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  • Developing and Applying TURBOMOLE

    Florian Weigend, Forschungszentrum Karlsruhe

    I. Quantumchemical Methods: HF, MP2, DFT

    II. The RI Method - Efficient Calculation of J, K and MP2

    ---------

    III. Applications: Calculations of Clusters

    (H2O)n-, Aun-, Mgn, PtnIrm

  • The Hartree-Fock Method

    Variation Principle: min ~~ !=ΨΨ= HE

    → HF equations: )()()(

    )(

    ))()(()( αϕαεαϕ

    α

    ααα iii i

    ii

    F

    KJh =��� ���� �

    −+

    ����� ������

    )(

    ...)2()1(

    ...)2()1( ~

    22

    11

    nnϕ ϕϕ ϕϕ

    ��=Ψ

    LCAO: ScFc εµµ β µ

    µ =→== e , 2r -�

    pcp

    > − +∇−

    − −=

    n nnN

    A

    n

    A

    h

    Z H

    α αβ αβα α

    α α

    α

    α

    rrrR 1

    2

    1 2

    ��

    Hamiltonian: ��� ���� ���� ��� �����

    K E

    DD

    J E

    DD

    E

    hDHE HFHFHF κλνµκλνµνµνµ µλνκκλνµ )|()|(

    1

    2 1−+=ΨΨ=→ ΨHF (i.e. c) and

    = −

    = i

    iii ccnDdd µννµλκµνκλνµ ,)()( 1

    )()()|( 22 21

    1121 rrrr rrrr

  • Applicability and Accuracy of HF

    • main group compounds:

    equilibrium distances: few pm

    • starting point for post-HF methods (at least if HF is

    not too bad)

    • delocalised systems (e.g. (metal) clusters)

    • appropriate basis sets: SV(P), check with TZVP

  • Defencies of HF: Dynamic Electron Correlation

    HF: variation principle → mean field description

    EHF

    He-atom (1S)

    Eexact

    ΨHF

    Ψexact

    electron 1

    -π π

    Ecorr

  • Improvement I: Perturbation Theory (MP2)

    + excellent if HF is good; fails, if HF fails

    + most economic way to account for dispersive intractions

    • appropriate bases: TZVPP, check by QZVPP, (SVP only qualitative)

    •d10 metals: QZVP, check by QZVPP

    { }

    −−+ −==→

    +Ψ+Ψ=Ψ

    Ψ−Ψ==+

    Ψ=Ψ+Ψ+Ψ+Ψ=Ψ−=

    −−=−=→−= <

    iajb baji

    ab ij

    iajb

    ab ijMP

    ia

    ab

    ijHF ab ij

    a

    iHF a i

    HFHF

    HFHF

    HFexactHFexactcorr

    jaibjbia tjbiatE

    tt

    EVEEEE

    EHVHH

    KJ r

    HHVEEE

    εεεε

    λλλ

    αα βα ααβ

    )|()|(2 ,)|(

    ... :Ansatz

    , :get to

    ˆin ...,ˆˆˆ :insert

    )()(ˆ 1ˆˆˆ

    2

    1

    11210

    2 2

    1

  • Improvement II: Density Functional Theory

    simple view:

    HF: E=E1+EJ+Ex

    DFT: E=E1+EJ+EXC

    ...

    gaselectron free

    )(),...)(),(( 33/43 +−=∇= rdkrdfEXC rrr ρρρ

    • EXC[ρ] exists, no prescription to get exact one

    good for (metal) clusters: Becke-Perdew 86

    • useful: almost universal, quite accurate, efficient

    • does not include dispersive interactions

    • appropriate bases: main groups: SVP, check with TZVP

    transition metals: (SVP), TZVPP, check by QZVP

  • Costs for HF, DFT and MP2

    DFT: E=E1+EJ+EXC → N4 Integrals

    N4 multiplications κλνµ κλνµ DDEJ )|(=

    HF: E=E1+EJ-EK → N4 Integrals

    N4 multiplications κλνµ µλνκ DDEK )|(21=

    MP2: →N4 Integrals (on disc)

    N5 multiplications =

    −−+ −

    =

    λκµν λκµκλνµ

    εεεε

    ,,,

    ,,, 2

    )|()|(

    )]|()|2(ia)[(|(

    bjavi

    bjai baji MP

    ccccjbia

    jaibjbjbia E

    − = )()(1)()()|( 22

    21 1121 rrrr

    rrrr λκµνκλνµ ddIntegrals:

  • II The RI Method

    Goal: Avoid N4 integral evaluations and N4 integral storage (MP2) and N5 integral transformations (MP2).

    J. L. Whitten; J. Chem. Phys. 58, 4496 (1973) B. I. Dunlap, J. W. D. Conolly, J. R. Sabin; J. Chem . Phys. 71, 3396 (1979) O. Vahtras, J. Almlöf, M.W. Feyereisen; Chem. Phys. Lett. 213, 514 (1993)

  • RI Approximation

    { } { }

    ==≈

    =

    =− −

    =≈=

    QPP

    P RI

    Q

    P

    P

    P

    PPQQPc

    PQQc

    dd

    Pc

    )|()|)(|()|()|()|(

    )|)(|(

    min)()(~ 1

    )()(~

    )()(~)()()(

    Identity" theof Resolution"

    1

    1

    2122 21

    11

    κλνµκλκλνµκλνµ

    νµ

    ρρρρ

    ρµνρ

    νµ

    νµ

    νµνµνµνµ

    νµνµνµ

    �� ��� ��

    rrrr rr

    rr

    rrrrrapproximate:

    minimize:

    to get:

    and finally:

    central problem: − = )()(1)()()|( 22

    21 1121 rrrr

    rrrr λκµνκλνµ dd

  • Taking Advantage of RI: Coulomb Matrix

    −=

    =

    κλ κλνµ

    κλ κλνµ

    κλνµ

    κλνµ

    QP

    NN

    N

    NN

    NN

    RI

    x

    x

    x

    x

    DPPQQJ

    DJ

    ,

    1

    2

    2

    2

    2

    )|()|()|(

    )|(

    Integrals and multiplications: N 4 → N 3

    K. Eichkorn, O. Treutler, H. Öhm, M. Häser, R. Ahlrichs; Chem. Phys. Lett. 240, 283 (1995).

    Note: E(J,RI)=JνµRID νµ≤ E(J,ex)

  • RI-J: Efficiency

    H(CH2)nH, SVP-basis

    number of basis functions 100 1000

    0,1

    1

    10

    100 J exact RI-J direct RI-J incore

    cpu/min

    n=10

    20

    30 40

    50

    640 1160 MB

  • Taking Advantage of RI: Exchange Matrix

    −−=

    ==

    κλ λκνµ

    λκκλ κλ

    κλνµ

    µ

    µλ

    ν

    νκ

    µλνκ

    iRQP ii

    RI

    i ii

    P i

    B

    cRRP

    P i

    B

    PQQcK

    ccDDK

    ,,,

    2/12/1 )|()|()|)(|(

    ,)|(

    Integrals: N 4 → N 3 , multiplications: NBF4 → NBF2 NX nocc

    → CPU times: conventional / RI ~ NBF/nocc

    F. Weigend; PCCP, 4(18) 4285 (2002).

    Note: E(K,RI)=KνµRID νµ≤ E(K,ex)

  • RI-K: Efficiency

    TZVPP

    cc-pVQZ

    cpu / min

    10

    100

    1000

    1

    10

    100

    exact HF RI-HF Exchange part

    10 20 30 40 50 60

    number of occupied orbitals

    (BnHn)2-

    200 400 600 800

    0

    2000

    1000 SVP

    TZVPP pVQZ

    Number of basis functions

    HF

    RI-HF

    27

    B3N3H6

    pV5Z(+sym)

    cpu / min

    n=4 5

    8 12

  • Taking Advantage of RI: MP2

    Disc space: N4 → N3

    CPU times: conventional / RI ~ NBF4nocc / Nxnvirt2 nocc2 ≈ NBF/nocc

    −−=

    =

    −−+ −=

    νµκλ

    λκµν

    λκµν νµκλ

    κλνµ

    κλνµ

    εεεε

    PQR bjaiRI

    bjai

    iajb baji MP

    P jb

    P ia BB

    ccRRPPQQccjbia

    ccccjbia

    jaibjbiajbia E

    ���� ����� ������ ����� ��

    )|()|()|)(|( )|( :RI

    )|( )|( :

    )]|()|(2)[|(

    2/12/1

    2

    F. Weigend and M. Häser; Theor. Chim. Acc.; 97 331 (1997).

    conventional

    Note: E(RI-MP2) is not variational.

  • RI-MP2: Efficiency

    +symmetry

    200 400 600 800

    0

    2000

    4000

    6000

    SVP TZVPP

    pVQZ

    MP2

    RI-MP2

    21

    B3N3H6

    pV5Z

    number of basis functions

    cpu/min

    100

    1

    10

    100 MP2

    RI-MP2

    cpu/min

    number of basis functions 200 300

    (Cu2S)n, SVP basis

    n=2

    3

    4

    5

    6

  • RI-MP2 Gradients

    1. Bilde (P|Q)-1/2

    2. LOOP I ( I ist eine Untermenge aktiver besetzter Orbitale) berechne (νµ|P) bilde 2/1)|)(|(

    −← QPPccB pi Q ip νµµν speichere

    P ipB

    ENDE LOOP I 3. LOOP I ( I ist eine Untermenge aktiver besetzter Orbitale)

    LOOP J ( J ist eine Untermenge aktiver besetzter Orbitale)

    bilde (ia|jb) ← Qjb Q ia BB (i ∈ I , j ∈ J )

    { } )/()|()|(2 gajiabij jaibjbiat εεεε −−+−← P jb

    ab ij

    P ia BtY ←

    )/()|( caji ab ijbc jciatP εεεε −−+←

    für "εI ≈εJ " 1) : )/()|( baki ab ijij kbiatP εεεε −−+←

    ENDE LOOP J P ip

    P iaap BYL ←

    "

    2/1)|( −←Γ PQY Qia P

    ia

    Q ia

    P iaPQ BΓ←γ~

    a P

    ia P

    i cνν Γ←Γ

    speichere PiνΓ und PQγ~ auf Festplatte ENDE LOOP I

    4. 2/1)|(~ −← QRPRPQ γγ

    ξξ γ )|(2 QPE PQMP ← 5. Loop Ξ (Untermenge von Basisfunktionen)

    )()|(2 Ξ∈Γ← ννµ ξ

    µν ξ PcE i

    P iMP

    p P

    iip cPL µννµ Γ← )|( ENDE LOOP Ξ

    6. Be