Theoretische Biophysikalische Chemie -...

0 28.01.2014 Christoph Jacob: Theoretische Biophysikalische Chemie DFG-Centrum für funktionelle Nanostrukturen

DFG-CENTRUM FÜR FUNKTIONELLE NANOSTRUKTUREN

Theoretische Biophysikalische ChemieQM/MM und Einbettungsverfahren

Christoph Jacob

KIT – Universität des Landes Baden-Württemberg undnationales Forschungszentrum in der Helmholtz-Gemeinschaft www.kit.edu

Quantum Chemistry


Electronic Energy: Eel = Eel(R1, . . . ,RN)

positions of nuclei R1, . . . ,RN

Quantum Mechanics

Eel =⟨Ψel|Hel|Ψel

⟩electronic wavefunction: Ψel = Ψel(r1, . . . , rN)

coordinates of electrons r1, . . . , rN

Quantum Chemistry

wave-function theory (WFT)density-functional theory (DFT)

Force Field Methods

→ neglect electrons and model Eel directly

QM/MM Embedding Methods


use quantum mechanics (QM) for “interesting” part(e.g., site of a chemical reaction or photoexcitation)

use force field / molecular mechanics (MM) for the remaining part(e.g., protein environment, solvent molecules)

Partitioning of Force Field Energy


Subsystem energies

E1 =subsystem 1

∑ “bonding terms” +subsystem 1

∑ “non-bonding terms”

E2 =subsystem 2

∑ “bonding terms” +subsystem 2

∑ “non-bonding terms”

Interaction energy

Eint =subsystem 1

∑I

subsystem 2

∑J

14πε0

qIqJ

|RI −RJ |

+subsystem 1

∑I

subsystem 2

∑J

4εIJ

[(σIJ

|RI −RJ |

)12

−(

σIJ

|RI −RJ |

)6]

QM/MM Embedding Methods


Mechanical Embedding

Etot = EQM + EMM + Eint,MM

EQM is calculated for the isolated QM system→ embedding only affect energy expression

Electronic Embedding

Etot = E ′QM + EMM + Eint,MM

E ′QM is calculated for the embedded QM system

Polarizable Embedding

Etot = E ′QM + E ′MM + Eint,MM

MM charges are polarized by the QM subsystem

QM/MM: Covalent Bonds


Interaction energy

Eint =subsystem 1

∑I

subsystem 2

∑J

14πε0

qIqJ

|RI −RJ |

+subsystem 1

∑I

subsystem 2

∑J

4εIJ

[(σIJ

|RI −RJ |

)12

−(

σIJ

|RI −RJ |

)6]

+ “bonding terms between subsystems”

Link Atoms

subtractive methods

Etot = E(QM)I+L − E(MM)

I+L + E(MM)tot

additive methods

Etot = E(QM)I+L + E(MM)

II + E(QM/MM)int

Quantum-Mechanical Subsystem Methods


Subsystem methods are based on a partitioning of the energy:

Etot = EI + EII + Eint

QM/MM methods

partition force field (MM) energy:

EMMtot = EMM

I + EMMII + EMM

int

replace force-field EMMI by EMM

I from quantum chemistry:

EQM/MMtot = EQM

I + EMMII + EMM

int

Quantum-mechanical subsystem methods

partition quantum-mechanical (QM) energy:

EQMtot = EQM

I + EQMII + EQM

int

Approximate Embedding Schemes


Kim−Gordon

electrostatic

discrete beyond purely electrostatic

non−discrete models

FDE responsecoupled

QM/MMpoint charges

point charges+ ECPs

couplingmechanical

couplingelectronic polarizable

embeddingpolarized response

unrelaxedFDE

ONIOM−EE

AIMP

polarizable QM/MM

shell models

EFP

frozen−density embedding

continuum models

typ

e o

f a

pp

roxim

atio

n

Bakowies−Thiel classification

energy−onlyQM/MM

COSMO, PCM, ...

ONIOM

freeze−thawFDE

discrete: only

Review: A. S. P. Gomes, Ch. R. Jacob, Annu. Rep. Prog. Chem. C 118, 222 (2012).

Density-Functional Theory (DFT) in a Nutshell


Electronic Schrödinger equation

Hel Ψel(r1, . . . , rN) = Eel Ψel(r1, . . . , rN)

electronic Hamiltonian: Hel = T + VNe + Vee + ENN

electronic wavefunction: Ψel = Ψel(r1, . . . , rN)

coordinates of electrons r1, . . . , rN

→ problem depends on 3N electronic coordinates

Density-functional theory (DFT)

Idea: use electron density ρ(r) instead of the wavefunction

ρ(r) =∫

Ψel(r , r2, . . . , rN)d3r2 · · ·d3rN

→ electron density depends on only 3 coordinates



Total energy functional

E [ρ] = Ts[ρ] + VNe[ρ] + J [ρ] + Exc[ρ] + ENN

electron–nuclear attraction:

VNe[ρ] =∫

ρ(r) vnuc(r) d3r

classical electron–electron Coulomb interaction:

J [ρ] =12

∫∫ρ(r)ρ(r ′)|r − r ′| d3rd3r ′

kinetic energy of noninteracting electrons with density ρ = ∑i |φi (r)|2

Ts[ρ] = − 12 ∑

i

∫φi ∗ (r)∆φi (r) d3r

everything else: exchange–correlation energy Exc[ρ]



⇒ derive equations for orbitals φi with ρ(r) = ∑i |φi (r)|2

Kohn–Sham equations[−∆

2+ vnuc(r) + vCoul[ρ](r) + vxc[ρ](r)

]φi (r) = εi φi (r)

classical Coulomb potential of the electrons:

vCoul[ρ](r) =12

∫ρ(r ′)|r − r ′| d3r ′

exchange–correlation potential

vxc[ρ](r) =δExc[ρ]

δρ(r)

exchange-correlation functional Exc[ρ] is not known

→ approximate functional have to be used

Partitioning the DFT Energy Functional


We want to partition the DFT energy functional into subsystems

E [ρtot] = Ts[ρtot] + VNe[ρtot] + J [ρtot] + Exc[ρtot] + ENN

⇒ Start from a partitioning of the electron density:

ρtot = ρI + ρII

ρI: system of interest ρII: environment



DFT total energy functional

E [ρtot] = Ts[ρtot] + VNe[ρtot] + J [ρtot] + Exc[ρtot] + ENN

Partitioned DFT total energy functional

E [ρtot] = E (I)NN + E (II)

NN + E (int)NN

+∫ (

ρI(r) + ρII(r))(

vnucI (r) + vnuc

II (r))

d3r

+12

∫ (ρI(r) + ρII(r)

)(ρI(r ′) + ρII(r ′)

)|r − r ′| d3rd3r ′

+ Exc[ρI] + Exc[ρII] + Enaddxc [ρI, ρII]

+ Ts[ρI] + Ts[ρII] + T nadds [ρI, ρII]

with the non-additive contributions

X nadd[ρI, ρII] = X [ρI + ρII]− X [ρI]− X [ρII]



DFT energy of subsystem I

E [ρI] = Ts[ρI] + VNe[ρI] + J [ρI] + Exc[ρI] + E (I)NN

DFT energy of subsystem II

E [ρII] = Ts[ρII] + VNe[ρII] + J [ρII] + Exc[ρII] + E (II)NN

DFT interaction energy

Eint[ρI, ρII] = E (int)NN +

∫ρI(r)v

nucII (r) d3r +

∫ρII(r)v

nucI (r) d3r

+∫

ρI(r)ρII(r ′)|r − r ′| d3rd3r ′ + Enadd

xc [ρI, ρII] + T nadds [ρI, ρII]

classical electrostatic interaction energynon-classical contributions from non-additive xc and kinetic energies

Frozen-Density Embedding


Partitioning of the electron density:

ρtot = ρI + ρII

ρII: environmentapproximate densityis kept frozen

ρI: system of interestembedded infrozen environment

Problem: For a given (frozen) ρII, determine ρI such that ρtot = ρI + ρII isthe “true” total electron density

Frozen-Density Embedding


⇒ KS-like equations for orbitals φ(I)i of subsystem I[

−∇2

2+ vKS

eff [ρI](r) + vembeff [ρI, ρII](r)

]φi (r) = εi φi (r)

Embedding potential

vembeff [ρI, ρII](r) = vnuc

II (r) +∫

ρII(r ′)|r − r ′|dr ′

+δT nadd

s [ρI, ρII]

δρI+

δEnaddxc [ρI, ρII]

δρI

embedding potential is in principle exact

local potential that depends only on the electron densities

⇒ this is the potential that QM/MM methods should approximate!

Solvent Effects on Excitation Energies


large systems: many solvent molecules have to be considered

dynamics: many different structures (snapshots) required

⇒ construct approximate electron density for the solvent

⇒ calculate embedding potential with approximate functional for T nadds

Example: aminocoumarin C151 in water and n-hexane

solvent shell of 900 atoms

400 snapshots from classical MD simulations

excitation energies calculated withTD-DFT and FDE embedding potential

calculated shift: 0.17 eV (exp.: 0.22 eV)

J. Neugebauer, CRJ et al., J. Phys. Chem. A 109, 7805 (2005).

Approximate Embedding Schemes


Kim−Gordon

electrostatic

discrete beyond purely electrostatic

non−discrete models

FDE responsecoupled

QM/MMpoint charges

point charges+ ECPs

couplingmechanical

couplingelectronic polarizable

embeddingpolarized response

unrelaxedFDE

ONIOM−EE

AIMP

polarizable QM/MM

shell models

EFP

frozen−density embedding

continuum models

typ

e o

f a

pp

roxim

atio

n

Bakowies−Thiel classification

energy−onlyQM/MM

COSMO, PCM, ...

ONIOM

freeze−thawFDE

discrete: only

Review: A. S. P. Gomes, Ch. R. Jacob, Annu. Rep. Prog. Chem. C 118, 222 (2012).

“Exact” Frozen-Density Embedding


Embedding Wave-Function Theory (WFT) in DFT

Start from full DFT calculation of complex chemical system

Partition electron density as ρtot(r) = ρI(r) + ρII(r)

Reconstruct embedding potential for subsystem I

Use this embedding potential in accurate wave-function calculation

Approximate WFT-in-DFT embedding: Actinunyls in Cs2UO2Cl4

electronic spectrum of NpO2+2

and UO2+2 in Cs2UO2Cl4

accurate methods required→ relativistic Fock-space

coupled cluster (FSCC)A. S. P. Gomes, Ch. R. Jacob, L. Visscher, Phys. Chem. Chem. Phys. 10, 5353 (2008).

A. S. P. Gomes, Ch. R. Jacob, et al., Phys. Chem. Chem Phys. 15 15153 (2013).

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