Post on 16-Oct-2019
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