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 iiii

ii

F

KJh =

−+

)(

...)2()1(

...)2()1(~

22

11

nnϕϕϕϕϕ

=Ψ

LCAO: ScFc εµµ β

µµ =→== e ,

2r -pcp

> −+∇−

−−=

n nnN

A

n

A

h

ZH

α αβ αβαα

α α

α

α

rrrR1

2

1 2

Hamiltonian:

KE

DD

JE

DD

E

hDHE HFHFHF κλνµκλνµνµνµ µλνκκλνµ )|()|(

1

21−+=ΨΨ=→ ΨHF (i.e. c) and

=−

=i

iii ccnDdd µννµλκµνκλνµ ,)()(1

)()()|( 2221

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

abij

iajb

abijMP

ia

ab

ijHFabij

a

iHFai

HFHF

HFHF

HFexactHFexactcorr

jaibjbiatjbiatE

tt

EVEEEE

EHVHH

KJr

HHVEEE

εεεε

λλλ

ααβα ααβ

)|()|(2,)|(

... :Ansatz

, :get to

ˆin ...,ˆˆˆ :insert

)()(ˆ1ˆˆˆ

2

1

11210

22

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 multiplicationsκλνµ κλνµ DDEJ )|(=

HF: E=E1+EJ-EK → N4 Integrals

N4 multiplicationsκλνµ µλνκ DDEK )|(2

1=

MP2: →N4 Integrals (on disc)

N5 multiplications=

−−+−

=

λκµνλκµκλνµ

εεεε

,,,

,,,2

)|()|(

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

bjavi

bjai bajiMP

ccccjbia

jaibjbjbiaE

−= )()(

1)()()|( 22

211121 rr

rrrrrr λκµνκλνµ ddIntegrals:

II The RI Method

Goal: Avoid N4 integral evaluationsand 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

PRI

Q

P

P

P

PPQQPc

PQQc

dd

Pc

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

)|)(|(

min)()(~1)()(~

)()(~)()()(

Identity" theof Resolution"

1

1

212221

11

κλνµκλκλνµκλνµ

νµ

ρρρρ

ρµνρ

νµ

νµ

νµνµνµνµ

νµνµνµ

rrrrrr

rr

rrrrrapproximate:

minimize:

to get:

and finally:

central problem: −= )()(

1)()()|( 22

211121 rr

rrrrrr λκµνκλνµ dd

Taking Advantage of RI: Coulomb Matrix

−=

=

κλκλνµ

κλκλνµ

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 functions100 1000

0,1

1

10

100 J exactRI-J directRI-J incore

cpu/min

n=10

20

3040

50

640 1160 MB

Taking Advantage of RI: Exchange Matrix

−−=

==

κλλκνµ

λκκλκλ

κλνµ

µ

µλ

ν

νκ

µλνκ

iRQPii

RI

iii

Pi

B

cRRP

Pi

B

PQQcK

ccDDK

,,,

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

,)|(

Integrals: N 4 → N 3 , multiplications: NBF4 → NBF

2 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 HFRI-HFExchange part

10 20 30 40 50 60

number of occupied orbitals

(BnHn)2-

200 400 600 800

0

2000

1000SVP

TZVPPpVQZ

Number of basis functions

HF

RI-HF

27

B3N3H6

pV5Z(+sym)

cpu / min

n=45

812

Taking Advantage of RI: MP2

Disc space: N4 → N3

CPU times: conventional / RI ~ NBF4nocc / Nxnvirt

2 nocc2 ≈ NBF/nocc

−−=

=

−−+−=

νµκλ

λκµν

λκµννµκλ

κλνµ

κλνµ

εεεε

PQRbjaiRI

bjai

iajb bajiMP

Pjb

Pia BB

ccRRPPQQccjbia

ccccjbia

jaibjbiajbiaE

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

SVPTZVPP

pVQZ

MP2

RI-MP2

21

B3N3H6

pV5Z

number of basis functions

cpu/min

100

1

10

100MP2

RI-MP2

cpu/min

number of basis functions200 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

Qip νµµν speichere P

ipB

ENDE LOOP I3. LOOP I ( I ist eine Untermenge aktiver besetzter Orbitale)

LOOP J ( J ist eine Untermenge aktiver besetzter Orbitale)

bilde (ia|jb) ← Qjb

Qia BB (i ∈ I , j ∈ J )

)/()|()|(2 gajiabij jaibjbiat εεεε −−+−←

Pjb

abij

Pia BtY ←

)/()|( cajiabijbc jciatP εεεε −−+←

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

ENDE LOOP JPip

Piaap BYL ←"

2/1)|( −←Γ PQY Qia

Pia

Qia

PiaPQ BΓ←γ~

aP

iaP

i cνν Γ←Γ

speichere PiνΓ und PQγ~ auf Festplatte

ENDE LOOP I

4. 2/1)|(~ −← QRPRPQ γγξξ γ )|(2 QPE PQMP ←

5. Loop Ξ (Untermenge von Basisfunktionen)

)()|(2 Ξ∈Γ← ννµ ξµν

ξ PcE iP

iMP

pP

iip cPL µννµ Γ← )|(

ENDE LOOP Ξ6. Berechne dieübrigen Elementevon Pij nach )(2/)( jiijjiij LLP εε −−=

Setze zusammen: )/(~fiifiiabpq LPP εεπ −⊕⊕←

und berechne qrapqrA π~ ( AO-Basis)

Löse pqlapqlaalbmalbmalla ALLZAZ πεε ~)( " −−=+−

7. alpqpq Z⊕← ππ ~

"21

21

21

2aqiqrslqrspq

pppq LLAW ⊕⊕⊕

+← ππ

εε

8. ξξξ εδδδδπ pqiiqippqpqiqippqMP SWhE

~)(2)(2 )(

2 +−+←

)~~

)((2 )(2

ξξξ δδπ pqpqiqippqMP SFE −+←

Auxiliary Basis Sets

-Goal: RI errors 1 order of magnitude smaller than energy changes due to basis set changes

-use atom centred Gaussians, but different sets for J, K, and MP2

-optimization: minimize at the atoms

J, K: J = | E(JRI)-E(Jex)|, same for K

MP2:

We require: J,K: J, K < 20 µH

MP2: I/|EMP2| < 10-6, and |EMP2-ERIMP2| < 20 µH

size: J: Nx< 3NBF, K, MP2: Nx< 4NBF

... and test at molecules...

( )−−+

−=∆

iajb baji

RIabijabij

Iεεεε

2 ||||

E(J) E(J,RI)

E(K) E(K,RI)

Some Details

J: 3-parameter even tempered set (→jbas)

K: Gradients for K(RI) with respect to exponents, relaxation (→jkbas)

MP2: Gradients for ∆I, relaxation (→cbas)

Example: auxiliary basis sets for B-F (TZVPP):

s p d f gbasis 62111 411 11 1jbas (RI-DFT) 3111111 111 111 1jkbas (RI-HF) 1111111111 1111111 11111 11 1cbas (RI-MP2) 11111111 111111 2111 111 1

Accuracy: Tests at Small Molecules

RI-DFT: E(electron-electron)=-E(XC)+E(J)

E(J)=E(J)-E(J,RI)

RI-MP2: E=|E(MP2)-E(MP2,RI)|

RI-HF: E(electron-electron)=-E(K)+E(J)

E(K)=E(K,RI)-E(K)

E(J): as above, but with auxbasis also used for K

Ca. 100 molecules (H-Br):

Al2O3 Al2S3 AlCl3 AlF3 AlH3 As4 AsCl3 AsCl6- AsH3 B2H6 B3N3H6 B4H4 BF3 BH3 BH3CO BH3NH3 Br2 BrCl BrO- BrO2- BrO3- BrO4- C2H2 C2H3N C2H4 C2H6 C4H4 C6H6 CF4 CH2O CH2O2CH3N CH3OH CH4 CO CO2 CS2 Cl2 ClF ClF3 F2 GaCl GaCl3 GaFGaH3 GeCl4 GeF4 GeH4 GeO GeO2 H2 H2CO3 H2O H2O2 H2SO4 H3PO4 HCN HCP HCl HF HNC HNO HNO2 HNO3 HSH HSSH N2 N2H2 N2H4 N4 NF3 NH3 NH4F OF2 P2 PF3 PF5 PH3 S2 S5 SF2 SF4 SF6 Se8 SeH2 SeO2 SiCl4 SiF4 SiH4 SiO2 SiS2

E(J) E(J,RI)

E(K) E(K,RI)

Errors of RI-J in DFT (TZVPP basis)

0

50

100

150

200

250

300

J/atom [ H]

H-F Al-Cl Ge-Br

∅

-σ

max

+σ

Errors of RI-J and RI-K in HF (TZVPP basis)

J and K both calculated with K-optimized auxiliary basis set

0

20

40

60

80

100

120

140K/atom [ H]

H-F Al-Cl Ge-Br

0

10

20

30

40

J/atom [ H]

H-F Al-Cl Ge-Br

Errors of RI-MP2 (TZVPP basis)

H-F Al-Cl Ge-Br

0

10

20

30

40

50

60MP2/atom [ H]

cc-pVQZ

RI-HF + RI-MP2 versus HF + MP2 (TZVPP basis)

H-F Al-Cl Ge-Br

0

20

40

60

80

100

120

140

|ERIMP2 ( RIHF)-EMP2( HF)| / atom [ H]

Errors in Properties for „Worst“ Cases

structure parameters:

distances (bonds) ca. 0.1 pm

angles: ca 0.1 °

dipole moments: ca. 0.01 Debye

Size Dependency of Accuracy

10 15 20 25 30 35 401

2

3

4

5

6

7

8

9

10

J

K

JK

Number of occupied orbitals

/atom [ H]

• erros per atom slightly decrease with molecular size

→ (partial) cancellation when calculating bond energies

• similar for MP2

(BnHn)2- TZVPP

cc-pVQZ

n=4

5 612

Summary: RI with Optimized Auxiliary Basis Sets

- Efficiency and accuracy of RI do not depend on a molecule‘s

- geometric structure

- electronic structure

- composition

- size (nearly)

- Efficiency (conventinonal/RI): RIDFT: ≈ 10

RI-HF(RIMP2): ~NBF/nocc, TZVPP ≈ 5(10)

RIMP2: disc space N4 → N3

- Accuracy:

-energy errors at least 1 o. M. smaller than energy changes due to changes of basis set

- structure parameters nearly unaffected

III Applications

-Anions of Water Clusters

-Anions of Au Clusters

-Mg Clusters

-Mixed metal Clusters (Pt-Ir)

Anions of Water Clusters

Expt.: H2O(g)Formation of (H2O)n

- clusters

MassSpectrum

and

VDESpectrum

Theory: (H2O)6-: structure ?

bonding of the excess electron:

cage-like („solvated electron“) or dipole bound?

+ electrons

F. Weigend and R. Ahlrichs, PCCP 1 4537 (1999).

Water Clusters: Quantumchemical Treatment

-optimization of many (ca. 40) isomers

-conditions to be fulfilled: 1. VDE(QC) =E(X-)-E(X) ≈ VDE(Expt.)

2. Stability

- MP2 is best choice (bond energy of (H2O)2 very close to experiment

- (possibly) dipole bound electron: floating basis

Cluster

µ e-

atoms → basis functions no atoms, no basis functions ?? → floating basis

Water Clusters: VDE Spectrum in Theory and Experiment

Conclusions:

1. The excess electron is dipole-bound.

2. (H2O)6- is metastable

Theory

Experiment

0

100

200

300

400

0 200 400 600

Rel

ativ

e E

nerg

y / m

eV

VDE / meV

Combining Theory and Experiment: Ions of Gold Clusters

clusterion source

drift cell(He) → Ω

TOF massspectrometer+ mass gate

masschargecross section ΩVDE (Aun

-)

VDE

Experiment

Theory

take Aun+/-

DFT:optimized structureenergy → stability

geometry → Ω

VDE=E(Aun

-)-E(Aun)

cross section ΩVDE (Aun

-)stability

S. Gilb, P. Weis, F. Furche, R. Ahlrichs, M. Kappes, J Chem Phys, 116, 4094 (2002)F. Furche, R. Ahlrichs, P. Weis, C. Jacob, S. Gilb, T. Bierweiler, M.Kappes, J Chem Phys, 117, 6982 (2002)

Way of Proceeding

• optimization of overall more than 100 isomers

• check minimum by calculation of force constants

• regard only the few most stable isomers (for each n)

• comparison of cross section (and VDE)

Gold Cluster Ions: Experimental Cross Sections

20

40

60

80

100

120

0 5 10 15 20

Number of Atoms

Cro

ss S

ectio

n (Å

2 )

anionscations

23/10 )(

3

4HeAu rrn +=Ω π

Anions of Gold Clusters

0.8

0.9

1

1.1

1.2

1.3

1.4

0 2 4 6 8 10 12 14 16

Number of Atoms

Rel

ativ

e C

ross

Sec

tion

10-I10-I

planar

Clusters of Magnesium – a DFT Study

• Simple Concepts: Shells of Atoms and Electrons

• Small Clusters: Stability

• Larger Clusters: Stability and Electronic Structure

A. Köhn, F. Weigend and R. Ahlrichs, PCCP 3, 711 (2001).

Mg Clusters: Simple Concepts

Mg: Mg2+-core plus 3s2 (+ empty 3p)

electronic shell model atomic shell model

Mg2+ → uniform background empty 3p: electron deficiency→ harmonic oscillator → geometrically closed structures, e.g.

Shell n(e-) sum Mgn

4p3f2h1k 72 240 120

4s3d2g1i 56 168 84

3p2f1h 42 112 56

3s2d1g 30 70 35

2p1f 20 40 20

2s1d 12 20 10

1p 6 8 4

1s 2 2 1

icosa octa cubocta deca tetra „hcp“

120 157

84 103

146 56 89

309 85 105 35 57

147 44 147 54 20 26

55 19 55 23 10 13

13 6 13 7 4 5

Mg Clusters: DFT Treatments

1.221.14De/eV

309.4310.3re/pm

DFT(BP86)CCSD(T)Mg4

→ DFT is o.k. (but take care for appropriate basis set!)

Mg4 : Effects of Electron Correlation

1. Hartree Fock (mean field)

2. MP2 perturbation theory

(V=Hexact-HHF)

→ perturbed wave function ΨMP2

→ changes in electron density due to electron correlation: ρ MP2 =|ΨMP2|2:

lower density near the nuclei, higher density in the middle of the tetrahedron

Stability of Small Clusters Mgn (n<23)

1. simulated annealing → local minima2. check by calculation of IR frequencies

nMgE natcoh /)(=ε

4 10 20

number of Mg atoms

Cohesive Energies of Large Mg Clusters

-no clear pictures, but rather preference of icosahedra than hcp -extrapolation to bulk: 1.38 eV (Expt.:1.51 eV, bulk DFT: 1.43)

13/23/1,

−−− ++= nanana corneredgesurfacebulkcohcoh εε

Validity of the Shell Model: DOS

Magic electron numbers: 2, 8, 20, 40, 70, 112

Shell Model: Anharmonic Distorsions and Subshells

Shell n(e-) sum

3p2f1h 42 112

3s2d1g 30 70

2p1f 20 40

2s1d 12 20

1p 6 8

1s 2 2

n(e-) sum ico cuboct trdec hcp

3p 6 112 2f 14 1061h 22 92 X (86) (88) (88)

3s 2 70 (64)2d 10 68 501g 18 58 X (54) (56)

2p 6 40 X1f 14 34 X X X X

2s 2 20 X X X X1d 10 18

1p 6 8 X X X X

1s 2 2 X X X X

harmonic distorted DFT

Mg147(Ih): Shell Model and DOS

Binary Metal Clusters: Pt13-nIrn as an Example

Problem: N-atomic cluster: 2N possibile distributions for metal A and B

Strategy: 1. Start with homoatomic clusters, find minima

2. First consider symmetric cases, substitute only one or a few atoms

3. Try to find (and understand) building principles

4. Apply them to predict distribution in general cases, i.e.

no (or low) symmetry, multiple substitutions

5. Calculate ‚best candidates‘

Method: DFT(BP86), TZVPP basis sets

Claudia Schrodt, Florian Weigend...

Pt13 and Ir13: High Symmetries, Homoatomic Cases

Ih Oh

Pt and Ir: Oh more stable than Ih by ca. 0.4 eV

IrnPt13-n: High Symmetry, Substitution of One Atom

[ ])Pt()13()Ir(13

1)PtIr( 1313n13n EnnEEE −+−=∆ −

0 1 12-2,0

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

2,5

#2

#1

#2: surface atom

#1: central atom

number of iridium atoms

13

OhOh

Single point calculation

for geometry of Pt13

In optimised geometry

- very similar for Ih

How to Make Plausible

•Start from Pt13: E=E(N-N)-E(N-e)+E(e-e)+E(e,kin)

•replace one Pt atom by Ir (Z → Z-1), do not change MOs and occupations

→ A) Lower N-N repulsion, but B) also lower N-e attraction

→ different size of cancellation of A) and B) for different sites

→ different change of energy for different sites

Valence orbitals valence density

(electron deficiency, e.g. Al12Si: central position for Si preferred by ca. 0.7 eV)

Larger / smaller gain in E(N-e)

ε,|ϕ|

r

Single Substitutions: Extrapolation from Homoatomic Case

•Start from Pt13: E=ENN-ENe+Eee+Ee,kin

• consider a specific site: change Z → Z+ δZ leads to change δE

• calculate ∆E for different sites, compare with uniform distribution of ∆Z/13 at all sites

→ Relative energies for Ir occupying all positions

[ ]

ZZ

EE

R

ZZZEE

I

∆=∆→

ΨΨ−−+=

*

)||)(E)(Z NNNN

δδ

δδδα α

Pt13-nIrn : Accuracy of Extrapolation

)M13(

ii

i ZZ

EE ∆

∂∂=∆

0 1 12-2,0

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

#2

#1

#2

#1

∆ E / eV

number of iridium atoms

13

Oh

#1: central atom#2: surface atom

Oh

Extrapolation from homoatomic system

Single point calculation

for geometry of Pt13

In optimised geometry

[ ])Pt()13()Ir(13

1)PtIr( 1313n13n EnnEEE −+−=∆ −

Pt13 and Ir13 : More Stable Isomer(s)

0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

IrPt

∆E/eVIh

Oh

OhIh

Pt13-nIrn: Low Symmetry, Substitution of One Atom

Extrapolation from homoatomic system

Single point calculation for geometry of Pt13

In optimised geometry

)M13(

ii

ZZ

EE ∆

∂∂=∆

-1

0

1

7,9

8,10

4

2,3

11

1

∆E/eV

[ ])Pt()13()Ir(13

1)PtIr( 1313n13n EnnEEE −+−=∆ −

Extrapolation From Homoatomic Case: Multiple Substitutions

• substitution of M atoms at positions I in an N-atomic system:

•δZI: additional charge δZ added at all positions I

• all (2N ) substitutions:

1. Only once: ∂ENe/ ∂Zi,

2. 2N times: ∆ENN, ∆ENe from ∂ENe/ ∂ZI

→ extrapolated values for all distributions

3. Final DFT calculation only for „best“ candidates

Z

R

Z

Z

E

NeE

M

Z

Z

E

NNE

M

Z

Z

ZEZZEE

I

I

I

Ne

I

I

I

NeINNIINN

δ

δ

δδ

δδ

δδ α α

ΨΨ=

∆

∆−

∆

∆−+=∆ ,**)()(

Pt13-nIrn: Low Symmetry, Multiple Substitutions

From perturbation theory

Single point calculation for geometry of Pt13

In optimised geometry

1 1,5,6,8,10,132 1,5,8-10,133 1,5-7,10,134 1,5,6,9,10,13

1 2 6-2

-1

0

1

68,10

# replaced

# replaced1 1,102 1,53 1,64 1,95 10,13

42,3

119

1

3,41,2

12

53,41,2

∆E/eV

number of Ir atoms

+73 others

+1712 others

Pt13-nIrn: The Most Stable of 213 Isomers

Pt6Ir7: ∆E=1.8 eV

+ very similar energies for Pt7Ir6 (6: Pt) and Pt8Ir5 (6,7: Pt)

Pt

Ir

6

7

Summary: Applications

• Calculation of stabilities by DFT and HF+MP2.

→ prediction of stable isomers

• Comparison of calculated and measured data

→ geometric structure

• Verification of (simple) models by calculations