MeV

4 6 8 10 12emax

-540

-500

-460

-420

-380

.

E[M

eV]

exp

4 6 8 10 12emax

56Ni~Ω = 28 MeV

Î

α= 0.02 fm4 α= 0.04 fm4 α= 0.08 fm4

Figure 5.10: Convergence of CCSD (open symbols) and ΛCCSD(T) ground-state energies with

respect to model-space size for the nuclei 16,24O, 40,48Ca and 56Ni for the NN+3N-

induced and NN+3N-full Hamiltonian in NO2B approximation with E3max = 14.

The calculations employed a HF basis. Figure taken from [92].



NN+3N-induced

CC

SD

ΛC

CS

D(T

)

-130

-128

-126

-124

-122

.

E[M

eV

]

NN+3N-full

CC

SD

ΛC

CS

D(T

)

16O

NN+3N-induced

CC

SD

ΛC

CS

D(T

)

-170

-165

-160

-155

.

E[M

eV

]

NN+3N-full

CC

SD

ΛC

CS

D(T

)

24O

NN+3N-induced

CC

SD

ΛC

CS

D(T

)

-380

-375

-370

-365

-360

-355

-350

.

E[M

eV

]

NN+3N-full

CC

SD

ΛC

CS

D(T

)

40Ca

NN+3N-induced

CC

SD

ΛC

CS

D(T

)

-460

-450

-440

-430

-420

-410

.

E[M

eV

]

NN+3N-full

CC

SD

ΛC

CS

D(T

)

48Ca

α= 0.02 fm4 α= 0.04 fm4 α= 0.08 fm4

Figure 5.11: Comparison of the CCSD and ΛCCSD(T) ground-state energies from Figure 5.10

for the emax = 12 model space.


Chapter 5. Results

NN+3N-induced

0

1

2

3

4

5

6

.

∆(Λ

CC

SD

(T))

[%]

16O 24O 40Ca 48Ca 56Ni

NN+3N-full

0

1

2

3

4

5

6

.

∆(Λ

CC

SD

(T))

[%]

16O 24O 40Ca 48Ca 56Ni

α= 0.02 fm4 α= 0.04 fm4 α= 0.08 fm4

Figure 5.12: Relative importance of the ΛCCSD(T) energy correction, normalized to the total en-

ergy E (ΛCCSD(T)).

all T3 amplitudes. Since there are too many of them – even in the spherical scheme– some kind of truncation has to be introduced. An obvious choice is an E3max(T3)

cut, analogous to the E3max used for three-body matrix elements. This way thepart of T3 that generates the energetically lowest excitations would be consideredwhich are also expected to be the most relevant for a ground-state description.Since δE (ΛCCSD(T)) is given by

δE (ΛCCSD(T)) =1

(3!)2

∑

ab ci j k

λi j k

ab c

1

εab ci j k

t ab ci j k

, (5.5)

where t ab ci j k are approximations to the amplitudes t ab c

i j k of T3, the effect of the E3max(T3)

cut can be simulated inΛCCSD(T) by constraining the t ab ci j k accordingly. IfΛCCSD(T)

is a good approximation to CCSDT, one may assume that both methods show asimilar E3max(T3)-dependence,

E (CCSDT)

E3max(T3)

≈ E (ΛCCSD(T))

E3max(T3)

(5.6)

and so E (ΛCCSD(T))(E3max(T3)) may be used to find the relevant E3max(T3) range. Fig-ure 5.13 shows the E3max(T3)-dependence of δE (ΛCCSD(T)) for 16O with the NN+3N-full Hamiltonian which is already very soft at α = 0.08 fm4. The energy correc-tion is sufficiently converged for E3max(T3)-values of about 25 which is beyondpresent capabilities to store the corresponding amplitudes. Very optimistic es-timates would allow for E3max(T3) = 20 calculations, which captures a significantportion of the correction but would not allow to detect convergence, even for thislight nucleus and large SRG flow parameter.



NN+3N full

10 15 20 25 30 35

E3 max(T3)

-3

-2.5

-2

-1.5

-1

-0.5

0

.

δE

(ΛC

CS

D(T

))[M

eV

] 16O

Figure 5.13: Dependence of δE (ΛCCSD(T)) for 16O and the NN+3N-full Hamiltonian, at α =

0.08 fm4 and in NO2B approximation with E3max = 12, on the E3max(T3) cut. The

calculations were performed using a HF basis with emax = 12.

5.6 The CR-CC(2,3) Energy Correction

Nuclear Coupled-Cluster calculations rely heavily on the spherical formulation ofthe theory. Therefore, as already discussed in Section 4.12, the presence of two-and three-body matrix elements of the effective Hamiltonian that enter the defi-nition of the denominator appearing in the CR-CC(2,3) energy expression (4.182)does not immediately allow a spherical formulation. However, as pointed out inSection 4.12, the use of projection-averaged matrix elements (4.184) and (4.186) isa promising way to overcome this problem.

Therefore, it is important to estimate the errors introduced by the approximatetreatment of the denominator. Figure 5.14 illustrates the accuracy of the proposedapproximation, where a m -scheme implementation, in which the denominatorcan be treated exactly, is used for comparing the results for the exact denomina-tors D(2) and D(3)with their projection-averaged counterparts D(2) and D(3). Theleft panels show for 16O and the NN-only as well as the NN+3N-full Hamilto-nian the CR-CC(2,3) energy corrections δE (CR−CC(2,3)) for the different denomina-tors including up to one- (D(1)), two- (D(2)), or three-body (D(3)) effective Hamil-tonian matrix elements. For the present discussion it is sufficient to note that thesize of δE (CR−CC(2,3)) is about –0.3 MeV and –1.5 MeV for the NN-only Hamilto-nian at emax= 2 and emax= 4, respectively, and about –0.8 MeV and –2.0 MeV forthe NN+3N-full Hamiltonian at emax= 2 and emax= 4, respectively. The error in-troduced by the projection average, defined as the difference of δE (CR−CC(2,3)) usingthe exact and the projection-averaged denominator, is much smaller, as shown


Chapter 5. Results

NN-only

emax = 2

D(1) D(2) D(3)

denominator

-0.32

-0.3

-0.28

-0.26

-0.24

δE

(CR−

CC

(2,3

))[M

eV]

16O

0.02 fm4

0.04 fm4

0.08 fm4

emax = 4

D(1) D(2) D(3)

denominator

-1.8

-1.6

-1.4

-1.2

-1

-0.8

NN-only

D(2) − D(2)

2 4emax

-0.0035

-0.003

-0.0025

-0.002

-0.0015

-0.001

dev

iati

on

[MeV

]

16O

0.02 fm4

0.04 fm4

0.08 fm4

D(3) − D(3)

2 4emax

NN+3N-full

emax = 2

D(1) D(2) D(3)

denominator

-0.85

-0.8

-0.75

-0.7

-0.65

δE

(CR−

CC

(2,3

))[M

eV

]

16O

0.08 fm4

0.02 fm4

0.04 fm4

emax = 4

D(1) D(2) D(3)

denominator

-3

-2.5

-2

-1.5

-1

NN+3N-full

D(2) − D(2)

2 4emax

-0.0055

-0.005

-0.0045

-0.004

-0.0035

-0.003

-0.0025

dev

iati

on

[MeV

]

16O

0.02 fm4

0.04 fm4

0.08 fm4

D(3) − D(3)

2 4emax

Figure 5.14: Left: Results for the CR-CC(2,3) energy correction for the NN-only and NN+3N-

full Hamiltonian, for different choices of the denominator D(k ), Eq. (4.181). Right:

Deviations introduced by using the angular-momentum-projection averaged vari-

ants D(2) and D(3) of the denominators D(2) and D(3). These deviations are com-

pletely negligible compared to the size of δE (CR−CC(2,3)), consequently, the CR-CC(2,3)

method can be accurately formulated in the spherical scheme. The calculations were

performed in a HF basis with ħhΩ = 24 MeV, and the 3N interactions were included

via NO2B with E3max = 14.



in the right panels of Figure 5.14, with deviations of about –0.02 MeV for theNN-only Hamiltonian and about –0.04 MeV for the NN+3N-full Hamiltonian, forboth model space sizes considered. In conclusion, the projection-averaged formof the denominator in (4.182) constitutes a legitimate approximation to the exacttreatment and opens the possibility for a spherical formulation of the CR-CC(2,3)method.

Figure 5.15 shows on the left a comparison of CR-CC(2,3) results to theΛCCSD(T)

energy correction. Both methods give comparable results but also show noticeabledeviations for harder interactions. For instance, for α = 0.02 fm4 these deviationsare about 1 MeV for 16O and 2 MeV for 24O, while the total ΛCCSD(T) energy cor-rection is –5.4 MeV and –8.2 MeV, respectively. The degree of deviation of bothmethods is not unexpected, considering the approximative nature of ΛCCSD(T)

compared to the CR-CC(2,3) approach. Furthermore, the observation that the re-sults for δE (ΛCCSD(T)) lie below δE (CR−CC(2,3)) is consistent to findings in quantum-chemistry, where ΛCCSD(T) tends to overshoot the exact triples correction [153].A similar comparison of ΛCCSD(T) and CR-CC(2,3) for heavier nuclei can be foundin Figure 5.26 in Section 5.9.3. On the right of Figure 5.15 the CR-CC(2,3) energycorrection, using different choices of the denominator, is compared to ΛCCSD(T)

in the emax= 12 model space. Most strikingly, the three-body effective Hamilto-nian matrix elements in the denominator have no measurable effect on the triplescorrection and may safely be neglected. The CR-CC(2,3) results using the denom-inator D(1), involving one-body matrix elements only, lie between the D(2) resultsand ΛCCSD(T). Thus, it may be speculated that one of the reasons why ΛCCSD(T)

overshoots the exact triples correction may be the absence of contributions com-parable to the two-body effective Hamiltonian matrix elements in the CR-CC(2,3)denominator. Additional comparisons of CR-CC(2,3) ground-state energies withΛCCSD(T) for heavier nuclei can be found in Section 5.9.3.


Chapter 5. Results

NN+3N-full NN+3N-full

4 6 8 10 12emax

-130

-125

-120

-115

.

E[M

eV

]16O

0.02 0.04 0.08

α [fm4]

-131

-130

-129

.

E[M

eV

]

16Oemax = 12 D(3)

D(2)

D(1)

Λ

NN+3N-full NN+3N-full

4 6 8 10 12emax

-170

-160

-150

-140

-130

.

E[M

eV

]

24O

0.02 0.04 0.08

α [fm4]

-172

-171

-170

-169

-168

-167

.

E[M

eV

]

24Oemax = 12 D(3)

D(2)

D(1)

Λ

Î

α= 0.02 fm4 α= 0.04 fm4 α= 0.08 fm4

Figure 5.15: Left: Comparison of the CR-CC(2,3) (full symbols) ground-state energies for 16O

and 24O to ΛCCSD(T) (open symbols) ground-state energies. CCSD energies for

emax = 12 are denoted as arrows. Right: Comparison of CR-CC(2,3) energies for

various denominators to the the ΛCCSD(T) results. All calculations were performed

in a HF basis with ħhΩ = 24 MeV, and 3N interactions were included via NO2B with

E3max = 14.


5.7. CCSD with Explicit 3N Interactions

5.7 CCSD with Explicit 3N Interactions

In previous sections the NO2B approximation to three-body interactions has beenused throughout for Coupled-Cluster calculations. This approximation drasti-cally accelerates the calculations, and the IT-NCSM results for light nuclei shownin Figure 1.5 suggest that the NO2B approximation already captures a large por-tion of the relevant 3N information. However, as the focus moves from lightto medium-mass or even heavy nuclei, the validity of the NO2B approximationshould be verified in these mass range. Another reason for performing Coupled-Cluster calculations with explicit 3N interactions rather than using the NO2B ap-proximation is simply that the explicit 3N calculations eliminate the errors intro-duced by the NO2B approximation which is relevant in cases where such errorsmay not be neglected, as discussed in the following sections.

Including explicit 3N interactions in Coupled-Cluster calculations results in adramatic increase of the computational expense. Not only the Coupled-Clusterequations get much more complex, it is mostly the large number of 3N matrixelements that quickly renders explicit 3N calculations unfeasible. To some ex-tent this is caused by special requirements of the Coupled-Cluster implementa-tion used in this work. Compared to the J T -coupled storage scheme [86] for 3Nmatrix elements in the HO basis, the 3N format (4.135) used here requires about 10times more fast memory. Since the J T -coupled storage for 3N matrix elementsrequires about 1 GB memory for E3max = 12, and about 5 GB for E3max = 14, thestorage scheme used in the Coupled-Cluster implementation requires 10 and 50GB fast memory to store the E3max = 12 and E3max = 14 matrix elements in the HObasis, respectively. In the HF basis representation the 3N matrix elements acquirean isospin dependence which translates into a 6 times larger storage requirementcompared to the HO basis. Therefore, for explicit 3N Coupled-Cluster calcula-tions in HF basis, the total set of E3max = 12 and E3max = 14 matrix elements requireabout 60 and 300 GB memory, respectively. Two ways to cope with this problemhave been implemented. The first way holds the matrix elements (4.135) in HObasis in memory and performs the HF transformation of individual matrix ele-ments on the fly when they are requested, and discards them afterwards. Thisreduces the memory requirements but is rather slow due to the six-fold sum overHF coefficients and HO matrix elements that result in a single 3N matrix elementin HF basis. Alternatively, the total index range of the 3N matrix elements maybe distributed over a range of computer nodes. Each node holds the J T -coupledor (4.135) matrix elements in HO basis and calculates the matrix elements of the


Chapter 5. Results

index range assigned to the nodes in HF basis when they are needed, but is nowalso able to store them for later re-use. In applications where individual matrixelements are needed several times during a calculation, such as the iterative solu-tion of Coupled-Cluster equations, the second strategy saves a significant amountof computing time. It should also be noted that in general not the full set of matrixelements is needed in the calculations and, therefore, only those that are actuallyused should be transformed to the HF basis. For example, in all CC applicationsdiscussed in this work, the largest set of 3N matrix elements ⟨ab c |w |d e f ⟩, withparticle orbitals only, does not enter anywhere. In order to keep the computa-tional runtime reasonable for Coupled-Cluster calculations using explicit 3N in-teractions, the cutoff E3max = 12 is used in the following. The first application ofCCSD for three-body Hamiltonians can be found in [63] but due to the use of an m -scheme implementation, these considerations were limited to proof-of-principlecalculations for 4He in the harmonic-oscillator basis and small model spaces. Thespherical scheme finally allows to move on to the medium-mass regime. Usingan E3max = 12 cut, medium-mass CCSD calculations at emax = 12 using three-bodyHamiltonians are comparable in cost to a ΛCCSD(T) NO2B calculation in the sameemax = 12 model space.

Figure 5.16 shows the convergence of CCSD ground-state energies with ex-plicit 3N (full symbols) and for the NO2B approximation (open symbols) for themedium-mass nuclei 16,24O, 40,48Ca and 56Ni using the NN+3N-induced and NN-+3N-full Hamiltonian. The agreement of the NO2B approximation with the ex-plicit 3N is remarkable. The normal-ordering approximation provides very accu-rate results and it seems that this accuracy is rather independent of the model-space size, the mass number, or even the SRG flow parameter. Furthermore, Fig-ure 5.17 indicates that the quality of the approximation is also independent ofthe oscillator frequency of the basis. The unnatural increase of the energies forsmaller values of ħhΩ in Figure 5.17 is due to the use of unsufficiently large SRGmodel spaces, as discussed in Section 5.9.3. However, for the optimal frequenciesdetermined from Figure 5.17, the effects of insufficient SRG model spaces are lessthan 1 % of the total energy.

The relative contribution of the residual 3N interaction normalized to the CCSDground-state energies according to

∆(3B) =|E (CCSD)− E

(CCSD)NO2B |

E (CCSD)/100% (5.7)

is shown in Figure 5.18. It should be stressed that E(CCSD)NO2B was calculated in a pure


5.7. CCSD with Explicit 3N Interactions

NN+3N-induced

-130

-120

-110

-100

.E

[MeV

]

exp

NN+3N-full

16O~Ω = 20 MeV

-180

-160

-140

-120

.

E[M

eV]

exp

24O~Ω = 20 MeV

-380

-340

-300

-260

.

E[M

eV]

exp

40Ca~Ω = 24 MeV

-450

-350

-250

.

E[M

eV]

exp

48Ca~Ω = 28 MeV

4 6 8 10 12emax

-550

-500

-450

-400

-350

.

E[M

eV]

exp

4 6 8 10 12emax

56Ni~Ω = 28 MeV

Î

α= 0.02 fm4 α= 0.04 fm4 α= 0.08 fm4

Figure 5.16: Comparison of CCSD with NO2B approximation (open symbols) and CCSD with

explicit 3N interaction (full symbols) for the nuclei 16,24O, 40,48Ca and 56Ni for

the NN+3N-induced and NN+3N-full Hamiltonian in NO2B approximation with

E3max = 12. The optimal oscillator frequencies ħhΩ have been determined from Fig-

ure 5.17. Figure taken from [92].


Chapter 5. Results

-170

-165

-160

-155

-150

.

E[M

eV

]

24O

-370

-360

-350

-340

-330. E

[MeV

]

40Ca

20 24 28 32 36

~Ω [MeV]

-460

-450

-440

-430

-420

-410

.

E[M

eV

]

48Ca

20 24 28 32 36

~Ω [MeV]

-530

-510

-490

-470

-450

.E[M

eV

]56Ni

Î

α= 0.02 fm4 α= 0.04 fm4 α= 0.08 fm4

Figure 5.17: Oscillator-frequency dependence of CCSD ground-state energies of 24O, 40,48Ca and56Ni for the NN+3N-full Hamiltonian with explicit 3N interactions (full symbols)

and in NO2B approximation (open symbols) with E3max = 12, using a Hartree-Fock

basis with emax = 12 model spaces. Figure taken from [92].

NO2B scheme, where residual 3N information entered in neither in the determi-nation of the cluster amplitudes nor in the energy expression. Therefore, ∆(3B)measures the total effect of the residual 3N interaction in the CCSD calculation,without discrimination between its effect on amplitudes or energy. This issue isfurther addressed in Section 5.8.1. There is no definite systematics of the relativecontribution with mass number or the SRG flow parameter. For all nuclei the rel-ative contribution is well below 1 % and in particular for the heavier nuclei it isa little smaller with values around 0.6 %. This confirms the earlier findings thatthe NO2B approximation seems to perform better for heavier nuclei. However,one reason for this might be that the ground-state energies for the heavier nucleiare not fully converged with respect to E3max and, therefore, not the full relevantinformation about the 3N interaction was used from the beginning. Furthermore,it seems that the relative contributions tend to be a little more important for largerflow parameters, but these effects lie in the range of 0.1 % and have no practicalsignificance.


5.8. ΛCCSD(T) with Explicit 3N Interactions

NN+3N-induced

0

0.2

0.4

0.6

0.8

1

.

∆(3

B)

[%]

16O 24O 40Ca 48Ca 56Ni

NN+3N-full

0

0.2

0.4

0.6

0.8

1

.

∆(3

B)

[%]

16O 24O 40Ca 48Ca 56Ni

α= 0.02 fm4 α= 0.04 fm4 α= 0.08 fm4

Figure 5.18: Relative contribution of the residual 3N interaction, normalized to the CCSD

ground-state energies using explicit 3N interactions. Parameters of the calculations

as in Figure 5.16.

5.8 ΛCCSD(T) with Explicit 3N Interactions

5.8.1 Benchmark of the NO2B Approximation

For CCSD, the contribution of the residual 3N interaction to the total ground-state energy of medium-mass nuclei was shown to be less than 1%. A similarresult for ΛCCSD(T) would be desirable in order to keep the error introduced bythe NO2B approximation at the 1% level. The alternative – routinely includingexplicit 3N interactions in ΛCCSD(T) calculations – is not an option due to theextreme computational costs.

A natural approach to assess the relevance of the residual normal-orderedthree-body interaction WN in CCSD and ΛCCSD(T) calculations is to modify thetotal energy expression


(ΛCCSD(T))NO2B + ∆E

(CCSD)3B + δE

(ΛCCSD(T))3B , (5.8)

to either include or not include the contributions ∆E(CCSD)3B and δE

(ΛCCSD(T))3B due to

WN . However, this discussion is complicated by the fact that the energy valuesare not only determined by their expressions in terms of the T (CCSD) and Λ(CCSD)

amplitudes, but also by the type of equation – with or without inclusion of the WN

terms – used to determine the amplitudes in the first place. This leads to variouspossible and reasonable combinations to consider.

In Figure 5.19, where for 16O, 24O, and 40Ca, and both, the NN+3N-induced


Chapter 5. Results

NN+3N-induced

-100

-80

-60

0.02 0.04 0.08

α [fm4]

-128

-126

-124

-122

-120

.

E[M

eV

]

≈ ≈

16O

Ere

fE

(CC

SD

)

NO

2B

E(C

CS

D)+δE

(T)

NO

2B+δE

(T)

3B

NN+3N-induced

-150

-100

-50

0.02 0.04 0.08

α [fm4]

-166

-164

-162

-160

-158

-156

-154

-152

.

E[M

eV

]

≈ ≈

24O

Ere

fE

(CC

SD

)

NO

2B

E(C

CS

D)+δE

(T)

NO

2B+δE

(T)

3B

NN+3N-induced

-300

-200

-100

0.02 0.04 0.08

α [fm4]

-380

-375

-370

-365

-360

-355

-350

-345

.

E[M

eV

]

≈ ≈

40CaE

ref

E(C

CS

D)

NO

2B

E(C

CS

D)+δE

(T)

NO

2B+δE

(T)

3B

NN+3N full

-100

-80

-60

0.02 0.04 0.08

α [fm4]

-132

-130

-128

-126

-124

.

E[M

eV

]

≈ ≈

16O

Ere

fE

(CC

SD

)

NO

2B

E(C

CS

D)+δE

(T)

NO

2B+δE

(T)

3B

NN+3N full

-150

-100

-50

0.02 0.04 0.08

α [fm4]

-172

-170

-168

-166

-164

-162

.

E[M

eV

]

≈ ≈

24O

Ere

fE

(CC

SD

)

NO

2B

E(C

CS

D)+δE

(T)

NO

2B+δE

(T)

3B

NN+3N full

-300

-200

-100

0.02 0.04 0.08

α [fm4]

-380

-375

-370

-365

-360

-355

-350

-345

.

E[M

eV

]

≈ ≈

40Ca

Ere

fE

(CC

SD

)

NO

2B

E(C

CS

D)+δE

(T)

NO

2B+δE

(T)

3B

Figure 5.19: Anatomy of the individual contributions from CCSD and ΛCCSD(T) to the total

binding energy of 16O, 24O and 40Ca for the two types of three-body Hamiltonians and

SRG flow parameters α= 0.02, 0.04, and 0.08 fm4. For 16O and 24O, a Hartree-Fock

basis with emax = 12 model space and oscillator frequency ħhΩ = 20 MeV was used,

whereas for 40Ca an emax = 10 model space with ħhΩ = 24 MeV was employed. The

shorthand notationδE(T)NO2B andδE

(T)3B is used to denoteδE

(ΛCCSD(T))NO2B andδE

(ΛCCSD(T))3B ,

respectively.



and NN+3N-full Hamiltonians results are shown for a series of increasingly com-plete calculations of the ground-state energies. The energy E

(CCSD)NO2B is calculated

in NO2B approximation, where the WN terms are neglected in both, the determi-nation of the cluster amplitudes as well as the calculation of the energy. For thecalculation of all other energies presented in Figure 5.19, the T (CCSD) and Λ(CCSD)

amplitudes were determined from their respective amplitude equations includingthe WN terms. Analogous to the discussion in Section 5.7, by comparing E

(CCSD)NO2B to

E (CCSD), a direct quantification of the combined effect of WN in the CCSD amplitudeequations and the energy expression can be made. It should be noted that here

E (CCSD)− E(CCSD)NO2B 6= E

(CCSD)3B , (5.9)

contrary to what (3.25) seems to imply, due to the use of different amplitudes forthe calculation of E

(CCSD)NO2B and E (CCSD), respectively. Contrary to this, the same am-

plitudes (obtained from solving the amplitude equations containing the WN terms)are used in the calculations of δE

(ΛCCSD(T))NO2B and δE

(ΛCCSD(T))3B . Therefore, using these

numbers it is only possible to quantify the importance of the WN contributions– simply given by δE

(ΛCCSD(T))3B itself – in the calculation of the total triples correc-

tion δE (ΛCCSD(T)). This allows to compare the results for the complete ΛCCSD(T)ground-state energy E (ΛCCSD(T)) to the simplified expression


(ΛCCSD(T))NO2B + ∆E

(CCSD)3B , (5.10)

in which the WN terms are included in the CCSD and ΛCCSD calculations but areomitted in the final calculation of the energy correction. However, particularly forthe calculation of δE

(ΛCCSD(T))NO2B , other choices of where to include the WN terms in the

amplitude equations seem reasonable, and this issue is addressed further below.However, it should already be mentioned that other choices of the amplitudesequations lead to practically the same results.

In the following, 16O and the NN+3N-full Hamiltonian at flow parameter val-ues α= 0.02 fm4 and 0.08 fm4 is considered as an example. For increasing α, moreand more of the binding energy is shifted to lower orders of the cluster expan-sion and the contributions from the higher orders consequently get smaller withthe SRG flow: The size of the reference energy Eref grows from –56.11 MeV to–101.67 MeV, while the CCSD correlation energy ∆E (CCSD) decreases from –69.03MeV to –26.52 MeV as the SRG evolution goes from α = 0.02 fm4 to α = 0.08 fm4

and the ΛCCSD(T) energy correction δE (ΛCCSD(T)), which, according to Section 5.5,is also considered as a measure for the contributions of the omitted cluster op-erators beyond the three-body level [95], decreases from –5.54 MeV to –2.34 MeV,


Chapter 5. Results

corresponding to 4.2 % and 1.8 % of the total binding energy. In the medium-mass regime, these uncertainties related to the cluster truncation are typically thelargest in the calculations for a given Hamiltonian, and, therefore, they set theoverall level of accuracy targeted at [95].

Examining the contributions from the residual 3N interaction to ∆E (CCSD) it isfound that, while the absolute value of ∆E (CCSD) decreases by about 30 MeV whenthe Hamiltonian is evolved from α = 0.02 fm4 further to α = 0.08 fm4, ∆E (CCSD) −∆E

(CCSD)NO2B is only subject to a slight increase from 0.54 MeV to 0.92 MeV, correspond-

ing to 0.4 % and 0.7 % of the total binding energy. Consequently, the relative aswell as the absolute importance of the residual 3N interaction to the CCSD corre-lation energy grows with the SRG flow.

Furthermore, while for the harder Hamiltonian at α = 0.02 fm4 the WN contri-butions to ∆E (CCSD) are about one order of magnitude smaller than the accuracylevel set by δE (ΛCCSD(T)), for the softer α = 0.08 fm4 Hamiltonian the WN contribu-tions have an comparable size of about 39 % of the triples correction. Therefore, inorder to keep different errors at a consistent level, for soft interactions the residual3N contributions should be included in CCSD if the triples correction is consid-ered as well.

For the ΛCCSD(T) triples correction δE (ΛCCSD(T)) itself, the WN contributionsδE

(ΛCCSD(T))3B , despite containing second-order MBPT contributions, have very small

values of about –15 keV. This effect is about one order of magnitude smaller thanthe targeted accuracy given by the size of δE (ΛCCSD(T)), and may therefore safely beneglected. From another perspective, the WN contributions to δE (ΛCCSD(T)) consti-tutes about 0.1 % of the total binding energy, which clearly is beyond the level ofaccuracy of any many-body method operating in the medium-mass regime today.

As is apparent from Figure 5.19, the discussion for the NN+3N-induced Hamil-tonian and the heavier nuclei 24O and 40Ca is similar. In the case of 40Ca, thesmaller emax = 10 model space is used in order to keep the computational costreasonable. In this model space the results are not fully converged with respect toemax, but since the quality of NO2B is largely independent of emax [95] this does notaffect the present discussion. For the NN+3N-induced Hamiltonian, for example,the relative contribution of WN to the CCSD correlation energy grows from 1.3 %for α = 0.02 fm4 to 4.2 % for α = 0.08 fm4, in both cases constituting about 0.6 % ofthe total binding energy. Again, as the SRG flow parameter increases, the contri-butions of WN to the CCSD correlation energy on the one hand, and the triples



correction on the other hand, become comparable, where ∆E (CCSD) −∆E(CCSD)NO2B is

about 18 % of the size of the triples correction at α = 0.02 fm4 and already about48 % at α = 0.08 fm4. The WN effect to the triples correction is again negligible,about one order of magnitude smaller than the triples correction itself, namelyabout 2 % of δE (ΛCCSD(T)) for α = 0.02 fm4 and about 11 % for α = 0.08 fm4, or 0.1 %and 0.2 % of the total binding energy E (ΛCCSD(T)).

In summary, as in Section 5.7, contributions from residual 3N interactions tothe CCSD correlation energy are found to be of the order of 1 % of the total bind-ing energy. For the triples correction the contributions are much smaller and maybe considered negligible. The fact that the residual 3N contributions are ratherinsensitive to the SRG flow parameter impacts the characterization of their im-portance. For hard interactions, the residual 3N effects to the CCSD correlationenergy E (CCSD) are rather small compared to the triples correction δE (ΛCCSD(T)), butthey become comparable as the triples contribution gets smaller for soft interac-tions. Therefore, when using soft interactions, the residual 3N interaction shouldbe included in CCSD if the desired accuracy level also demands inclusion of triplesexcitation effects. For the triples correction, on the other hand, the residual 3Ninteraction only plays an insignificant role, providing contributions that are shad-owed by the considerably larger uncertainties stemming, e.g., from the clustertruncation. This motivates the use of the truncated energy expression E (ΛCCSD(T)),Eq. (5.10), instead of the full form E (ΛCCSD(T)), resulting in only negligible losses inaccuracy.

5.8.2 Approximation Schemes for the Amplitudes

The above considerations indicate that the residual 3N interaction may be ne-glected in calculating the ΛCCSD(T) energy correction δE (ΛCCSD(T)) without signif-icantly affecting the overall accuracy, leading to Eq. (5.10) as an approximate, yetaccurate, form for E (ΛCCSD(T)). From a practitioner’s point of view, discarding theWN contributions to δE (ΛCCSD(T)), Eqs. (3.46) and (3.55), already leads to significantsavings in the implementational effort and computing time, but one still has tosolve the CCSD equations determining the T (CCSD) amplitudes t a

i and t abi j , as well

as the ΛCCSD equations determining the Λ(CCSD) amplitudes λia

and λi j

ab , with fullincorporation of WN . Particularly solving the ΛCCSD equations, for which the ef-fective Hamiltonian contributions given in Figures 3.10–3.12 have to be evaluated,consumes most of the computing time in practical calculations. Therefore, it isalso worthwhile to investigate how much of the residual 3N interaction informa-


Chapter 5. Results

tion has to be included in solving for the T (CCSD) and Λ(CCSD) amplitudes that enterthe energy expressions, in order to obtain accurate results at the lowest possiblecomputational cost.

In order to distinguish between different approximation schemes the followingnotation is introduced in which for energy quantities that only depend on T (CCSD)

amplitudes the label in brackets denote if the T (CCSD) amplitudes were determinedfrom the amplitude equations with (3B) or without residual 3N interaction (2B).Similarly, for quantities that depend on both, T (CCSD) and Λ(CCSD) amplitudes, thefirst label denotes the type of equation used to determine the T (CCSD) amplitudesand the second specifies the ΛCCSD equations. For example, E (ΛCCSD(T))(3B, 2B)

refers to the energy expression (5.10), calculated using T (CCSD) amplitudes deter-mined from the equations including the WN terms, while the Λ(CCSD) amplitudesare determined using the NO2B approximation.

The following approximation schemes are considered, in which the WN contri-butions δE

(ΛCCSD(T))3B to the triples correction are always neglected: For the “NO2B”

scheme, all WN terms are discarded in both, the determination of the T and Λ am-plitudes and the energy E

(ΛCCSD(T))NO2B ,

E (NO2B) = E(ΛCCSD(T))NO2B (2B, 2B) . (5.11)

This of course corresponds to an ordinary ΛCCSD(T) calculation in NO2B approx-imation. For scheme “A”, the energy E

(ΛCCSD(T))NO2B is computed as in the NO2B case

and ∆E(CCSD)3B , calculated T (CCSD) amplitudes obtained from the NO2B CCSD equa-

tions, is added,

E (A) = E(ΛCCSD(T))NO2B (2B, 2B)+∆E

(CCSD)3B (2B) . (5.12)

This represents the simplest and most economic way to include WN information,where it only enters in the expression for the energy contribution ∆E

(CCSD)3B , but

not in the considerably more complex equations that determinate the amplitudes.In scheme “B”, full WN information is included in the calculation of the CCSDcorrelation energy, keeping the WN terms in the amplitude equations as well as inthe energy expression. The triples correction, however, is calculated without anyWN information,

E (B) = E (CCSD)(3B)+δE(ΛCCSD(T))NO2B (2B, 2B) . (5.13)

This way, consistency is kept between the T (CCSD) and Λ(CCSD) amplitudes that enterthe triples correction, while capturing all residual 3N effects in the CCSD energy



∆E (CCSD). In scheme “C”, an inconsistency is introduced between the T (CCSD) andΛ(CCSD) amplitudes by solving for T (CCSD) with the WN terms present, while we solvefor Λ(CCSD) without WN terms and the energy expression is given by E (ΛCCSD(T)),

E (C) = E (ΛCCSD(T))(3B, 2B) . (5.14)

This variant is reasonable since one typically has to solve for the T (CCSD) ampli-tude equations with WN terms anyway in order to obtain the comparatively large∆E

(CCSD)3B contribution to the energy while one would like to avoid to solve for the

Λ(CCSD) amplitudes in this manner. Finally, in scheme “D”, in which the residual3N interaction terms are neglected only in the expression for δE (ΛCCSD(T)), the fullWN -containing equations are used to solve for the T (CCSD) and Λ(CCSD) amplitudesand the energy is determined via Eq. (5.10),

E (D) = E (ΛCCSD(T))(3B, 3B) . (5.15)

As in the discussion of Figure 5.19, by comparing with scheme “C”, this variantallows to estimate the importance of WN for the Λ(CCSD) amplitudes.

In Figure 5.20, for the case of 16O, 40Ca and the NN+3N-full Hamiltonian, thedeviations introduced by the aforementioned approximation schemes are com-pared to the complete 3N calculations. For 24O and the NN+3N-induced Hamil-tonian very similar results are obtained and, therefore, not presented here. Asexpected, the “NO2B” scheme shows the largest deviations because the contribu-tions of WN to CCSD are completely missing. Including the WN terms in the energyexpression for the CCSD correlation energy but evaluating it using T (CCSD) ampli-tudes without WN information in scheme “A” virtually does not change the result.Therefore, we can conclude that it is the WN effect on the T (CCSD) amplitudes thatis most important for CCSD, rather than the additional terms ∆E

(CCSD)3B . In these

calculations, the best approximation to the complete calculations is provided byscheme “B”, where the full WN information is used to determine the CCSD cor-relation energy, but otherwise no WN information enters at all in the calculationof the triples correction. However, approximation schemes “B”,“C” and “D” givevery similar results, again hinting at the WN effect on the T (CCSD) amplitudes tobe the most important ingredient in the inclusion of residual 3N interactions inCCSD and ΛCCSD(T) calculations.


Chapter 5. Results

NN+3N-full

0.0

0.2

0.4

0.6

0.8

1.0

.

dev

iati

on

[%]

16O

NO

2B

A

B CD

0.02 0.04 0.08

α [fm4]

NN+3N-full

0.0

0.2

0.4

0.6

0.8

1.0

.

dev

iati

on

[%]

40Ca

NO

2B

A

BC

D

0.02 0.04 0.08

α [fm4]

Figure 5.20: Comparison of the deviations introduced by the different approximation schemes de-

scribed in the text from the full inclusion of the residual 3N interaction in all steps

involving CCSD and ΛCCSD(T) calculations for three-body Hamiltonians. Param-

eters of the calculations as in Figure 5.19.

5.9 Ab Initio Description of Heavy Nuclei

The previous sections gave an overview of various aspects of Coupled-Clustercalculations in the medium-mass regime using chiral interactions, from which thefollowing conclusions may be drawn about how to obtain the accurate results forheavier nuclei in the present framework:

The normal-ordered two-body approximation works very well and allows toalmost completely include the relevant 3N interaction into the calculations at verymuch reduced computational cost. The error introduced by this approximationis only about 1 % of the total binding energy, which is absolutely acceptable forab initio calculations in the heavy regime considering other sources of uncertaintypresent in the calculations, such as the omission of SRG-induced four- and multi-nucleon interactions.

The E3max cut emerged as one of the main limiting factors in the calculation ofnuclear properties in the mass range A > 60. Therefore, significant improvementover the E3max = 14 cut is required in order to obtain accurate results for nucleibeyond the medium-mass regime considered so far. However, full sets of 3N ma-trix elements with E3max > 14 are not easy to generate and to store. Additionally,the many-body methods often cannot handle such large sets of explicit 3N ma-trix elements and, therefore, there is no need for them except for the computation


5.9. Ab Initio Description of Heavy Nuclei

of the normal-ordered matrix elements. Computing only those matrix elements thatare used in the normal-ordering saves computing time and finally allows to go tolarger values of E3max in the normal-ordering procedure. However, it is importantto retain consistency between the E3max used in the Hartree-Fock calculation thatdetermines the reference state and the E3max used in the normal-ordering proce-dure, as described in Section 5.9.1.

Another source of uncertainty for the nuclei considered so far is given by thecluster truncation. There is not much room for practical improvements on themany-body side, but the uncertainties may be reduced by the use of softer in-teractions. Furthermore, the uncertainties due to the cluster truncation seem notto increase with the mass number and so accurate calculations should be possibleusing ΛCCSD(T) or CR-CC(2,3) in the A > 60 region.

The cluster truncation motivates the use of soft interactions at flow parameterssuch as 0.08 fm4, for which the triples correction only contribute about 2 % to theenergy. However, at this level of accuracy other sources of uncertainties, such asthe error introduced by the NO2B approximation become relevant. Explicit 3Ncalculations are very expensive for E3max > 12 but, nevertheless, the error intro-duced by omitting residual 3N contributions may be reduced by using a schemein which the 3N matrix elements are included explicitly up to some parameterE

explicit3max , and 3N matrix elements with E NO2B

3max > Eexplicit3max enter the calculation only

through the normal-ordering.

How large the SRG flow-parameter dependence due to omitted many-bodyforces in medium-mass nuclei really is cannot be decisively determined from theprevious calculations, because E3max and the cluster truncation are also sourcesof flow-parameter dependence. However, as mentioned above, the uncertaintiesdue to E3max will largely be reduced by going to sufficient large values of E3max.

Finally, the insufficient SRG model spaces, as mentioned in Section 5.7, needto be addressed. The strategies pursued in Section 5.9.2 are straightforward en-largements of the model spaces and a frequency conversion technique.

By virtue of all the developments above it will then be possible to extend therange of accurate ab initio calculations into the heavy nuclear regime.


Chapter 5. Results

5.9.1 Self-Consistent Hartree-Fock Reference Normal-Ordering

In order to perform accurate calculations of nuclei beyond the medium-mass re-gime considered in Section 5.9.3, the normal-ordering procedure has to performedfor larger values of E3max than the ones used so far. For a given reference state,computing the normal-ordered matrix elements corresponding to E3max values forwhich no full sets of 3N matrix elements can be stored any more can be achieved bydistributing the workload over many independently operating computing nodeswhich calculate (and temporarily store) the required matrix elements on the fly.However, using full sets of 3N matrix elements with large E3max in this mannerin many-body calculations is not a preferred option; this also includes the HFmethod from which the reference state for the normal-ordering is computed. There-fore, a different strategy that avoids using explicit 3N matrix elements with largeE3max in the HF calculations is pursued in the following. To this end, the role ofthe HF reference state in the normal-ordering procedure needs to be investigated.

A first attempt of going beyond current normal-ordering capabilities is to per-form a HF calculation which determines the HF reference state |Φ⟩ using 3N ma-trix elements with an E

|Φ⟩3max cut for which full sets of 3N matrix elements can be

handled. Afterwards, this reference state may be used in the normal orderingof 3N matrix elements for a larger E NO

3max. Obviously, in this case the referencestate used is not fully appropriate since it only contains E

|Φ⟩3max < E NO

3max information.Furthermore, the Hartree-Fock basis will also be not consistent to the normal-ordered matrix elements because, again, the construction of the HF basis onlyused E

|Φ⟩3max < E NO

3max information while the matrix elements also contain informa-tion up to E NO

3max. This means that the reference state used in the normal orderingand which also enters the many-body calculations is actually no longer the properreference state from the point of view of the obtained normal-ordered interaction.

To demonstrate the effects of using reference states that do not correspond tothe employed interaction, in Figure 5.21 CCSD ground-state energies are shownfor 16O and 40Ca, using inconsistent E3max values in the Hartree-Fock calculationand the subsequent normal-ordering, with E

|Φ⟩3max = 8 and E NO

3max = 14. The leftmostbars show the results for this combination of E3max values while the rightmost barsrepresent the result obtained for the consistent values E

|Φ⟩3max = E NO

3max = 14 which iscalled exact in the following. In the case of 16O, for instance, the correspondingresults disagree, for the α= 0.02 fm4 Hamiltonian about 3 MeV and for α= 0.08 fm4



NN+3N-full

0 1 2 3 exact

iteration

-130

-126

-122

.

E[M

eV

]

16O

NN+3N-full

0 1 2 3 exact

iteration

-375

-365

-355

-345

.

E[M

eV

]

40Ca

α= 0.02 fm4 α= 0.04 fm4 α= 0.08 fm4

Figure 5.21: CCSD ground-state energies from emax = 12 model spaces and ħhΩ = 24 MeV, for 16O

and 40Ca for the NN+3N-full Hamiltonian in NO2B approximation with E3max = 14

using different HF reference states (see text). The iterative HF normal ordering con-

verges immediately to the results obtained for consistent HF and normal ordering.

about 700 keV. These results demonstrate the importance of consistency betweenthe E3max of the reference state and of the normal-ordered interaction.

To correct for this inconsistency, the normal-ordered interaction can now beused in yet another HF calculation in order to determine a corrected version ofthe reference state 3. This reference state now contains also E NO

3max information atthe NO2B level and will be close to the exact reference state corresponding toE NO

3max, due to the good performance of the NO2B approximation. Therefore, usingthis updated reference state in a subsequent normal-ordering for E NO

3max will thenyield a more consistent combination of reference state and normal-ordered matrixelements. This procedure can then be iterated until consistency is achieved.

The CCSD ground-state energies calculated using normal-ordered interactionsfrom reference states obtained from these additional iterative HF calculations andnormal orderings are also shown in Figure 5.21. From there it is apparent thatalready after the first iteration the reference state is typically close enough to theexact reference state such that the CCSD results become indistinguishable fromthe exact case. As mentioned above, this fast convergence can be attributed to the

3This calculation only involves the normal-ordered zero-, one-, and two-body matrix elementsand can be performed very efficiently.


Chapter 5. Results

capability of the NO2B approximation to capture most of the 3N information inthe lower-rank normal-ordered matrix elements.

5.9.2 Role of the SRG Model Space

For practical applications, the SRG operator flow equation (1.29) has to be con-verted into matrix representation. To that end, resolutions of the identity in formof infinite summations are inserted between adjacent operators, as it is done to ob-tain (1.32). Since these infinite summations have to be truncated to finite sums, er-rors are inevitably introduced in the evolution. The energy and momentum rangethat the SRG model space spans depends on the oscillator frequency of the HOstates in which the flow equation is represented. Particularly for small frequen-cies this range may not be sufficient for the model space sizes that are accessiblein practical computations. As is discussed in more detail in [86, 156], this issuemay be overcome by solving the SRG flow equations at a large enough parent fre-quency and subsequently transforming the obtained matrix elements to smallertarget frequencies through a basis transformation. The latter step is facilitated bythe fact that the evolved matrix has a band-diagonal structure which makes thetransformation numerically more accurate.

Figure 5.22 illustrates the importance of the frequency conversion in a seriesof CCSD ground-state calculations for nuclei ranging from 40Ca to 78Ni, for whichthe frequency-converted matrix elements have been generated from the parent fre-quency ħhΩ= 36 MeV. In the cases where no frequency conversion has been applied,i.e., where the matrix elements corresponding to a specific value of ħhΩ have beencomputed from SRG evolutions in a model space spanned by HO states of samefrequency ħhΩ, the insufficient energy span of the model space causes an artificialincrease of the ground-state energies at smaller frequencies. The ground-stateenergies obtained using frequency-converted matrix elements, however, show amuch more natural behavior. Since for the considered nuclei the energy minimumis located at smaller frequencies, accurately evolved matrix elements are particu-larly important in this frequency range. In fact, as can be seen in Figure 5.22,the artificial increase of the energies obtained using the non-converted matrix ele-ments shifts the energy minima to higher frequencies and, thus, to smaller bindingenergies.

The use of the frequency-conversion technique mentioned above allows to cir-cumvent limitations of the SRG due to the insufficiency of the three-body SRGmodel space for small frequencies by converting matrix elements corresponding



NN+3N-induced NN+3N-full

-380

-360

-340

-320

.

E[M

eV

] 40Ca

-450

-400

-350

.

E[M

eV

] 48Ca

-550

-500

-450

-400

.

E[M

eV

] 56Ni

-650

-600

-550

.

E[M

eV

] 68Ni

24 28 32 36 40

~Ω [MeV]

-700

-600

-500

-400

.

E[M

eV

]

24 28 32 36 40

~Ω [MeV]

78Ni

Î

α= 0.02 fm4 α= 0.04 fm4 α= 0.08 fm4

Figure 5.22: Comparison of CCSD ground-state energies corresponding to matrix elements ob-

tained with (full symbols) and without (open symbols) frequency conversion. The

calculations were performed in a HF basis with emax = 12 and using the NO2B ap-

proximation with E3max = 12. The matrix elements were evolved in the ramp40C

SRG model space at parent frequency ħhΩ = 36 MeV.


Chapter 5. Results

to a higher parent frequency to lower frequencies. Consequently, the SRG modelspace has at least to be sufficiently large in order to accurately evolve the Hamil-tonian at the parent frequency. Specifically, the SRG model space is parametrizedby the way the infinite summations in the resolution of the identity in (1.32) aretruncated. In the present case, these summations are truncated using a truncationESRG in the energy quantum number of the three-body Jacobi states in which theidentity is resolved [86], i.e., in a schematic notation,

1(3) ≈

E (|φ(3)p ⟩)≤ESRG∑

p

|φ(3)p⟩⟨φ(3)

p| . (5.16)

Since the SRG evolution can be performed for each total angular momentum J

of the Jacobi states separately, and larger J are expected to be less relevant, thismotivates the definition of a J -dependent truncation parameter ESRG.

Figure 5.23 (top) presents different parametrizations of the SRG model spacesin terms of ESRG, which will in the following be referred to as ramps. The parame-trization used so far in this work corresponds to ramp 40C, in which the maximumvalue ESRG = 40 is used for the smallest angular momenta J = 1/2, . . . , 5/2. Then, thetruncation parameter is linearly ramped down to ESRG = 24 at J = 13/2, where itremains for all higher values of J . The largest SRG model space considered inthe following corresponds to ramp 40J, and the effects of different choices of thetruncation parameter ESRG, such as the 38G, 40C, 40F, 40K or 40L 4 ramps from Fig-ure 5.23, are assessed by comparing CCSD ground-state energies correspondingto these ramps to ground-state energies corresponding to the 40J ramp.

Figure 5.23 shows at the bottom a comparison of the 40C ramp to the 40J rampbased on CCSD ground-state energies for nuclei ranging from 36Ca up to 142Sn,where the experimental ground-state energies are shown as well. As mass num-ber grows, increasing deviations of the 40C from the 40J results are clearly visible,indicating that the 40C ramp becomes less sufficient for heavier nuclei. For the40C ramp, the heavier nuclei are even almost unbound.

These deviations are considered in more detail in Figure 5.24, where at the topthe 40C ramp is once again compared to the 40J ramp, this time in terms of theabsolute deviation of the corresponding CCSD results. While for the lighter nucleithere are moderate differences in the ground-state energies related to the different

4The number in the name of the ramp denotes the starting ESRG at small angular momenta,while the letter has no further meaning.



40F38G

40J

40K

40C

40L

12

32

52

72

92

112

132

152

172

192

212

24

28

32

36

40

J

ESR

G

36Ca40Ca

48Ca52Ca

54Ca48Ni

56Ni60Ni

62Ni66Ni

68Ni78Ni

88Sr90Zr

100Sn106Sn

114Sn118Sn

120Sn132Sn

142Sn-12

-10

-8

-6

-4

-2

0

.

E/A

[MeV

]

: 40C : 40J : exp

Figure 5.23: Top: Various choices of parametrizations of the SRG model space in terms of the

truncation parameter ESRG. Bottom: Experimental ground-state energies compared

to CCSD results from a HF basis with emax = 12, obtained from the 40C and 40J ramp

and the NN+3N-full Hamiltonian at α= 0.08 fm4 and in NO2B approximation with

E3max = 14, using frequency-converted matrix elements at ħhΩ = 24 MeV with parent

frequency of 36 MeV.


Chapter 5. Results

36Ca40Ca

48Ca52Ca

54Ca48Ni

56Ni60Ni

62Ni66Ni

68Ni78Ni

88Sr90Zr

100Sn106Sn

114Sn118Sn

120Sn132Sn

142Sn0

2

4

6

8

.

∆E/A

[MeV

]

36Ca40Ca

48Ca52Ca

54Ca48Ni

56Ni60Ni

62Ni66Ni

68Ni78Ni

88Sr90Zr

100Sn106Sn

114Sn118Sn

120Sn132Sn

142Sn0

0.05

0.1

0.15

0.2

0.25

0.3

.

∆E/A

[MeV

] : 40C

: 38G : 40F : 40K : 40L

Figure 5.24: Deviations of CCSD ground-state energies (per nucleon) corresponding to ramp

40J to the other SRG ramps from Figure 5.23, for the NN+3N-full Hamiltonian at

α = 0.08 fm4 in NO2B approximation with E3max = 14, using frequency-converted

matrix elements at ħhΩ = 24 MeV with parent frequency of 36 MeV. While ramp 40C

is completely inappropriate in the heavy regime, the small deviations for ramp 40K

and 40L suggest that ramp 40J allows for accurate calculations. The calculations

employed a HF basis with emax = 12.



choices of SRG model spaces, for 56Ni these deviations already reach 0.5 MeV pernucleon. From there, the deviations increase quickly, reaching values up to 7 MeVper nucleon for 120Sn. Therefore, ramp 40C is far from sufficient for calculationsof nuclei with mass numbers A > 60. There are distinct jumps in the deviationplot, occurring, for instance, between nuclei 68,78Ni or 120,132Sn. Such jumps occureach time a new high-momentum single-particle orbit in Figure 4.1 is occupiedin the reference state. For example, the reference configurations of 68Ni and 78Nidiffer by the 1g9/2 shell, and the configurations of 120Sn and 132Sn differ by the1h11/2 shell. This observation hints at a growing importance of the large-J part ofthe SRG model space, which is subject to much stronger truncation for ramp 40Cthan the low-J part is.

Consequently, the other ramps presented in Figure 5.23 perform much better,as they imprint less truncations on the large-J SRG model space than ramp 40Cdoes. This is illustrated in the bottom plot of Figure 5.24. Among these alternativeramps, even the one corresponding to the smallest model space, ramp 38G, showsdeviations of less than 0.25 MeV per nucleon from the 40J ramp. By comparingthe 38G ramp with 40F, the effect of the ESRG cut for the small angular momentaJ ≤ 5/2 can be probed, which turns out to have only a minor effect. Therefore, itmay be concluded that for these small J the SRG model space is sufficiently largeat ESRG = 40. Similar comparisons suggest that, with respect to 40C, increasingESRG for large angular momenta is crucial. Ground-state energies correspondingto ramps 40K and 40J then only differ by at most 50 keV over the whole mass range.This indicates that ramp 40J is an appropriate choice for nuclear calculations upto mass numbers A = 150, with uncertainties related to the SRG model-space trun-cation below 100 keV per nucleon.

The use of larger SRG model spaces also has a decisive impact on the previ-ously observed E3max-dependence of the ground-state energies. Figure 5.25 com-pares the E3max dependence of ground-state energies, measured by their differenceat the two E3max values of 12 and 14, corresponding to frequency-converted matrixelements obtained from ramps 40C and 40J. The striking observation is that forthe larger SRG model space the E3max dependence does not increase, while it doesfor the smaller model space. This indicates that much of the previously observedE3max dependence was induced by the use of insufficient SRG model spaces andthat heavier nuclei may be accessible with manageable values of E3max.


Chapter 5. Results

36Ca40Ca

48Ca52Ca

54Ca48Ni

56Ni60Ni

62Ni66Ni

68Ni78Ni

88Sr90Zr

0

0.2

0.4

0.6

0.8

1

1.2

.

∆E/A

[MeV

]

: 40C, α= 0.04 fm4

: 40C, α= 0.08 fm4

: 40J, α= 0.04 fm4

: 40J, α= 0.08 fm4

Figure 5.25: Comparison of the E3max-dependence of CCSD ground-state energies for the NN+3N-

full Hamiltonian in NO2B approximation evolved in the SRG model spaces cor-

responding to ramp 40C and 40J. The ground-state energies are evaluated for

E3max = 12 and E3max = 14, and the energy difference (per nucleon) is shown in the

plot. The results clearly indicate an enhancement of the E3max-dependence due to the

insufficient model space corresponding to ramp 40C, which increases with the nuclear

mass number. Other parameters of the calculations as in Figure 5.24.



5.9.3 Results for Heavy Nuclei

The discussion of heavier nuclei will be based on SRG-transformed Hamiltoni-ans at flow parameters α = 0.04 fm4 and 0.08 fm4 for which the Coupled-Clustermethod provides nearly converged results. Examples for convergence patterns ofground-state energies of nuclei ranging from 48Ni to 132Sn are shown in Figure 5.26.Except for the heaviest nucleus 132Sn considered in Figure 5.26, all other calcula-tions are reasonably well converged within the model spaces up to emax = 12. Thecluster expansion also converges quickly. For 100Sn and the NN+3N-full Hamil-tonian at α= 0.04 fm4, for instance, the reference energy is –590.8 MeV, the CCSDcorrelation energy amounts to –375.5 MeV and the CR-CC(2,3) triples correctioncontributes –26.3 MeV, which is less than 3 % of the total binding energy. Forα = 0.08 fm4 the convergence is naturally faster, with reference energy of –767.2MeV, CCSD correlation energy of –218.8 MeV and the triples correction of –17.7MeV. These numbers suggest that even in the regime of heavy nuclei the Coupled-Cluster method is expected to provide, for a given Hamiltonian at fixed α, thecorresponding nuclear ground-state energies up to an accuracy of few percent.

In a next step it is necessary to identify the values of E3max required for calcu-lations of heavy nuclei. Figure 5.27 presents CCSD ground-state energies of vari-ous nuclei for E3max values ranging from 10 to 18. For all but the heaviest nucleus132Sn convergence is achieved at E3max = 18, and for most of the lighter nuclei al-ready at much smaller values of E3max. It is noteworthy that for the energies usingSRG-evolved matrix elements from ramp-40J model spaces the E3max dependenceis larger for the softer interaction, which is opposite to the observations made forramp 40C and lighter nuclei. This may hint at the growing importance of SRG-induced 3N interactions.

Another interesting aspect of the E3max truncation is its impact on the ħhΩ de-pendence. In Figure 5.28 the ħhΩ dependence of CCSD ground-state energies ofnuclei 40Ca, 56Ni and 132Sn is studied at various values of E3max. For 40Ca, the ħhΩdependence decreases with increasing E3max up to the point where it is practicallyflat for E3max = 16. The fact that the curves have positive slope for smaller valuesof E3max implies that for nuclei like 40Ca, too small values of E3max shift the optimaloscillator frequency to smaller frequencies, which is also confirmed by the 56Ni re-sults. For 132Sn, the situation is reversed. Here, the optimal frequency is shifted tolarger frequencies if a too small E3max is used. In fact, the E3max = 16 results suggestthat the optimal frequency for 132Sn lies even somewhat below ħhΩ = 24 MeV.

A systematic survey of CR-CC(2,3) ground-state energies of medium-mass and


Chapter 5. Results

-8

-7.5

-7

.

E/A

[MeV

]

48Ni

-10

-9.5

-9

.

E/A

[MeV

]

68Ni

6 8 10 12

emax

-10

-9.5

-9

.

E/A

[MeV

]

100Sn

6 8 10 12

emax

-10

-9

-8

.

E/A

[MeV

]

132Sn

Î

α= 0.04 fm4 α= 0.08 fm4

Figure 5.26: Convergence and ΛCCSD(T) (open symbols) and CR-CC(2,3) (full symbols) ground-

state energies of 48Ni, 68Ni, 100Sn and 132Sn for the NN+3N-full Hamiltonian in

NO2B approximation with E3max = 14 and using a Hartree-Fock basis with ħhΩ = 24

MeV. The arrows indicate the CCSD results from the emax = 12 model space.




-12

-11

-10

-9

-8

-7

.

E/A

[MeV

] 48Ca

-12

-11

-10

-9

-8

.

E/A

[MeV

] 68Ni

-12

-11

-10

-9

-8

.

E/A

[MeV

] 90Zr

-12

-11

-10

-9

-8

.

E/A

[MeV

] 120Sn

10 12 14 16 18

E3max

-12

-11

-10

-9

-8

.

E/A

[MeV

]

10 12 14 16 18

E3max

132Sn

Î

α= 0.04 fm4 α= 0.08 fm4

Figure 5.27: Convergence of CCSD ground-state energies for medium-mass and heavy nuclei

ranging from 48Ca to 132Sn, for the NN+3N-induced and NN+3N-full Hamiltonian

in NO2B approximation, with respect to E3max. All calculations are performed in a

HF basis with oscillator frequency ħhΩ = 24 MeV and at emax = 12.


Chapter 5. Results

NN+3N-full

-9.4

-9.2

-9

-8.8

.

E/A

[MeV

]

40Ca

-9.6

-9.4

-9.2

-9

.

E/A

[MeV

]

56Ni

24 28 32 36

~Ω [MeV]

-11

-10

-9

-8

.

E/A

[MeV

]

132Sn

E3max = 10 (), 12 ( ), 14 (Î), 16 (), 18 ()

Figure 5.28: Frequency dependence of CCSD ground-state energies of 40Ca, 56Ni and 132Sn for the

NN+3N-full Hamiltonian in NO2B approximation for various values of E3max. The

calculations employed a HF basis at emax = 12.



heavy nuclei ranging from 16O to 132Sn with emphasis on the Ca, Ni, and Sn iso-topic chains is presented in Figure 5.29. All results are obtained using E3max = 18

and for the two SRG flow parametersα= 0.04 fm4 and α = 0.08 fm4, which are usedto quantify the flow-parameter dependence of the results. In panel (a), the NN+3N-induced Hamiltonian is employed. The corresponding ground-state energies showa significant increase in the flow-parameter dependence, rising from about 0.1MeV per nucleon for 16O to about 1 MeV per nucleon for 132Sn, indicating grow-ing contributions of SRG-induced 4N (and multi-nucleon) interactions out of theinitial NN interaction. From the direction in which the ground-state energiesmove for smaller values of the flow parameter it can be concluded that the in-duced many-body interactions have an attractive net effect. In order to confirmthat this is a general property of chiral NN Hamiltonians, in panel (a) also theN2LO-optimized chiral NN interaction is used [159]. The flow-parameter depen-dence is very similar to the former calculations for light and medium-mass nu-clei, and is reduced for heavy nuclei, resulting in a flow-parameter dependence ofabout 0.5 MeV per nucleon for 132Sn. Even with the reduced amount of induced4N and multi-nucleon interactions for the N2LO-optimized interaction, inducedmany-body interactions out of the initial NN interaction are a new challenge in theheavy-mass regime which, if not addressed, will prevent any attempt to reliablycalculate ground-state energies of heavy nuclei based on chiral interactions. Inorder to emphasize that the observed flow-parameter dependencies are indeed tobe attributed to induced many-body interactions and not truncations in the many-body treatment such as the cluster truncation, in panel (b) the contributions of theCR-CC(2,3) triples correction are shown.

Considering the large flow-parameter dependence of the NN+3N-induced re-sults, the much reduced flow-parameter dependence of about 0.1 MeV per nu-cleon for the results shown in panel (c) using the NN+3N-full Hamiltonian withthe Λ3N = 400 MeV regular cutoff is remarkable. This reduced flow-parameterdependence has to be the consequence of a delicate cancellation of the attractiveinduced 4N contributions from the initial NN interaction with additional repul-sive 4N contributions originating from the initial 3N interaction. The directionof the flow-parameter dependence of the ground-state energies is reversed to theNN+3N-induced case, indicating that the attractive induced interactions are infact slightly overcompensated. Since for the NN+3N-induced case the contribu-tions of induced many-body interactions grow with the nuclear mass, while theNN+3N-full results exhibit a virtually constant flow-parameter dependence overthe whole considered mass range, the contributions from the repulsive induced


Chapter 5. Results

-10 -9 -8 -7 -6

NN+

3N

-ind

uced

N

3L

O

N

2L

Oo

pt

(a)

exp

-0.5

0.5

(b)

-10 -9 -8 -7

.

E/A [MeV]

NN+

3N

-full

Λ

3N=

400

MeV/c

Λ

3N=

350

MeV/c

(c)

exp

16O

24O

36C

a40C

a

48C

a52C

a

54C

a48N

i

56N

i60N

i

62N

i66N

i

68N

i78N

i

88S

r90Z

r

100S

n106S

n

108S

n114S

n

116S

n118S

n

120S

n132S

n

-0.5

0.5

(d)

Figure

5.29:G

round-state

energies

fromC

R-C

C(2,3)

for(a)

theN

N+

3N-in

duced

Ham

iltonian

starting

fromthe

N3L

Oan

dN

2LO

-

optimized

NN

interaction

and

(c)the

NN

+3N

-full

Ham

iltonian

withΛ

3N=

40

0M

eV

andΛ

3N=

35

0M

eV

.T

heboxes

representthe

spreadofthe

results

fromα=

0.0

4fm

4toα=

0.0

8fm

4,and

thetip

points

into

thedirection

ofsmaller

values

ofα.A

lsoshow

nare

thecon

tribution

softhe

CR

-CC

(2,3)triples

correctionto

the(b)

NN

+3N

-indu

cedan

d(d)

NN

+3N

-full

results.

All

results

employħhΩ=

24

Me

Van

d3N

interaction

sw

ithE

3m

ax=

18

inN

O2B

approximation

and

full

inclu

sion

ofthe

3Nin

teractionin

CC

SD

up

toE

3m

ax=

12.

Experim

ental

bindin

gen

ergies[160]

areshow

nas

blackbars.



16O 40Ca 48Ca 60Ni 62Ni 88Sr 114Sn 116Sn 118Sn 120Sn2

3

4

5

.

Rch

[fm

]


Figure 5.30: Charge radii obtained from Hartree-Fock calculations for the NN+3N-induced and

NN+3N-full Hamiltonian with parameters as in Figure 5.29. Experimental val-

ues [161] are shown as black bars.

many-body interaction grow the same way with the the nuclear masses as the con-tributions from the attractive induced many-body interactions do. This statementcan be confirmed by reducing the initial 3N cutoff from Λ3N = 400 MeV to 350 MeV.For light nuclei this cutoff reduction is known to weaken the repulsive 4N com-ponent originating from the initial 3N interaction [86]. Consequently, the flow-parameter dependence of the ground-state energies for the 350 MeV NN+3N-fullHamiltonian points in the same direction as for the NN+3N-induced Hamilto-nian, because the repulsive contributions are no longer able to overcompensatethe attractive contributions. Furthermore, the 350 MeV NN+3N-full Hamiltonianresults exhibit a flow-parameter dependence that increases with mass number.This shows that in addition to the fact that the repulsive contributions are notonly smaller for the Λ3N = 350 MeV NN+3N-full Hamiltonian compared to the400 MeV case, but that they also grow slower with the nuclear masses. Therefore,an almost constant flow-parameter dependence of the NN+3N-full results overthe whole mass range is not a general property of NN+3N-full Hamiltonians andmay only be achieved in a small window of initial 3N regular momentum cutoffsaround Λ3N = 400 MeV.

Taking advantage of the robust cancellation of SRG-induced 4N terms for theNN+3N-full Hamiltonian with Λ3N = 400 MeV, resulting in a very small α depen-dence, the obtained ground-state energies may be compared to experiment. Theground-state energies of the oxygen isotopes are very well reproduced. Startingfrom the Ca isotopes, a systematic and slowly increasing deviation of the theoret-


Chapter 5. Results

ical results from experiment is visible, which is of the order of 1 MeV per nucleon.Apart from this constant energy shift, the experimental trend of the binding en-ergies is nicely reproduced. Therefore, these results represent a first confirmationthat chiral Hamiltonians that have been determined in the few-body sector arealso capable of qualitatively describe heavy nuclei. Furthermore, since the many-body problem can be solved very accurately using ab initio methods, potentialshortcomings of the input Hamiltonian may be identified. In the present case,the overbinding of the NN+3N-full results are well beyond theoretical uncertain-ties and, thus, may be considered a deficiency of the current chiral NN+3N-fullHamiltonian. This outcome is of course not surprising given the inconsistent chi-ral perturbation order of the employed NN and 3N interaction and the neglect ofthe chiral 4N force at N3LO.

While chiral Hamiltonians provide good results for binding energies with de-viations from experiment that might be resolved using improved chiral Hamil-tonians, charge radii come out significantly too small. Figure 5.30 shows chargeradii on the HF level for nuclei for which the experimental value is known [161].An increasing deviation from experiment is apparent, about 0.3 fm or 10 % for16O, up to about 1 fm or 20 % for 132Sn. It should be noted that the radius oper-ator has not been SRG evolved. However, neither the consistent evolution of theradius nor beyond-HF correlation effects are expected to have a significant impacton these findings.


Chapter 6

Conclusion

Chapter 6. Conclusion

The two foundations of ab initio nuclear structure theory – the fundamental de-scription of the nuclear interaction, and methods that provide accurate solutionsto the many-body problem along with some estimate of the errors involved – haveboth seen impressive progress in recent years. The goal of the present work wasto combine and advance these developments into a theoretical and computationalframework capable of performing accurate ab initio nuclear structure calculationsof medium-mass and heavy nuclei.

The main focus of this work is the treatment of nucleon correlations in themany-body problem. Here, the Coupled-Cluster method with iterative single-and double-excitations contributions combined with a non-iterative treatment oftriply excited clusters constitutes a framework that is both, efficient and suffi-ciently accurate in solving the many-body Schrödinger equation in order to makeacceptable estimates for nuclear ground-state energies. The spherical formula-tion of Coupled-Cluster theory was one key element to the success of this work.However, while Coupled-Cluster theory in its m -scheme formulation is alreadynotorious for the human effort required to derive and implement the method, thespherical formulation multiplies this demand. Nevertheless, even under thesecircumstances, spherical Coupled-Cluster theory has highly appealing propertiesthat make implementing the method worthwhile in the long run. For one, it ex-tends the reach of ab initio Coupled-Cluster calculations to large mass numbersway beyond the reach of an m -scheme formulation. Furthermore, the sphericalscheme is also sufficiently efficient to allow calculations including explicit 3N in-teractions which makes the Coupled-Cluster method a favorable tool to test ap-proximation schemes for 3N interactions, such as the normal-ordering approxi-mation. Through the inclusion of triples excitation effects in form of a posteriori

non-iterative energy corrections, the degree of convergence of the cluster expan-sion at the level of the triply excited clusters can be estimated. Using SRG-softenedinteractions, the results suggest fast convergence of the cluster expansion. How-ever, for harder interactions or even the bare nuclear interaction, the quality oftriples-corrected CCSD ground-state energies is questionable. Since full CCSDTcalculations cannot be done at present time, two different triples-correction meth-ods have been considered in order to estimate their quality. The respective resultslie sufficiently close, motivating the claim that both represent reasonable approx-imations to the exact triples contributions to the energy.

Three-nucleon forces play a central role in present-time nuclear structure the-ory and applications. Chiral three-nucleon forces are known to be important forthe description of nuclear properties, and the framework of chiral effective field


theory provides nuclear physicists with a convenient approach to the construc-tion of QCD-based nuclear interactions and electro-weak currents, due to its abil-ity to assign power-counting orders to the many possible operator structures ofthe interactions, which in turn allows to identify the most relevant ones of theseoperator structures. This is a particularly important feature of the chiral pertur-bation approach for the derivation of three- and more-nucleon interactions, be-cause their treatment is much more complicated in a purely phenomenologicalapproach. The generation of chiral interactions still is a dynamic field, even moretoday after the great potential of chiral interactions has been realized in nuclearmany-body and reaction calculations. The most obvious improvement over thecurrent status of chiral interactions concerns the availability of the 3N interactionat N3LO in order to achieve consistency between the NN and 3N interactions in thechiral expansion. This will further be a next step towards assessing convergence ofnuclear observables with respect to the chiral expansion parameter which, how-ever, would in principle also require the consideration of the chiral 4N interactionat this order. The inclusion of the ∆ degree of freedom, which also causes a shiftof certain diagrams to lower power-counting orders, is another exciting prospectfor the future.

But even without chiral three-nucleon interactions, nuclear-structure calcula-tions that rely on SRG-evolved or otherwise renormalized interactions, are in-evitably confronted with induced three-nucleon forces. Regardless their origin,the inclusion of three-nucleon interactions into the many-body calculations posesa challenge which has to be met with caution in order to avoid the introductionof severe sources of uncertainty, and lots of research has been dedicated to thisissue. This work demonstrates that a proper inclusion of three-nucleon interac-tions is possible for many-body calculations operating up to nuclear masses ofabout A ≈ 150, using manageably large SRG model spaces and reachable valuesof E3max. It was shown that an approximate treatment of 3N interactions in themany-body calculations via the normal-ordering two-body approximation is sat-isfactory at the level of overall accuracy targeted at in ab initio nuclear structurecalculations. A remarkable outcome of this work is that a single Hamiltonian,more precisely the SRG-evolved NN+3N-full Hamiltonian with reduced cutoff-momentum of 400 MeV provides, within the computational framework used inthis work, a qualitative description of nuclei ranging from 16O to 132Sn, and maybeeven beyond. Given the still quite preliminary status of the chiral interactions, al-ready these results are sufficiently encouraging to conclude that it is worthwhileto proceed further along this path of research.


Chapter 6. Conclusion

In conclusion, ab initio nuclear structure theory is a vivid branch of physicsand the results of this work indicate great potential of the first-principle descrip-tion of the nuclear many-body problem. Future research will have to address fur-ther observables besides (ground-state) energies. For example, although energiesare reasonably reproduced by the chiral interactions, radii come out too small.However, when using SRG-transformed Hamiltonians, the observables have toundergo analogous transformations, and electromagnetic observables require theinclusion of chiral electro-weak currents into the calculations. As already men-tioned above, the chiral 3N interaction at N3LO will be a next step towards thedescription of nuclear properties using interactions at consistent power countingorders, giving insight in the convergence properties of the chiral expansion andmay enable uncertainty estimates. In the long run, nuclear theory will have to dealwith 4N interactions, either in form of the initial chiral 4N interaction at N3LO, or,unless novel generators are found that reduce the amount of SRG-induced 4Ncontributions, with these induced 4N interactions.


Appendix A

Excited Nuclear States

Appendix A. Excited Nuclear States

The spherical EOM-CCSD method considered in this work enables calculationsof excited states of closed-shell nuclei. The low-energy spectra of such nucleitypically consist of simple particle-hole excitations and collective rotational or vi-brational states. The current method of choice for collective states has been theRandom Phase Approximation (RPA) [78], in which excited states are described vialinear 1p1h (de-) excitations on a ground state which contains certain correlations.This approach has also been extended to include 2p2h (de-) excitations, known assecond RPA [162,163]. Since the quasi-Boson approximation that enters the deriva-tion of the RPA equations works better for collective states than for single-nucleonexcitations, RPA methods have difficulties to describe the latter type of excitednuclear states. The EOM-CCSD approach also employs linear excitations up tothe 2p2h level. However, since these excitations act on the fully correlated CCSDground state, EOM-CCSD is considered superior to the RPA approaches and may,therefore, be able to describe single-nucleon excitations as well as collective states.

Typically, those excited states are rather high-lying, beyond the neutron sepa-ration threshold, and no longer bound. These states are, therefore, not expected tobe highly accurate reproduced in calculations in which the continuum is not prop-erly taken into account. Additionally, some of the low-lying 0+ states in closed-shell nuclei, such as 16O, are suspected to haveα-cluster structure [164] and would,consequently, require 4p4h excitations for an accurate description, which is be-yond the scope of EOM-CCSD.

This section focuses mainly on the computational aspects of EOM-CCSD cal-culations rather than a thorough physical discussion, although it will address thequestion raised above as to whether EOM-CCSD is able to describe single-nucleonexcitations as well as collectivity within the same framework. The EOM-CCSDeigenvalue problem constitutes a non-Hermitean eigenvalue problem, which canreliably be solved using non-symmetric Lanczos methods provided by the ArnoldiPackage [165]. Since the matrix-vector multiplications can be distributed via MPI,these multiplications can be evaluated quickly using multiple computing nodes.This way, matrices of linear dimensions of hundreds of millions can be diagonal-ized. However, for large linear dimensions the orthogonalizations of the Lanczosvectors, which are performed on a single computing node only, does in practicalcalculations eventually spoil the scaling.

When calculating excited states it is important to take spurious center-of-massexcitations into account. While ground states are approximately free from center-of-mass contaminations [152], these contaminations cannot be ignored for the ex-

226 Coupled Cluster Theory for Nuclear Structure


cited states. One way to probe the degree of center-of-mass contamination is touse the Hamiltonian (1.3),

H =1

A

A∑

i<j

(p i − p j )2

2m+

A∑

i<j

V NNi j

+

A∑

i<j<k

V 3Ni j k+ λCM HCM , (A.1)

augmented by the center-of-mass Hamiltonian λCM HCM with

HCM =1

2 A mP

2

CM+

1

2(A m ΩCM)

2 R CM −3

2ħhΩCM , (A.2)

and to study the dependence of the eigenvalues on the parameter λCM. In the exactcase the nuclear wavefunction factorizes into an intrinsic and a center-of-masspart, and the intrinsic energies are clearly independent of λCM. Furthermore, forλCM 6= 0 the center of mass will then be in a HCM eigenstate |n⟩ with eigenenergy

ECM,n = λCM n ħhΩCM (A.3)

that scales linearly with the parameter λCM. At large enough values of λCM onlythe center-of-mass ground states |0⟩ will be visible at the lower end of the energyspectrum. In the following calculations the oscillator frequency ΩCM of the center-of-mass potential is chosen to coincide with the HO basis frequency Ω, but otherchoices are possible as well [152].

Figure A.1 shows the λCM-dependence of CCSD ground-state energies of 16,24Oand 40,48Ca for the chiral NN-only and NN+3N-full Hamiltonian, at two values ofthe SRG flow parameter. The energies change only very little when λCM is variedfrom 0.0 to 1.0, about 250 keV for 16O and about 500 keV for 40Ca. Excited states,however, exhibit a much stronger dependence onλCM, as can be seen in Figures A.2and A.3, where the lowest 10 J π = 0+ and J π = 2+ states of 16O and 40Ca are shown.In Figures A.2 and A.3 it is clearly visible how spurious excited states are linearlyshifted upwards linearly with λCM. Typically at values of λCM around 1.0 the low-energy spectra have become stable with respect the λCM variations.

Another important information is the rate of convergence of the EOM-CCSDcalculations with respect to model space size. The convergence of the 3 lowest J π =

0+,2+, and 4+ states of 16,24O and 40,48Ca is considered in Figures A.4 and A.5. Thecalculations used λCM = 1.0. Regarding convergence, the results are encouraging.Most states, and in particular the low-lying ones, are converged already in smallerCoupled-Cluster model spaces. This allows a clear identification of high- and low-lying states.

Coupled Cluster Theory for Nuclear Structure 227


16O, NN-only

-154.75

-154.50

-154.25

0.0 0.2 0.4 0.6 0.8 1.0

λCM

-169.25

-169.00

-168.75. E

[MeV

]40Ca, NN-only

-540.5

-540.0

-539.5

0.0 0.2 0.4 0.6 0.8 1.0

λCM

-619.5

-619.0

-618.5. E

[MeV

]

16O, NN+3N-full

-127.00

-126.75

-126.50

0.0 0.2 0.4 0.6 0.8 1.0

λCM

-128.50

-128.25

-128.00. E

[MeV

]

40Ca, NN+3N-full

-369.5

-369.0

-368.5

0.0 0.2 0.4 0.6 0.8 1.0

λCM

-372.0

-371.5

-371.0. E

[MeV

]

Î

α= 0.04 fm4 α= 0.08 fm4

Figure A.1: Dependence of 16O and 40Ca CCSD ground-state energies on λCM using a HO basis

with emax = 10, and at ħhΩ = 20 MeV.

Figures A.2 and A.3 show that beyond some threshold energy, EOM-CCSDyields a bulk of excitation energies that lie closely together. These spectra stronglyresemble the results for collective excitations obtained from RPA. Apart from that,EOM-CCSD is also capable of providing some low-lying excitations, as is appar-ent from Figures A.4 and A.5. For 24O, two low-lying 2+ states are obtained, thelowest at an excitation energy of 6.5 MeV. This might be a candidate for the ex-perimentally known 2+ state at 4.8 MeV (experimental values from [166]). Thecalculations also show a low-lying 4+ state at about 9.4 MeV, for which there is noexperimental evidence. For 48Ca, there are also low-lying 2+ and 4+ states. Thelowest 2+ state is at 5.0 MeV excitation energy, which is a candidate for the exper-imentally observed state at 3.8 MeV. For the calculated 4+ state at 5.6 MeV thereis also a possible match in the observed 48Ca excitation spectrum at 4.5 MeV. Nolow-lying 0+, 2+, or 4+ states are obtained for 16O and 40Ca. However, the 0+ and



16O

0

10

20

30

40

50

.

Ex

[MeV

]

0+

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

λCM

0

10

20

30

40

.

Ex

[MeV

]

2+

Figure A.2: Dependence of the 10 lowest J π = 0+ and J π = 2+ states of 16O on the parameter λCM.

The energies are obtained from EOM-CCSD using the NN+3N-full Hamiltonian in

NO2B approximation with E3max = 14 and atα= 0.08 fm4. The calculations employed

a HO basis with emax = 10 and ħhΩ = 20 MeV.



40Ca

0

5

10

15

20

25

30

.

Ex

[MeV

]

0+

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

λCM

0

5

10

15

20

25

.

Ex

[MeV

]

2+

Figure A.3: As in Figure A.2, for 40Ca.



2+ spectra are less converged than the low-lying states for other nuclei, indicatingthat the 16O states are rather complicated. Experiments show low-lying 0+ statesat 6.1 and 3.3 MeV for 16O and 40Ca, respectively, but these are potentially α-clusterstates [164] and, therefore, expected to be out of reach of the EOM-CCSD. Thereare also low-lying 2+ states at 6.9 and 3.9 MeV for 16O and 40Ca, respectively. Sincethere is also no evidence for these in the calculations, it may be speculated theyalso have a complicated structure, such as multi-particle-hole excitations, whichEOM-CCSD is simply not able to describe with linear 2p2h excitations.



16O 24O

0

10

20

30

40

.

Ex

[MeV

]

0+

0

10

20

30

.

Ex

[MeV

]

2+

4 6 8 10

emax

0

10

20

30

.

Ex

[MeV

]

4+

4 6 8 10

emax

Figure A.4: Convergence of the 3 lowest J π = 0+ , J π = 2+, and J π = 4+ with respect to emax.

Other parameters as in Figure A.2, with λCM = 1.0.



40Ca 48Ca

0

10

20

30

40

.

Ex

[MeV

]

0+

0

10

20

30

40

.

Ex

[MeV

]

2+

4 6 8 10

emax

0

10

20

30

40

.

Ex

[MeV

]

4+

4 6 8 10

emax

Figure A.5: As in Figure A.5, for 40Ca and 48Ca.


Appendix B

Trapped Neutrons

Appendix B. Trapped Neutrons

As an application of Coupled-Cluster theory beyond common atomic nuclei, theground-state energies of neutrons trapped in an external potential are consideredin the following. Since pure neutron systems are not bound, the external potentialis required to prevent the neutrons from moving apart. Thus, the Hamiltonian isof the form

H =

A∑

i

p 2i

2m+

A∑

i

Uext(ri ) +

A∑

i<j

V NNi j

+

A∑

i<j<k

V 3Ni j k

(B.1)

where the external potential considered is a harmonic-oscillator potential withfrequency Ωtrap

Uext(ri ) =1

2m Ω2

trapr 2

i. (B.2)

Unlike for the nuclear case, in (B.1) the total, rather than the intrinsic kinetic energyis used because the external potential also acts on the center-of-mass coordinates.One motivation for considering neutron drops is that they provide a very simplemodel of neutron-rich nuclei in which the core is approximated by an externalwell acting on the valence neutrons [167]. Additionally, it allows to investigateproperties of the interaction in the neutron-neutron or three-neutron sector.

First, basic properties of the many-body calculations are considered. All cal-culations employ SRG-evolved matrix elements from ramp-40C model spaces,and with applied frequency conversion from parent frequency ħhΩ = 28 MeV. Fig-ures B.1 and B.2 show for various neutron drops and interactions, and at fixedtrap potential frequency ħhΩtrap = 10 MeV, the dependence of the CCSD ground-state energy on the harmonic-oscillator frequency. For the NN-only, the NN+3N-induced, and the NN+3N-full (Λ3N = 400 MeV) Hamiltonians, the CCSD energiesfor SRG flow parameters α= 0.04 fm4 and α= 0.08 fm4 are sufficiently flat in the fre-quency range ħhΩ = 16, 20, 24 MeV so that any of these frequencies may be chosenas optimum. The results for α= 0.02 fm4, where available, suggest an optimal fre-quency at around ħhΩ = 28 MeV, however, for reason discussed below, an optimalfrequency of ħhΩ= 16 MeV is more convenient and will be chosen in the following.

The results for the NN+3N-full (Λ3N = 500 MeV) Hamiltonian show an odd fre-quency dependence, where for the heavier neutron drops the CCSD energies keepon decreasing with increasing frequency, which clearly hints at a defect in the in-teraction matrix elements. Indeed, this unnatural behavior may be explained byan insufficient E3max cut. Figure B.3 shows the frequency dependence of the CCSDenergies for the heavier neutron drops for the two values E3max = 12 and 14. From



NN-only NN+3N-inducedNN+3N-full NN+3N-full

Λ3N = 500 MeV Λ3N = 400 MeV

133

134

135

136

.

E[M

eV]

8 neutrons

325

330

335

340

.

E[M

eV]

16 neutrons

16 20 24 28 32

~Ω [MeV]

420

430

440

.

E[M

eV]

20 neutrons

16 20 24 28 32

~Ω [MeV]

16 20 24 28 32

~Ω [MeV]

16 20 24 28 32

~Ω [MeV]

Î

α= 0.02 fm4 α= 0.04 fm4 α= 0.08 fm4

Figure B.1: Frequency-dependence of CCSD ground-state energies for various Hamiltonians and

various neutron drops and interactions in an external harmonic-oscillator potential

with frequency ħhΩtrap = 10 MeV. The 3N interactions are included via NO2B approx-

imation with E3max = 14, and a HF basis with emax = 12 is employed.




Λ3N = 500 MeV Λ3N = 400 MeV

660

680

700

.

E[M

eV]

28 neutrons

1000

1050

1100

.

E[M

eV]

40 neutrons

16 20 24 28 32

~Ω [MeV]

1400

1500

.

E[M

eV]

50 neutrons

16 20 24 28 32

~Ω [MeV]

16 20 24 28 32

~Ω [MeV]

16 20 24 28 32

~Ω [MeV]

Î

α= 0.02 fm4 α= 0.04 fm4 α= 0.08 fm4

Figure B.2: As in Figure B.1 for the neutron drops 28n, 40n, and 50n.



NN+3N-fullΛ3N = 500 MeV

Eref E(CCSD)

680

690

700

.

E[M

eV

]

28 neutrons

1025

1050

1075

1100

.

E[M

eV

]

40 neutrons

16 20 24 28 32

~Ω [MeV]

1400

1450

1500

.

E[M

eV

]

16 20 24 28 32

~Ω [MeV]

50 neutrons

Î

α= 0.04 fm4 α= 0.08 fm4

Figure B.3: Dependence on the HO frequency of the CCSD ground-state energy for various neu-

tron drops for the NN+3N-full Hamiltonian with chiral 3N cutoff Λ3N = 500 MeV for

E3max = 12 (open symbols, dashed lines) and E3max = 14 (full symbols, solid lines).

Parameters of the calculations as in Figure B.1.



16n α = 0.04 fm4 α = 0.08 fm4 50n α = 0.04 fm4 α = 0.08 fm4

Eref [MeV] 340.8 339.1 1467.0 1459.8∆E (CCSD) [MeV] – 10.2 – 8.0 – 28.1 – 17.3δE (ΛCCSD(T)) [MeV] – 0.7 – 0.5 – 1.3 – 1.0

Table B.1: Contributions from different orders of the cluster truncation to the ground-state energies

of 16n and 50n for the NN+3N-full (Λ3N = 400 MeV) Hamiltonian. Parameters of the

calculations as in Fig. B.4, with emax = 12.

this it becomes apparent that the problematic frequency dependence is enhancedfor smaller values of E3max and may vanish for sufficient large values. However, forthe frequency ħhΩ= 16 MeV chosen as optimum, there is virtually no E3max effect onthe energy scales considered in Figure B.3, while the effect increases with the oscil-lator frequency. From these observations it may be concluded that is ħhΩ= 16 MeV

is the most appropriate choice for the NN+3N-full (Λ3N = 500 MeV) Hamiltonian.As in the case of nuclei, the insufficiency of the ramp-40C SRG model spaces forheavier neutron drops is expected to cause the problems mentioned above.

Next, the convergence properties of the calculations and the size of the differ-ent contributions of the cluster expansion are discussed. In Figure B.4, the emax-dependence of the reference energy, as well as the CCSD and ΛCCSD(T) ground-state energy is depicted. The α = 0.04 fm4 and 0.08 fm4 results are well converged.This is not quite the case for α = 0.02 fm4 results, showing a more linear, ratherthan exponential, convergence pattern which would also not allow for reliable ex-trapolations to infinite model-space sizes. Therefore, the α = 0.02 fm4 results willnot be considered in the following.

In Table B.1 the contributions from different orders of the cluster expansionto the ground-state energies of 16n and 50n for the NN+3N-full (Λ3N = 400 MeV)Hamiltonian are listed. These numbers show that the beyond-HF contributionsare very small. For the α = 0.04 fm4 Hamiltonian, the CCSD correlation energycontributes only 3.1 % to the ground-state energy of 16n, and 2.0 % to the ground-state energy of 50n. The ΛCCSD(T) triples correction is practically negligible, con-tributing about 0.2 % and 0.1 % to the ground-state energy of 16n and 50n, respec-tively. This, not surprisingly, indicates that the most part of the binding energystems from the external potential while the neutron interact only weakly withinthe trap.

The E3max cut does not pose a problem in most of the present calculation of




Λ3N = 500 MeV Λ3N = 400 MeV

320

330

340

350

.

E[M

eV]

16 neutrons

660

680

700

720

.

E[M

eV]

28 neutrons

4 6 8 10 12emax

1400

1500

1600

.

E[M

eV]

50 neutrons

4 6 8 10 12emax

4 6 8 10 12emax

4 6 8 10 12emax

Î

α= 0.02 fm4 α= 0.04 fm4 α= 0.08 fm4

Figure B.4: Reference (dashed line), CCSD (dotted line) and ΛCCSD(T) (solid line) ground-state

energy for various neutron drops and interactions in an external harmonic-oscillator

potential with frequency ħhΩtrap = 10 MeV. Parameters of the calculations as in Fig-

ure B.1, with HO basis frequency ħhΩ = 16 MeV.



αNN+3N-ind.

NN+3N-full NN+3N-full[fm4] Λ3N = 500 MeV Λ3N = 400 MeV

8n0.04 0.00 0.00 0.000.08 0.00 0.00 0.00

20n0.04 0.07 –0.11 0.000.08 0.05 –0.03 0.03

40n0.04 0.82 –0.58 0.270.08 0.79 –0.08 0.50

50n0.04 3.96 0.53 4.130.08 4.01 1.25 4.47

Table B.2: The E3max effect in the ΛCCSD(T) ground-state energies, measured by (B.3), for various

neutron drops and obtained from emax = 12 modes spaces. Other parameters of the

calculations as in Fig. B.4.

neutron drops at HO basis frequency ħhΩ = 16 MeV. This can be seen in Table B.2where the differences in the ΛCCSD(T) ground-state energies

E (ΛCCSD(T))(E3max = 14)− E (ΛCCSD(T))(E3max = 12) (B.3)

are listed. In the calculations of the lighter neutron drops the E3max effect is com-pletely negligible. For 40n, the absolute E3max effect ranges from 0.3 MeV to about0.8 MeV, which is still small regarding the large energy scales involved. For 50n,the E3max effect rises dramatically. This is most likely a signature of a problem inthe input matrix elements, and it may well be caused by the 40C ramp used forthe SRG evolution which also led to the observations made in Figure B.3. In con-clusion, apart from 50n, the Coupled-Cluster results are expected to be accuratewith uncertainties well below 1 %. The full set of ΛCCSD(T) ground-state energiesis presented in Table B.3.

It should be noted that in the calculations the external potential is not SRG-evolved. However, the SRG is not expected to have a large effect there becausethe SRG alters mainly the short-range behavior while the external potential haslong-range character. Apart from the NN-only results, the ground-state energiesshow very little flow-parameter dependence, even for the NN+3N-full Hamilto-nian with Λ3N = 500 MeV. This outcome may indicate that the (T , MT ) = (3/2, 3/2)

isospin channels of the chiral NN and 3N interactions are not the driving forcebehind SRG-induced 4N (and beyond) contributions. Therefore, pure neutronsystems give the opportunity to compare the results for Λ3N = 500 MeV NN+3N-full Hamiltonian to the Λ3N = 400 MeV variant. As can be seen in Figure B.4, the



αNN-only NN+3N-ind.

NN+3N-full NN+3N-full[fm4] Λ3N = 500 MeV Λ3N = 400 MeV

8n0.04 133.6 134.2 135.1 135.50.08 133.5 134.4 135.3 135.6

16n0.04 326.7 330.0 332.0 335.10.08 326.0 330.5 332.5 335.6

20n0.04 421.8 427.1 432.8 436.50.08 420.3 427.9 433.3 437.2

28n0.04 662.8 674.7 680.7 690.80.08 659.4 676.0 681.3 691.8

40n0.04 1013.2 1042.6 1058.3 1080.20.08 1005.3 1045.1 1058.3 1081.7

50n0.04 1365.9 1437.6 1449.1 1492.30.08 1354.2 1441.5 1448.1 1494.2

Table B.3: Ground-state energies from ΛCCSD(T) in MeV for various neutron drops in a

harmonic-oscillator trap with ħhΩtrap = 10 MeV. The calculations used frequency-

converted matrix elements with ħhΩ = 16 MeV, obtained from the parent frequency

ħhΩ = 28 MeV. The 3N interactions were included via NO2B approximation with

E3max = 14. The calculations were performed in a HF basis with emax = 12.

ground-state energies change noticeably when the cutoff-momentum is loweredfrom 500 MeV to 400 MeV. The initial 3N contributions are in any case repulsive,as is apparent from the comparison of the NN+3N-full energies with the NN+3N-induced results. However, lowering the cutoff in the initial chiral 3N interactionleads to an enhancement of these repulsive contributions, which is also found togrow with the neutron number, as can be seen in Table B.3.

Finally, the Coupled-Cluster results using chiral interaction are compared toGreen’s function Monte Carlo (GFMC) calculations [168] using the Argonne V8′

(AV8′) potential [169], either alone or in combination with the Urbana IX (UIX) [169]or the Illinois-7 (IL7) [170] three-nucleon interaction. The comparison is shown inFigure B.5. For 8n, all methods and interactions give the same result on the en-ergy scales considered in Figure B.5. However, for larger neutron numbers theresults increasingly deviate from each other. Up to 40n, both Hamiltonians that donot contain initial 3N forces, the chiral NN+3N-induced and the AV8′, give verysimilar results. Even closer to the AV8′ results is the NN+3N-full Hamiltonianwith 500 MeV cutoff momentum. As already discussed above, the NN+3N-full



Hamiltonian with 400 MeV cutoff shows noticeable deviations from the 500 MeVresults, due to an enhanced repulsion. Even more repulsion is produced by theUIX three-body interaction, such that for 40n, the AV8′+UIX results are clearly dis-tinguishable from the other ones. For example, the AV8′+UIX ground-state energyfor 40n differs to the NN+3N-full (Λ3N = 500 MeV) result by about 1.5 MeV per neu-tron, and to the NN+3N-full (Λ3N = 400 MeV) by about 0.9 MeV per neutron. Onthe other hand, the AV8′ potential in combination with the IL7 three-nucleon in-teraction produces significantly less repulsion than the other interactions. Thisdemonstrates once more that different current 3N interactions lead to very differ-ent results in many-body calculations.

In conclusion, neutron drops represent convenient systems to test the extreme-isospin properties of nuclear interactions. The Coupled-Cluster framework con-sidered here is capable to provide very accurate energies, and the effects of SRG-induced many-body interactions are very limited. Therefore, the uncertainties in-volved are much reduced compared to nuclear calculations in such a mass range.Particularly for heavier neutron drops, differences in the interactions become visi-ble. This may help to understand the behavior of these interactions in the calcula-tion of neutron-rich nuclei. A particularly interesting observation in the context ofthis work are the different results for the NN+3N-full Hamiltonians for differentregulator cutoff momentum.



8n 16n 20n 28n 40n

15

20

25

30

.

E/A

[MeV

]

Î

ΛCCSD(T) ΛCCSD(T) ΛCCSD(T) GFMC GFMC GFMC

NN+3N-ind.NN+3N-full NN+3N-full

AV8′AV8′ AV8′

Λ3N = 500 MeV Λ3N = 400 MeV + Urbana IX + Illinois-7

Figure B.5: Comparison of ground-state energies per neutron of neutron drops obtained from

ΛCCSD(T) and Green’s function Monte Carlo calculations employing various inter-

actions. The Coupled-Cluster calculations used the chiral NN as well as two variants

of the chiral NN+3N interactions in form of the NN+3N-induced, the NN+3N-full

(Λ3N = 500 MeV), and the NN+3N-full (Λ3N = 400 MeV) Hamiltonian. The GFMC

calculations used either the AV8′ two-nucleon potential, or the AV8′ potential together

with the Urbana IX model 3N interaction [168]. The Coupled-Cluster results can be

found in Table B.3.


Appendix C

CCSD Diagrams and SphericalExpressions

Appendix C. CCSD Diagrams and Spherical Expressions

C.1 Diagrams

b b b l b b

EA EB EC

b l b l b b b b b l

SA SBa SBb SBc SCa

b l b b b b b b

SCb SCc SDa SDb

b b b l b b b b b b

SDc SEa SEb SEc SF

Figure C.1: CCSD correlation energy and T1 diagrams.



b b b l b l b b

DA DBa DBb DBc

b b b b b b b b

DBd DBe DCa DCb

b b b b b b b b

DCc DCd DDa DDb

l b l b b b b b

DEa DEb DEc DEd

b b b b b b b b

DEe DEf DEg DEh

b b b b b b b b

DFa DFb DFc DGa

b b b b b b b b

DGb DGc DGd DGe

b b b b b b

DHa DHb DI

Figure C.2: CCSD T2 diagrams.



C.2 Spherical Equations

(SA)

+ ⟨a | f |i ⟩

00

(SB a )

+ −1a

∑

c k

−1c

∑

J

J 2 ⟨a c |t2|i k ⟩J M J M

⟨k | f |c ⟩

00

(SBb )

− 12−1

a

∑

c d k

∑

J

J 2 ⟨a k |v |c d ⟩

J M J M

⟨c d |t2|i k ⟩

J M J M

(SBc )

+ 12−1

a

∑

c k l

∑

J

J 2 ⟨a c |t2|k l ⟩J M J M

⟨k l |v |i c ⟩

J M J M

(SC a )

− −1a

∑

c

⟨a | f |c ⟩

00

⟨c |t1|i ⟩

00

(SC b )

+ −1a

∑

k

⟨a |t1|k ⟩

00

⟨k | f |i ⟩

00

(SC c )

+ −1a

∑

c k

−1c

∑

J

J 2 ⟨a k |v |i c ⟩

J M J M

⟨c |t1|k ⟩

00

(SDa )

− 12−2

a

∑

c d k l

∑

J

J 2 ⟨a d |t2|k l ⟩

J M J M

⟨k l |v |c d ⟩

J M J M

⟨c |t1|i ⟩

00

(SDb )

− 12−2

a

∑

c d k l

∑

J

J 2 ⟨a |t1|k ⟩

00

⟨k l |v |c d ⟩

J M J M

⟨c d |t2|i l ⟩

J M J M

(SDc )

+ −1a

∑

c d k l

−2d−1

k

∑

J J ′J 2 ( J ′)2 ⟨c |t1|k ⟩

00

⟨k l |v |c d ⟩

J M J M

⟨d a |t2|l i ⟩

J ′M ′ J ′M ′

δjd j l

(SE a )

− −2a

∑

c k

⟨a |t1|k ⟩

00

⟨k | f |c ⟩

00

⟨c |t1|i ⟩

00

(SEb )

− −2a

∑

c d k

−1k

∑

J

J 2 ⟨a k |v |c d ⟩

J M J M

⟨c |t1|i ⟩

00

⟨d |t1|k ⟩

00

(SE c )

+ −2a

∑

c k l

−1c

∑

J

J 2 ⟨k l |v |i c ⟩

J M J M

⟨a |t1|k ⟩

00

⟨c |t1|l ⟩

00

(SF )

− −3a

∑

c d k l

−1d

∑

J

J 2 ⟨k l |v |c d ⟩

J M J M

⟨a |t1|k ⟩

00

⟨d |t1|l ⟩

00

⟨c |t1|i ⟩

00

= 0 , ∀ a , i

Figure C.3: Spherical expressions for the CCSD T1 amplitude equations.



Pab (J ) Pi j (J )

¨(DA)

+ 14⟨ab |v |i j ⟩

J M J M

(D B a )

− 12−1

b

∑

c

⟨b | f |c ⟩

00


(D Bb )

+ 12−1

j

∑

k

⟨k | f |j ⟩

00

⟨ab |t2|i k ⟩

J M J M

(D Bc )

+ 18

∑

c d

⟨ab |v |c d ⟩

J M J M

⟨c d |t2|i j ⟩

J M J M

(D Bd )

+ 18

∑

k l

⟨ab |t2|k l ⟩

J M J M

⟨k l |v |i j ⟩

J M J M«

(D B e )

− CCAtoStd(A)

ab

i jJ ; J ′

∑

c k

(−1)jc+jk−J ′ ⟨a c |t2|i k ⟩

J ′M ′

J ′M ′

⟨k b |v |c j ⟩

J ′M ′

J ′M ′

(DC a )

+ 116

Pab (J ) Pi j (J )∑

c d k l

⟨ab |t2|k l ⟩

J M J M

⟨k l |v |c d ⟩

J M J M

⟨c d |t2|i j ⟩

J M J M

(DC b )

+ CCAtoStd(A)

ab

i jJ ; J ′

12

∑

c d k l

(−1)jc+jd+jk+j l ⟨a c |t2|i k ⟩

J ′M ′

J ′M ′

⟨d b |t2|l j ⟩

J ′M ′

J ′M ′

⟨k l |v |c d ⟩

J ′M ′

J ′M ′

+ Pab (J ) Pi j (J )

¨(DC c )

− 14−2

i

∑

c d k l

∑

J ′( J ′)2 ⟨ab |t2|l j ⟩

J M J M

⟨k l |v |c d ⟩

J ′M ′ J ′M ′

⟨c d |t2|k i ⟩

J ′M ′ J ′M ′

δj i j l

(DC d )

− 14−2

a

∑

c d k l

∑

J ′( J ′)2 ⟨c a |t2|k l ⟩

J ′M ′ J ′M ′

⟨k l |v |c d ⟩

J ′M ′ J ′M ′

⟨d b |t2|i j ⟩

J M J M

δja jd

(DDa )

− 12−1

i

∑

c

⟨c |t1|i ⟩

00

⟨ab |v |c j ⟩

J M J M

(DDb )

+ 12−1

a

∑

k

⟨a |t1|k ⟩

00

⟨kb |v |i j ⟩

J M J M

(DE a )

− 12−2

i

∑

c k

⟨c |t1|i ⟩

00

⟨k | f |c ⟩

00

⟨ab |t2|k j ⟩

J M J M«

+ . . .

Figure C.4: Spherical expressions for the CCSD T2 amplitude equations.



(DEb )

− 12

Pab (J ) Pi j (J ) −2a

∑

c k

⟨a |t1|k ⟩

00

⟨k | f |c ⟩

00

⟨cb |t2|i j ⟩

J M J M

(DE c )

+ CCAtoStd(A)

ab

i jJ ; J ′

−1i

∑

c d k

(−1)jd+jk−J ′ ⟨c |t1|i ⟩

00

⟨a k |v |c d ⟩

J ′M ′

J ′M ′

⟨d b |t2|k j ⟩

J ′M ′

J ′M ′

(DE d )

− CCAtoStd(A)

ab

i jJ ; J ′

−1a

∑

c k l

(−1)jc+j l−J ′ ⟨a |t1|k ⟩

00

⟨k l |v |i c ⟩

J ′M ′

J ′M ′

⟨c b |t2|l j ⟩

J ′M ′

J ′M ′


¨(DE e )

+ 14−1

a

∑

c d k

⟨a |t1|k ⟩

00

⟨kb |v |c d ⟩

J M J M

⟨c d |t2|i j ⟩

J M J M

(DE f )

− 14−1

i

∑

c k l

⟨c |t1|i ⟩

00

⟨k l |v |c j ⟩

J M J M

⟨ab |t2|k l ⟩

J M J M

(DE g )

− 12−2

a

∑

c d k

−1c

∑

J ′( J ′)2 ⟨c |t1|k ⟩

00

⟨k a |v |c d ⟩

J ′M ′ J ′M ′

⟨d b |t2|i j ⟩

J M J M

δja jd

(DE h)

+ 12−2

i

∑

c k l

−1c

∑

J ′( J ′)2 ⟨c |t1|k ⟩

00

⟨ab |t2|l j ⟩

J M J M

⟨k l |v |c i ⟩

J ′M ′ J ′M ′

δj i j l

(DFa )

+ 14−1

i −1j

∑

c d

⟨c |t1|i ⟩

00

⟨d |t1|j ⟩

00

⟨ab |v |c d ⟩

J M J M

(DF b )

+ 14−1

a −1b

∑

k l

⟨a |t1|k ⟩

00

⟨b |t1|l ⟩

00

⟨k l |v |i j ⟩

J M J M

(DF c )

− −1a

−1i

∑

c k

⟨a |t1|k ⟩

00

⟨c |t1|i ⟩

00

⟨kb |v |c j ⟩

J M J M

(DG a )

+ 18−1

a −1b

∑

c d k l

⟨a |t1|k ⟩

00

⟨b |t1|l ⟩

00

⟨k l |v |c d ⟩

J M J M

⟨c d |t2|i j ⟩

J M J M«

+ . . .

Figure C.5: Spherical expressions for the CCSD T2 amplitude equations, continued.



(DG b )

+ 18

Pab (J ) Pi j (J ) −1i −1

j

∑

c d k l

⟨c |t1|i ⟩

00

⟨d |t1|j ⟩

00

⟨ab |t2|k l ⟩

J M J M

⟨k l |v |c d ⟩

J M J M

(DG c )

+ CCAtoStd(A)

ab

i jJ ; J ′

−1a −1

i

∑

c d k l

(−1)jd+j l−J ′ ⟨a |t1|k ⟩

00

⟨c |t1|i ⟩

00

⟨d b |t2|l j ⟩

J ′M ′

J ′M ′

⟨k l |v |c d ⟩

J ′M ′

J ′M ′

(DG d )

− CCAtoStd(A)

ab

i jJ ; J ′

12−3

i

∑

c d k l

−1c

∑

J ′′( J ′′)2 ⟨c |t1|k ⟩

00

⟨d |t1|i ⟩

00

⟨a b |t2|l j ⟩

J ′M ′

J ′M ′

⟨k l |v |c d ⟩

J ′′M ′′ J ′′M ′′

δjd j l


¨

(DG e )

− 12−3

a

∑

c d k l

−1c

∑

J ′( J ′)2 ⟨a |t1|l ⟩

00

⟨c |t1|k ⟩

00

⟨d b |t2|i j ⟩

J M J M

⟨k l |v |c d ⟩

J ′M ′ J ′M ′

δja jd

(DHa )

+ 12−1

a −1i −1

j

∑

c d k

⟨a |t1|k ⟩

00

⟨c |t1|i ⟩

00

⟨d |t1|j ⟩

00

⟨kb |v |c d ⟩

J M J M

(DHb )

− 12−1

a −1b−1

i

∑

c k l

⟨a |t1|k ⟩

00

⟨b |t1|l ⟩

00

⟨c |t1|i ⟩

00

⟨k l |v |c j ⟩

J M J M

(DI )

+ 14−1

a −1b

∑

c d k l

−1c −1

d⟨a |t1|k ⟩

00

⟨b |t1|l ⟩

00

⟨c |t1|i ⟩

00

⟨d |t1|j ⟩

00

⟨k l |v |c d ⟩

J M J M«

= 0 , ∀ a ,b , i , j , J , M

Figure C.6: Spherical expressions for the CCSD T2 amplitude equations, continued.



C.3 Diagrams for Three-Body Hamiltonians

b b b b b b

ED EE


T1Aa T1Ab T1Ba T1Bb


T1Bc T1Bd T1Be T1Bf


T1Ca T1Cb T1Cc T1Cd

b b b b b b b b b

T1Ce T1Cf T1Cg

Figure C.7: CCSD correlation energy and T1 diagrams for three-body Hamiltonians.




T2Aa T2Ba T2Bb T2Bc


T2Bd T2Ca T2Cb T2Cc


T2Cd T2Ce T2Cf T2Cg


T2Ch T2Ci T2Cj T2Da


T2Db T2Dc T2Dd T2De


T2Df T2Dg T2Dh T2Di


T2Dj T2Dk T2Dl T2Dm


T2Dn T2Do T2Dp T2Dq

Figure C.8: CCSD T2 diagrams for three-body Hamiltonians.



b b b b b b b b b

T2Dr T2Ea T2Eb

b b b b b b b b b

T2Ec T2Ed T2Ee

b b b b b b b b b

T2Ef T2Eg T2Eh

b b b b b b b b b

T2Ei T2Ej T2Ek

b b b b b b b b b

T2El T2Em T2En

b b b b b b b b b

T2Eo T2Ep T2Eq

b b b

T2Er

Figure C.9: CCSD T2 diagrams for three-body Hamiltonians, continued.



C.4 Spherical Equations for Three-Body Hamiltonians

(T1Aa)+ 1

2

∑

c d k l

−1k−1

l

∑

J

J ⟨k l a ||w ||c d i ⟩

J J

0

⟨c |t1|k ⟩

00

⟨d |t1|l ⟩

00

(T1Ab)+ 1

4

∑

c d k l

∑

J

J ⟨k l a ||w ||c d i ⟩

J J

0

⟨c d |t2|k l ⟩

J M J M

(T1Ba)− 1

2−1

i

∑

c d e k l

−1k−1

l

∑

J

J ⟨k l a ||w ||c d e ⟩

J J

0

⟨c |t1|k ⟩

00

⟨d |t1|l ⟩

00

⟨e |t1|i ⟩

00

(T1Bb)− 1

4−1

i

∑

c d e k l

∑

J

J ⟨k l a ||w ||c d e ⟩

J J

0

⟨c d |t2|k l ⟩

J M J M

⟨e |t1|i ⟩

00

(T1Bc)− 1

2−1

a

∑

c d e k l

∑

J

J ⟨a k l ||w ||c d e ⟩

J J

0

⟨c d |t2|i k ⟩

J M J M

⟨e |t1|l ⟩

00

(T1Bd)+ 1

2−1

a

∑

c d k l m

−1k−1

l

∑

J

J ⟨k l m ||w ||c d i ⟩

J J

0

⟨c |t1|k ⟩

00

⟨d |t1|l ⟩

00

⟨a |t1|m ⟩

00

(T1Be)+ 1

4−1

a

∑

c d k l m

∑

J

J ⟨k l m ||w ||c d i ⟩

J J

0

⟨c d |t2|k l ⟩

J M J M

⟨a |t1|m ⟩

00

(T1Bf)+ 1

2−1

i

∑

c d k l m

∑

J

J ⟨k l m ||w ||c i d ⟩

J J

0

⟨c a |t2|k l ⟩J M J M

⟨d |t1|m ⟩

00

+ . . .

Figure C.10: Spherical expressions for the CCSD T1 amplitude equations for three-body Hamilto-

nians.



(T1Ca)+ 1

2−2

a

∑

c d e k l m

−1k−1

l

∑

J

J ⟨k l m ||w ||c d e ⟩

J J

0

⟨c |t1|k ⟩

00

⟨d |t1|l ⟩

00

⟨e |t1|i ⟩

00

⟨a |t1|m ⟩

00

(T1Cb)− 1

4−2

a

∑

c d e k l m

∑

J

J ⟨k l m ||w ||c d e ⟩

J J

0

⟨c d |t2|k l ⟩

J M J M

⟨a |t1|m ⟩

00

⟨e |t1|i ⟩

00

(T1Cc)− 1

2−2

a

∑

c d e k l m

∑

J

J ⟨k l m ||w ||c d e ⟩

J J

0

⟨c d |t2|k i ⟩

J M J M

⟨a |t1|l ⟩

00

⟨e |t1|m ⟩

00

(T1Cd)− 1

2−2

a

∑

c d e k l m

∑

J

J ⟨k l m ||w ||c d e ⟩

J J

0

⟨c a |t2|k l ⟩J M J 0

⟨d |t1|i ⟩

00

⟨e |t1|m ⟩

00

(T1Ce)+ 1

2

∑

c d e k l m

−1k−1

l

∑

J

J ⟨k l m ||w ||c d e ⟩

J J

0

⟨c |t1|k ⟩

00

⟨d |t1|l ⟩

00

⟨e a |t2|m i ⟩

00

00

(T1Cf)+ 1

4

∑

c d e k l m

∑

J

J ⟨k l m ||w ||c d e ⟩

J J

0

⟨c d |t2|k l ⟩

J M J M

⟨e a |t2|m i ⟩

00

00

(T1Cg)− 1

4−1

a

∑

c d e k l m

∑

J J ′ J ′′(−1)J+J ′+J ′′ J J ′ J ′′

¦J J ′ J ′′j i je jm

©

6j

× ⟨k l m ||w ||c d e ⟩

J J ′

J ′′

⟨c d |t2|i m ⟩

J ′M ′ J ′M ′

⟨e a |t2|k l ⟩J M J M

+ ⟨a |t1|i ⟩

00

[NO2B] = 0 , ∀ a , i


nians, continued.



Pab (J ) Pi j (J )

¨(T2Aa)+ 1

4J−1

∑

c k

⟨ab k ||w ||i j c ⟩

J J

0

⟨c |t1|k ⟩

00

(T2Ba)+ 1

2J−1 −1

j (−1)j i+j j−J∑

c d k

⟨ab k ||w ||c i d ⟩

J J

0

⟨c |t1|j ⟩

00

⟨d |t1|k ⟩

00

(T2Bb)+ 1

2J−1 −1

b

∑

c k l

⟨a k l ||w ||i j c ⟩

J J

0

⟨b |t1|k ⟩

00

⟨c |t1|l ⟩

00

(T2Bc)− 1

4J−1

∑

c d k

∑

J ′ J ′′(−1)J+J ′+J ′′ J ′ J ′′

nJ ′ J ′′ Jj i j j jk

o

6j⟨ab k ||w ||c d i ⟩

J J ′

J ′′

⟨c d |t2|j k ⟩

J ′M ′ J ′M ′

(T2Bd)− 1

4J−1 (−1)ja+jb−J

∑

c k l

∑

J ′ J ′′(−1)J+J ′+J ′′ J ′ J ′′

¦J ′ J ′′ Jja jb jc

©

6j

× ⟨k l a ||w ||i j c ⟩

J ′ J

J ′′

⟨cb |t2|k l ⟩

J ′M ′ J ′M ′

(T2Ca)+ 1

4J−1 −1

i −1j

∑

c d e k

⟨ab k ||w ||c d e ⟩

J J

0

⟨c |t1|i ⟩

00

⟨d |t1|j ⟩

00

⟨e |t1|k ⟩

00

(T2Cb)− J−1 −1

b−1

j

∑

c d k l

⟨a k l ||w ||i c d ⟩

J J

0

⟨b |t1|k ⟩

00

⟨c |t1|j ⟩

00

⟨d |t1|l ⟩

00«

+ . . .


nians.



Pab (J ) Pi j (J )

¨(T2Cc)+ 1

4J−1 −1

a −1b

∑

c k l m

⟨k l m ||w ||i j c ⟩

J J

0

⟨a |t1|k ⟩

00

⟨b |t1|l ⟩

00

⟨c |t1|m ⟩

00

(T2Cd)+ 1

8J−1

∑

c d e k

⟨ab k ||w ||c d e ⟩

J J

0

⟨e |t1|k ⟩

00

⟨c d |t2|i j ⟩

J M J M

(T2Ce)+ 1

4J−1 −1

l

∑

c d e k

∑

J ′ J ′′(−1)J+J ′+J ′′ J ′ J ′′

nJ J ′ J ′′j j jk j i

o

6j

× ⟨ab k ||w ||c d e ⟩

J J ′

J ′′

⟨c d |t2|j k ⟩

J ′M ′ J ′M ′

⟨e |t1|i ⟩

00«

(T2Cf)+ CCAtoStd(A)

ab

i jJ ; J ′

12( J ′)−1 −1

j

∑

c d k l

∑

J ′′ J ′′′J ′′ ˆJ ′′′

nJ ′ J ′′ J ′′′jd j j jb

o

6j

× ⟨k l a ||w ||c d i ⟩

J ′′ J ′′′

J ′

⟨d b |t2|k l ⟩

J ′′M ′′ J ′′M ′′

⟨c |t1|j ⟩

00

(T2Cg)− 1

2Pab (J ) Pi j (J ) J−1 −1

b

∑

c d k l

∑

J ′ J ′′(−1)J+J ′+J ′′ J ′ J ′′

nJ ′ J ′′ Jj i j j j l

o

6j

× ⟨a k l ||w ||c d i ⟩

J J ′

J ′′

⟨b |t1|k ⟩

00

⟨c d |t2|j l ⟩

J ′M ′ J ′M ′

(T2Ch)− CCAtoStd(A)

ab

i jJ ; J ′

( J ′)−1∑

c d k l

−1l

∑

J ′ J ′′(−1)J

′+J ′′+J ′′′ J ′′ ˆJ ′′′

×¦

J ′′ J ′′′ J ′jd jk j l

©

6j⟨k l a ||w ||c d i ⟩

J ′′ J ′′′

J ′

⟨b d |t2|j k ⟩

J ′M ′

J ′M ′

⟨c |t1|l ⟩

00

+ . . .


nians, continued.



Pab (J ) Pi j (J )

¨(T2Ci)+ 1

8J−1

∑

c k l m

⟨k l m ||w ||i j c ⟩

J J

0

⟨ab |t2|k l ⟩

J M J M

⟨c |t1|m ⟩

00

(T2Cj)− 1

4J−1 −1

a (−1)ja+jb−J∑

c k l m

∑

J ′ J ′′(−1)J+J ′+J ′′ J ′ J ′′

¦J ′ J ′′ J

jm jb jc

©

6j

× ⟨k l m ||w ||i j c ⟩

J ′ J

J ′′

⟨cb |t2|k l ⟩

J ′M ′ J ′M ′

⟨a |t1|m ⟩

00

(T2Da)+ 1

2J−1 −1

b−1

i −1j

∑

c d e k l

⟨a k l ||w ||c d e ⟩

J J

0

⟨b |t1|k ⟩

00

⟨c |t1|i ⟩

00

⟨d |t1|j ⟩

00

⟨e |t1|l ⟩

00

(T2Db)+ 1

2J−1 −1

a −1b−1

j (−1)ja+jb−J

×∑

c d k l m

⟨k l m ||w ||i c d ⟩

J J

0

⟨a |t1|l ⟩

00

⟨b |t1|k ⟩

00

⟨c |t1|j ⟩

00

⟨d |t1|m ⟩

00

(T2Dc)− 1

4J−1 −1

i −1j (−1)ja+jb−J

∑

c d e k l

∑

J ′ J ′′(−1)J+J ′+J ′′ J ′ J ′′

¦J ′ J ′′ Jja jb je

©

6j

× ⟨k l a ||w ||c d e ⟩

J ′ J

J ′′

⟨c |t1|i ⟩

00

⟨d |t1|j ⟩

00

⟨e b |t2|k l ⟩

J ′M ′ J ′M ′

(T2Dd)+ 1

2J−1 −1

b−1

i

∑

c d e k l

∑

J ′ J ′′(−1)J+J ′+J ′′ J ′ J ′′


o

6j

× ⟨a k l ||w ||c d e ⟩

J J ′

J ′′

⟨c d |t2|j l ⟩

J ′M ′ J ′M ′

⟨b |t1|k ⟩

00

⟨e |t1|i ⟩

00«

+ . . .


nians, continued.



(T2De)− CCAtoStd(A)

ab

i jJ ; J ′

( J ′)−1 −1i

∑

c d e k l

−1k(−1)jc+j l−J ′

×∑

J ′′ J ′′′J ′′ J ′′′

¦J ′ J ′′ J ′′′jk jc j l

©

6j⟨k l a ||w ||c d e ⟩

J ′′ J ′′′

J ′

⟨c b |t2|l j ⟩

J ′M ′

J ′M ′

⟨d |t1|k ⟩

00

⟨e |t1|i ⟩

00

Pab (J ) Pi j (J )

¨(T2Df)+ 1

4J−1 −1

b

∑

c d e k l

⟨a k l ||w ||c d e ⟩

J J

0

⟨c d |t2|i j ⟩

J M J M

⟨b |t1|k ⟩

00

⟨e |t1|l ⟩

00

(T2Dg)− 1

4−1

a

∑

c d e k l

−1k−1

l

∑

J ′J ′ ⟨k l a ||w ||c d e ⟩

J ′ J ′

0

⟨c |t1|k ⟩

00

⟨d |t1|l ⟩

00

⟨e b |t2|i j ⟩

J M J M

(T2Dh)− 1

4J−1 −1

a −1b

∑

c d k l m

∑

J ′ J ′′(−1)J+J ′+J ′′ J ′ J ′′

nJ ′ J ′′ Jj i j j jm

o

6j

× ⟨k l m ||w ||c d i ⟩

J J ′

J ′′

⟨a |t1|k ⟩

00

⟨b |t1|l ⟩

00

⟨c d |t2|j m ⟩

J ′M ′ J ′M ′

(T2Di)+ 1

2J−1 −1

a −1j (−1)ja+jb−J

∑

c d k l m

∑

J ′ J ′′(−1)J+J ′+J ′′ J ′ J ′′

¦J J ′ J ′′

jd jm jb

©

6j

× ⟨k l m ||w ||i c d ⟩

J ′ J

J ′′

⟨d b |t2|k l ⟩

J ′M ′ J ′M ′

⟨a |t1|m ⟩

00

⟨c |t1|j ⟩

00«

(T2Dj)− CCAtoStd(A)

ab

i jJ ; J ′

( J ′)−1 −1a

∑

c d k l m

−1l

∑

J ′′ J ′′′(−1)J

′+J ′′+J ′′′ J ′′ J ′′′¦

J ′ J ′′ J ′′′j l jd jk

©

6j

× ⟨k l m ||w ||c d i ⟩

J ′′ J ′′′

J ′

⟨d b |t2|k j ⟩

J ′M ′

J ′M ′

⟨a |t1|m ⟩

00

⟨c |t1|l ⟩

00

+ . . .


nians, continued.



Pab (J ) Pi j (J )

¨

(T2Dk)+ 1

4J−1 −1

j (−1)j i+j j−J∑

c d k l m

⟨k l m ||w ||c i d ⟩

J J

0

⟨c |t1|j ⟩

00

⟨d |t1|m ⟩

00

⟨ab |t2|k l ⟩

J M J M

(T2Dl)− 1

8−1

a

∑

c d e k l

∑

J ′J ′ ⟨k l a ||w ||c d e ⟩

J ′ J ′

0

⟨e b |t2|i j ⟩

J M J M

⟨c d |t2|k l ⟩

J ′M ′ J ′M ′

(T2Dm)− 1

8J−1 (−1)ja+jb−J

∑

c d e k l

∑

J ′ J ′′(−1)J+J ′+J ′′ J ′ J ′′

¦J ′ J ′′ Jja jb je

©

6j

× ⟨k l a ||w ||c d e ⟩

J ′ J

J ′′

⟨c d |t2|i j ⟩

J M J M

⟨e b |t2|k l ⟩

J ′M ′ J ′M ′«

(T2Dn)+ CCAtoStd(A)

ab

i jJ ; J ′

12( J ′)−1 (−1)ja+j i−J ′

∑

c d e k l

(−1)je+j l−J ′

∑

J ′′ J ′′′(−1)J

′+J ′′+J ′′′ J ′′ J ′′′¦

J ′ J ′′ J ′′′jk ja j i

©

6j⟨a k l ||w ||c d e ⟩

J ′′′ J ′′

J ′

⟨c d |t2|k i ⟩

J ′′M ′′ J ′′M ′′

⟨b e |t2|j l ⟩

J ′M ′

J ′M ′

(T2Do)− 1

8Pab (J ) Pi j (J )

−1i (−1)j i+j j−J

×∑

c d k l m

∑

J ′J ′ ⟨k l m ||w ||c d i ⟩

J ′ J ′

0

⟨ab |t2|j m ⟩

J M J M

⟨c d |t2|k l ⟩

J ′M ′ J ′M ′

+ . . .


nians, continued.



(T2Dp)− 1

8Pab (J ) Pi j (J ) J−1

∑

c d k l m

∑

J ′ J ′′(−1)J+J ′+J ′′ J ′ J ′′

nJ ′ J ′′ Jj i j j jm

o

6j

× ⟨k l m ||w ||c d i ⟩

J J ′

J ′′

⟨ab |t2|k l ⟩

J M J M

⟨c d |t2|j m ⟩

J ′M ′ J ′M ′

(T2Dq)+ CCAtoStd(A)

ab

i jJ ; J ′

12( J ′)−1

∑

c d k l m

(−1)jd+jm−J ′

×∑

J ′′ J ′′′J ′′ ˆJ ′′′

¦J ′ J ′′ J ′′′jc j i ja

©

6j⟨k l m ||w ||i c d ⟩

J ′′ J ′′′

J ′

⟨b d |t2|j m ⟩

J ′M ′

J ′M ′

⟨c a |t2|k l ⟩J ′′M ′′ J ′′M ′′

(T2Dr)− Pab (J ) Pi j (J )

¨

14−1

i (−1)j i+j j−J

×∑

c d k l m

−1k−1

l

∑

J ′J ′ ⟨k l m ||w ||c d i ⟩

J ′ J ′

0

⟨ab |t2|j m ⟩

J M J M

⟨c |t1|k ⟩

00

⟨d |t1|l ⟩

00

(T2Ra)+ 1

4J−1 −1

a −1b−1

i −1j

×∑

c d e k l m

⟨k l m ||w ||c d e ⟩

J J

0

⟨a |t1|k ⟩

00

⟨b |t1|l ⟩

00

⟨c |t1|i ⟩

00

⟨d |t1|j ⟩

00

⟨e |t1|m ⟩

00

(T2Eb)− 1

4−2

a

∑

c d e k l m

−1k−1

l

∑

J ′J ′⟨k l m ||w ||c d e ⟩

J ′ J ′

0

⟨c |t1|k ⟩

00

⟨d |t1|l ⟩

00

⟨a |t1|m ⟩

00

⟨e b |t2|i j ⟩

J M J M

(T2Ec)+ 1

4−2

i (−1)j i+j j−J∑

c d e k l m

−1k−1

l

∑

J ′J ′

× ⟨k l m ||w ||c d e ⟩

J ′ J ′

0

⟨c |t1|k ⟩

00

⟨d |t1|l ⟩

00

⟨e |t1|i ⟩

00

⟨ab |t2|j m ⟩

J M J M«

+ . . .


nians, continued.



(T2Ed)+ 1

8Pab (J ) Pi j (J ) J−1 −1

a −1b

×∑

c d e k l m

⟨k l m ||w ||c d e ⟩

J J

0

⟨a |t1|k ⟩

00

⟨b |t1|l ⟩

00

⟨e |t1|m ⟩

00

⟨c d |t2|i j ⟩

J M J M

(T2Ee)− CCAtoStd(A)

ab

i jJ ; J ′

( J ′)−1 −1a −1

i

∑

c d e k l m

−1k(−1)je+jm−J ′

×∑

J ′′ J ′′′J ′′ J ′′′

¦J ′ J ′′ J ′′′jk jc j l

©

6j⟨k l m ||w ||c d e ⟩

J ′′ J ′′′

J ′

⟨e b |t2|m j ⟩

J ′M ′

J ′M ′

⟨a |t1|l ⟩

00

⟨c |t1|i ⟩

00

⟨d |t1|k ⟩

00

Pab (J ) Pi j (J )

¨

(T2Ef)+ 1

8J−1 −1

i −1j

∑

c d e k l m

⟨k l m ||w ||c d e ⟩

J J

0

⟨ab |t2|k l ⟩

J M J M

⟨c |t1|i ⟩

00

⟨d |t1|j ⟩

00

⟨e |t1|m ⟩

00

(T2Eg)− 1

4J−1 −1

a −1i −1

j (−1)ja+jb−J∑

c d e k l m

∑

J ′ J ′′(−1)J+J ′+J ′′ J ′ J ′′

×¦

J J ′ J ′′je jm jb

©

6j⟨k l m ||w ||c d e ⟩

J ′ J

J ′′

⟨e b |t2|k l ⟩

J ′M ′ J ′M ′

⟨a |t1|m ⟩

00

⟨c |t1|i ⟩

00

⟨d |t1|j ⟩

00

(T2Eh)+ 1

4J−1 −1

a −1b−1

i

∑

c d e k l m

∑

J ′ J ′′(−1)J+J ′+J ′′ J ′ J ′′

nJ J ′ J ′′

jm j i j j

o

6j

× ⟨k l m ||w ||c d e ⟩

J J ′

J ′′

⟨c d |t2|j m ⟩

J ′M ′ J ′M ′

⟨a |t1|k ⟩

00

⟨b |t1|l ⟩

00

⟨e |t1|i ⟩

00«

+ . . .


nians, continued.



(T2Ei)− CCAtoStd(A)

ab

i jJ ; J ′

12( J ′)−1 −1

j

∑

c d e k l m

(−1)je+jm−J ′

×∑

J ′′ J ′′′J ′′ J ′′′

nJ ′ J ′′ J ′′′jd j j jb

o

6j⟨k l m ||w ||c d e ⟩

J ′′ J ′′′

J ′

⟨a e |t2|i m ⟩

J ′M ′

J ′M ′

⟨d b |t2|k l ⟩

J ′′M ′′ J ′′M ′′

⟨c |t1|j ⟩

00

Pab (J ) Pi j (J )

¨(T2Ej)+ 1

8J−1 −1

i

∑

c d e k l m

∑

J ′ J ′′(−1)J+J ′+J ′′ J ′ J ′′

×n

J J ′ J ′′jm j i j j

o

6j⟨k l m ||w ||c d e ⟩

J J ′

J ′′

⟨c d |t2|j m ⟩

J ′M ′ J ′M ′

⟨ab |t2|k l ⟩

J M J M

⟨e |t1|i ⟩

00

(T2Ek)− 1

8−2

j

∑

c d e k l m

∑

J ′J ′ ⟨k l m ||w ||c d e ⟩

J ′ J ′

0

⟨ab |t2|i m ⟩

J M J M

⟨c d |t2|k l ⟩

J ′M ′ J ′M ′

⟨e |t1|j ⟩

00

(T2El)− 1

8−2

a

∑

c d e k l m

∑

J ′( J ′)−1 ⟨k l m ||w ||c d e ⟩

J ′ J ′

0

⟨c d |t2|k l ⟩

J ′M ′ J ′M ′

⟨e b |t2|i j ⟩

J M J M

⟨a |t1|m ⟩

00

(T2Em)− 1

8J−1 −1

a (−1)ja+jb−J∑

c d e k l m

∑

J ′ J ′′(−1)J+J ′+J ′′ J ′ J ′′

¦J J ′ J ′′je jm jb

©

6j

× ⟨k l m ||w ||c d e ⟩

J ′ J

J ′′

⟨c d |t2|i j ⟩

J M J M

⟨e b |t2|k l ⟩

J ′M ′ J ′M ′

⟨a |t1|m ⟩

00«

(T2En)− CCAtoStd(A)

ab

i jJ ; J ′

12( J ′)−1 −1

b

∑

c d e k l m

(−1)je+jm−J ′

×∑

J ′′ J ′′′J ′′ J ′′′

nJ ′ J ′′ J ′′′jk j l j j

o

6j⟨k l m ||w ||c d e ⟩

J ′′′ J ′′

J ′

⟨a e |t2|i m ⟩

J ′M ′

J ′M ′

⟨c d |t2|j k ⟩

J ′′M ′′ J ′′M ′′

⟨b |t1|l ⟩

00

+ . . .


nians, continued.



Pab (J ) Pi j (J )

¨

(T2Eo)− 1

4

∑

c d e k l m

∑

J ′ J ′′J ′ ( J ′′)2

¦J ′ J ′′ Jja jb jd

©

6j

¦J ′ J ′′ Jja jc jd

©

6j

× ⟨k l m ||w ||c d e ⟩

J ′ J ′

0


⟨b d |t2|k l ⟩

J ′M ′ J ′M ′

⟨e |t1|m ⟩

00

(T2Ep)+ 1

16J−1

∑

c d e k l m

⟨k l m ||w ||c d e ⟩

J J

0

⟨c d |t2|i j ⟩

J M J M

⟨ab |t2|k l ⟩

J M J M

⟨e |t1|m ⟩

00«

(T2Eq)+ CCAtoStd(A)

ab

i jJ ; J ′

12( J ′)−1

∑

c d e k l m

−1l(−1)je+jm−J ′

∑

J ′′ J ′′′

× (−1)J′+J ′′+J ′′ J ′′ J ′′′

¦J ′ J ′′ J ′′′j l jd jk

©

6j⟨k l m ||w ||c d e ⟩

J ′′ J ′′′

J ′

⟨a d |t2|i k ⟩

J ′M ′

J ′M ′

⟨b e |t2|j m ⟩

J ′M ′

J ′M ′

⟨c |t1|l ⟩

00

(T2Er)− 1

4Pab (J ) Pi j (J )

∑

c d e k l m

∑

J ′ J ′′J ′ ( J ′′)2

nJ J ′ J ′′j l j i j j

o

6j

¦J J ′ J ′′j l j i jk

©

6j

× ⟨k l m ||w ||c d e ⟩

J ′ J ′

0

⟨ab |t2|i k ⟩

J M J M

⟨c d |t2|j l ⟩

J ′M ′ J ′M ′

⟨e |t1|m ⟩

00

+ ⟨ab |t2|i j ⟩

J M J M

[NO2B] = 0 , ∀ a ,b , i , j , J , M


nians, continued.


Appendix D

Effective Hamiltonian Diagrams andSpherical Expressions

Appendix D. Effective Hamiltonian Diagrams and Spherical Expressions

D.1 Spherical Equations

⟨ı |H1|a ⟩

00

= ⟨ı | f |a ⟩

00

+∑

c k

−1i −1

k

∑

J

J 2 ⟨c |t1|k ⟩

00

⟨i k |v |a c ⟩

J M J M

⟨a |H1|b ⟩

00

= ⟨a | f |b ⟩

00

+ −1a

∑

c k

−1k

∑

J

J 2 ⟨a k |v |b c ⟩

J M J M

⟨c |t1|k ⟩

00

+ −1a

∑

k

⟨a |t1|k ⟩

00

⟨k |H1|b ⟩

00

+ 12−1

a

∑

c k l

∑

J

J 2 ⟨a c |t2|k l ⟩J M J M

⟨k l |v |b c ⟩

J M J M

⟨ı |H1|j ⟩

00

= ⟨ı |χ ′|j ⟩

00

− −1i

∑

c

⟨ı |H1|c ⟩

00

⟨c |t1|j ⟩

00

⟨a |H1|i ⟩

00

= 0

⟨i j |H2|ab ⟩

J M J M

= ⟨i j |v |ab ⟩

J M J M

⟨a i |H2|b c ⟩

J M J M

= ⟨a i |v |b c ⟩

J M J M

+ −1a

∑

k

⟨a |t1|k ⟩

00

⟨k i |v |b c ⟩

J M J M

⟨i k |H2|j a ⟩

J M J M

= ⟨i k |v |j a ⟩

J M J M

− −1j

∑

c

⟨c |t1|j ⟩

00

⟨i k |v |c a ⟩

J M J M

⟨ab |H2|c d ⟩

J M J M

= ⟨ab |χ ′|c d ⟩

J M J M

+ 12

∑

k l

⟨ab |t2|k l ⟩

J M J M

⟨k l |v |c d ⟩

J M J M

Figure D.1: Spherical expressions for the effective Hamiltonian matrix elements.



⟨i j |H2|k l ⟩

J M J M

= ⟨i j |v |k l ⟩

J M J M

− Pk l (J ) −1l

∑

c

⟨i j |χ ′|k c ⟩

J M J M

⟨c |t1|l ⟩

00

+ 12

∑

c d

⟨i j |v |c d ⟩

J M J M

⟨c d |t2|k l ⟩

J M J M

⟨a j |H2|ib ⟩

J M J M

= ⟨a j |χ ′′′|ib ⟩

J M J M

− 12−1

i

∑

c

⟨c |t1|i ⟩

00

⟨a j |H2|cb ⟩

J M J M

− CCAtoStd

a j

ibJ ; J ′

∑

c k

(−1)jc+jk−J ′ ⟨a c |t2|i k ⟩

J ′M ′

J ′M ′

⟨k |v |cb ⟩

J ′M ′

J ′M ′

⟨i a |H2|j k ⟩

J M J M

= ⟨i a |χ ′′|j k ⟩

J M J M

− −1i

∑

c

⟨ı |H1|c ⟩

00

⟨c a |t2|j k ⟩J M J M

⟨ab |H2|i j ⟩

J M J M

= 0

⟨ab |H2|c i ⟩

J M J M

= ⟨ab |v |c i ⟩

J M J M

− −1i

∑

d

⟨ab |v |c d ⟩

J M J M

⟨d |t1|i ⟩

00

− Pab (J ) −1b

∑

k

(−1)j i+jc−J ⟨a k |χ ′|i c ⟩

J M J M

⟨b |t1|k ⟩

00

+ −1c

∑

k

⟨k |H1|c ⟩

00

⟨ab |t2|k i ⟩

J M J M

− CCAtoStd(A)

ab

c iJ ; J ′

|Tc i =0

∑

d k

(−1)jd+jk−J ′ ⟨a k |H2|c d ⟩

J ′M ′

J ′M ′

⟨d b |t2|k i ⟩

J ′M ′

J ′M ′

− 12

∑

k l

(−1)jc+j i−J ⟨ab |t2|k l ⟩

J M J M

⟨k l |H2|i c ⟩

J M J M

Figure D.2: Spherical expressions for the effective Hamiltonian matrix elements.



⟨ı |χ ′|j ⟩

00

= ⟨ı | f |j ⟩

00

+ −1i

∑

c k

−1k

∑

J

J 2 ⟨c |t1|k ⟩

00

⟨i k |v |j c ⟩

J M J M

− 12−1

i

∑

c d k

∑

J

J 2 ⟨i k |v |c d ⟩

J M J M

⟨c d |t2|j k ⟩

J M J M

⟨a i |χ ′|b c ⟩

J M J M

= ⟨a i |v |b c ⟩

J M J M

+ 12−1

a

∑

k

⟨a |t1|k ⟩

00

⟨k i |v |b c ⟩

J M J M

⟨i k |χ ′|j a ⟩

J M J M

= ⟨i k |v |j a ⟩

J M J M

− 12−1

j

∑

c

⟨c |t1|j ⟩

00

⟨i k |v |c a ⟩

J M J M

⟨ab |χ ′|c d ⟩

J M J M

= ⟨ab |v |c d ⟩

J M J M

+ Pab (J ) −1b

∑

k

⟨a k |χ ′|c d ⟩

J M J M

⟨b |t1|k ⟩

00

⟨a j |χ ′|ib ⟩

J M J M

= ⟨a j |v |ib ⟩

J M J M

+ 12−1

a

∑

k

⟨a |t1|k ⟩

00

⟨k j |v |ib ⟩

J M J M

− −1i

∑

c

⟨c |t1|i ⟩

00

⟨a j |χ ′|cb ⟩

J M J M

⟨a j |χ ′′|ib ⟩

J M J M

= ⟨a j |v |ib ⟩

J M J M

+ 12−1

a

∑

k

⟨a |t1|k ⟩

00

⟨k j |v |ib ⟩

J M J M

− 12−1

i

∑

c

⟨c |t1|i ⟩

00

⟨a j |χ ′|cb ⟩

J M J M

Figure D.3: Spherical expressions for the intermediates used in the calculations of the effective

Hamiltonian matrix elements.



⟨a j |χ ′′′|ib ⟩

J M J M

= ⟨a j |v |ib ⟩

J M J M

+ −1a

∑

k

⟨a |t1|k ⟩

00

⟨k j |v |ib ⟩

J M J M

− 12−1

i

∑

c

⟨c |t1|i ⟩

00

⟨a j |H2|cb ⟩

J M J M

⟨i a |χ ′|j k ⟩

J M J M

= ⟨i a |v |j k ⟩

J M J M

+ 12−1

a

∑

l

⟨i l |v |j k ⟩

J M J M

⟨a |t1|l ⟩

00

⟨i a |χ ′′|j k ⟩

J M J M

= ⟨i a |v |j k ⟩

J M J M

+ −1a

∑

l

⟨i l |v |j k ⟩

J M J M

⟨a |t1|l ⟩

00

+ Pi j (J ) −1k

∑

c

(−1)ja+j i−J ⟨a i |χ ′′′|j c ⟩

J M J M

⟨c |t1|k ⟩

00

− CCAtoStd(A)

i a

j kJ ; J ′

|Ti a=0

∑

c l

(−1)jc+j l−J ′ ⟨c a |t2|l k ⟩

J ′M ′

J ′M ′

⟨ı l |H2|j c ⟩

J ′M ′

J ′M ′

− 12

∑

c d

(−1)ja+j i−J ⟨a i |H2|c d ⟩

J M J M

⟨c d |t2|j k ⟩

J M J M

⟨ab |χ ′|c i ⟩

J M J M

= ⟨ab |v |c i ⟩

J M J M

− 12−1

i

∑

d

⟨ab |v |c d ⟩

J M J M

⟨d |t1|i ⟩

00

− Pab (J ) −1b(−1)j i+jc−J

∑

k

⟨a k |χ ′′|i c ⟩

J M J M

⟨b |t1|k ⟩

00

Figure D.4: Spherical expressions for the intermediates used in the calculations of the effective

Hamiltonian matrix elements.



D.2 Diagrams for Three-Body Hamiltonians

b b b b b b

b b b b b b

H hp A H h

p B H pp A H p

p B


H pp C H p

p D H pp E H h

hA


H hh

B H hh

C H hh

D H hh

Eb b b b b b

b b b b b b

H hhpp A H ph

pp A H phpp B H ph

pp Cb b b

b b b b b b b b b

H hhhp

A H hhhp

B H hhhp

C H pppp A


H pppp B H pp

pp C H pppp D H pp

pp E


H pppp F H hh

hhA H hh

hhB H hh

hhC


H hhhh

D H hhhh

E H hhhh

F H ph

hpA

Figure D.5: Effective Hamiltonian diagrams for three-body Hamiltonians.




H ph

hpB H ph

hpC H ph

hpD H ph

hpE


H ph

hpF H ph

hpG H ph

hpH H ph

hpI


H hp

hhA H hp

hhB H hp

hhC H hp

hhD


H hp

hhE H hp

hhF H hp

hhG H hp

hhH


H hp

hhI H hp

hhJ H hp

hhK H hp

hhL


H hp

hhM H hp

hhN H hp

hhO H hp

hhP


H hp

hhQ H hp

hhR H hp

hhS H pp

phA


H pp

phB H pp

phC H pp

phD H pp

phE

Figure D.6: Effective Hamiltonian diagrams for three-body Hamiltonians, continued.




H pp

phF H pp

phG H pp

phH H pp

phI


H pp

phJ H pp

phK H pp

phL H pp

phM


H pp

phN H pp

phO H pp

phP H pp

phQ


H pp

phR H pp

phS H hhp

hphA H hhp

hphB


H hhp

hphC H hhp

hphD H hhp

hphE H hhp

hphF


H hhp

hphG H hhp

hphH H hhp

hphI H hhp

hphJ


H hhp

hphK H php

pphA H php

pphB H php

pphC


H php

pphD H php

pphE H php

pphF H php

pphG





H php

pphH H php

pphI H php

pphJ H php

pphK


H pph

hhpA H pph

hhpB H pph

hhpC H pph

hhpD

b b b b b b b b b

H pph

hhpE H pph

hhpF H pph

hhpG


H pph

hhpH H pph

hhpI H pph

hhpJ H pph

hhpK


H pph

hhpL H pph

hhpM H pph

hhpN H pph

hhpO


H pph

hhpP H pph

hhpQ H pph

hhpR H pph

hhpS

b b b b b b b b b

H pph

hhpT H pph

hhpU H pph

hhpV

b b b b b b b b b

H pph

hhpW H pph

hhpX H pph

hhpY




b b b b b b b b b

H pph

hhpZ H pph

hhpAA H hhhh

pppp A

b b b b b b b b b

H hhhhpppp B H hhhh

pppp C H hhhhpppp D




D.3 Spherical Equations for Three-Body Hamiltonians

⟨ı |H |a ⟩

00

=(H h

p A)+ 1

4

∑

c d k l

∑

J

J ⟨k l ı ||w ||c d a ⟩

J J

0

⟨c d |t2|k l ⟩

J M J M

(H hp B)+ 1

2

∑

c d k l

−1c −1

d

∑

J

J ⟨k l ı ||w ||c d a ⟩

J J

0

⟨c |t1|k ⟩

00

⟨d |t1|l ⟩

00

⟨a |H |b ⟩

00

=(H

pp A)+ 1

4

∑

c d k l

∑

J

J ⟨k l a ||w ||c d b ⟩

J J

0

⟨c d |t2|k l ⟩

J M J M

(Hp

p B)+ 1

4−1

a

∑

c d k l m

∑

J

J ⟨k l m ||w ||c d b ⟩

J J

0

⟨c d |t2|k l ⟩

J M J M

⟨a |t1|m ⟩

00

(Hp

p C)+ 1

2−1

a

∑

c d k l m

∑

J

J ⟨k l m ||w ||b c d ⟩

J J

0

⟨a c |t2|k l ⟩J M J M

⟨d |t1|m ⟩

00

(Hp

p D)+ 1

2

∑

c d k l

−1c −1

d

∑

J

J ⟨k l a ||w ||c d b ⟩

J J

0

⟨c |t1|k ⟩

00

⟨d |t1|l ⟩

00

(Hp

p E)+ 1

2−1

a

∑

c d k l m

−1c −1

d

∑

J

J ⟨k l m ||w ||c d b ⟩

J J

0

⟨c |t1|k ⟩

00

⟨d |t1|l ⟩

00

⟨a |t1|m ⟩

00

Figure D.10: Spherical expressions for the effective Hamiltonian for three-body Hamiltonians.



⟨ı |H |j ⟩

00

=(H h

hA)+ 1

4

∑

c d k l

∑

J

J ⟨k l ı ||w ||c d j ⟩

J J

0

⟨c d |t2|k l ⟩

J M J M

(H hh

B)− 1

4−1

i

∑

c d e k l

∑

J

J ⟨k l ı ||w ||c d e ⟩

J J

0

⟨c d |t2|k l ⟩

J M J M

⟨e |t1|j ⟩

00

(H hh

C)− 1

2−1

i

∑

c d e k l

∑

J

J ⟨k i l ||w ||c d e ⟩

J J

0

⟨c d |t2|k j ⟩

J M J M

⟨e |t1|l ⟩

00

(H hh

D)+ 1

2

∑

c d k l

−1c −1

d

∑

J

J ⟨k l ı ||w ||c d j ⟩

J J

0

⟨c |t1|k ⟩

00

⟨d |t1|l ⟩

00

(H hh

E)− 1

2−1

i

∑

c d e k l

−1c −1

d

∑

J

J ⟨k l ı ||w ||c d e ⟩

J J

0

⟨c |t1|k ⟩

00

⟨d |t1|l ⟩

00

⟨e |t1|j ⟩

00

⟨i j |H |ab ⟩

J M J M

=(H hh

pp A)+ J−1

∑

c k

⟨i j k ||w ||ab c ⟩

J J

0

⟨c |t1|k ⟩

00

⟨a i |H |b c ⟩

J M J M

=

H phpp A

+ 12

∑

d k l

∑

J ′ J ′′(−1)J+J ′+J ′′ J−1 J ′ J ′′

¦J J ′ J ′′

jd j i ja

©

6j

× ⟨k l ı ||w ||b c d ⟩

J ′ J

J ′′

⟨d a |t2|k l ⟩

J ′M ′ J ′M ′

H phpp B

+ J−1∑

d l

⟨a i l ||w ||b c d ⟩

J J

0

⟨d |t1|l ⟩

00

H phpp C

+ −1a J−1

∑

d k l

⟨k i l ||w ||b c d ⟩

J J

0

⟨a |t1|k ⟩

00

⟨d |t1|l ⟩

00

Figure D.11: Spherical expressions for the effective Hamiltonian for three-body Hamiltonians,

continued.



⟨i k |H |j a ⟩

J M J M

=

H hhhp

A

+ 12(−1)ja+j j−J

∑

c d l

∑

J ′ J ′′(−1)J+J ′+J ′′ J−1 J ′ J ′′

×n

J J ′ J ′′j l ja j j

o

6j⟨i k l ||w ||c d a ⟩

J J ′

J ′′

⟨c d |t2|j l ⟩

J ′M ′ J ′M ′

H hhhp

B

+ J−1∑

c l

⟨i k l ||w ||j a c ⟩

J J

0

⟨c |t1|l ⟩

00

H hhhp

C

− −1j J−1

∑

c d l

⟨i k l ||w ||c a d ⟩

J J

0

⟨d |t1|l ⟩

00

⟨c |t1|j ⟩

00

⟨ab |H |c d ⟩

J M J M

=(H

pppp A)− 1

2Pab (J ) (−1)ja+jb−J

∑

e k l

∑

J ′ J ′′(−1)J+J ′+J ′′ J−1 J ′ J ′′

×¦

J J ′ J ′′je ja jb

©

6j⟨k l a ||w ||c d e ⟩

J ′ J

J ′′

⟨e b |t2|k l ⟩

J ′M ′ J ′M ′

(Hpp

pp B)+ J−1

∑

e k

⟨ab k ||w ||c d e ⟩

J J

0

⟨e |t1|k ⟩

00

(Hpp

pp C)+ 1

2J−1

∑

e k l m

⟨k l m ||w ||c d e ⟩

J J

0

⟨ab |t2|k l ⟩

J M J M

⟨e |t1|m ⟩

00

(Hpp

pp D)− 1

2Pab (J )

−1a

∑

e k l m

∑

J ′ J ′′(−1)je+jm−J ′′ J−1 J ′ J ′′

¦J J ′ J ′′je jm jb

©

6j

× ⟨k l m ||w ||c d e ⟩

J ′ J

J ′′

⟨b e |t2|k l ⟩

J ′M ′ J ′M ′

⟨a |t1|m ⟩

00

(Hpp

pp E)− −1

bJ−1

∑

e k l

⟨a k l ||w ||c d e ⟩

J J

0

⟨b |t1|k ⟩

00

⟨e |t1|l ⟩

00

(Hpp

pp F)+ 1

2Pab (J )

−1a −1

bJ−1

∑

e k l m

⟨k l m ||w ||c d e ⟩

J J

0

⟨a |t1|k ⟩

00

⟨b |t1|l ⟩

00

⟨e |t1|m ⟩

00


continued.



⟨i j |H |k l ⟩

J M J M

=(H hh

hhA)− 1

2Pk l (J )

∑

c d m

∑

J ′ J ′′(−1)J+J ′+J ′′ J−1 J ′ J ′′

×¦

J J ′ J ′′jm jk j l

©

6j⟨i j m ||w ||c d k ⟩

J J ′

J ′′

⟨c d |t2|l m ⟩

J ′M ′ J ′M ′

(H hhhh

B)+ J−1

∑

c m

⟨i j m ||w ||k l c ⟩

J J

0

⟨c |t1|m ⟩

00

(H hhhh

C)+ 1

2J−1

∑

c d e m

⟨i j m ||w ||c d e ⟩

J J

0

⟨c d |t2|k l ⟩

J M J M

⟨e |t1|m ⟩

00

(H hhhh

D)+ 1

2Pk l

−1k

∑

c d e m

∑

J ′ J ′′J−1 J ′ J ′′

¦J J ′ J ′′

jm je j l

©

6j⟨i j m ||w ||c d e ⟩

J J ′

J ′′

× ⟨c d |t2|l m ⟩

J ′M ′ J ′M ′

⟨e |t1|k ⟩

00

(H hhhh

E)+ −1

lJ−1

∑

c d m

⟨i j m ||w ||k c d ⟩

J J

0

⟨c |t1|l ⟩

00

⟨d |t1|m ⟩

00

(H hhhh

F)+ 1

2Pk l (J )

−1k−1

lJ−1

∑

c d e m

⟨i j m ||w ||c d e ⟩

J J

0

⟨c |t1|k ⟩

00

⟨d |t1|l ⟩

00

⟨e |t1|m ⟩

00

⟨a j |H |ib ⟩

J M J M

=

H phhp

A

+ 12(−1)jb+j i−J

∑

c d k

∑

J ′ J ′′(−1)J+J ′+J ′′ J−1 J ′ J ′′

×¦

J J ′ J ′′jk jb j i

©

6j⟨a j k ||w ||c d b ⟩

J J ′

J ′′

⟨c d |t2|i k ⟩

J ′M ′ J ′M ′

H phhp

B

+ 12

∑

c k l

∑

J ′ J ′′(−1)J+J ′+J ′′ J−1 J ′ J ′′

nJ J ′ J ′′jc j j ja

o

6j⟨k l ||w ||ib c ⟩

J ′ J

J ′′

⟨c a |t2|k l ⟩J ′M ′ J ′M ′

H phhp

C

+ J−1∑

c k

⟨a j k ||w ||ib c ⟩

J J

0

⟨c |t1|k ⟩

00

+ . . .


continued.



H phhp

D

+ CCAtoStd

a i

j bJ ; J ′

12−1

a

∑

c d k l

∑

J ′′ J ′′′( J ′)−1 J ′′ J ′′′

¦J ′ J ′′ J ′′′jk j l j i

©

6j

× ⟨k l ||w ||c d b ⟩

J ′′′ J ′′

J ′

⟨c d |t2|i k ⟩

J ′′M ′′ J ′′M ′′

⟨a |t1|l ⟩

00

H phhp

E

+ CCAtoStd

a i

j bJ ; J ′

12−1

i

∑

c d k l

∑

J ′′ J ′′′( J ′)−1 J ′′ J ′′′

¦J ′ J ′′ J ′′′jd jc ja

©

6j

× ⟨k l ||w ||c d b ⟩

J ′′ J ′′′

J ′

⟨d a |t2|k l ⟩

J ′′M ′′ J ′′M ′′

⟨c |t1|i ⟩

00

H phhp

F

− CCAtoStd

a i

j bJ ; J ′

12

∑

c d k l

−1l

∑

J ′′ J ′′′(−1)J

′+J ′′+J ′′′ ( J ′)−1 J ′′ J ′′′

×¦

J ′ J ′′ J ′′′jd jc jk

©

6j⟨k l ||w ||d cb ⟩

J ′′ J ′′′

J ′

⟨a c |t2|i k ⟩

J ′M ′

J ′M ′

⟨d |t1|l ⟩

00

H phhp

G

− −1i J−1

∑

c d k

⟨a j k ||w ||cb d ⟩

J J

0

⟨c |t1|i ⟩

00

⟨d |t1|k ⟩

00

H phhp

H

− −1a J−1

∑

c k l

⟨k j l ||w ||ib c ⟩

J J

0

⟨a |t1|k ⟩

00

⟨c |t1|l ⟩

00

H phhp

I

+ CCAtoStd

a i

j bJ ; J ′

−1a −1

i

∑

c d k l

−1k

∑

J ′′ J ′′′( J ′)−1 J ′′ J ′′′

×¦

J ′ J ′′ J ′′′jd jc j l

©

6j⟨k l ||w ||c d b ⟩

J ′′ J ′′′

J ′

⟨d |t1|k ⟩

00

⟨c |t1|i ⟩

00

⟨a |t1|l ⟩

00


continued.



⟨ab |H |c i ⟩

J M J M

=

H ppph

A

+ J−1∑

d k

⟨ab k ||w ||c i d ⟩

J J

0

⟨d |t1|k ⟩

00

H ppph

B

− 12

Pab (J ) −1b

∑

d e k l

∑

J ′ J ′′(−1)J+J ′+J ′′ J−1 J ′ J ′′

¦J J ′ J ′′j l jc j i

©

6j

× ⟨a k l ||w ||d e c ⟩

J J ′

J ′′

⟨d e |t2|i l ⟩

J ′M ′ J ′M ′

⟨b |t1|k ⟩

00

H ppph

C

− 12

Pab (J ) −1i

∑

d e k l

∑

J ′ J ′′(−1)J+J ′+J ′′ J−1 J ′ J ′′

¦J J ′ J ′′je ja jb

©

6j

× ⟨k l a ||w ||c d e ⟩

J ′ J

J ′′

⟨e b |t2|k l ⟩

J ′M ′ J ′M ′

⟨d |t1|i ⟩

00

H ppph

D

− CCAtoStd(A)

ab

c iJ ; J ′

|Tc i =0

∑

d e k l

−1l

∑

J ′′ J ′′′(−1)J

′+J ′′+J ′′′

× ( J ′)−1 J ′′ J ′′′¦

J ′ J ′′ J ′′′jd je jk

©

6j⟨k l a ||w ||d e c ⟩

J ′′ J ′′′

J ′

⟨e b |t2|k i ⟩

J ′M ′

J ′M ′

⟨d |t1|l ⟩

00

H ppph

E

+ 12

J−1∑

d k l m

⟨k l m ||w ||c i d ⟩

J J

0

⟨ab |t2|k l ⟩

J M J M

⟨d |t1|m ⟩

00

H ppph

F

− 12

Pab (J ) (−1)ja+jb−J −1a

∑

d k l m

∑

J ′ J ′′(−1)J+J ′+J ′′ J−1 J ′ J ′′

×¦

J J ′ J ′′jd jm jb

©

6j⟨k l m ||w ||c i d ⟩

J ′ J

J ′′

⟨d b |t2|k l ⟩

J ′M ′ J ′M ′

⟨a |t1|m ⟩

00

+ . . .


continued.



H ppph

G

− −1i J−1

∑

d e k

⟨ab k ||w ||c d e ⟩

J J

0

⟨e |t1|k ⟩

00

⟨d |t1|i ⟩

00

H ppph

H

− Pab (J ) (−1)ja+jb−J −1a J−1

∑

d k l

⟨b k l ||w ||c i d ⟩

J J

0

⟨a |t1|k ⟩

00

⟨d |t1|l ⟩

00

H ppph

I

+ 12

∑

d e k l m

−1c −1

d−1

e

∑

J ′J ′ ⟨k l m ||w ||d e c ⟩

J ′ J ′

0

⟨ab |t2|m i ⟩

J M J M

⟨d |t1|k ⟩

00

⟨e |t1|l ⟩

00

H ppph

J

− CCAtoStd(A)

ab

c iJ ; J ′

|Tc i =0

−1a

∑

d e k l m

−1l

∑

J ′′ J ′′′(−1)J

′+J ′′+J ′′′

× ( J ′)−1 J ′′ J ′′′¦

J ′ J ′′ J ′′′j l je jk

©

6j⟨k l m ||w ||d e c ⟩

J ′′ J ′′′

J ′

⟨e b |t2|k i ⟩

J ′M ′

J ′M ′

⟨d |t1|l ⟩

00

⟨a |t1|m ⟩

00

H ppph

K

+ 14

Pab (J ) (−1)ja+jb−J −1a −1

b

∑

d e k l m

∑

J ′ J ′′(−1)J+J ′+J ′′ J−1 J ′ J ′′

×¦

J J ′ J ′′jm jc j i

©

6j⟨k l m ||w ||d e c ⟩

J J ′

J ′′

⟨d e |t2|i m ⟩

J ′M ′ J ′M ′

⟨a |t1|k ⟩

00

⟨b |t1|l ⟩

00

H ppph

L

− 12

Pab (J ) −1b−1

i

∑

d e k l m

∑

J ′ J ′′(−1)J+J ′+J ′′ J−1 J ′ J ′′

¦J J ′ J ′′je jm ja

©

6j

× ⟨k l m ||w ||c d e ⟩

J ′ J

J ′′

⟨e a |t2|k l ⟩J ′M ′ J ′M ′

⟨b |t1|m ⟩

00

⟨d |t1|i ⟩

00

H ppph

M

+ 12−1

i J−1∑

d e k l m

⟨k l m ||w ||c d e ⟩

J J

0

⟨ab |t2|k l ⟩

J M J M

⟨e |t1|m ⟩

00

⟨d |t1|i ⟩

00

+ . . .


continued.



H ppph

N

− Pab (J ) −1b−1

i J−1∑

d e k l

⟨a k l ||w ||c d e ⟩

J J

0

⟨e |t1|l ⟩

00

⟨b |t1|k ⟩

00

⟨d |t1|i ⟩

00

H ppph

O

− 12

Pab (J ) (−1)ja+jb−J −1a −1

bJ−1

×∑

d k l m

⟨k l m ||w ||c i d ⟩

J J

0

⟨d |t1|m ⟩

00

⟨b |t1|k ⟩

00

⟨a |t1|l ⟩

00

H ppph

P

+ 12

Pab (J ) (−1)ja+jb−J −1a −1

b−1

i J−1

×∑

d e k l m

⟨k l m ||w ||c d e ⟩

J J

0

⟨b |t1|k ⟩

00

⟨d |t1|i ⟩

00

⟨e |t1|m ⟩

00

⟨a |t1|l ⟩

00

H ppph

Q

+ 14

∑

d e k l m

−1c

∑

J ′J ′ ⟨k l m ||w ||d e c ⟩

J ′ J ′

0

⟨ab |t2|m i ⟩

J M J M

⟨d e |t2|k l ⟩

J ′M ′ J ′M ′

H ppph

R

− 14

∑

d e k l m

∑

J ′ J ′′(−1)J+J ′+J ′′ J−1 J ′ J ′′

×¦

J J ′ J ′′jm jc j i

©

6j⟨k l m ||w ||d e c ⟩

J J ′

J ′′

⟨ab |t2|k l ⟩

J M J M

⟨d e |t2|i m ⟩

J ′M ′ J ′M ′

H ppph

S

+ CCAtoStd(A)

ab

c iJ ; J ′

|Tc i =0

12

∑

d e k l m

(−1)je+jm−J ′∑

J ′′ J ′′′

× ( J ′)−1 J ′′ J ′′′¦

J ′ J ′′ J ′′′jd jc ja

©

6j⟨k l m ||w ||c d e ⟩

J ′′ J ′′′

J ′

⟨d a |t2|k l ⟩

J ′′M ′′ J ′′M ′′

⟨e b |t2|m i ⟩

J ′M ′

J ′M ′


continued.



⟨i a |H |j k ⟩

J M J M

=

H hphh

A

+ J−1∑

c l

⟨i a l ||w ||j k c ⟩

J J

0

⟨c |t1|l ⟩

00

H hphh

B

+ 12

Pj k (J ) −1k

∑

c d l m

∑

J ′ J ′′(−1)j i+jd−J ′′ J−1 J ′ J ′′

¦J J ′ J ′′

jd j i ja

©

6j

× ⟨l m ı ||w ||j c d ⟩

J ′ J

J ′′

⟨a d |t2|l m ⟩

J ′M ′ J ′M ′

⟨c |t1|k ⟩

00

H hphh

C

− 12

Pj k (J ) −1a

∑

c d l m

∑

J ′ J ′′(−1)J+J ′+J ′′ J−1 J ′ J ′′

nJ J ′ J ′′

jm j j jk

o

6j

× ⟨i l m ||w ||c d j ⟩

J J ′

J ′′

⟨c d |t2|k m ⟩

J ′M ′ J ′M ′

⟨a |t1|l ⟩

00

H hphh

D

− CCAtoStd(A)

i a

j kJ ; J ′

|Ti a=0

∑

c d l m

−1l

∑

J ′′ J ′′′(−1)J

′+J ′′+J ′′′

× ( J ′)−1 J ′′ J ′′′¦

J ′ J ′′ J ′′′jc jd jm

©

6j⟨m l ı ||w ||c d j ⟩

J ′′ J ′′′

J ′

⟨a d |t2|k m ⟩

J ′M ′

J ′M ′

⟨c |t1|l ⟩

00

H hphh

E

− 12

J−1∑

c d e l

⟨i a l ||w ||c d e ⟩

J J

0

⟨c d |t2|j k ⟩

J M J M

⟨e |t1|l ⟩

00

H hphh

F

+ 12

Pj k (J ) −1j

∑

c d e l

∑

J ′ J ′′(−1)J+J ′+J ′′ J−1 J ′ J ′′

¦J J ′ J ′′j l je jk

©

6j

× ⟨i a l ||w ||c d e ⟩

J J ′

J ′′

⟨c d |t2|k l ⟩

J ′M ′ J ′M ′

⟨e |t1|j ⟩

00


continued.



H hphh

G

+ −1a J−1

∑

c l m

⟨i l m ||w ||j k c ⟩

J J

0

⟨c |t1|m ⟩

00

⟨a |t1|l ⟩

00

H hphh

H

+ Pj k (J ) −1j J−1

∑

c d l

⟨i a l ||w ||k c d ⟩

J J

0

⟨c |t1|j ⟩

00

⟨d |t1|l ⟩

00

H hphh

I

− 12−1

i

∑

c d e l m

−1c −1

d

∑

J ′J ′ ⟨l m ı ||w ||c d e ⟩

J ′ J ′

0

⟨e a |t2|j k ⟩J M J M

⟨c |t1|l ⟩

00

⟨d |t1|m ⟩

00

H hphh

J

+ CCAtoStd(A)

i a

j kJ ; J ′

|Ti a=0

−1j

∑

c d e l m

−1l

∑

J ′′ J ′′′(−1)J

′+J ′′+J ′′′

× ( J ′)−1 J ′′ J ′′′¦

J ′ J ′′ J ′′′jc jd jm

©

6j⟨m l ı ||w ||c d e ⟩

J ′′ J ′′′

J ′

⟨a d |t2|k m ⟩

J ′M ′

J ′M ′

⟨c |t1|l ⟩

00

⟨e |t1|j ⟩

00

H hphh

K

− 12

Pj k (J ) −1j −1

k

∑

c d e l m

∑

J ′ J ′′(−1)j i+je−J ′′ J−1 J ′ J ′′

×¦

J J ′ J ′′je j i ja

©

6j⟨m l ı ||w ||c d e ⟩

J ′ J

J ′′

⟨a e |t2|l m ⟩J ′M ′ J ′M ′

⟨c |t1|j ⟩

00

⟨d |t1|k ⟩

00

H hphh

L

− 12

Pj k (J ) −1a −1

k(−1)jk+j j−J

∑

c d e l m

∑

J ′ J ′′(−1)J+J ′+J ′′ J−1 J ′ J ′′

×n

J J ′ J ′′jm je j j

o

6j⟨i l m ||w ||c d e ⟩

J J ′

J ′′

⟨c d |t2|j m ⟩

J ′M ′ J ′M ′

⟨e |t1|k ⟩

00

⟨a |t1|l ⟩

00

+ . . .


continued.



H hphh

M

− CCAtoStd(A)

i a

j kJ ; J ′

|Ti a=0

−1j

∑

c d e l m

−1l(−1)je+jm−J ′

∑

J ′′ J ′′′

× ( J ′)−1 J ′′ J ′′′¦

J ′ J ′′ J ′′′jd jc j i

©

6j⟨l i m ||w ||c d e ⟩

J ′′ J ′′′

J ′

⟨a e |t2|k m ⟩

J ′M ′

J ′M ′

⟨d |t1|l ⟩

00

⟨c |t1|j ⟩

00

H hphh

N

− Pj k (J ) −1a −1

kJ−1

∑

c d l m

⟨i l m ||w ||j c d ⟩

J J

0

⟨c |t1|k ⟩

00

⟨d |t1|m ⟩

00

⟨a |t1|l ⟩

00

H hphh

O

+ 12

Pj k (J ) −1j −1

kJ−1

∑

c d e l

⟨i a l ||w ||c d e ⟩

J J

0

⟨c |t1|j ⟩

00

⟨d |t1|k ⟩

00

⟨e |t1|l ⟩

00

H hphh

P

− 12

Pj k (J ) (−1)j j+jk−J −1a J−1

∑

c d e l m

−1c −1

d

× ⟨i l m ||w ||c d e ⟩

J J

0

⟨c |t1|k ⟩

00

⟨d |t1|j ⟩

00

⟨e |t1|m ⟩

00

⟨a |t1|l ⟩

00

H hphh

Q

− 14−1

i

∑

c d e l m

∑

J ′J ′ ⟨l m ı ||w ||c d e ⟩

J ′ J ′

0

⟨e a |t2|j k ⟩J M J M

⟨c d |t2|l m ⟩

J ′M ′ J ′M ′

H hphh

R

− 14

∑

c d e l m

∑

J ′ J ′′(−1)je+j i−J ′′ J−1 J ′ J ′′

¦J J ′ J ′′je j i ja

©

6j

× ⟨l m ı ||w ||c d e ⟩

J ′ J

J ′′

⟨c d |t2|j k ⟩

J M J M

⟨a e |t2|l m ⟩J ′M ′ J ′M ′

+ . . .


continued.



H hphh

S

− CCAtoStd(A)

i a

j kJ ; J ′

|Ti a=0

12

∑

c d e l m

(−1)je+jm−J ′∑

J ′′ J ′′′

× ( J ′)−1 J ′′ J ′′′n

J ′ J ′′ J ′′′j l j i j j

o

6j⟨l i m ||w ||c d e ⟩

J ′′′ J ′′

J ′

⟨c d |t2|j l ⟩

J ′′M ′′ J ′′M ′′

⟨e a |t2|m k ⟩

J ′M ′

J ′M ′


continued.


Appendix E

ΛCCSD Diagrams and SphericalExpressions

Appendix E. ΛCCSD Diagrams and Spherical Expressions

E.1 Diagrams

b b b b b

Λ1A Λ1B Λ1C Λ1D

b b b b b b b b

Λ1E Λ1F Λ1G Λ1H

b b

b

b b b b

Λ2A Λ2B Λ2C Λ2D

b b b b b b

Λ2E Λ2F Λ2G Λ2H

b b b b b b

Λ2I Λ2J Λ2K

Figure E.1: ΛCCSD daiagrams for the Λ1 and Λ2 amplitude equations.



E.2 Spherical Equations

(Λ1A)+ ⟨ı |H1|a ⟩

00

(Λ1B)− −1

i

∑

b

⟨ı |λ1|b ⟩

00

⟨b |H1|a ⟩

00

(Λ1C)+ −1

i

∑

k

⟨ı |H1|k ⟩

00

⟨k |λ1|a ⟩

00

(Λ1D)+ −1

i

∑

c k

−1c

∑

J

J 2 ⟨c i |H2|k a ⟩

J M J M

⟨k |λ1|c ⟩

00

(Λ1E)− 1

2−1

i

∑

c d k

∑

J

J 2 ⟨i k |λ2|c d ⟩

J M J M

⟨c d |H2|a k ⟩

J M J M

(Λ1F)+ 1

2−1

i

∑

c k l

∑

J

J 2 ⟨i c |H2|k l ⟩

J M J M

⟨k l |λ2|a c ⟩

J M J M

(Λ1G)− 1

2−1

i

∑

c d e k l

−2d

∑

J J ′J 2 ( J ′)2 ⟨d i |H2|e a ⟩

J ′M ′J ′M ′

⟨c e |t2|k l ⟩J M J M

⟨k l |λ2|c d ⟩

J M J M

δjd je

(Λ1H)+ 1

2−1

i

∑

c d k l m

−2l

∑

J J ′J 2 ( J ′)2 ⟨k l |λ2|c d ⟩

J M J M

⟨c d |t2|k m ⟩

J M J M

⟨m i |H2|l a ⟩

J ′M ′ J ′M ′

δjm j l

= 0 , ∀ a , i

Figure E.2: Spherical expressions for the ΛCCSD Λ1 amplitude equations.



(Λ2A)+ ⟨i j |H2|ab ⟩

J M J M

(Λ2B)+ CCAtoStd(A)

i j

abJ ; J ′

⟨ı |H1|a ⟩

00

⟨ |λ1|b ⟩

00

δJ ′0

(Λ2C)− 1

2Pab (J ) Pi j (J )

−1i

∑

c

⟨ı |λ1|c ⟩

00

⟨c j |H1|ab ⟩

J M J M

δj i jc

(Λ2D)+ 1

2Pab (J ) Pi j (J )

−1a

∑

k

⟨k |λ1|a ⟩

00

⟨i j |H2|kb ⟩

J M J M

δja jk

(Λ2E)− 1

2Pab (J ) Pi j (J )

−1b

∑

c

⟨i j |λ2|a c ⟩

J M J M

⟨c |H1|b ⟩

00

δjb jc

(Λ2F)+ 1

2Pab (J ) Pi j (J )

−1j

∑

k

⟨i k |λ2|ab ⟩

J M J M

⟨ |H1|k ⟩

00

δj j jk

(Λ2G)− CCAtoStd(A)

i j

abJ ; J ′

∑

c k

(−1)jc+jk−J ′ ⟨ı k |λ2|a c ⟩

J ′M ′

J ′M ′

⟨c |H2|kb ⟩

J ′M ′

J ′M ′

(Λ2H)+ 1

8Pab (J ) Pi j (J )

∑

c d

⟨i j |λ2|c d ⟩

J M J M

⟨c d |H1|ab ⟩

J M J M

(Λ2I)+ 1

8Pab (J ) Pi j (J )

∑

k l

⟨i j |H2|k l ⟩

J M J M

⟨k l |λ2|ab ⟩

J M J M

(Λ2J)− 1

4Pab (J ) Pi j (J )

−2a

∑

c d k l

∑

J ′( J ′)2 ⟨c d |t2|k l ⟩

J ′M ′ J ′M ′

⟨k l |λ2|c a ⟩

J ′M ′J ′M ′

⟨i j |v |d b ⟩

J M J M

δja jd

(Λ2K)− 1

4Pab (J ) Pi j (J )

−2i

∑

c d k l

∑

J ′( J ′)2 ⟨k i |λ2|c d ⟩

J ′M ′ J ′M ′

⟨c d |t2|k l ⟩

J ′M ′ J ′M ′

⟨l j |v |ab ⟩

J M J M

δj i j l

= 0 , ∀ a ,b , i , j , J , M

Figure E.3: Spherical expressions for the ΛCCSD Λ2 amplitude equations.



E.3 Spherical Equations for Three-Body Hamiltonians

(Λ3B1

A)+ 1

4

∑

c d k l

∑

J

J ⟨c d ı ||w ||k l a ⟩

J J

0

⟨k l |λ2|c d ⟩

J M J M

(Λ3B1

B)+ 1

8

∑

c d e f k l

∑

J

J ⟨c d ı ||w ||e f a ⟩

J J

0

⟨k l |λ2|c d ⟩

J M J M

⟨e f |t2|k l ⟩

J M J M

(Λ3B1

C)+ −1

i

∑

c d e k l m

∑

J J ′ J ′′(−1)jc+jk−J (−1)jd+j l−J (−1)je+jm−J J J ′ J ′′

×¦

J J ′ J ′′ja j l jd

©

6j⟨i d m ||w ||l a e ⟩

J ′ J ′′

J

⟨k l |λ2|c d ⟩

J M

J M

⟨c e |t2|k m ⟩

J M

J M

(Λ3B1

D)+ 1

8

∑

c d k l m n

∑

J

J ⟨m n ı ||w ||k l a ⟩J J

0

⟨k l |λ2|c d ⟩

J M J M

⟨c d |t2|m n⟩

J M J M

(Λ3B1

E)− 1

4

∑

c d e f k l m n

∑

J J ′ J ′′J J ′ ( J ′′)2

nJ J ′ J ′′jc j f jd

o

6j

nJ J ′ J ′′jc j f je

o

6j

× ⟨m n ı ||w ||e f a ⟩J J

0

⟨k l |λ2|c d ⟩

J M J M

⟨d f |t2|m n⟩

J M J M

⟨c e |t2|k l ⟩J ′M ′ J ′M ′

(Λ3B1

F)+ 1

2−1

i

∑

c d e f k l m n

∑

J J ′ J ′′(−1)jc+jk−J (−1)jd+j l−J (−1)j f +jn−J (−1)J+J ′+J ′′

× J J ′ J ′′¦

J J ′ J ′′j i je jm

©

6j⟨m i n ||w ||a e f ⟩

J ′ J ′′

J

⟨k l |λ2|c d ⟩

J M

J M

⟨c e |t2|k m ⟩

J M

J M

⟨d f |t2|l n⟩

J M

J M

+ . . .

Figure E.4: Spherical expressions for the ΛCCSD Λ1 amplitude equations for three-body Hamilto-

nians.



(Λ3B1

G)+ 1

16

∑

c d e f k l m n

∑

J

J ⟨m n ı ||w ||e f a ⟩J J

0

⟨k l |λ2|c d ⟩

J M J M

⟨c d |t2|m n⟩

J M J M

⟨e f |t2|k l ⟩

J M J M

(Λ3B1

H)− 1

4

∑

c d e f k l m n

∑

J J ′ J ′′J 2 J ′ ( J ′′)2

×¦

J J ′ J ′′jn jk j l

©

6j

¦J J ′ J ′′

jn jk jm

©

6j⟨m n ı ||w ||e f a ⟩

J ′ J ′

0

⟨k l |λ2|c d ⟩

J M J M

⟨c d |t2|k m ⟩

J M J M

⟨e f |t2|l n⟩

J M J M

(Λ3B1

I)− 1

2

∑

c d e k l

−1k

∑

J

J ⟨c d ı ||w ||e l a ⟩

J J

0

⟨k l |λ2|c d ⟩

J M J M

⟨e |t1|k ⟩

00

(Λ3B1

J)+ 1

2

∑

c d k l m

−1c

∑

J

J ⟨m d ı ||w ||k l a ⟩

J J

0

⟨k l |λ2|c d ⟩

J M J M

⟨c |t1|m ⟩

00

(Λ3B1

K)+ 1

4

∑

c d e f k l m

−1c

∑

J

J ⟨m d ı ||w ||e f a ⟩

J J

0

⟨k l |λ2|c d ⟩

J M J M

⟨e f |t2|k l ⟩

J M J M

⟨c |t1|m ⟩

00

(Λ3B1

L)+ −1

i

∑

c d e f k l m

−1k

∑

J J ′ J ′′(−1)jd+j l−J (−1)j f +jm−J (−1)J+J ′+J ′′

× J J ′ J ′′¦

J J ′ J ′′j i jk jc

©

6j⟨c i m ||w ||a e f ⟩

J ′′ J ′

J

⟨k l |λ2|c d ⟩

J M

J M

⟨d f |t2|l m ⟩

J M

J M

⟨e |t1|k ⟩

00

(Λ3B1

M)− 1

4

∑

c d e k l m n

−1k

∑

J

J ⟨m n ı ||w ||e l a ⟩J J

0

⟨k l |λ2|c d ⟩

J M J M

⟨c d |t2|m n⟩

J M J M

⟨e |t1|k ⟩

00

+ . . .


nians, continued.



(Λ3B1

N)− −1

i

∑

c d e k l m n

−1c

∑

J J ′ J ′′(−1)jd+j l−J (−1)je+jn−J

× J J ′ J ′′¦

J J ′ J ′′j i jk jm

©

6j⟨i m n ||w ||a k e ⟩

J ′ J ′′

J

⟨k l |λ2|c d ⟩

J M

J M

⟨d e |t2|l n⟩

J M

J M

⟨c |t1|m ⟩

00

(Λ3B1

O)− 1

4−1

i

∑

c d e f k l m

−1c −1

m

∑

J J ′J 2 J ′

× ⟨i m c ||w ||a e f ⟩

J ′ J ′

0

⟨k l |λ2|c d ⟩

J M J M

⟨ f d |t2|k l ⟩

J M J M

⟨e |t1|m ⟩

00

(Λ3B1

P)+ 1

2−1

i

∑

c d e k l m n

−1l−1

m

∑

J J ′J 2 J ′

× ⟨i m n ||w ||a e l ⟩

J ′ J ′

0

⟨k l |λ2|c d ⟩

J M J M

⟨c d |t2|k n⟩

J M J M

⟨e |t1|m ⟩

00

(Λ3B1

Q)+ 1

4

∑

c d e f k l

−1e −1

f

∑

J

J ⟨c d ı ||w ||e f a ⟩

J J

0

⟨k l |λ2|c d ⟩

J M J M

⟨e |t1|k ⟩

00

⟨ f |t1|l ⟩

00

(Λ3B1

R)−

∑

c d e k l m

−1c −1

k

∑

J

J ⟨m d ı ||w ||e l a ⟩

J J

0

⟨k l |λ2|c d ⟩

J M J M

⟨c |t1|m ⟩

00

⟨e |t1|k ⟩

00

(Λ3B1

S)+ 1

4

∑

c d k l m n

−1c −1

d

∑

J

J ⟨m n ı ||w ||k l a ⟩J J

0

⟨k l |λ2|c d ⟩

J M J M

⟨c |t1|m ⟩

00

⟨d |t1|n⟩

00

(Λ3B1

T)+ 1

8

∑

c d e f k l m n

−1c −1

d

∑

J

J ⟨m n ı ||w ||e f a ⟩J J

0

⟨k l |λ2|c d ⟩

J M J M

⟨e f |t2|k l ⟩

J M J M

⟨c |t1|m ⟩

00

⟨d |t1|n⟩

00

+ . . .


nians, continued.



(Λ3B1

U)+

∑

c d e f k l m n

−1c −1

e −1i

∑

J J ′ J ′′J J ′ J ′′ (−1)J+J ′+J ′′ (−1)jd+j l−J (−1)j f +jn−J

×¦

J J ′ J ′′j i je jm

©

6j⟨m i n ||w ||a e f ⟩

J ′ J ′′

J

⟨k l |λ2|c d ⟩

J M

J M

⟨ f d |t2|nl ⟩

J M

J M

⟨c |t1|m ⟩

00

⟨e |t1|k ⟩

00

(Λ3B1

V)+ 1

8

∑

c d e f k l m n

−1e −1

f

∑

J

J

× ⟨m n ı ||w ||e f a ⟩J J

0

⟨k l |λ2|c d ⟩

J M J M

⟨c d |t2|m n⟩

J M J M

⟨e |t1|k ⟩

00

⟨ f |t1|l ⟩

00

(Λ3B1

W)+ 1

2

∑

c d e f k l m n

−1c −1

f−1

m −1n

∑

J J ′J 2 J ′

× ⟨m n ı ||w ||e f a ⟩J ′ J ′

0

⟨k l |λ2|d c ⟩

J M J M

⟨d f |t2|k l ⟩

J M J M

⟨c |t1|n⟩

00

⟨e |t1|m ⟩

00

(Λ3B1

X)− 1

2

∑

c d e f k l m n

−1k−1

m −2f

∑

J J ′J 2 J ′

× ⟨m n ı ||w ||e f a ⟩J ′ J ′

0

⟨k l |λ2|c d ⟩

J M J M

⟨c d |t2|nl ⟩

J M J M

⟨e |t1|m ⟩

00

⟨ f |t1|k ⟩

00

(Λ3B1

Y)+ 1

2

∑

c d e f k l m

−1c −1

e −1f

∑

J

J

× ⟨m d ı ||w ||e f a ⟩

J J

0

⟨k l |λ2|c d ⟩

J M J M

⟨c |t1|m ⟩

00

⟨e |t1|k ⟩

00

⟨ f |t1|l ⟩

00

+ . . .


nians, continued.



(Λ3B1

Z)− 1

2

∑

c d e k l m n

−1k−1

c −1d

∑

J

J

× ⟨m n ı ||w ||e l a ⟩J J

0

⟨k l |λ2|c d ⟩

J M J M

⟨c |t1|m ⟩

00

⟨d |t1|n⟩

00

⟨e |t1|k ⟩

00

(Λ3B1

AA)+ 1

4

∑

c d e f k l m n

−1c −1

d−1

e −1f

∑

J

J

× ⟨m n ı ||w ||e f a ⟩J J

0

⟨k l |λ2|c d ⟩

J M J M

⟨c |t1|m ⟩

00

⟨d |t1|n⟩

00

⟨e |t1|k ⟩

00

⟨ f |t1|l ⟩

00

+ ⟨ı |λ1|a ⟩

00

[NO2B] = 0 , ∀ a , i


nians, continued.



Λ3B2,J

A

− 14

Pab (J ) Pi j (J ) (−1)ja+jb−J∑

c k l

∑

J ′ J ′′(−1)J+J ′+J ′′ J−1 J ′ J ′′

×¦

J J ′ J ′′jc jb ja

©

6j⟨i j c ||w ||k l b ⟩

J J ′

J ′′

⟨k l |λ2|a c ⟩

J ′M ′J ′M ′

Λ3B2,J

B

− 12


c d k l m

∑

J ′ J ′′(−1)jc+j l−J ′ (−1)jd+jm−J ′ (−1)J+J ′+J ′′

× J−1 J ′ J ′′¦

J J ′ J ′′jk jb ja

©

6j⟨i j m ||w ||b k d ⟩

J J ′′

J ′

⟨k l |λ2|a c ⟩

J ′M ′

J ′M ′

⟨c d |t2|l m ⟩

J ′M ′

J ′M ′

Λ3B2,J

C

− 18


c d e k l

∑

J ′ J ′′(−1)J+J ′+J ′′ J−1 J ′ J ′′

×¦


©

6j⟨k l |λ2|a c ⟩

J ′M ′J ′M ′

⟨i j c ||w ||d e b ⟩

J J ′

J ′′

⟨d e |t2|k l ⟩

J ′M ′ J ′M ′

Λ3B2,J

D

+ 12


c d k l

−1k

∑

J ′ J ′′(−1)J+J ′+J ′′ J−1 J ′ J ′′

×¦


©

6j⟨i j c ||w ||d l b ⟩

J J ′

J ′′

⟨k l |λ2|a c ⟩

J ′M ′J ′M ′

⟨d |t1|k ⟩

00

Λ3B2,J

E

− 14


c k l m

−1c

∑

J ′ J ′′(−1)J+J ′+J ′′ J−1 J ′ J ′′

×¦

J J ′ J ′′jm jb ja

©

6j⟨i j m ||w ||k l b ⟩

J J ′

J ′′

⟨k l |λ2|a c ⟩

J ′M ′J ′M ′

⟨c |t1|m ⟩

00

+ . . .


nians.



Λ3B2,J

F

− 14


c d e k l m

∑

J ′ J ′′J−1 ( J ′)2 ( J ′′)2

¦J J ′ J ′′jc jb ja

©

6j

¦J J ′ J ′′jc jb jd

©

6j

× ⟨i j m ||w ||d b e ⟩

J J

0

⟨k l |λ2|c a ⟩

J ′M ′J ′M ′

⟨c d |t2|k l ⟩

J ′M ′ J ′M ′

⟨e |t1|m ⟩

00

Λ3B2,J

G

+ 12


c d e k l m

−1k

∑

J ′ J ′′(−1)jc+j l−J ′ (−1)je+jm−J ′ (−1)J+J ′+J ′′

× J−1 J ′ J ′′¦

J J ′ J ′′jk jb ja

©

6j⟨i j m ||w ||b d e ⟩

J J ′′

J ′

⟨k l |λ2|a c ⟩

J ′M ′

J ′M ′

⟨c e |t2|l m ⟩

J ′M ′

J ′M ′

⟨d |t1|k ⟩

00

Λ3B2,J

H

− 18


c d e k l m

−1c

∑

J ′ J ′′(−1)J+J ′+J ′′ J−1 J ′ J ′′

×¦


©

6j⟨i j m ||w ||d e b ⟩

J J ′

J ′′

⟨k l |λ2|a c ⟩

J ′M ′J ′M ′

⟨d e |t2|k l ⟩

J ′M ′ J ′M ′

⟨c |t1|m ⟩

00

Λ3B2,J

I

+ 12


c d k l m

−1c −1

d

∑

J ′ J ′′(−1)J+J ′+J ′′ J−1 J ′ J ′′

×¦


©

6j⟨i j m ||w ||d l b ⟩

J J ′

J ′′

⟨k l |λ2|a c ⟩

J ′M ′J ′M ′

⟨d |t1|k ⟩

00

⟨c |t1|m ⟩

00

Λ3B2,J

J

− 14


c d e k l

−1d−1

e

∑

J ′ J ′′(−1)J+J ′+J ′′ J−1 J ′ J ′′

×¦


©

6j⟨i j c ||w ||d e b ⟩

J J ′

J ′′

⟨k l |λ2|a c ⟩

J ′M ′J ′M ′

⟨d |t1|k ⟩

00

⟨e |t1|l ⟩

00

+ . . .

Figure E.10: Spherical expressions for the ΛCCSD Λ2 amplitude equations for three-body Hamil-

tonians, continued.



Λ3B2,J

K

− 14


c d e k l m

−1k−1

l−1

m

∑

J ′ J ′′(−1)J+J ′+J ′′ J−1 J ′ J ′′

×¦


©

6j⟨i j m ||w ||d e b ⟩

J J ′

J ′′

⟨k l |λ2|a c ⟩

J ′M ′J ′M ′

⟨d |t1|k ⟩

00

⟨e |t1|l ⟩

00

⟨c |t1|m ⟩

00

Λ3B2,K

A

+ 14


c d k

∑

J ′ J ′′(−1)J+J ′+J ′′ J−1 J ′ J ′′

nJ J ′ J ′′

jk j j j i

o

6j

× ⟨c d ||w ||ab k ⟩

J ′ J

J ′′

⟨k i |λ2|c d ⟩

J ′M ′ J ′M ′

Λ3B2,K

B

+ 12

Pab (J ) Pi j (J ) (−1)j i+j j−J∑

c d e k l

∑

J ′ J ′′(−1)jd+jk−J ′ (−1)je+j l−J ′

× J−1 J ′ J ′′n

J J ′ J ′′jc j j j i

o

6j⟨c j l ||w ||ab e ⟩

J ′′ J

J ′

⟨i k |λ2|c d ⟩

J ′M ′

J ′M ′

⟨d e |t2|k l ⟩

J ′M ′

J ′M ′

Λ3B2,K

C

+ 18


c d k l m

∑

J ′ J ′′(−1)J+J ′+J ′′ J−1 J ′ J ′′

×n

J J ′ J ′′jk j j j i

o

6j⟨l m ||w ||ab k ⟩

J ′ J

J ′′

⟨k i |λ2|c d ⟩

J ′M ′ J ′M ′

⟨c d |t2|l m ⟩

J ′M ′ J ′M ′

Λ3B2,K

D

− 12

Pab (J ) Pi j (J ) (−1)j i+j j−J∑

c d k l

−1c

∑

J ′ J ′′(−1)jd+jk−J ′ (−1)J+J ′+J ′′

× J−1 J ′ J ′′n

J J ′ J ′′j l j j j i

o

6j⟨l j d ||w ||ab k ⟩

J ′′ J

J ′

⟨ı k |λ2|c d ⟩

J ′M ′

J ′M ′

⟨c |t1|l ⟩

00

+ . . .


tonians, continued.



Λ3B2,K

E

− 14


c d e k

−1k

∑

J ′ J ′′(−1)J+J ′+J ′′ J−1 J ′ J ′′

×n


o

6j⟨c d ||w ||ab e ⟩

J ′ J

J ′′

⟨k i |λ2|c d ⟩

J ′M ′ J ′M ′

⟨e |t1|k ⟩

00

Λ3B2,K

F

− 14

Pab (J ) Pi j (J ) −1i J−1

∑

c d e k l m

−1l

∑

J ′( J ′)2

× ⟨l j m ||w ||ab e ⟩

J J

0

⟨i k |λ2|c d ⟩

J ′M ′ J ′M ′

⟨c d |t2|l k ⟩

J ′M ′ J ′M ′

⟨e |t1|m ⟩

00

Λ3B2,K

G

+ 12

Pab (J ) Pi j (J ) (−1)j i+j j−J∑

c d e k l m

−1c

∑

J ′ J ′′(−1)je+jm−J ′ (−1)jd+jk−J ′

× (−1)J+J ′+J ′′ J−1 J ′ J ′′n

J J ′ J ′′j l j j j i

o

6j⟨l j m ||w ||ab e ⟩

J ′′ J

J ′

⟨ı k |λ2|c d ⟩

J ′M ′

J ′M ′

⟨d e |t2|k m ⟩

J ′M ′

J ′M ′

⟨c |t1|l ⟩

00

Λ3B2,K

H

− 18


c d e k l m

−1k

∑

J ′ J ′′(−1)J+J ′+J ′′ J−1 J ′ J ′′

×n


o

6j⟨l m ||w ||ab e ⟩

J ′ J

J ′′

⟨k i |λ2|c d ⟩

J ′M ′ J ′M ′

⟨c d |t2|l m ⟩

J ′M ′ J ′M ′

⟨e |t1|k ⟩

00

Λ3B2,K

I

+ 12

Pab (J ) Pi j (J ) (−1)j i+j j−J∑

c d e k l

−1k−1

l

∑

J ′ J ′′(−1)jk+jd−J ′

× (−1)J+J ′+J ′′ J−1 J ′ J ′′n

J J ′ J ′′jc j j j i

o

6j⟨c j l ||w ||ab e ⟩

J ′′ J

J ′

⟨ı k |λ2|c d ⟩

J ′M ′

J ′M ′

⟨d |t1|l ⟩

00

⟨e |t1|k ⟩

00

+ . . .


tonians, continued.



Λ3B2,K

J

+ 14


c d k l m

−1c −1

d

∑

J ′ J ′′(−1)J+J ′+J ′′ J−1 J ′ J ′′

nJ J ′ J ′′

jk j j j i

o

6j

× ⟨l m ||w ||ab k ⟩

J ′ J

J ′′

⟨k i |λ2|c d ⟩

J ′M ′ J ′M ′

⟨c |t1|l ⟩

00

⟨d |t1|m ⟩

00

Λ3B2,K

K

− 14


c d e k l m

−1k−1

l−1

m

∑

J ′ J ′′(−1)J+J ′+J ′′ J−1 J ′ J ′′

×n


o

6j⟨l m ||w ||ab e ⟩

J ′ J

J ′′

⟨k i |λ2|c d ⟩

J ′M ′ J ′M ′

⟨c |t1|l ⟩

00

⟨d |t1|m ⟩

00

⟨e |t1|k ⟩

00

Λ3B2,L

A

+ 14

Pab (J ) Pi j (J ) J−1∑

c k

⟨i j c ||w ||ab k ⟩

J J

0

⟨k |λ1|c ⟩

00

Λ3B2,L

B

− 14

Pab (J ) Pi j (J ) J−1∑

c d k

−1c ⟨i j c ||w ||ab d ⟩

J J

0

⟨k |λ1|c ⟩

00

⟨d |t1|k ⟩

00

Λ3B2,L

C

+ 14

Pab (J ) Pi j (J ) J−1∑

c k l

−1c ⟨i j l ||w ||ab k ⟩

J J

0

⟨k |λ1|c ⟩

00

⟨c |t1|l ⟩

00

Λ3B2,L

D

+ 14

Pab (J ) Pi j (J ) J−1∑

c d k l

−1c −1

k⟨i j l ||w ||ab d ⟩

J J

0

⟨k |λ1|c ⟩

00

⟨c |t1|l ⟩

00

⟨d |t1|k ⟩

00

Λ3B2,L

E

+ 14

Pab (J ) Pi j (J ) J−1∑

c d k l

⟨i j l ||w ||ab d ⟩

J J

0

⟨k |λ1|c ⟩

00

⟨c d |t2|k l ⟩

00

00

+ . . .


tonians, continued.



Λ3B2,M

A

− 18


c d e k l

−1c

∑

J ′J−1 ( J ′)2 ⟨i j c ||w ||ab e ⟩

J J

0

⟨k l |λ2|c d ⟩

J ′M ′ J ′M ′

⟨e d |t2|k l ⟩

J ′M ′ J ′M ′

Λ3B2,M

B

+ 18


c d k l m

−1k

∑

J J ′J−1 ( J ′)2 ⟨i j m ||w ||ab k ⟩

J J

0

⟨k l |λ2|c d ⟩

J ′M ′ J ′M ′

⟨c d |t2|m l ⟩

J ′M ′ J ′M ′

Λ3B2,M

C

− 18


c d e k l m

−1c −1

m

∑

J ′J−1 ( J ′)2

× ⟨i j m ||w ||ab e ⟩

J J

0

⟨k l |λ2|c d ⟩

J ′M ′ J ′M ′

⟨e d |t2|k l ⟩

J ′M ′ J ′M ′

⟨c |t1|m ⟩

00

Λ3B2,M

D

− 18


c d e k l m

−1e −1

k

∑

J ′J−1 ( J ′)2

× ⟨i j m ||w ||ab e ⟩

J J

0

⟨k l |λ2|c d ⟩

J ′M ′ J ′M ′

⟨c d |t2|m l ⟩

J ′M ′ J ′M ′

⟨e |t1|k ⟩

00

+ ⟨i j |λ2|ab ⟩

J M J M

[NO2B] = 0 , ∀ a ,b , i , j , J , M


tonians, continued.


Appendix F

Spherical Reduced Density Matrix

Appendix F. Spherical Reduced Density Matrix

⟨ı |γ(1)N |a ⟩

00

=

γhpN

A

+ ⟨ı |λ1|a ⟩

00

⟨a |γ(1)N |i ⟩

00

=

γphN

A

+ ⟨a |t1|i ⟩

00

γphN

B

+∑

b j

⟨a b |t2|i j ⟩

00

00

⟨ |λ2|b ⟩

00

γphN

C

− −2a

∑

b j

⟨a |t1|j ⟩

00

⟨ |λ1|b ⟩

00

⟨b |t1|i ⟩

00

γphN

D

− 12−2

a

∑

b c j k

∑

J

J 2 ⟨a |t1|k ⟩

00

⟨j k |λ2|b c ⟩

J M J M

⟨b c |t2|j i ⟩

J M J M

γphN

E

− 12−2

a

∑

b c j k

∑

J

J 2 ⟨b a |t2|j k ⟩

J M J M

⟨j k |λ2|b c ⟩

J M J M

⟨c |t1|i ⟩

00

⟨b |γ(1)N |a ⟩

00

=(γ

ppN

A)− −1

a

∑

j

⟨b |t1|j ⟩

00

⟨ |λ2|a ⟩

00

(γppN

B)− 1

2−1

a

∑

c j k

∑

J

J 2 ⟨cb |t2|j k ⟩

J M J M

⟨j k |λ2|c a ⟩

J M J M

⟨ |γ(1)N |i ⟩

00

=(γhh

NA)+ −1

i

∑

j

⟨ |λ1|b ⟩

00

⟨b |t1|i ⟩

00

(γhhN

B)+ 1

2−1

i

∑

b c k

∑

J

J 2 ⟨k j |λ2|b c ⟩

J M J M

⟨b c |t2|k i ⟩

J M J M

Figure F.1: Spherical expressions for the CCSD one-body response density matrix elements.



⟨ab |γ(2)N |c i ⟩

J M J M

=

γppphA

− −1c

∑

k

⟨ab |t2|k i ⟩

J M J M

⟨k |λ1|c ⟩

00

γppphB

− Pab (J ) −1a −1

b

∑

k

−1c ⟨a |t1|k ⟩

00

⟨b |t1|i ⟩

00

⟨k |λ1|c ⟩

00

+ CCAtoStd(A)

ab

c iJ ; J ′

|Tc i =0

¨

γppphC

− 12δJ ′0

−1a

∑

d k l

∑

J ′′( J ′′)2 ⟨d a |t2|k l ⟩

J ′′M ′′ J ′′M ′′

⟨k l |λ2|d c ⟩

J ′′M ′′ J ′′M ′′

⟨b |t1|i ⟩

00

γppphD

+ −1a

∑

d k l

(−1)jd+j l−J ′ ⟨b d |t2|i l ⟩

J ′M ′

J ′M ′

⟨k l |λ2|c d ⟩

J ′M ′

J ′M ′

⟨a |t1|k ⟩

00«

γppphE

+ 12−1

i

∑

d k l

⟨ab |t2|k l ⟩

J M J M

⟨k l |λ2|c d ⟩

J M J M

⟨d |t1|i ⟩

00

γppphF

+ Pab (J ) −1a −1

b−1

i

∑

d k l

⟨k l |λ2|c d ⟩

J M J M

⟨a |t1|k ⟩

00

⟨b |t1|l ⟩

00

⟨d |t1|i ⟩

00

γppphG

+ CCAtoStd(A)

ab

c iJ ; J ′

|Tc i =0

−1a

∑

d k l

(−1)jd+jk−J ′ ⟨b d |t2|i k ⟩

J ′M ′

J ′M ′

⟨k l |λ2|d c ⟩

J ′M ′

J ′M ′

⟨a |t1|l ⟩

00

Figure F.2: Spherical expressions for the CCSD two-body response density matrix elements.



⟨ab |γ(2)N |i j ⟩

J M J M

=(γ

pphhA)+ ⟨ab |t2|i j ⟩

J M J M

(γpphhB)+ Pab (J ) Pi j (J )

−1a −1

b⟨a |t1|i ⟩

00

⟨b |t1|j ⟩

00

(γpphhC)+ CCAtoStd(A)

ab

i jJ ; J ′

δJ ′0

∑

c k

⟨c a |t2|k i ⟩

00

00

⟨b |t1|j ⟩

00

⟨k |λ1|c ⟩

00

(γpphhD)+ Pi j (J ) (−1)j i+j j−J −1

i

∑

c k

−1c ⟨ab |t2|j k ⟩

J M J M

⟨c |t1|i ⟩

00

⟨k |λ1|c ⟩

00

(γpphhE)− Pab (J )

−1a

∑

c k

−1k⟨cb |t2|i j ⟩

J M J M

⟨a |t1|k ⟩

00

⟨k |λ1|c ⟩

00

(γpphhF)− Pab (J ) Pi j (J )

−1a −1

b−2

i

∑

c k

⟨a |t1|k ⟩

00

⟨b |t1|j ⟩

00

⟨c |t1|i ⟩

00

⟨k |λ1|c ⟩

00

(γpphhG)+ CCAtoStd(A)

ab

i jJ ; J ′

∑

c d k l

(−1)jc+jk−J ′ (−1)jd+j l−J ′ ⟨c a |t2|k i ⟩

J ′M ′

J ′M ′

⟨b d |t2|j l ⟩

J ′M ′

J ′M ′

⟨k l |λ2|c d ⟩

J ′M ′

J ′M ′

(γpphhH)− 1

2Pi j (J )

−2i

∑

c d k l

∑

J ′( J ′)2 ⟨c d |t2|i k ⟩

J ′M ′ J ′M ′

⟨ab |t2|l j ⟩

J M J M

⟨l k |λ2|c d ⟩

J ′M ′ J ′M ′

(γpphhI)− 1

2Pab (J )

−2a

∑

c d k l

∑

J ′( J ′)2 ⟨c a |t2|k l ⟩

J ′M ′ J ′M ′

⟨d b |t2|i j ⟩

J M J M

⟨k l |λ2|c d ⟩

J ′M ′ J ′M ′

Figure F.3: Spherical expressions for the CCSD two-body response density matrix elements, con-

tinued.



(γpphhJ)+ 1

4

∑

c d k l

⟨ab |t2|k l ⟩

J M J M

⟨c d |t2|i j ⟩

J M J M

⟨k l |λ2|c d ⟩

J M J M

+ CCAtoStd(A)

ab

i jJ ; J ′

¨(γ

pphhK)− 1

2δJ ′0

−2i

∑

c d k l

×∑

J ′′( J ′′)2 ⟨c d |t2|k i ⟩

J ′′M ′′ J ′′M ′′

⟨k l |λ2|c d ⟩

J ′′M ′′ J ′′M ′′

⟨a |t1|l ⟩

00

⟨b |t1|j ⟩

00

(γpphhL)− 1

2δJ ′0

−2i

∑

c d k l

∑

J ′′( J ′′)2 ⟨c a |t2|k l ⟩

J ′′M ′′ J ′′M ′′

⟨k l |λ2|c d ⟩

J ′′M ′′ J ′′M ′′

⟨d |t1|i ⟩

00

⟨b |t1|j ⟩

00

(γpphhM)− −1

a −1i

∑

c d k l

(−1)jd+j l+J ′ (−1)jc+jk+J ′ ⟨b d |t2|j l ⟩

J ′M ′

J ′M ′

⟨k l |λ2|c d ⟩

J ′M ′

J ′M ′

⟨a |t1|k ⟩

00

⟨c |t1|i ⟩

00«

(γpphhN)+ 1

2Pi j (J )

−1i −1

j

∑

c d k l

⟨ab |t2|k l ⟩

J M J M

⟨k l |λ2|c d ⟩

J M J M

⟨c |t1|i ⟩

00

⟨d |t1|j ⟩

00

(γpphhO)+ 1

2Pab (J )

−1a −1

b

∑

c d k l

⟨c d |t2|i j ⟩

J M J M

⟨k l |λ2|c d ⟩

J M J M

⟨a |t1|k ⟩

00

⟨b |t1|l ⟩

00

(γpphhP)+ Pab (J ) Pi j (J )

−1a −1

b−1

i −1j

∑

c d k l

⟨k l |λ2|c d ⟩

J M J M

⟨a |t1|k ⟩

00

⟨c |t1|i ⟩

00

⟨b |t1|l ⟩

00

⟨d |t1|j ⟩

00


tinued.



⟨i a |γ(2)N |b j ⟩

J M J M

=

γhpphA

+ −1a −1

b⟨a |t1|j ⟩

00

⟨ı |λ1|b ⟩

00

CCAtoStd

i a

b jJ ; J ′

¨

γhpphB

−∑

c k

(−1)jc+jk−J ′ ⟨c a |t2|k j ⟩

J ′M ′

J ′M ′

⟨k ı |λ2|cb ⟩

J ′M ′

J ′M ′

γhpphC

− −1a −1

j

∑

c k

(−1)jc+jk−J ′ ⟨ı k |λ2|b c ⟩

J ′M ′

J ′M ′

⟨a |t1|k ⟩

00

⟨c |t1|j ⟩

00«

⟨i j |γ(2)N |ab ⟩

J M J M

=(γhh

ppA)+ ⟨i j |λ2|ab ⟩

J M J M

⟨ab |γ(2)N |c d ⟩

J M J M

=(γ

ppppA)+ 1

2

∑

k l

⟨ab |t2|k l ⟩

J M J M

⟨k l |λ2|c d ⟩

J M J M

(γppppB)+ Pab (J )

−1a −1

b

∑

k l

⟨k l |λ2|c d ⟩

J M J M

⟨a |t1|k ⟩

00

⟨b |t1|l ⟩

00

⟨i j |γ(2)N |k l ⟩

J M J M

=(γhh

hhA)+ 1

2

∑

c d

⟨c d |t2|k l ⟩

J M J M

⟨i j |λ2|c d ⟩

J M J M

(γhhhhB)+ Pk l (J )

−1k−1

l

∑

c d

⟨i j |λ2|ab ⟩

J M J M

⟨a |t1|k ⟩

00

⟨b |t1|l ⟩

00

⟨a i |γ(2)N |b c ⟩

J M J M

=

γphppA

− −1a

∑

k

⟨k i |λ2|b c ⟩

J M J M

⟨a |t1|k ⟩

00

⟨i j |γ(2)N |k a ⟩

J M J M

=

γhhhpA

+ −1k

∑

c

⟨i j |λ2|c a ⟩

J M J M

⟨c |t1|k ⟩

00


tinued.



⟨i a |γ(2)N |j k ⟩

J M J M

=

γhphhA

+ −1i

∑

c

⟨c a |t2|j k ⟩J M J M

⟨ı |λ2|c ⟩

00

γhhhpB

+ −1a −2

i

∑

c

⟨c |t1|j ⟩

00

⟨a |t1|k ⟩

00

⟨ı |λ2|c ⟩

00

+ CCAtoStd(A)

i a

j kJ ; J ′

|Ti a=0

¨

γhhhpC

+ 12δJ ′0

−1i

∑

c d l

∑

J ′′( J ′′)2 ⟨c d |t2|l j ⟩

J ′′M ′′ J ′′M ′′

⟨l i |λ2|c d ⟩

J ′′M ′′ J ′′M ′′

⟨a |t1|k ⟩

00

γhhhpD

− −1j

∑

c d l

(−1)jd+j l−J ′ ⟨a d |t2|k l ⟩

J ′M ′

J ′M ′

⟨ı l |λ2|c d ⟩

J ′M ′

J ′M ′

⟨c |t1|j ⟩

00«

γhhhpE

− 12−1

a

∑

c d l

⟨c d |t2|j k ⟩

J M J M

⟨i l |λ2|c d ⟩

J M J M

⟨a |t1|l ⟩

00

γhhhpF

− Pj k (J ) −1a

∑

c d l

−1c −1

d⟨i l |λ2|c d ⟩

J M J M

⟨a |t1|l ⟩

00

⟨c |t1|j ⟩

00

⟨d |t1|k ⟩

00

γhhhpG

− CCAtoStd(A)

i a

j kJ ; J ′

|Ti a=0

−1j

∑

c d l

(−1)jc+j l−J ′ ⟨a c |t2|k l ⟩

J ′M ′

J ′M ′

⟨ı l |λ2|d c ⟩

J ′M ′

J ′M ′

⟨d |t1|j ⟩

00


tinued.


Appendix G

ΛCCSD(T) Spherical Expressions

Appendix G. ΛCCSD(T) Spherical Expressions

G.1 Spherical Equations

(λA1)− (−1)ja+jb−J ′∑

d

¦J J ′ J ′′

jb jd ja

©

X⟨d k |v |b c ⟩

J ′′M ′′

J ′′M ′′

⟨i j |λ2|a d ⟩

J M J M

λATab

+∑

d

¦J J ′ J ′′

ja jd jb

©

X⟨d k |v |a c ⟩

J ′′M ′′

J ′′M ′′

⟨i j |λ2|b d ⟩

J M J M

(λATa c )− (−1)J+J ′+J ′′

∑

d

¦J J ′ J ′′

jk jc jd

©

X⟨d k |v |ab ⟩

J ′M ′ J ′M ′

⟨i j |λ2|c d ⟩

J M J M

λATi k

−∑

d

∑

J ′J ′′

nJ ′ J ′′ Jj j j i jd

o

X

jb jc J ′ja jk J ′′J ′ J ′′ J

X

⟨d i |v |cb ⟩

J ′M ′ J ′M ′

⟨ k |λ2|d a ⟩

J ′′M ′′

J ′′M ′′

λATj k

+ (−1)j i+j j−J∑

d

∑

J ′J ′′

nJ ′ J ′′ Jj i j j jd

o

X


X

× ⟨d j |v |cb ⟩

J ′M ′ J ′M ′

⟨ı k |λ2|d a ⟩

J ′′M ′′

J ′′M ′′

λATab Ti k

+ (−1)jc+jk−J ′′∑

d

∑

J ′J ′′(−1)J

′+J ′+J ′′nJ ′ J ′′ J ′

jb ja jd

o

X

×

j i jc J ′j j jk J ′′J J ′′ J ′

X

⟨d ı |v |a c ⟩

J ′M ′

J ′M ′

⟨j k |λ2|b d ⟩

J ′′M ′′ J ′′M ′′

+ . . .

Figure G.1: Spherical expressions for the ˆt and ˆλ amplitudes of the ΛCCSD(T) energy correction.

It should be noted that the X variant of the Wigner 6j and 9j symbols (4.35) and (4.34)

are used here and in the following.



λATab Tj k

− (−1)j i+j j−J (−1)jc+jk−J ′′∑

d

∑

J ′J ′′(−1)J

′+J ′+J ′′nJ ′ J ′′ J ′

jb ja jd

o

X

×

j j jc J ′j i jk J ′′J J ′′ J ′

X

⟨d |v |a c ⟩

J ′M ′

J ′M ′

⟨i k |λ2|b d ⟩

J ′′M ′′ J ′′M ′′

λATa c Ti k

+∑

d

nJ J ′ J ′′

jd j j j i

o

X⟨d i |v |ab ⟩

J ′M ′ J ′M ′

⟨k |λ2|c d ⟩

J ′′M ′′

J ′′M ′′

λATa c Tj k

− (−1)j i+j j−J∑

d

nJ J ′ J ′′

jd j i j j

o

X⟨d j |v |ab ⟩

J ′M ′ J ′M ′

⟨k ı |λ2|c d ⟩

J ′′M ′′

J ′′M ′′

(λB1)−∑

l

nJ J ′ J ′′j l j j j i

o

X⟨ k |v |l c ⟩

J ′′M ′′

J ′′M ′′

⟨l i |λ2|ab ⟩

J ′M ′ J ′M ′

(λBTa c )+

∑

l

∑

J ′J ′′


o

X

jk ja J ′jc jb J ′′J ′′ J ′ J

X

⟨ k |v |l a ⟩

J ′M ′

J ′M ′

⟨l i |λ2|cb ⟩

J ′′M ′′ J ′′M ′′

λBTb c

− (−1)jc+jk−J ′′∑

l

∑

J ′J ′′(−1)J+J

′+J ′′nJ ′ J ′′ Jj i j j j l

o

X

jb jk J ′ja jc J ′′J ′ J ′′ J

X

× ⟨ k |v |l b ⟩

J ′M ′

J ′M ′

⟨l i |λ2|c a ⟩

J ′′M ′′ J ′′M ′′

λBTi j

+ (−1)j i+j j−J∑

l


o

X⟨ı k |v |l c ⟩

J ′′M ′′

J ′′M ′′

⟨l j |λ2|ab ⟩

J ′M ′ J ′M ′

+ . . .

Figure G.2: Spherical expressions for the ˆt and ˆλ amplitudes of the ΛCCSD(T) energy correction,

continued.



λBTi k

− (−1)jc+jk−J ′′∑

l

¦J J ′ J ′′

jk jc j l

©

X⟨i j |v |l c ⟩

J M J M

⟨k l |λ2|ab ⟩

J ′M ′ J ′M ′

λBTa c Ti j

− (−1)j i+j j−J∑

l

∑

J ′J ′′

nJ ′ J ′′ Jj j j i j l

o

X


X

⟨ı k |v |l a ⟩

J ′M ′

J ′M ′

⟨l j |λ2|cb ⟩

J ′′M ′′ J ′′M ′′

λBTa c Ti k

+ (−1)J+J ′+J ′′∑

l

(−1)jb+j l−J ′′¦

J J ′ J ′′jb j l ja

©

X⟨i j |v |l a ⟩

J M J M

⟨k l |λ2|cb ⟩

J ′′M ′′

J ′′M ′′

λBTb c Ti j

+ (−1)j i+j j−J (−1)ja+jb−J ′∑

l

∑

J ′J ′′

nJ ′ J ′′ Jj j j i j l

o

X

×

jk jb J ′jc ja J ′′J ′′ J ′ J

X

⟨ı k |v |l b ⟩

J ′M ′

J ′M ′

⟨l j |λ2|c a ⟩

J ′′M ′′ J ′′M ′′

λBTb c Ti k

−∑

l

¦J J ′ J ′′

ja j l jb

©

X⟨i j |v |b l ⟩

J M J M

⟨k l |λ2|c a ⟩

J ′′M ′′

J ′′M ′′

(λC1)

+ (−1)ja+jb−J ′ −1i

nJ J ′ J ′′

jb j j ja

o

X⟨ k |v |b c ⟩

J ′′M ′′

J ′′M ′′

⟨ı |λ1|a ⟩

00

λCTab

− −1i

nJ J ′ J ′′

ja j j jb

o

X⟨ k |v |a c ⟩

J ′′M ′′

J ′′M ′′

⟨ı |λ1|b ⟩

00

(λCTa c )+ −1

i (−1)J+J ′+J ′′n

J J ′ J ′′jk jc j j

o

X⟨j k |v |ab ⟩

J ′M ′ J ′M ′

⟨ı |λ1|c ⟩

00

+ . . .


continued.



λCTi j

− (−1)j i+j j−J (−1)ja+jb−J ′ −1a

¦J J ′ J ′′

jb j i ja

©

X⟨ı k |v |b c ⟩

J ′′M ′′

J ′′M ′′

⟨ |λ1|a ⟩

00

λCTi k

− (−1)ja+jb−J ′ −1a (−1)J+J ′+J ′′

¦J J ′ J ′′

ja jc jb

©

X⟨i j |v |cb ⟩

J M J M

⟨k |λ1|a ⟩

00

λCTab Ti j

+ (−1)j i+j j−J −1b

¦J J ′ J ′′

ja j i jb

©

X⟨ı k |v |a c ⟩

J ′′M ′′

J ′′M ′′

⟨ |λ1|b ⟩

00

λCTab Ti k

+ −1b(−1)J+J ′+J ′′

¦J J ′ J ′′

jb jc ja

©

X⟨i j |v |c a ⟩

J M J M

⟨k |λ1|b ⟩

00

λCTa c Ti j

− (−1)j i+j j−J −1j (−1)J+J ′+J ′′

¦J J ′ J ′′

jk jc j i

©

X⟨i k |v |ab ⟩

J ′M ′ J ′M ′

⟨ |λ1|c ⟩

00

λCTa c Ti k

+ δJ J ′ δJ ′′0 J ⟨i j |v |ab ⟩

J M J M

⟨k |λ1|c ⟩

00

(λD1)

+ . . .

λDTa c Ti k

+

(t3A1)

+∑

d

¦J J ′ J ′′

jb jd ja

©

X⟨b c |v |d k ⟩

J ′′M ′′

J ′′M ′′

⟨d a |t2|i j ⟩

J M J M

t3ATab

− (−1)ja+jb−J ′∑

d

¦J J ′ J ′′

ja jd jb

©

X⟨a c |v |d k ⟩

J ′′M ′′

J ′′M ′′

⟨d b |t2|i j ⟩

J M J M

(t3ATa c )+ (−1)jc+jk−J ′′

∑

d

¦J J ′ J ′′

jk jc jd

©

X⟨ab |v |d k ⟩

J ′M ′ J ′M ′

⟨c d |t2|i j ⟩

J M J M

+ . . .

Figure G.4: Spherical expressions for the ˆt and ˆλ amplitudes of the ΛCCSD(T) energy correc-

tion, continued. The (λDX ) contributions are obtained from the corresponding (λCX )

contributions by Λ1 → F and replacing the matrix elements of the normal-ordered

two-body Hamiltonian by Λ2 matrix elements.



t3ATi k

−∑

d

∑

J ′J ′′

nJ ′ J ′′ J ′

ja jb jd

o

X

jc j i J ′jk j j J ′′J ′′ J J ′

X

⟨b c |v |d i ⟩

J ′M ′

J ′M ′

⟨d a |t2|k j ⟩

J ′′M ′′ J ′′M ′′

t3ATj k

+ (−1)j i+j j−J∑

d

∑

J ′J ′′

nJ ′ J ′′ J ′

ja jb jd

o

X

jc j j J ′jk j i J ′′J ′′ J J ′

X

× ⟨b c |v |d j ⟩

J ′M ′

J ′M ′

⟨d a |t2|k i ⟩

J ′′M ′′ J ′′M ′′

t3ATab Ti k

+ (−1)ja+jb−J ′∑

d

∑

J ′J ′′

nJ ′ J ′′ J ′

jb ja jd

o

X


X

× ⟨a c |v |d i ⟩

J ′M ′

J ′M ′

⟨d b |t2|k j ⟩

J ′′M ′′ J ′′M ′′

t3ATab Tj k

− (−1)j i+j j−J (−1)ja+jb−J ′∑

d

nJ ′ J ′′ J ′

jb ja jd

o

X

jc j j J ′jk j i J ′′J ′′ J J ′

X

× ⟨a c |v |d j ⟩

J ′M ′

J ′M ′

⟨d b |t2|k i ⟩

J ′′M ′′ J ′′M ′′

t3ATa c Ti k

− (−1)j i+j j−J (−1)J+J ′+J ′′∑

d

nJ J ′ J ′′

jd j j j i

o

X⟨ab |v |i d ⟩

J ′M ′ J ′M ′

⟨c d |t2|k j ⟩

J ′′M ′′

J ′′M ′′

t3ATa c Tj k

+∑

d

nJ J ′ J ′′

jd j i j j

o

X⟨ab |v |j d ⟩

J ′M ′ J ′M ′

⟨c d |t2|k i ⟩

J ′′M ′′

J ′′M ′′

+ . . .


continued.



(t3B1)

+ (−1)j i+j j−J∑

l

nJ J ′ J ′′j l j j j i

o

X⟨l c |v |j k ⟩

J ′′M ′′

J ′′M ′′

⟨ab |t2|i l ⟩

J ′M ′ J ′M ′

(t3BTa c )+

∑

l

∑

J ′J ′′

nJ ′ J ′′ J ′

jb ja j l

o

X

j j jk J ′j i jc J ′′J J ′′ J ′

X

⟨l a |v |k j ⟩

J ′M ′ J ′M ′

⟨c b |t2|i l ⟩

J ′′M ′′

J ′′M ′′

t3BTb c

− (−1)j i+j j−J (−1)ja+jb−J ′ (−1)jc+jk−J ′′∑

l

∑

J ′J ′′(−1)J+J

′+J ′′

×nJ ′ J ′′ Jj i j j j l

o

X

jb jk J ′ja jc J ′′J ′ J ′′ J

X

⟨l b |v |j k ⟩

J ′M ′

J ′M ′

⟨a c | t2 |i l ⟩

J ′′M ′′ J ′′M ′′

t3BTi j

−∑

l


o

X⟨l c |v |i k ⟩

J ′′M ′′

J ′′M ′′

⟨ab |t2|j l ⟩

J ′M ′ J ′M ′

t3BTi k

+ (−1)J+J ′+J ′′∑

l

¦J J ′ J ′′

jk jc j l

©

X⟨l c |v |i j ⟩

J M J M

⟨ab |t2|k l ⟩

J ′M ′ J ′M ′

t3BTa c Ti j

− (−1)jc+jk−J ′′∑

l

∑

J ′J ′′(−1)J+J

′+J ′′nJ ′ J ′′ Jj j j i j l

o

X

ja jk J ′jb jc J ′′J ′ J ′′ J

X

× ⟨l a |v |i k ⟩

J ′M ′

J ′M ′

⟨b c |t2|j l ⟩

J ′′M ′′J ′′M ′′

t3BTa c Ti k

−∑

l

¦J J ′ J ′′

jb j l ja

©

X⟨l a |v |i j ⟩

J M J M

⟨c b |t2|k l ⟩

J ′′M ′′

J ′′M ′′

+ . . .


continued.



t3BTb c Ti j

+ (−1)j i+j j−J (−1)ja+jb−J ′∑

l

∑

J ′J ′′

nJ ′ J ′′ J ′

ja jb j l

o

X

j i jk J ′j j jc J ′′J J ′′ J ′

X

× ⟨l b |v |k i ⟩

J ′M ′ J ′M ′

⟨c a |t2|j l ⟩

J ′′M ′′

J ′′M ′′

t3BTb c Ti k

+ (−1)ja+jb−J ′∑

l

¦J J ′ J ′′

ja j l jb

©

X⟨l b |v |i j ⟩

J M J M

⟨a c |t2|l k ⟩

J ′′M ′′

J ′′M ′′


continued.



G.2 Spherical Equations for Three-Body Hamiltonians

⟨i j k ||ˆλ||ab c ⟩

J J ′

J ′′

= ⟨i j k ||ˆλ||ab c ⟩

J J ′

J ′′

[NO2B](LA1)

+ −1c

∑

l

⟨i j k ||w ||ab l ⟩

J J ′

J ′′

⟨l |λ1|c ⟩

00

LATa cTb c

+ Pab (J′) −1

a

∑

l

⟨i j k ||w ||l b c ⟩

J J ′

J ′′

⟨l |λ1|a ⟩

00

(LB1)− −1k

∑

d

⟨i j d ||w ||ab c ⟩

J J ′

J ′′

⟨k |λ1|d ⟩

00

LBTi kTj k

− Pi j (J ) −1i

∑

d

⟨d j k ||w ||ab c ⟩

J J ′

J ′′

⟨ı |λ1|d ⟩

00

(LC1)

+ 12

∑

d e

⟨d e k ||w ||ab c ⟩

J J ′

J ′′

⟨i j |λ2|d e ⟩

J M J M

LCTi kTj k

+ 12

Pi j (J ) (−1)jc+jk−J ′′∑

d e

∑

J ′J ′′(−1)J

′+J ′+J ′′

j i jc J ′′j j jk J ′J J ′′ J ′

X

× ⟨d e ı ||w ||ab c ⟩

J ′ J ′

J ′′

⟨j k |λ2|d e ⟩

J ′M ′ J ′M ′

(LD1)

+ 12

∑

l m

⟨i j k ||w ||l m c ⟩

J J ′

J ′′

⟨l m |λ2|ab ⟩

J ′M ′ J ′M ′

LDTa cTb c

+ 12

Pab (J′)∑

l m

∑

J ′J ′′(−1)jb+jc−J ′′


X

⟨i j k ||w ||l m a ⟩

J J ′′

J ′

⟨l m |λ2|b c ⟩

J ′′M ′′ J ′′M ′′

+ . . .

Figure G.8: Spherical expressions for the ˆλ amplitudes of the ΛCCSD(T) energy correction for

three-body Hamiltonians.



(LE1)−∑

d l

(−1)jd+j l−J ′′ ⟨i j d ||w ||ab l ⟩

J J ′

J ′′

⟨k l |λ2|c d ⟩

J ′′M ′′

J ′′M ′′

LETa cTb c

+ Pab (J′)∑

d l

∑

J ′J ′′(−1)jd+j l−J ′′


X

⟨i j d ||w ||cb l ⟩

J J ′

J ′′

⟨k l |λ2|a d ⟩

J ′′M ′′

J ′′M ′′

LETi kTj k

− Pi j (J ) (−1)j i+j j−J∑

d l

∑

J ′J ′′(−1)jd+j l−J ′′


X

× ⟨j k d ||w ||ab l ⟩

J ′ J ′

J ′′

⟨ı l |λ2|c d ⟩

J ′′M ′′

J ′′M ′′

LETi k ,Tj kTa c ,Tb c

+ Pab (J′) Pi j (J ) (−1)j i+j j−J

∑

d l

∑

J ′J ′′J ′′′(−1)J

′+J ′′′+J ′′

× J J ′ ˆJ ′ ( ˆJ ′′)2 ˆJ ′′′nJ ′ J ′′ J ′

ja jb j l

o

6j

nJ ′′ J ′′′ Jj j j i jd

o

6j

nJ J ′ J ′′J ′ J ′′′ J ′′

o

6j

× ⟨d j k ||w ||b l c ⟩

J ′′′ J ′

J ′′

⟨l ı |λ2|a d ⟩

J ′′M ′′

J ′′M ′′

(LF)+ ⟨i j k ||w ||ab c ⟩

J J ′

J ′′

Figure G.9: Spherical expressions for the ˆλ amplitudes of the ΛCCSD(T) energy correction for

three-body Hamiltonians, continued.




J ′ J

J ′′

= ⟨ab c ||ˆt ||i j k ⟩

J ′ J

J ′′

[NO2B](MA1)

+ −1c

∑

l

⟨ab l ||w ||i j k ⟩

J ′ J

J ′′

⟨c |t1|l ⟩

00

MATa cTb c

+ Pab (J′) −1

a

∑

l

⟨l b c ||w ||i j k ⟩

J ′ J

J ′′

⟨a |t1|l ⟩

00

(MB1)− −1k

∑

d

⟨ab c ||w ||i j d ⟩

J ′ J

J ′′

⟨d |t1|k ⟩

00

MBTi kTj k

− Pi j (J ) −1i

∑

d

⟨ab c ||w ||d j k ⟩

J ′ J

J ′′

⟨d |t1|i ⟩

00

(MC1)

+ 12

∑

d e

⟨ab c ||w ||d e k ⟩

J ′ J

J ′′

⟨d e |t2|i j ⟩

J M J M

MCTi kTj k

− 12

Pi j (J )∑

d e

∑

J ′J ′′


X

⟨ab c ||w ||d e i ⟩

J ′ J ′′

J ′

⟨d e |t2|k j ⟩

J ′′M ′′ J ′′M ′′

(MD1)

+ 12

∑

l m

⟨l m c ||w ||i j k ⟩

J ′ J

J ′′

⟨ab |t2|l m ⟩

J ′M ′ J ′M ′

MDTa cTb c

+ 12

Pab (J′) (−1)jc+jk−J ′′

∑

l m

∑

J ′J ′′(−1)J+J

′+J ′′

ja jk J ′jb jc J ′′J ′ J ′′ J

X

× ⟨l m a ||w ||i j k ⟩

J ′′ J

J ′

⟨b c |t2|l m ⟩

J ′′M ′′J ′′M ′′

+ . . .

Figure G.10: Spherical expressions for the ˆt amplitudes of the ΛCCSD(T) energy correction for

three-body Hamiltonians.



(ME1)−∑

d l

(−1)jd+j l−J ′′ ⟨ab l ||w ||i j d ⟩

J ′ J

J ′′

⟨c d |t2|k l ⟩

J ′′M ′′

J ′′M ′′

METi kTj k

+ Pi j (J )∑

d l

∑

J ′J ′′(−1)jd+j l−J ′′


X

⟨ab l ||w ||k j d ⟩

J ′ J ′

J ′′

⟨c d |t2|i l ⟩

J ′′M ′′

J ′′M ′′

METa cTb c

− Pab (J′) (−1)ja+jb−J ′

∑

d l

(−1)jd+j l−J ′′


X

× ⟨b c l ||w ||i j d ⟩

J ′ J

J ′′

⟨a d |t2|k l ⟩

J ′′M ′′

J ′′M ′′

METi k ,Tj kTa c ,Tb c

+ Pab (J′) Pi j (J ) (−1)ja+jb−J ′

∑

d l

∑

J ′J ′′J ′′′(−1)J

′′+J ′+J ′′′

× J J ′ ˆJ ′ ( ˆJ ′′)2 ˆJ ′′′nJ ′ J ′′ J ′

ja jb j l

o

6j

nJ ′′ J ′′′ Jj j j i jd

o

6j

nJ J ′ J ′′J ′ J ′′ J ′′′

o

6j

× ⟨l b c ||w ||j d k ⟩

J ′ J ′′′

J ′′

⟨a d |t2|l i ⟩

J ′′M ′′

J ′′M ′′

(MF)+ ⟨ab c ||w ||i j j ⟩

J ′ J

J ′′

Figure G.11: Spherical expressions for the ˆt amplitudes of the ΛCCSD(T) energy correction for

three-body Hamiltonians, continued.


Appendix H

EOM-CCSD Diagrams and SphericalExpressions

Appendix H. EOM-CCSD Diagrams and Spherical Expressions

H.1 Diagrams

b b b b b b b

R0A R0B R1A R1B R1C

b b b b b b b

R1D R1E R1F R2A R2B

b b b b b b b b

R2C R2D R2E R2F R2G

b b b b b b b b

R2H R2I R2J R2K

Figure H.1: EOM-CCSD R diagrams.



H.2 Spherical Equations

ωµ R(J )0 =

(R0A)− δJ 0

∑

a i

⟨a ||R(J )1 ||i ⟩ ⟨ı |H1|a ⟩

00

(R0B)+ δJ 0

14

∑

ab i j

∑

J ′J ′ ⟨ab ||R(J )2 ||i j ⟩

J ′ J ′

⟨i j |H2|ab ⟩

J ′M ′ J ′M ′

⟨a ||

H R(J )

C||i ⟩

=(R1A)− −1

a

∑

c

⟨a |H1|c ⟩

00

⟨c ||R(J )1 ||i ⟩

(R1B)+ −1

i

∑

k

⟨a ||R(J )1 ||k ⟩ ⟨k |H1|i ⟩

00

(R1C)−∑

c k

(−1)jc+jk−J ⟨a k |H2|i c ⟩

J M

J M

⟨c ||R(J )1 ||k ⟩

(R1D)+∑

c k

−1c

∑

J ′ J ′′(−1)J+J ′+J ′′ J ′ J ′′

¦J ′ J ′′ Jj i ja jk

©

6j⟨c a ||R(J )2 ||i k ⟩

J ′ J ′′

⟨k |H1|c ⟩

00

(R1E)− 1

2

∑

c d k

∑

J ′ J ′′(−1)J+J ′+J ′′ J ′ J ′′

¦J ′ J ′′ Jj i ja jk

©

6j⟨k a |H2|c d ⟩

J ′M ′ J ′M ′

⟨c d ||R(J )2 ||i k ⟩

J ′ J ′′

(R1F)+ 1

2

∑

c k l

∑

J ′ J ′′(−1)J+J ′+J ′′ J ′ J ′′

¦J ′ J ′′ Jj i ja jc

©

6j⟨c a ||R(J )2 ||k l ⟩

J ′ J ′′

⟨k l |H2|i c ⟩

J ′′M ′′ J ′′M ′′

Figure H.2: Spherical expressions for the EOM-CCSD R0 and R1 amplitude equations.



⟨ab ||

H R(J )

C||i j ⟩

Jab Ji j

= Pab (J ) Pi j (J )

¨(R2A)− 1

2(−1)Jab+Ji j+J Jab Ji j

×∑

c

nJ Jab Ji j

j j j i jc

o

6j⟨c ||R(J )1 ||i ⟩ ⟨ab |H2|j c ⟩

Jab M ab Jab M ab

(R2B)− 1

2Jab Ji j

∑

k

(−1)ja+jk−Jn

J Ji j Jab

jb ja jk

o

6j⟨a ||R(J )1 ||k ⟩ ⟨b k |H2|i j ⟩


(R2C)− 1

2−1

b

∑

c

⟨a c ||R(J )2 ||i j ⟩

JabJi j

⟨b |H1|c ⟩

00

(R2D)+ 1

2−1

j

∑

k

⟨ab ||R(J )2 ||i k ⟩

Jab Ji j

⟨k |H1|j ⟩

00

(R2E)+ 1

8

∑

c d

⟨ab |H2|c d ⟩

Jab M ab Jab M ab

⟨c d ||R(J )2 ||i j ⟩

Jab Ji j

(R2F)+ 1

8

∑

k l

⟨ab ||R(J )2 ||k l ⟩

Jab Ji j

⟨k l |H2|i j ⟩

Ji j M i j Ji j M i j«

(R2G)− CCAtoStd(A)

ab Jab ; Ja i

i j Ji j ; Jb j

∑

c k

(−1)jc+jk−Jb j ⟨a c ||R(J )2 ||i k ⟩

Ja i

Jb j

⟨b k |H2|j c ⟩

Jb j Mb j

Jb j Mb j

(R2H)+ 1

2Pab (J ) Pi j (J ) (−1)J+Jab+Ji j Jab Ji j

×∑

c d k

(−1)jd+jk−Jn

J Ji j Jab

ja jb jc

o

6j⟨c a |t2|i j ⟩

Ji j M i jJi j M i j

⟨b k |H2|c d ⟩

J M

J M

⟨d ||R(J )1 ||k ⟩ + . . .

Figure H.3: Spherical expressions for the EOM-CCSD R2 amplitude equations.




¨

(R2I)+ 1

4(−1)ja+jb−Ji j Jab Ji j

∑

c d k l

∑

J ′ J ′′J ′ J ′′ (−1)J

′+J ′′

×n

J Ji j Jab

jb ja jc

o

6j

¦J ′ J ′′ Jjc ja jd

©

6j⟨d a ||R(J )2 ||k l ⟩

J ′ J ′′

⟨k l |v |c d ⟩

J ′′M ′′ J ′′M ′′

⟨cb |t2|i j ⟩


(R2J)+ 1

2(−1)j i+j j−Ji j (−1)Jab+Ji j Jab Ji j

×∑

k l c

(−1)jc+j l

nJ Jab Ji j

j i j j jk

o

6j⟨ab | t2 |i k ⟩

Jab M ab Jab M ab

⟨k l |H2|j c ⟩

J M

J M

⟨c ||R(J )1 ||l ⟩

(R2K)− 1

4(−1)Jab+Ji j Jab Ji j

×∑

k l c d

∑

J ′ J ′′J ′ J ′′ (−1)J

′+J ′′n

J Jab Ji j

j j j i jk

o

6j

¦J ′ J ′′ Jjk j i j l

©

6j

× ⟨c d ||R(J )2 ||i l ⟩

J ′′ J ′

⟨l k |v |c d ⟩

J ′′M ′′ J ′′M ′′

⟨ab | t2 |j k ⟩

Jab M ab Jab M ab«

Figure H.4: Spherical expressions for the EOM-CCSD R2 amplitude equations, continued.



⟨i || L(J ) Hopen ||a ⟩

=(L1B)− −1

a

∑

c

⟨i ||L(J )1 ||c ⟩ ⟨c |H1|a ⟩

00

(L1C)+ −1

i

∑

k

⟨ı |H1|k ⟩

00

⟨k ||L(J )1 ||a ⟩

(L1D)−∑

c k

(−1)jc+jk−J ⟨c ı |H2|k a ⟩

J M

J M

⟨k ||L(J )1 ||c ⟩

(L1E)− 1

2

∑

c d k

∑

J ′ J ′′(−1)J+J ′+J ′′ J ′ J ′′

¦J ′ J ′′ Jja j i jk

©

6j⟨k i ||L(J )2 ||c d ⟩

J ′ J ′′

⟨c d |H2|a k ⟩

J ′′M ′′ J ′′M ′′

(L1F)+ 1

2

∑

c k l

∑

J ′ J ′′(−1)J+J ′+J ′′ J ′ J ′′

¦J ′ J ′′ Jja j i jc

©

6j⟨c i |H2|k l ⟩

J ′M ′ J ′M ′

⟨k l ||L(J )2 ||a c ⟩

J ′ J ′′

(L1G)− 1

2

∑

c d e k l

∑

J ′ J ′′J ′ J ′′

¦J ′ J ′′ Jjd je jc

©

6j⟨e c |t2|k l ⟩J ′M ′ J ′M ′

⟨d ı |H2|e a ⟩

J M

J M

⟨k l ||L(J )2 ||c d ⟩

J ′ J ′′

(L1H)+ 1

2

∑

c d k l m

∑

J ′ J ′′J ′ J ′′

¦J ′ J ′′ J

jm j l jk

©

6j⟨l k ||L(J )2 ||c d ⟩

J ′ J ′′

⟨c d |t2|k m ⟩

J ′′M ′′ J ′′M ′′

⟨m ı |H2|l a ⟩

J M

J M

Figure H.5: Spherical expressions for the EOM-CCSD L1 amplitude equations.



⟨i j || L(J ) Hopen ||ab ⟩

Ji j Jab


¨

(L2B)− (−1)J Ji j Jab (−1)j i+jb−Ji j −1

i

nJab Ji j J

j j jb ja

o

6j⟨ı |H1|a ⟩

00

⟨j ||L(J )1 ||b ⟩

(L2C)+ 1

2Ji j Jab (−1)j i+j j−J (−1)Jab

∑

c

nJ Jab Ji j

j j j i jc

o

6j⟨i ||L(J )1 ||c ⟩ ⟨c j |H2|ab ⟩

Jab M ab Jab M ab

(L2D)+ 1

2(−1)Jab+Ji j+J Ji j Jab

∑

k

nJ Ji j Jab

jb ja jk

o

6j⟨k ||L(J )1 ||a ⟩ ⟨i j |H2|b k ⟩


(L2E)− 1

2−1

b

∑

c

⟨i j ||L(J )2 ||a c ⟩

Ji j Jab

⟨c |H1|b ⟩

00

(L2F)− 1

2(−1)j i+j j−Ji j −1

j

∑

k

⟨k i ||L(J )2 ||ab ⟩

Ji j Jab

⟨ |H1|k ⟩

00«

(L2G)− CCAtoStd(A)

i j Ji j ; Ja i

ab Jab ; Jb j

∑

c k

(−1)jc+jk−Jb j ⟨ı k ||L(J )2 ||a c ⟩

Ja i

Jb j

⟨c |H2|kb ⟩

Jb j Mb j

Jb j Mb j


¨

(L2H)+ 1

8

∑

c d

⟨i j ||L(J )2 ||c d ⟩

Ji j Jab

⟨c d |H2|ab ⟩

Jab M ab Jab M ab

(L2I)+ 1

8

∑

k l

⟨i j |H2|k l ⟩


⟨k l ||L(J )2 ||ab ⟩

Ji j Jab«

+ . . .

Figure H.6: Spherical expressions for the EOM-CCSD L2 amplitude equations.




¨

(L2J)− 1

4(−1)Ji j+Jab Ji j Jab

×∑

c d k l

∑

J ′ J ′′(−1)J

′+J ′′ J ′ J ′′n

J Ji j Jab

jb ja jd

o

6j

¦J ′ J ′′ Jja jd jc

©

6j

× ⟨c d |t2|k l ⟩

J ′M ′ J ′M ′

⟨k l ||L(J )2 ||a c ⟩

J ′ J ′′

⟨i j |v | b d ⟩


(L2K)+ 1

4(−1)j i+j j−Jab Jab Ji j

×∑

c d k l

∑

J ′ J ′′J ′ J ′′ (−1)J

′+J ′′n

J Jab Ji j

j j j i j l

o

6j

¦J ′ J ′′ Jj i j l jk

©

6j

× ⟨c d |t2|l k ⟩

J ′M ′ J ′M ′

⟨k i ||L(J )2 ||c d ⟩

J ′′ J ′

⟨l j |v | ab ⟩

Jab M ab Jab M ab«

Figure H.7: Spherical expressions for the EOM-CCSD L2 amplitude equations, continued.



H.3 Spherical Equations (Scalar)

ω R(0) =

(R0A)−∑

a i

⟨a ||R(0)1 ||i ⟩ ⟨ı |H1|a ⟩

00

(R0B)+ 1

4

∑

ab i j

∑

J

J ⟨ab ||R(0)2 ||i j ⟩

J J

⟨i j |H2|ab ⟩

J M J M

⟨a ||

H R(0)

C||i ⟩

=(R1A)− −1

a

∑

c

⟨a |H1|c ⟩

00

⟨c ||R(0)1 ||i ⟩

(R1B)+ −1

a

∑

k

⟨a ||R(0)1 ||k ⟩ ⟨k |H1|i ⟩

00

(R1C)+∑

c k

⟨a k |H2|i c ⟩

00

00

⟨c ||R(0)1 ||k ⟩

(R1D)−∑

c k

⟨a c ||R(0)2 ||i k ⟩

0

0

⟨k |H1|c ⟩

00

(R1E)+ 1

2−1

a

∑

c d k

∑

J

J ⟨a k |H2|c d ⟩

J M J M

⟨c d ||R(0)2 ||i k ⟩

J J

(R1F)− 1

2−1

a

∑

c k l

∑

J

J ⟨a c ||R(0)2 ||k l ⟩J J

⟨k l |H2|i c ⟩

J M J M

Figure H.8: Spherical expressions for the scalar EOM-CCSD R0 and R2 amplitudes.



⟨ab ||

H R(0)

C||i j ⟩

J J


¨(R2A)+ 1

2J −1

i

∑

c

⟨c ||R(0)1 ||i ⟩ ⟨ab |H2|c j ⟩

J M J M

(R2B)− 1

2J −1

a

∑

k

⟨a ||R(0)1 ||k ⟩ ⟨kb |H2|i j ⟩

J M J M

(R2C)− 1

2−1

b

∑

c

⟨a c ||R(0)2 ||i j ⟩J J

⟨b |H1|c ⟩

00

(R2D)+ 1

2−1

j

∑

k

⟨ab ||R(0)2 ||i k ⟩

J J

⟨k |H1|j ⟩

00

(R2E)+ 1

8

∑

c d

⟨ab |H2|c d ⟩

J M J M

⟨c d ||R(0)2 ||i j ⟩

J J

(R2F)+ 1

8

∑

k l

⟨ab ||R(0)2 ||k l ⟩

J J

⟨k l |H2|i j ⟩

J M J M«

(R2G)− CCAtoStd(A)

ab

i jJ ; J ′

× J ( J ′)−1∑

c k

(−1)jc+jk−J ′⟨a c ||R(0)2 ||i k ⟩

J ′

J ′

⟨b k |H2|j c ⟩

J ′M ′

J ′M ′


¨

(R2H)+ 1

2J −2

b

∑

c d k

−1d

∑

J ′( J ′)2 ⟨a c |t2|i j ⟩

J M J M

⟨b k |H2|c d ⟩

J ′M ′ J ′M ′

⟨d ||R(0)1 ||k ⟩ δjb jc

(R2I)− 1

4J −2

a

∑

c d k l

∑

J ′J ′ ⟨a d ||R(0)2 ||k l ⟩

J ′ J ′

⟨k l |v |c d ⟩

J ′M ′ J ′M ′

⟨cb |t2|i j ⟩

J M J M

δja jc

(R2J)− 1

2J −2

j

∑

c k l

−1c

∑

J ′( J ′)2 ⟨c ||R(0)1 ||l ⟩ ⟨ab |t2|i k ⟩

J M J M

⟨k l |H2|j c ⟩

J ′M ′ J ′M ′

δjk j j

(R2K)− 1

4J −2

i

∑

c d k l

∑

J ′J ′ ⟨k l |v |c d ⟩

J ′M ′ J ′M ′

⟨c d ||R(0)2 ||i l ⟩

J ′ J ′

⟨ab |t2|k j ⟩

J M J M

δj i jk

«

Figure H.9: Spherical expressions for the scalar EOM-CCSD R2 amplitude equations.



⟨i || L(0) Hopen ||a ⟩

=(L1B)− −1

i

∑

c

⟨i ||L(0)1 ||c ⟩ ⟨c |H1|a ⟩

00

(L1C)+ −1

i

∑

k

⟨ı |H1|k ⟩

00

⟨k ||L(0)1 ||a ⟩

(L1D)+∑

c k

⟨c ı |H2|k a ⟩

00

00

⟨k ||L(0)1 ||c ⟩

(L1E)+ 1

2−1

i

∑

c d k

∑

J

J ⟨i k ||L(0)2 ||c d ⟩

J J

⟨c d |H2|a k ⟩

J M J M

(L1F)− 1

2−1

i

∑

c k l

∑

J

J ⟨i c |H2|k l ⟩

J M J M

⟨k l ||L(0)2 ||a c ⟩

J J

(L1G)+ 1

2

∑

c d e k l

−1d

∑

J

J ⟨d ı |H2|e a ⟩

00

00

⟨c e |t2|k l ⟩J M J M

⟨k l ||L(0)2 ||c d ⟩

J J

(L1H)− 1

2

∑

c d k l m

−1l

∑

J

J ⟨k l ||L(0)2 ||c d ⟩

J J

⟨c d |t2|k m ⟩

J M J M

⟨m ı |H2|l a ⟩

00

00

Figure H.10: Spherical expressions for the scalar EOM-CCSD L1 amplitude equations.



⟨i j || L(0) Hopen ||ab ⟩

J J

(L2B)− CCAtoStd(A)

ab

i jJ ; J ′

J ⟨ı |H1|a ⟩

00

⟨j ||L(0)1 ||b ⟩ δJ ′0

Pab (J ) Pi j (J )

¨

(L2C)+ 1

2J −1

i

∑

c

⟨i ||L(0)1 ||c ⟩ ⟨c j |H2|ab ⟩

J M J M

(L2D)− 1

2J −1

a

∑

k

⟨k ||L(0)1 ||a ⟩ ⟨i j |H2|kb ⟩

J M J M

(L2E)− 1

2−1

b

∑

c

⟨i j ||L(0)2 ||a c ⟩

J J

⟨c |H1|b ⟩

00

(L2F)+ 1

2−1

j

∑

k

⟨i k |L(0)2 |ab ⟩

J J

⟨ |H1|k ⟩

00«

(L2G)− CCAtoStd(A)

i j

abJ ; J ′

× J ( J ′)−1∑

c k

(−1)jc+jk−J ′ ⟨ı k ||L(0)2 ||a c ⟩

J ′

J ′

⟨c |H2|kb ⟩

J ′M ′

J ′M ′

Pab (J ) Pi j (J )

¨

(L2H)+ 1

8

∑

c d

⟨i j ||L(0)2 ||c d ⟩

J J

⟨c d |H2|ab ⟩

J M J M

(L2I)+ 1

8

∑

k l

⟨i j |H2|k l ⟩

J M J M

⟨k l ||L(0)2 ||ab ⟩

J J

(L2J)− 1

4J −2

a

∑

c d k l

∑

J ′J ′ ⟨c d |t2|k l ⟩

J ′M ′ J ′M ′

⟨k l ||L(0)2 ||c a ⟩

J ′ J ′

⟨i j |v |d b ⟩

J M J M

δja jd

(L2K)− 1

4J −2

i

∑

c d k l

∑

J ′J ′ ⟨k i ||L(0)2 ||c d ⟩

J ′ J ′

⟨c d |t2|k l ⟩

J ′M ′ J ′M ′

⟨l j |v |ab ⟩

J M J M

δj i j l

«

Figure H.11: Spherical expressions for the scalar EOM-CCSD L2 amplitude equations.


Appendix I

Publications

Appendix I. Publications

The following articles originated during the course of this dissertation:

Peer-Reviewed Journals:

1. R. Roth, J. Langhammer, A. Calci, S. Binder, P. Navrátil:"Similarity-Transformed Chiral NN+3N Interactions for the Ab Initio Description

of 12C and 16O"

Phys. Rev. Lett. 107, 072501 (2011)

2. R. Roth, S. Binder, K. Vobig, A. Calci, J. Langhammer, P. Navrátil:"Ab Initio Calculations of Medium-Mass Nuclei with Normal-Ordered Chiral NN+3N

Interactions",Phys. Rev. Lett 109, 052501 (2012)

3. S. Binder, J. Langhammer, A. Calci, P. Navrátil, R. Roth:"Ab Initio Calculations of Medium-Mass Nuclei with Explicit Chiral 3N Interac-

tions"

Phys. Rev. C 87, 021303(R) (2013) - Editors’ Suggestion

4. H. Hergert, S. K. Bogner, S. Binder, A. Calci, J. Langhammer, R. Roth, A.Schwenk:"In-Medium Similarity Renormalization Group with Chiral Two- Plus Three-Nucleon

Interactions"

Phys. Rev. C 87, 034307 (2013)

5. H. Hergert, S. Binder, A. Calci, J. Langhammer, R. Roth:"Ab Initio Calculations of Even Oxygen Isotopes with Chiral Two- Plus Three-

Nucleon Interactions"

Phys. Rev. Lett. 110, 242501 (2013)

6. S. Binder, P. Piecuch, A. Calci, J. Langhammer, P. Navrátil, R. Roth:"Extension of coupled-cluster theory with a non-iterative treatment of connected

triply excited clusters to three-body Hamiltonians"

Phys. Rev. C 88, 054319 (2013)



7. R. Roth, A. Calci, J. Langhammer, S. Binder:"Evolved Chiral NN+3N Hamiltonians for Ab Initio Nuclear Structure

Calculations"

submitted to Phys. Rev. C

8. S. Binder, J. Langhammer, A. Calci, R. Roth:"Ab Initio Path to Heavy Nuclei"

submitted to Phys. Rev. Lett.

Peer-Reviewed Conference Proceedings:

1. R. Roth, J. Langhammer, A. Calci, S. Binder, Petr Navrátil:"Ab Initio Nuclear Structure Theory with Chiral NN+3N Interactions"

Prog. Theor. Phys. Suppl. 196, 131 (2012)

2. R. Roth, J. Langhammer, S. Binder, A. Calci:"New Horizons in Ab Initio Nuclear Structure Theory"

J. Phys.: Conf. Ser. 403, 012020 (2012)

3. R. Roth, J. Langhammer, A. Calci, S. Binder:"From Chiral EFT Interactions to Ab Initio Nuclear Structure"

PoS(CD12) 015 (2013) Proceedings of the 7th International Workshop on Chi-ral Dynamics, August 6 - 10, 2012, Jefferson Lab, Newport News, VA, USA

4. P. Maris, H. M. Aktulga, S. Binder, A. Calci, Ü. Catalyürek, J. Langhammer,E. Ng, E. Saule, R. Roth, J. Vary, C. Yang:"No Core CI Calculations for Light Nuclei with Chiral 2- and 3-body Forces"

J. Phys.: Conf. Ser. 454, 012063 (2013) Proceedings of the 24th IUPAP Con-ference on Computational Physics (IUPAP-CCP 2012), October 14 - 18, 2012,Kobe, Japan

5. R. Roth, A. Calci, J. Langhammer, S. Binder:"Ab Initio Nuclear Structure Theory: From Few to Many"

Proceedings of the 22nd European Conference on Few-Body Problems inPhysics (EFB22), September 9-13, 2013, Krakow, Poland



6. R. Roth, A. Calci, J. Langhammer, S. Binder:"Towards New Horizons in Ab Initio Nuclear Structure Theory"

Proceedings of the 25th International Nuclear Physics Conference 2013(INPC2013), June 2 - 7, 2013, Firenze, Italy

7. D. Oryspayev, H. Potter, P. Maris, M. Sosonkina, J. P. Vary, S. Binder, A. Calci,J. Langhammer, R. Roth:"Leveraging GPUs in Ab Initio Nuclear Physics Calculations"

IEEE 27th Parallel and Distributed Processing Symposium Workshops &PhD Forum (2013)


Bibliography


Bibliography

[1] R. A. Bryan, B. L. Scott; Nucleon-Nucleon Scattering from One-Boson-Exchange

Potentials; Phys. Rev. 135 (1964) B434.

[2] R. Machleidt; High-precision, charge-dependent Bonn nucleon-nucleon potential;Phys. Rev. C 63 (2001) 024001.

[3] S. Weinberg; Phenomenological Lagrangians; Physica A: Statistical Mechanicsand its Applications 96 (1979) 327.

[4] S. Weinberg; Nuclear forces from chiral lagrangians; Phys. Lett. B 251 (1990)288.

[5] S. Weinberg; Effective chiral lagrangians for nucleon-pion interactions and nuclear

forces; Nucl. Phys. B 363 (1991) 3.

[6] J. Gasser, H. Leutwyler; Chiral perturbation theory to one loop; Annals ofPhysics 158 (1984) 142.

[7] J. Gasser, H. Leutwyler; Chiral perturbation theory: Expansions in the mass of

the strange quark; Nuclear Physics B 250 (1985) 465.

[8] U. van Kolck; Effective field theory of nuclear forces; Prog. Part. Nucl. Phys. 43

(1999) 337.

[9] U. van Kolck; Few-nucleon forces from chiral Lagrangians; Phys. Rev. C 49 (1994)2932.

[10] R. Machleidt, D. R. Entem; Chiral effective field theory and nuclear forces; Phys.Rep. 503 (2011) 1.

[11] E. Epelbaum, H.-W. Hammer, U.-G. Meißner; Modern theory of nuclear forces;Rev. Mod. Phys. 81 (2009) 1773.

[12] E. Epelbaum; Nuclear forces from chiral effective field theory: A primer;arXiv:1001.3229 [nucl-th] (2010).

[13] R. Roth, H. Hergert, P. Papakonstantinou, et al.; Matrix Elements and Few-

Body Calculations within the Unitary Correlation Operator Method; Phys. Rev. C72 (2005) 034002.

[14] R. B. Wiringa, V. G. J. Stoks, R. Schiavilla; An Accurate Nucleon–Nucleon Po-

tential with Charge–Indenpendence Breaking; Phys. Rev. C 51 (1995) 38.


Bibliography

[15] I. Talmi; Simple Models of Complex Nuclei; Harwood Academic Publishers(1993).

[16] E. D. Jurgenson, P. Navrátil, R. J. Furnstahl; Evolution of Nuclear Many-Body

Forces with the Similarity Renormalization Group; Phys. Rev. Lett. 103 (2009)082501.

[17] R. Roth, J. Langhammer, A. Calci, et al.; Similarity-Transformed Chiral NN+3N

Interactions for the Ab Initio Description of 12C and 16O; Phys. Rev. Lett. 107

(2011) 072501.

[18] E. D. Jurgenson, P. Maris, R. J. Furnstahl, et al.; Structure of p-shell nuclei using

three-nucleon interactions evolved with the similarity renormalization group; Phys.Rev. C 87 (2013) 054312.

[19] P. Maris, J. P. Vary, A. M. Shirokov; Ab initio no-core full configuration calcula-

tions of light nuclei; Phys. Rev. C 79 (2009) 014308.

[20] R. Roth; Ab initio nuclear structure calculations with transformed realistic inter-

actions; Eur. Phys. J. Special Topics 156 (2008) 191.

[21] R. Roth; Importance Truncation for Large-Scale Configuration Interaction Ap-

proaches; Phys. Rev. C 79 (2009) 064324.

[22] F. Coester; Bound states of a many-particle system; Nuclear Physics 7 (1958) 421.

[23] F. Coester, H. Kümmel; Short-range correlations in nuclear wave functions; Nu-clear Physics 17 (1960) 477.

[24] J. Čížek; On the Correlation Problem in Atomic and Molecular Systems. Calcula-

tion of Wavefunction Components in Ursell-Type Expansion Using Quantum-Field

Theoretical Methods; The Journal of Chemical Physics 45 (1966) 4256.

[25] T. D. Crawford, H. F. Schaefer; An Introduction to Coupled Cluster Theory for

Computational Chemists; Reviews in computational chemistry 14 (2000) 33.

[26] R. J. Bartlett I. Shavitt; Many-Body Methods in Chemistry and Physics: MBPT

and Coupled-Cluster Theory; Cambridge Molecular Science (2009).

[27] G. Hagen, T. Papenbrock, D. J. Dean, M. Hjorth-Jensen; Ab initio coupled-

cluster approach to nuclear structure with modern nucleon-nucleon interactions;Phys. Rev. C82 (2010) 034330.


Bibliography

[28] K. Tsukiyama, S. K. Bogner, A. Schwenk; In-Medium Similarity Renormaliza-

tion Group For Nuclei; Phys. Rev. Lett. 106 (2011) 222502.

[29] H. Hergert, S. K. Bogner, S. Binder, et al.; In-medium similarity renormalization

group with chiral two- plus three-nucleon interactions; Phys. Rev. C 87 (2013)034307.

[30] H. Hergert, S. Binder, A. Calci, et al.; Ab Initio Calculations of Even Oxygen

Isotopes with Chiral Two-Plus-Three-Nucleon Interactions; Phys. Rev. Lett. 110

(2013) 242501.

[31] R. Schneider; Analysis of the projected coupled cluster method in electronic struc-

ture calculation; Numerische Mathematik 113 (2009) 433.

[32] K. Heyde; The Nuclear Shell Model; Springer (1990).

[33] J. Suhonen; From Nucleons to Nucleus: Concepts of Microscopic Nuclear Theory;Springer (2007).

[34] P. Navrátil, J. P. Vary, B. R. Barrett; Properties of 12C in the Ab Initio Nuclear

Shell Model; Phys. Rev. Lett. 84 (2000) 5728.

[35] P. Navrátil, G. P. Kamuntavicius, B. R. Barrett; Few-nucleon systems in a trans-

lationally invariant harmonic oscillator basis; Phys. Rev. C 61 (2000) 044001.

[36] P. Navrátil, J. P. Vary, B. R. Barrett; Large-basis ab initio no-core shell model and

its application to 12C; Phys. Rev. C 62 (2000) 054311.

[37] E. Caurier, P. Navrátil, W. E. Ormand, J. P. Vary; Intruder states in 8Be; Phys.Rev. C 64 (2001) 051301.

[38] P. Navrátil, W. E. Ormand; Ab Initio Shell Model Calculations with Three-Body

Effective Interactions for p-Shell Nuclei; Phys. Rev. Lett. 88 (2002) 152502.

[39] B. R. Barrett, P. Navrátil, J. P. Vary; Large-basis no-core shell model; NuclearPhysics A 704 (2002) 254 .

[40] E. Caurier, P. Navrátil, W. E. Ormand, J. P. Vary; Ab initio shell model for A = 10

nuclei; Phys. Rev. C 66 (2002) 024314.

[41] P. Navrátil, W. E. Ormand; Ab initio shell model with a genuine three-nucleon

force for the p-shell nuclei; Phys. Rev. C 68 (2003) 034305.


Bibliography

[42] H. Zhan, A. Nogga, B. R. Barrett, et al.; Extrapolation method for the no-core

shell model; Phys. Rev. C 69 (2004) 034302.

[43] M. A. Hasan, J. P. Vary, P. Navrátil; Hartree-Fock approximation for the ab initio

no-core shell model; Phys. Rev. C 69 (2004) 034332.

[44] J. P. Vary, B. R. Barrett, R. Lloyd, et al.; Shell model in a first principles approach;Nuclear Physics A 746 (2004) 123.

[45] I. Stetcu, B. R. Barrett, P. Navrátil, J. P. Vary; Effective operators within the ab

initio no-core shell model; Phys. Rev. C 71 (2005) 044325.

[46] J. P. Vary, O. V. Atramentov, B. R. Barrett, et al.; Ab initio No-Core Shell

Model –Recent results and future prospects; The European Physical Journal A- Hadrons and Nuclei 25 (2005) 475.

[47] I. Stetcu, B. R. Barrett, P. Navrátil, J. P. Vary; Long- and short-range correlations

in the ab-initio no-core shell model; Phys. Rev. C 73 (2006) 037307.

[48] A. Nogga, P. Navrátil, B. R. Barrett, Vary J. P.; Spectra and binding energy pre-

dictions of chiral interactions for 7Li; Phys. Rev. C 73 (2006) 064002.

[49] B. R. Barrett, I. Stetcu, P. Navrátil, J. P. Vary; From non-Hermitian effective op-

erators to large-scale no-core shell model calculations for light nuclei; J. Phys. A 39

(2006) 9983.

[50] P. Navrátil, V. G. Gueorguiev, J. P. Vary, et al.; Structure of A1013 nuclei with

two- plus three-nucleon interactions from chiral field theory; Phys. Rev. Lett. 99

(2007) 042501.

[51] R. Roth, P. Navrátil; Ab Initio Study of 40Ca with an Importance-Truncated No-

Core Shell Model; Phys. Rev. Lett. 99 (2007) 092501.

[52] C. Forssén, J. P. Vary, E. Caurier, P. Navrátil; Converging sequences in the ab

initio no-core shell model; Phys. Rev. C 77 (2008) 024301.

[53] E. Caurier, G. Martínez-Pinedo, F. Nowacki, et al.; The shell model as a unified

view of nuclear structure; Rev. Mod. Phys. 55 (2005) 427.

[54] P. Navrátil, S. Quaglioni, I. Stetcu, B. Barrett; Recent developments in no-core

shell-model calculations; J. Phys. G: Nucl. Part. Phys. 36 (2009) 083101.


Bibliography

[55] J. Čížek; On the Use of the Cluster Expansion and the Technique of Diagrams in

Calculations of Correlation Effects in Atoms and Molecules; Adv. Chem. Phys 14

(1969) 35.

[56] J. Paldus, J. Čížek; Time-Independent Diagrammatic Approach to Perturbation

Theory of Fermion Systems; Adv. Quantum Chem. 9 (1975) 105.

[57] A. G. Taube, R. J. Bartlett; Improving upon CCSD(T): ΛCCSD(T). I. Potential

energy surfaces; The Journal of Chemical Physics 128 (2008) 044110.

[58] A. G. Taube, R. J. Bartlett; Improving upon CCSD(T): ΛCCSD(T). II. Station-

ary formulation and derivatives; The Journal of Chemical Physics 128 (2008)044111.

[59] Y. J. Bomble, J. F. Stanton, M. Kallay, J. Gauss; Coupled-cluster methods includ-

ing noniterative corrections for quadruple excitations; The Journal of ChemicalPhysics 123 (2005) 054101.

[60] J. H. Heisenberg, B. Mihaila; Ground state correlations and mean field in 16O;Phys. Rev. C 59 (1999) 1440.

[61] D. J. Dean, J. R. Gour, G. Hagen, et al.; Nuclear Structure Calculations with

Coupled Cluster Methods from Quantum Chemistry; Nucl. Phys. A 752 (2005)299.

[62] G. Hagen, D. J. Dean, M. Hjorth-Jensen, et al.; Benchmark calculations for 3H,4He, 16O, and 40Ca with ab initio coupled-cluster theory; Phys. Rev. C 76 (2007)044305.

[63] G. Hagen, T. Papenbrock, D. J. Dean, et al.; Coupled-cluster theory for three-body

Hamiltonians; Phys. Rev. C 76 (2007) 034302.

[64] G. Hagen, D. J. Dean, M. Hjorth-Jensen, T. Papenbrock; Complex coupled-

cluster approach to an ab-initio description of open quantum systems; Physics Let-ters B 656 (2007) 169.

[65] G. Hagen, T. Papenbrock, D. J. Dean, M. Hjorth-Jensen; Medium-Mass Nuclei

from Chiral Nucleon-Nucleon Interactions; Phys. Rev. Lett. 101 (2008) 092502.

[66] G. Hagen, T. Papenbrock, D. J. Dean, et al.; Ab initio computation of neutron-

rich oxygen isotopes; Phys. Rev. C 80 (2009) 021306.


Bibliography

[67] G. Hagen, T. Papenbrock, M. Hjorth-Jensen; Ab Initio Computation of the 17F

Proton Halo State and Resonances in A = 17 Nuclei; Phys. Rev. Lett. 104 (2010)182501.

[68] Ø. Jensen, G. Hagen, T. Papenbrock, et al.; Computation of spectroscopic factors

with the coupled-cluster method; Phys. Rev. C 82 (2010) 014310.

[69] G. R. Jansen, M. Hjorth-Jensen, G. Hagen, T. Papenbrock; Toward open-shell

nuclei with coupled-cluster theory; Phys. Rev. C 83 (2011) 054306.

[70] G. Hagen, M. Hjorth-Jensen, G. R. Jansen, et al.; Continuum Effects and Three-

Nucleon Forces in Neutron-Rich Oxygen Isotopes; Phys. Rev. Lett. 108 (2012)242501.

[71] D. A. Pigg, G. Hagen, H. Nam, T. Papenbrock; Time-dependent coupled-cluster

method for atomic nuclei; Phys. Rev. C 86 (2012) 014308.

[72] G. Hagen, N. Michel; Elastic proton scattering of medium mass nuclei from

coupled-cluster theory; Phys. Rev. C 86 (2012) 021602.

[73] S. Bacca, N. Barnea, G. Hagen, et al.; First Principles Description of the Giant

Dipole Resonance in 16O; Phys. Rev. Lett. 111 (2013) 122502.

[74] G. Hagen, P. Hagen, H.-W. Hammer, L. Platter; Efimov Physics Around the

Neutron-Rich 60Ca Isotope; Phys. Rev. Lett. 111 (2013) 132501.

[75] G. Baardsen, A. Ekström, G. Hagen, M. Hjorth-Jensen; Coupled Cluster stud-

ies of infinite nuclear matter; arXiv:1306.5681 [nucl-th] (2013).

[76] J. F. Stanton, R. J. Bartlett; The equation of motion coupled-cluster method. A

systematic biorthogonal approach to molecular excitation energies, transition prob-

abilities, and excited state properties; The Journal of Chemical Physics 98 (1993)7029.

[77] S. Veerasamy, W. N. Polyzou; Momentum-space Argonne V18 interaction; Phys.Rev. C 84 (2011) 034003.

[78] P. Ring, P. Schuck; The Nuclear Many-Body Problem; Springer Verlag, NewYork (1980).

[79] W. N. Polyzou, W. Glöckle; Three-body interactions and on-shell equivalent two-

body interactions; Few-Body Systems 9 (1990) 97.


Bibliography

[80] H. Yukawa; On the Interaction of Elementary Particles. I; Proceedings of thePhysico-Mathematical Society of Japan. 3rd Series 17 (1935) 48.

[81] S. Ogawa, S. Sawasa, T. Ueda, et al.; One-Boson-Exchange Model; Prog. Theor.Phys. Suppl. 39 (1967).

[82] E. Jenkins, A. V. Manohar; Baryon chiral perturbation theory using a heavy

fermion lagrangian; Physics Letters B 255 (1991) 558.

[83] V. Bernard, N. Kaiser, J. Kambor, U.-G. Meißner; Chiral structure of the nu-

cleon; Nuclear Physics B 388 (1992) 315.

[84] E. Epelbaum, U.-G. Meißner; On the Renormalization of the One–Pion Exchange

Potential and the Consistency of Weinberg’s Power Counting; Few-Body Systems(2012) 1.

[85] S. K. Bogner, R. J. Furnstahl, A. Schwenk; From low-momentum interactions to

nuclear structure; Prog. Part. Nucl. Phys. 65 (2010) 94.

[86] R. Roth, A. Calci, J. Langhammer, S. Binder; Evolved Chiral NN+3N Hamil-

tonians for Ab Initio Nuclear Structure Calculations; arXiv:1311.3563 [nucl-th](2013).

[87] S. K. Bogner, T. T. S. Kuo, A. Schwenk; Model-independent low momentum nu-

cleon interaction from phase shift equivalence; Phys. Rep. 386 (2003) 1.

[88] S. Kehrein; The Flow Equation Approach to Many-Particle Systems; vol. 217 ofSpringer Tracts in Modern Physics; Springer, Berlin (2006).

[89] R. Roth, J. Langhammer, S. Binder, A. Calci; New Horizons in Ab Initio Nuclear

Structure Theory; Journal of Physics: Conference Series 403 (2012) 012020.

[90] I. Mayer; Simple Theorems, Proofs, and Derivations in Quantum Chemistry;Kluwer Academic (2003).

[91] A. Zapp; Kernstruktur mit effektiven Dreiteilchenpotentialen; Master’s thesis;TU Darmstadt (2006).

[92] R. Roth, S. Binder, K. Vobig, et al.; Medium-Mass Nuclei with Normal-Ordered

Chiral N N+3N Interactions; Phys. Rev. Lett. 109 (2012) 052501.


Bibliography

[93] J. D. Holt, T. Otsuka, A. Schwenk, T. Suzuki; Three-body forces and shell struc-

ture in calcium isotopes; Journal of Physics G: Nuclear and Particle Physics 39

(2012) 085111.

[94] K. Vobig; B.Sc. Thesis (2001).

[95] S. Binder, J. Langhammer, A. Calci, et al.; Ab initio calculations of medium-mass

nuclei with explicit chiral 3N interactions; Phys. Rev. C 87 (2013) 021303(R).

[96] S. Binder, P. Piecuch, A. Calci, et al.; Extension of coupled-cluster theory with a

noniterative treatment of connected triply excited clusters to three-body Hamiltoni-

ans; Phys. Rev. C 88 (2013) 054319.

[97] R. Roth, J. R. Gour, P. Piecuch; Center-of-mass problem in truncated configura-

tion interaction and coupled-cluster calculations; Phys. Lett. B 679 (2009) 334.

[98] M. A. Caprio, P. Maris, J. P. Vary; Coulomb-Sturmian basis for the nuclear many-

body problem; Phys. Rev. C 86 (2012) 034312.

[99] H. Kümmel, K. H. Lührmann, J. G. Zabolitzky; Many-fermion theory in expS-

(or coupled cluster) form; Phys. Rep. 36 (1978) 1.

[100] J. Čížek; On the Use of the Cluster Expansion and the Technique of Diagrams in

Calculations of Correlation Effects in Atoms and Molecules; Adv. Chem. Phys.14 (1969) 35.

[101] J. Čížek, J. Paldus; Correlation problems in atomic and molecular systems III. Red-

erivation of the coupled-pair many-electron theory using the traditional quantum

chemical methods; Int. J. Quantum Chem. 5 (1971) 359.

[102] J. Paldus, I. Shavitt, J. Čížek; Correlation Problems in Atomic and Molecular

Systems. IV. Extended Coupled-Pair Many-Electron Theory and Its Application to

the BH3 Molecule; Phys. Rev. A 5 (1972) 50.

[103] R. F. Bishop; Microscopic Quantum Many-Body Theories and Their Applications;in J. Navarro, A. Polls (editors), Microscopic Quantum Many-Body Theories

and Their Applications; vol. 510 of Lecture Notes in Physics; 119–206; Springer,Berlin (1998).

[104] M. L. Goldberger, K. M. Watson; Collision Theory; Dover Publications (2004).

[105] G. D. Purvis, R. J. Bartlett; A full coupled-cluster singles and doubles model: The

inclusion of disconnected triples; J. Chem. Phys. 76 (1982) 1910.


Bibliography

[106] J. M. Cullen, M. C. Zerner; The linked singles and doubles model: An approximate

theory of electron correlation based on the coupled-cluster ansatz; J. Chem. Phys.77 (1982) 4088.

[107] G. E. Scuseria, A. C. Scheiner, T. J. Lee, et al.; The closed-shell coupled cluster

single and double excitation (CCSD) model for the description of electron correla-

tion. A comparison with configuration interaction (CISD) results; J. Chem. Phys.86 (1987) 2881.

[108] P. Piecuch, J. Paldus; Orthogonally spin-adapted coupled-cluster equations in-

volving singly and doubly excited clusters. Comparison of different procedures for

spin-adaptation; Int. J. Quantum Chem. 36 (1989) 429.

[109] K. Kowalski, D. J. Dean, M. Hjorth-Jensen, et al.; Coupled Cluster Calculations

of Ground and Excited States of Nuclei; Phys. Rev. Lett. 92 (2004) 132501.

[110] D. J. Dean, M. Hjorth-Jensen; Coupled-cluster approach to nuclear physics; Phys.Rev. C 69 (2004) 054320.

[111] M. Wloch, D. J. Dean, J. R. Gour, et al.; Ab-Initio Coupled-Cluster Study of 16O;Phys. Rev. Lett. 94 (2005) 212501.

[112] M. Włoch, J. R. Gour, P. Piecuch, et al.; Coupled-cluster calculations for ground

and excited states of closed- and open-shell nuclei using methods of quantum chem-

istry; J. Phys. G: Nucl. Part. Phys. 31 (2005) S1291.

[113] T. Papenbrock, D. J. Dean, J. R. Gour, et al.; Coupled-Cluster Theory for Nuclei;Int. J. Mod. Phys. B 20 (2006) 5338.

[114] M. Horoi, J. R. Gour, M. Włoch, et al.; Coupled-Cluster and Configuration-

Interaction Calculations for Heavy Nuclei; Phys. Rev. Lett. 98 (2007) 112501.

[115] P. Piecuch, R. J. Bartlett; EOMXCC: A new Coupled-Cluster Method for Elec-

tronic Excited States; Adv. Quantum Chem. 34 (1999) 295.

[116] J. Gauss; Encyclopedia of Computational Chemistry; in P. v. R. Schleyer, N. L.Allinger, T. Clark, et al. (editors), Encyclopedia of Computational Chemistry;vol. 1; 615–636; Wiley, Chichester (1998).

[117] P. Piecuch, M. Włoch, M. Lodriguito, J. R. Gour; Recent Advances in the Theory

of Chemical and Physical Systems; in S. Wilson, J.-P. Julien, J. Maruani, et al. (ed-itors), Recent Advances in the Theory of Chemical and Physical Systems; vol. 15


Bibliography

of Progress in Theoretical Chemistry and Physics; 45–106; Springer, Dordrecht(2006).

[118] R. J. Bartlett, M. Musiał; Coupled-cluster theory in quantum chemistry; Rev.Mod. Phys. 79 (2007) 291.

[119] P. Piecuch, M. Wloch; Renormalized coupled-cluster methods exploiting left

eigenstates of the similarity-transformed Hamiltonian; The Journal of ChemicalPhysics 123 (2005) 224105.

[120] R. Roth, J. R. Gour, P. Piecuch; Ab initio coupled-cluster and configuration inter-

action calculations for [sup 16]O using the V[sub UCOM] interaction; Phys. Rev.C 79 (2009) 054325.

[121] J. Shen, P. Piecuch; Biorthogonal moment expansions in coupled-cluster theory:

Review of key concepts and merging the renormalized and active-space coupled-

cluster methods; Chem. Phys. 401 (2012) 180.

[122] M. Urban, J. Noga, S. J. Cole, R. J. Bartlett; Towards a full CCSDT model for

electron correlation; J. Chem. Phys. 83 (1985) 4041.

[123] P. Piecuch, J. Paldus; Coupled cluster approaches with an approximate account of

triexcitations and the optimized inner projection technique; Theor. Chem. Acc. 78

(1990) 65.

[124] K. Raghavachari, G. W. Trucks, J. A. Pople, M. Head-Gordon; A fifth-order

perturbation comparison of electron correlation theories; Chem. Phys. Lett. 157

(1989) 479.

[125] S. A. Kucharski, R. J. Bartlett; An efficient way to include connected quadruple

contributions into the coupled cluster method; J. Chem. Phys. 108 (1998) 9221.

[126] S. A. Kucharski, R. J. Bartlett; Noniterative energy corrections through fifth-order

to the coupled cluster singles and doubles method; J. Chem. Phys. 108 (1998) 5243.

[127] M. Musiał, R. J. Bartlett; Improving upon CCSD(TQf) for potential energy sur-

faces: ΛCCSD(TQf) models; J. Chem. Phys. 133 (2010) 104102.

[128] S. R. Gwaltney, M. Head-Gordon; A second-order correction to singles and dou-

bles coupled-cluster methods based on a perturbative expansion of a similarity-

transformed Hamiltonian; Chem. Phys. Lett. 323 (2000) 21.


Bibliography

[129] S. R. Gwaltney, M. Head-Gordon; A second-order perturbative correction to the

coupled-cluster singles and doubles method: CCSD(2); J. Chem. Phys. 115 (2001)2014.

[130] S. Hirata, M. Nooijen, I. Grabowski, R. J. Bartlett; Perturbative corrections to

coupled-cluster and equation-of-motion coupled-cluster energies: A determinantal

analysis; J. Chem. Phys. 114 (2001) 3919; 115, 3967 (2001) [Erratum].

[131] S. Hirata, P.-D. Fan, A. A. Auer, et al.; Combined coupled-cluster and many-body

perturbation theories; The Journal of Chemical Physics 121 (2004) 12197.

[132] P. Piecuch, K. Kowalski; Computational Chemistry: Reviews of Current Trends;in J. Leszczyński (editor), Computational Chemistry: Reviews of Current Trends;vol. 5; 1–104; World Scientific, Singapore (2000).

[133] K. Kowalski, P. Piecuch; The method of moments of coupled-cluster equations and

the renormalized CCSD[T], CCSD(T), CCSD(TQ), and CCSDT(Q) approaches; J.Chem. Phys. 113 (2000) 18.

[134] K. Kowalski, P. Piecuch; Renormalized CCSD(T) and CCSD(TQ) approaches:

Dissociation of the N2 triple bond; J. Chem. Phys. 113 (2000) 5644.

[135] P. Piecuch, K. Kowalski, I. S. O. Pimienta, M. J. McGuire; Recent advances in

electronic structure theory: Method of moments of coupled-cluster equations and

renormalized coupled-cluster approaches; Int. Rev. Phys. Chem. 21 (2002) 527.

[136] P. Piecuch, K. Kowalski, I. S. O. Pimienta, et al.; Method of moments of coupled-

cluster equations: a new formalism for designing accurate electronic structure

methods for ground and excited states; Theor. Chem. Acc. 112 (2004) 349.

[137] M. Włoch, J. R. Gour, P. Piecuch; Extension of the Renormalized Coupled-Cluster

Methods Exploiting Left Eigenstates of the Similarity-Transformed Hamiltonian to

Open-Shell Systems: A Benchmark Study; J. Phys. Chem. A 111 (2007) 11359.

[138] P. Piecuch, J. R. Gour, M. Włoch; Left-eigenstate completely renormalized

equation-of-motion coupled-cluster methods: Review of key concepts, extension

to excited states of open-shell systems, and comparison with electron-attached and

ionized approaches; International Journal of Quantum Chemistry 109 (2009)3268.

[139] P. Piecuch, M. Włoch, A. J. C Varandas; Application of renormalized coupled-

cluster methods to potential function of water; Theor. Chem. Acc. 120 (2008) 59.


Bibliography

[140] K. Kowalski, P. Piecuch; Extensive generalization of renormalized coupled-cluster

methods; J. Chem. Phys. 122 (2005) 074107.

[141] Basic Linear Algebra Subprograms; URL http://www.netlib.org/blas.

[142] K. Kowalski, P. Piecuch; New type of noniterative energy corrections for excited

electronic states: Extension of the method of moments of coupled-cluster equations

to the equation-of-motion coupled-cluster formalism; The Journal of ChemicalPhysics 115 (2001) 2966.

[143] K. Kowalski, P. Piecuch; Extension of the method of moments of coupled-cluster

equations to excited states: The triples and quadruples corrections to the equation-

of-motion coupled-cluster singles and doubles energies; The Journal of ChemicalPhysics 116 (2002) 7411.

[144] K. Jankowski, J. Paldus, P. Piecuch; ; Theor. Chim. Acta 80 (1991) 223.

[145] J. W. Demmel; Applied Numerical Linear Algebra; Siam (1997).

[146] A. Bohr, B. Mottelson; Struktur der Atomkerne; Carl Hanser Verlag (1979).

[147] D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii; Quantum Theory of

Angular Momentum; World Scientific Publishing Company (1988).

[148] T. T. S. Kuo, J. Shurpin, K. C. Tam, et al.; A simple method for evaluating

Goldstone diagrams in an angular momentum coupled representation; Annals ofPhysics 132 (1981) 237.

[149] A. Baran, A. Bulgac, M. McNeil Forbes, et al.; Broyden’s method in nuclear

structure calculations; Phys. Rev. C 78 (2008) 014318.

[150] H. Walker, P. Ni; Anderson Acceleration for Fixed-Point Iterations; SIAM Journalon Numerical Analysis 49 (2011) 1715.

[151] D. D. Johnson; Modified Broyden’s method for accelerating convergence in self-

consistent calculations; Phys. Rev. B 38 (1988) 12807.

[152] G. Hagen, T. Papenbrock, D. J. Dean; Solution of the Center-Of-Mass Problem

in Nuclear Structure Calculations; Phys. Rev. Lett. 103 (2009) 062503.

[153] A. G. Taube; Alternative perturbation theories for triple excitations in coupled-

cluster theory; Molecular Physics 108 (2010) 2951.


Bibliography

[154] D. J. Thouless; Stability conditions and nuclear rotations in the Hartree-Fock the-

ory; Nuclear Physics 21 (1960) 225.

[155] P. Navrátil; Local three-nucleon interaction from chiral effective field theory; FewBody Syst. 41 (2007) 117.

[156] A. Calci; Evolved Chiral Hamiltonians at the Three-Body Level and Beyond; Ph.D.thesis; TU Darmstadt (2014).

[157] J. Langhammer; Chiral Three-Nucleon Interactions in Ab-Initio Nuclear Struc-

ture and Reactions; Ph.D. thesis; TU Darmstadt (2014).

[158] A. Cipollone, C. Barbieri, P. Navrátil; Isotopic Chains Around Oxygen from

Evolved Chiral Two- and Three-Nucleon Interactions; Phys. Rev. Lett. 111 (2013)062501.

[159] A. Ekström, G. Baardsen, C. Forssén, et al.; Optimized Chiral Nucleon-

Nucleon Interaction at Next-to-Next-to-Leading Order; Phys. Rev. Lett. 110

(2013) 192502.

[160] M. Wang, G. Audi, A.H. Wapstra, et al.; The Ame2012 atomic mass evaluation;Chinese Physics C 36 (2012) 1603.

[161] H. de Vries, C. W. de Jager, C. de Vries; Nuclear charge-density-distribution

parameters from elastic electron scattering; At. Data Nucl. Data Tables 36 (1987)495.

[162] J. Speth, J. Wambach; Electric and Magnetic Giant Electric and Magnetic Giant

Resonances in Nuclei; World Scientific, Singapore (1991).

[163] P. Papakonstantinou, R. Roth; Large-scale second random-phase approximation

calculations with finite-range interactions; Phys. Rev. C 81 (2010) 024317.

[164] A. Tohsaki, H. Horiuchi, P. Schuck, G. Röpke; Alpha Cluster Condensation in12C and 16O; Phys. Rev. Lett. 87 (2001) 192501.

[165] R. B. Lehoucq, D. C. Sorensen, C. Yang; ARPACK Users’ Guide: Solution of

Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods; Siam(1998).

[166] National Nuclear Data Center; URL http://www.nndc.bnl.gov.


Bibliography

[167] P. Maris, J. P. Vary, S. Gandolfi, et al.; Properties of trapped neutrons interacting

with realistic nuclear Hamiltonians; Phys. Rev. C 87 (2013) 054318.

[168] S. Gandolfi, J. Carlson, S. C. Pieper; Cold Neutrons Trapped in External Fields;Phys. Rev. Lett. 106 (2011) 012501.

[169] B. S. Pudliner, V. R. Pandharipande, J. Carlson, et al.; Quantum Monte Carlo

calculations of nuclei with A < 7; Phys. Rev. C 56 (1997) 1720.

[170] S. C. Pieper; The Illinois Extension to the Fujita-Miyazawa Three-Nucleon Force;AIP Conference Proceedings 1011 (2008) 143.


Acknowledgements

I would like to thank Prof. Robert Roth for giving me the opportunity of doingresearch as part of his tnp++ group.

Many thanks go to Prof. Jochen Wambach who kindly agreed to be the secondreviewer of this work.

A great pleasure has been collaborating with Prof. Piotr Piecuch, who showedincomparable interest in my work and, with great patience, provided crucial in-formation on the triples correction methods.

Helpful discussions about Coupled-Cluster theory with Prof. Thomas Papenbrock,Gaute Hagen and Gustav Jansen are very much appreciated.

Special thanks go to the few people brave enough to enter the 2nd-floor Gomor-rhean office, in particular Joachim Langhammer, Angelo Calci, Thomas Krüger,Heiko Hergert, and Eskendr Gebrerufael.

And of course I would like to thank Nadja for her patience and support over thepast years.


Lebenslauf

Zur PersonName Sven BinderGeburtstag 15.12.1982Geburtsort MannheimNationalität deutsch

Bildungsweg

1989–1993 Grundschule, Goetheschule, Lampertheim1993–1999 Gymnasium, Lessing-Gymnasium, Lampertheim1999–2003 Gymnasium, Albertus-Magnus Schule, Viernheim2003–2007 Bachelorstudium Physik, TU Darmstadt2007–2010 Masterstudium Physik, TU Darmstadt2010–2014 Promotionsstudium Physik, TU Darmstadt

Erklärung zur Dissertation

Hiermit versichere ich, die vorliegende Dissertation ohne Hilfe Dritter nur mitden angegebenen Quellen und Hilfsmitteln angefertigt zu haben. Alle Stellen dieaus Quellen entnommen wurden sind als solche kenntlich gemacht. Diese Arbeithat in gleicher oder ähnlicher Form noch keiner Prüfungsbehörde vorgelegen.

Darmstadt, den 05. Februar 2014


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