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

    DFG-CENTRUM FR FUNKTIONELLE NANOSTRUKTUREN

    Theoretische Biophysikalische ChemieQM/MM und Einbettungsverfahren

    Christoph Jacob

    KIT Universitt des Landes Baden-Wrttemberg undnationales Forschungszentrum in der Helmholtz-Gemeinschaft www.kit.edu

  • Quantum Chemistry

    1 28.01.2014 Christoph Jacob: Theoretische Biophysikalische Chemie DFG-Centrum fr funktionelle Nanostrukturen

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

    positions of nuclei R1, . . . ,RN

    Quantum Mechanics

    Eel =el|Hel|el

    electronic wavefunction: el = el(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

    2 28.01.2014 Christoph Jacob: Theoretische Biophysikalische Chemie DFG-Centrum fr funktionelle Nanostrukturen

    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)

  • Force Field Approximation

    3 28.01.2014 Christoph Jacob: Theoretische Biophysikalische Chemie DFG-Centrum fr funktionelle Nanostrukturen

    Standard Force Field

    Eel(R1, . . . ,RN) =12

    bonds

    i

    ki (ri r0i )2 +

    12

    angles

    i

    kj (j 0j )2

    +12

    torsions

    n

    cos(nnn n)

    + I

    J>I

    14e0

    qIqJ|RI RJ |

    + I

    J>I

    4eIJ

    [(IJ

    |RI RJ |

    )12(

    IJ|RI RJ |

    )6]

  • Partitioning of Force Field Energy

    4 28.01.2014 Christoph Jacob: Theoretische Biophysikalische Chemie DFG-Centrum fr funktionelle Nanostrukturen

    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

    14e0

    qIqJ|RI RJ |

    +subsystem 1

    I

    subsystem 2

    J

    4eIJ

    [(IJ

    |RI RJ |

    )12(

    IJ|RI RJ |

    )6]

  • QM/MM Embedding Methods

    5 28.01.2014 Christoph Jacob: Theoretische Biophysikalische Chemie DFG-Centrum fr funktionelle Nanostrukturen

    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 + EMM + Eint,MM

    MM charges are polarized by the QM subsystem

  • QM/MM: Covalent Bonds

    6 28.01.2014 Christoph Jacob: Theoretische Biophysikalische Chemie DFG-Centrum fr funktionelle Nanostrukturen

    Interaction energy

    Eint =subsystem 1

    I

    subsystem 2

    J

    14e0

    qIqJ|RI RJ |

    +subsystem 1

    I

    subsystem 2

    J

    4eIJ

    [(IJ

    |RI RJ |

    )12(

    IJ|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

    7 28.01.2014 Christoph Jacob: Theoretische Biophysikalische Chemie DFG-Centrum fr funktionelle Nanostrukturen

    Subsystem methods are based on a partitioning of the energy:

    Etot = EI + EII + Eint

    QM/MM methods

    partition force field (MM) energy:

    EMMtot = EMMI + E

    MMII + E

    MMint

    replace force-field EMMI by EMMI from quantum chemistry:

    EQM/MMtot = EQMI + E

    MMII + E

    MMint

    Quantum-mechanical subsystem methods

    partition quantum-mechanical (QM) energy:

    EQMtot = EQMI + E

    QMII + E

    QMint

  • Approximate Embedding Schemes

    8 28.01.2014 Christoph Jacob: Theoretische Biophysikalische Chemie DFG-Centrum fr funktionelle Nanostrukturen

    KimGordon

    electrostatic

    discrete beyond purely electrostatic

    nondiscrete models

    FDE responsecoupled

    QM/MMpoint charges

    point charges+ ECPs

    couplingmechanical

    couplingelectronic polarizable

    embeddingpolarized response

    unrelaxedFDE

    ONIOMEE

    AIMP

    polarizable QM/MM

    shell models

    EFP

    frozendensity embedding

    continuum models

    typ

    e o

    f a

    pp

    roxim

    atio

    n

    BakowiesThiel classification

    energyonlyQM/MM

    COSMO, PCM, ...

    ONIOM

    freezethawFDE

    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

    9 28.01.2014 Christoph Jacob: Theoretische Biophysikalische Chemie DFG-Centrum fr funktionelle Nanostrukturen

    Electronic Schrdinger equation

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

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

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

    coordinates of electrons r1, . . . , rN

    problem depends on 3N electronic coordinates

    Density-functional theory (DFT)

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

    (r) =

    el(r , r2, . . . , rN)d3r2 d3rN

    electron density depends on only 3 coordinates

  • Density-Functional Theory (DFT) in a Nutshell

    10 28.01.2014 Christoph Jacob: Theoretische Biophysikalische Chemie DFG-Centrum fr funktionelle Nanostrukturen

    Total energy functional

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

    electronnuclear attraction:

    VNe[] =

    (r) vnuc(r) d3r

    classical electronelectron Coulomb interaction:

    J [] =12

    (r)(r )|r r | d

    3rd3r

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

    Ts[] = 12 i

    i (r)i (r) d3r

    everything else: exchangecorrelation energy Exc[]

  • Density-Functional Theory (DFT) in a Nutshell

    11 28.01.2014 Christoph Jacob: Theoretische Biophysikalische Chemie DFG-Centrum fr funktionelle Nanostrukturen

    derive equations for orbitals i with (r) = i |i (r)|2

    KohnSham equations[

    2+ vnuc(r) + vCoul[](r) + vxc[](r)

    ]i (r) = ei i (r)

    classical Coulomb potential of the electrons:

    vCoul[](r) =12

    (r )|r r | d

    3r

    exchangecorrelation potential

    vxc[](r) =Exc[](r)

    exchange-correlation functional Exc[] is not known

    approximate functional have to be used

  • Partitioning the DFT Energy Functional

    12 28.01.2014 Christoph Jacob: Theoretische Biophysikalische Chemie DFG-Centrum fr funktionelle Nanostrukturen

    We want to partition the DFT energy functional into subsystems

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

    Start from a partitioning of the electron density:

    tot = I + II

    I: system of interest II: environment

  • Partitioning the DFT Energy Functional

    13 28.01.2014 Christoph Jacob: Theoretische Biophysikalische Chemie DFG-Centrum fr funktionelle Nanostrukturen

    DFT total energy functional

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

    Partitioned DFT total energy functional

    E [tot] = E(I)NN + E

    (II)NN + E

    (int)NN

    + (

    I(r) + II(r))(

    vnucI (r) + vnucII (r)

    )d3r

    +12

    (I(r) + II(r))(I(r ) + II(r ))|r r | d

    3rd3r

    + Exc[I] + Exc[II] + Enaddxc [I, II]

    + Ts[I] + Ts[II] + Tnadds [I, II]

    with the non-additive contributions

    X nadd[I, II] = X [I + II] X [I] X [II]

  • Partitioning the DFT Energy Functional

    14 28.01.2014 Christoph Jacob: Theoretische Biophysikalische Chemie DFG-Centrum fr funktionelle Nanostrukturen

    DFT energy of subsystem I

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

    DFT energy of subsystem II

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

    DFT interaction energy

    Eint[I, II] = E(int)NN +

    I(r)v

    nucII (r) d

    3r +

    II(r)vnucI (r) d

    3r

    +

    I(r)II(r )|r r | d

    3rd3r + Enaddxc [I, II] + Tnadds [I, II]

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

  • Frozen-Density Embedding

    15 28.01.2014 Christoph Jacob: Theoretische Biophysikalische Chemie DFG-Centrum fr funktionelle Nanostrukturen

    Partitioning of the electron density:

    tot = I + II

    II: environmentapproximate densityis kept frozen

    I: system of interestembedded infrozen environment

    Problem: For a given (frozen) II, determine I such that tot = I + II isthe true total electron density

  • Frozen-Density Embedding

    16 28.01.2014 Christoph Jacob: Theoretische Biophysikalische Chemie DFG-Centrum fr funktionelle Nanostrukturen

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

    2

    2+ vKSeff [I](r) + v

    embeff [I, II](r)

    ]i (r) = ei i (r)

    Embedding potential

    vembeff [I, II](r) = vnucII (r) +

    II(r )|r r |dr

    +T nadds [I, II]

    I+

    Enaddxc [I, II]I

    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

    17 28.01.2014 Christoph Jacob: Theoretische Biophysikalische Chemie DFG-Centrum fr funktionelle Nanostrukturen

    large s