Alexander A. Auer - MPI CEC · Alexander A. Auer Max-Planck-Institute for Chemical Energy Conversion, Mulheim 2.9.2015. ... distinction of exchange and correlation in DFT arti cial

Electron CorrelationAccuracy in Electronic Structure Methods

Alexander A. Auer

Max-Planck-Institute for Chemical Energy Conversion, Mulheim

2.9.2015

Hartree-Fock theory and accuracy

Hartree-Fock bond lengths [a.u.]

sto-3g 4-31G 6-31G* 6-31G** exp.CH4 2.047 2.043 2.048 2.048 2.050NH3 1.952 1.873 1.897 1.897 1.913H2O 1.871 1.797 1.791 1.782 1.809HF 1.812 1.742 1.722 1.703 1.733

minimum structure of C20 - ring (R), bowl (B) or cage (C) ?(energy difference in kcal/mol)

HF R 20 B 78 C“correct” B 3 C 44 R

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Hartree-Fock theory: a recommendation

• qualitative investigations of electronic structure

• interpretation of electron density at one electron level

• ”good guess” for molecular properties

• relatively fast basis set convergence

• standard : HF-SCF/SV(P)

• note: DFT usually provides better results at comparable cost !!

Textbook recommendation :“A Chemist’s guide to DFT” Holthausen, Koch, Wiley VCH

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Hartree-Fock theory: interaction of electrons

Interaction of two electrons in HF theory :coulomb and exchange∫

φ∗p(1)φ∗q(2)1

r12φp(1)φq(2)dτ1 dτ2

=

∫ρp(1)

1

r12ρq(2)dτ1 dτ2

BUT : particles only interact through their charge distribution mean fieldapproximation - no explicit correlation of electron movement !

in the HF Wavefunction two electrons will never ”see” each other, onlytheir charge clouds !→ follows from one electron orbital approximation in HF

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Hartree-Fock theory: interaction of electrons

Interaction of two electrons in HF theory :coulomb and exchange∫

φ∗p(1)φ∗q(2)1

r12φp(1)φq(2)dτ1 dτ2

=

∫ρp(1)

1

r12ρq(2)dτ1 dτ2

BUT : particles only interact through their charge distribution mean fieldapproximation - no explicit correlation of electron movement !

in the HF Wavefunction two electrons will never ”see” each other, onlytheir charge clouds !→ follows from one electron orbital approximation in HF

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Electron correlation

Hartree Fock : Mean field approximation

no explicit correlation of electrons

definition (!) of correlation energy :

Ecorr = Eexact − EHF

general Ansatz : improve HF method

→ post Hartree-Fock methods

often needed even for qualitative agreement with experiment !

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Hierarchy of quantum chemical methods

“Pople Diagram”

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A completely different approach

basic idea proven by Hohenberg and Kohn :

All molecular properties can be expressed as a functional E [ρ]of the electron density ρ alone

based on the energy functional, a variational method can be devised toconstruct the exact electron density(Kohn-Sham equations - SCF procedure)

→ Density Functional Theory (DFT)no wavefunction, just electron density is sufficient for exact solution !

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Density Functional Theory - DFT

So what is the form of this functional ?

E [ρ] has to contain functional forms for :

• one electron contributions (T , V )

• exchange contribution

• coulomb and correlation contribution

These expressions can be constructed easily ...(... actually not so easy) for the uniform electron gas !

→ Local Density Approximation (LDA)

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So what is the form of this functional ?E [ρ] has to contain functional forms for :




These expressions can be constructed easily ...

(... actually not so easy) for the uniform electron gas !


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So what is the form of this functional ?E [ρ] has to contain functional forms for :




These expressions can be constructed easily ...(... actually not so easy) for the uniform electron gas !


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LDA good for molecules ?

wrong asymptotic behavior for electron density !

The general, exact functional cannot be derived !

construct functionals based on physical intuition

promising approach :

• give up first principles approach

• include (molecule independent) parameters and fit them according to

• accurate experimental or theoretical values (energies, geometries ...)

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better functionals

Becke : include gradient of density in Ex [ρ]

ε[ρ] = −βρ1/3 x2

[1 + 6βxsinh−1(x)]x =|∆ρ|ρ4/3

β = 0.0042

Lee, Yang and Parr :include gradient and Laplacian of density in Ec [ρ]

→ B-LYP functional :Dirac exchange + Becke correction + LYP correlation(Generalized Gradient Approximation : GGA functionals)

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better functionals ?

why use approximate functionals when exact exchange is known fromHartree-Fock ?

distinction of exchange and correlation in DFT artificial

solution : mix DFT and HF exchange → Hybrid functional

example : famous B3LYP functional

AEDirac + (1− A)EHFx + B∆EBecke88

x + (1− C )EVWNv + CELYP

c

A=0.80, B=0.72, C=0.81

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DFT - functional zoo

• basic functionals (LDA) - plain E [ρ]

• Generalized Gradient Approximation GGA (PBE) - include gradient

• Hybrid Functionals (B3-LYP) - include HF exchange

• Double Hybrids (B2PLYP) - combine DFT and MP2-like correlation

• Exact Exchange Functionals and Random Phase Approximationhierarchy of methods beyond one-particle approach

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DFT - pitfalls

• dispersion interaction

• charge-Transfer excitations

• self interaction error

• single determinant approximation

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DFT - pitfalls, self-interaction error

electron-electron interaction in HF and other methods :

coulomb and exchange∫φ∗i (1)φ∗j (2)

1

r12φi (1)φj(2)dτ −

∫φ∗i (1)φ∗j (2)

1

r12φi (2)φj(1)dτ

for a one electron systems these exactly cancel !

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DFT - pitfalls, self-interaction error

electron-electron interaction in DFT :

coulomb and exchange-correlation functional∫ρ(1)

1

r12ρ(2)dr1dr2 −

∫ρ(1)hxc(r1, r2)

1

r12dr1dr2

→ coulomb and exchange-correlation will not cancel !consequence:very poor results for ionization energies, properties and various bondingsituations, in particular open shell systems !(especially LDA and GGAs)

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DFT - pitfalls, charge-transfer excitationscalculation of excitation spectra using TD-DFT :

A. Dreuw, M. Head-Gordon, JACS, 126 (12), 4007, 2004

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DFT - a recommendation

• this really is (currently) the workhorse for quantum chemistry

• use standard functionals (like B3-LYP) with moderate basis sets

• apply dispersion correction

• compare results from different functionals

• be critical about the results, be aware of pitfalls !

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“Pople Diagram”

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ab-initio methods : perturbation theory

divide System into unperturbed part H0 and small perturbation H ′

H = H0 + λ H ′

expand energy and wavefunction in series

E = E (0) + λE (1) + λ2E (2) + ... Ψ = Ψ(0) + λΨ(1) + λ2Ψ(2) + ...

sort in powers of λ to obtain energy corrections :

E [0] = < Ψ(0)|H0|Ψ(0) >

E [1] = < Ψ(0)|H ′|Ψ(0) >

E [2] = < Ψ(0)|H ′|Ψ(1) >

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define unperturbed and perturbed Hamiltonian for our problem :

H = H0 + H ′ H0 =∑i

F (i)

H ′ as difference between exact electron electroninteraction and mean field interaction :

H ′ =∑i<j

1

rij−∑i

(Ji − Ki )

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Rayleigh-Schrodinger perturbation theory leads to :

E [0] =∑i

εi

E [1] = −1

2

∑ij

〈ij | |ij〉

zeroth and first order give HF solution

first correction at 2nd order : Møller-Plesset PT - MP2

E [2] =1

4

∑ij

∑ab

|〈ij ||ab〉|2

εi + εj − εa − εb

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Rayleigh-Schrodinger perturbation theory leads to :

E [0] =∑i

εi

E [1] = −1

2

∑ij

〈ij | |ij〉

zeroth and first order give HF solutionfirst correction at 2nd order : Møller-Plesset PT - MP2

E [2] =1

4

∑ij

∑ab

|〈ij ||ab〉|2


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bond lengths [a.u.]: HF and MP2

sto-3g 4-31G 6-31G* 6-31G** exp.CH4 2.047 2.043 2.048 2.048 2.050

2.077 2.065 2.060 2.048NH3 1.952 1.873 1.897 1.897 1.913

1.997 1.907 1.922 1.912H2O 1.871 1.797 1.791 1.782 1.809

1.916 1.842 1.831 1.816HF 1.812 1.742 1.722 1.703 1.733

1.842 1.790 1.765 1.740

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Problem : convergence of perturbation series

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EMP2 =1

4

∑ijab

|〈ij ||ab〉|2


• MP2 often improves HF results significantly

• note: integrals also over virtual orbitals

• higher order expressions lead to better correction

• BUT : investigation shows that MP series usually does not converge !

• computational effort: HF/DFT N2/3/4, MP2 N5 )

• recommendation : use other methods beyond MP2

• standard : RI-MP2 / TZ (note: RI approximation !)

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EMP2 =1

4

∑ijab

|〈ij ||ab〉|2









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EMP2 =1

4

∑ijab

|〈ij ||ab〉|2









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EMP2 =1

4

∑ijab

|〈ij ||ab〉|2









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EMP2 =1

4

∑ijab

|〈ij ||ab〉|2









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EMP2 =1

4

∑ijab

|〈ij ||ab〉|2









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EMP2 =1

4

∑ijab

|〈ij ||ab〉|2









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ab-initio methods : configuration interaction

exact wavefunction is a many electron function !

idea :expand exact wavefunction in complete basisof many electron functions

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many electron function : HF-SCF slater determinant

complete basis of many electron functions :

all possible determinants from occupied and virtuals

→ Configuration Interaction (CI) :

linear expansion, one parameter per determinant,optimized variationally

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wavefunction expression as linear CI expansion :

|ΨCI 〉 = |Ψ0〉+∑ia

cai |Ψai 〉+

1

4

∑ijab

cabij

∣∣∣Ψabij

⟩+ ...

|ΨCI 〉 = |Ψ0〉+ C1 |Ψ0〉+ C2 |Ψ0〉+ ...

diagonalization of Hamiltonian matrix yields energies for ground andexcited states

〈ΨCI | H |ΨCI 〉

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wavefunction expression as linear CI expansion :

|ΨCI 〉 = |Ψ0〉+∑ia

cai |Ψai 〉+

1

4

∑ijab

cabij

∣∣∣Ψabij

⟩+ ...

|ΨCI 〉 = |Ψ0〉+ C1 |Ψ0〉+ C2 |Ψ0〉+ ...

diagonalization of Hamiltonian matrix yields energies for ground andexcited states

〈ΨCI | H |ΨCI 〉

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use all determinants that can be constructed→ expansion in complete basis(number of determinants grows exponentially !!)

→ exact solution of electronic problem in given AO basis(Full Configuration Interaction FCI)

in practice only truncated expansion feasible :CIS : only singly substituted determinantsCISD : singly and double substituted determinantsCISDT : single, double and triple substitutions...

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correlation energy contributions from different levels of excitations in BeH2

excitation level ECI (DZ basis) [a.u.]1+2 -0.074033

3 -0.0004284 -0.0014395 -0.0000116 -0.000006

total -0.075917exact -0.14

0.1 a.u.= 262.6 kJ/mol = 2.7 eV = 21947 cm−1

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correlation energy contributions from different levels of excitations in BeH2

excitation level ECI (DZ basis) [a.u.]1+2 -0.074033

3 -0.0004284 -0.0014395 -0.0000116 -0.000006

total -0.075917exact -0.14

0.1 a.u.= 262.6 kJ/mol = 2.7 eV = 21947 cm−1

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“Pople Diagram”

30/55

configuration interaction and size consistency

example : H2 molecule in minimal basis - 2 electrons, 2 orbitals :

CISD = FCI !

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example : two noninteracting hydrogen molecules (H2)2

CISDTQ = FCI !

CISD does not allow for simultaneous double excitations

at the CISD level of theory a single hydrogen molecule is treated moreaccurately than an isolated hydrogen molecule in a system of noninteracting molecules !

32/55



CISDTQ = FCI !



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CISDTQ = FCI !



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CISDTQ = FCI !



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configuration interaction and size constancy

unphysical description leads to wrong asymptoticof energy for extended systems

extensivity of energy not given !!(CI is not size-extensive)

↓

the larger the system, the smaller the fraction of correlation energy from atruncated CI

recommendation : use CI with caution !

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↓



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↓



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improving configuration interaction

not only include C1, C2 but also products of excitationsC2

1, C22 for FCI result !

instead of a linear Ansatz, a product separable Ansatzis required for size extensivity

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coupled cluster theory

Ansatz :instead of linear expansion, use exponential expansion !

|ΨCC 〉 = eT |Ψ0〉

cluster operator T analogous to CI-Operator

T = T1 + T2 + T3 + ...

coefficients are called amplitudes:

eT = 1 + T +1

2!T 2 +

1

3!T 3 + ...

35/55


product terms resulting from exponential expansion

|ΨCC 〉 = (1 + T1 +1

2!T 2

1 +1

3!T 3

1 + ...

+T2 +1

2!T 2

2 +1

3!T 3

2 + ...

+T1T2 +1

2!T 2

1 T2 +1

2!T1T

22 + ...

+ ...) |Ψ0〉

36/55


Energy equations :

〈Ψ0| e−T HeT |Ψ0〉 = E

Amplitude equations :

〈 Ψab..ij .. |e−T HeT |Ψ0〉 = 0

energy as expectation value of similarity transformed Hamiltonian

“projection trick” to derive amplitude equations

iterative solution of nonlinear set of equations !

state selective (ground state)

quite expensive !

37/55


• CI based method with inclusion of higher excitations

• strictly size consistent

• faster convergence towards FCI

• highly accurate !

approximate methods :CCSD : Coupled Cluster Singles and DoublesCCSD(T):Singles, Doubles + perturbative approximation to TriplesCCSDT : Singles, Doubles and Triples...

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methods and basis sets

“Pople Diagram”

39/55

benchmark studies

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benchmark studies

K. Bak, et. al J. Chem. Phys. 114, 6548 (2001).

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pitfalls in electronic structure methods

up to now all methods based on orbital expansions(ground state wavefunction or density represented by a single slaterdeterminant)

consider Ozone :· O - O - O · O = O+ - O− O− - O+ = O

more than one determinant needed to describe electronic structure evenqualitatively !

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HF is not a good basis to start from

→ breakdown of PT, CI, CC etc. !!

DFT no good due to one determinant representation in Kohn-Shamequations

→ multireference problem !

often observed :

excited states, radicals, multiply charged ions, non-minimum structures,bond formation and breaking of chemical bonds ...

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often observed :


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often observed :


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typical multireference case

H2O molecule - at minimum geometry and at stretched geometry

r=Re r=2Re

E-EFCI W E-EFCI WRHF 0.2178 0.9410 0.3639 0.5897CISD 0.0120 0.9980 0.0720 0.9487CISDT 0.0090 0.9985 0.0561 0.9591CISDTQ 0.0003 0.9999 0.0058 0.9987CISDTQ5 0.0001 0.9999 0.0022 0.9999

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multireference problems

multireference treatment : optimize determinant coefficients and orbitalsparameters at the same time !

artificial distinction :CI - dynamic correlation MR - static correlation

Methods :MCSCF multi configuration SCFCASSCF complete active space SCFCASPT2 CASSCF + 2nd order PTNEV-PT or NEV-CIMRCI multi reference CI

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multireference problems

multireference treatment : optimize determinant coefficients and orbitalsparameters at the same time !

artificial distinction :CI - dynamic correlation MR - static correlation

Methods :MCSCF multi configuration SCFCASSCF complete active space SCFCASPT2 CASSCF + 2nd order PTNEV-PT or NEV-CIMRCI multi reference CI

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using basis sets in practice

• H-Atom: slater orbitals

• but: basis sets constructed from Gaussian orbitals

• contracted function: linear combination with fixed exponents

• two parameters: exponents and contraction coefficients

• optimize parameters variationally for atoms

• construction and choice : alchemy ....

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slater vs. contracted gaussians

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4

x

Slater functionGaussian function

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4

x

Slater functionGaussian function

resulting sum

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• different basis set families

• Pople basis sets : 3-21G, 6-31G, 6-31G* ...

• Ahlrichs basis sets : SV(P), DZP, TZP, TZPP ...

• Dunning basis sets : cc-pVDZ, cc-pVTZ ....

• Pople and Ahlrichs basis sets : optimized for HF

• Dunning : optimized for CI

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cc-pVXZ basis sets

-1 0 1 2 3 4 5 6 7

log (exp)

cc-pVDZ-J

cc-pVTZ-J

cc-pVQZ-J

cc-pVDZ

s-functionsadditional s-functions

p-functionsadditional p-function

d-functionsadditional d-function

49/55

minimal vs. optimal - electron density of CO

CO− STO-3G aug-cc-pVTZ

2s1p 5s4p3d2f

50/55

basis sets completeness

Note that∑

i |φi >< φi | = 1this can be inserted into an overlap integral for functions of varyingexponents, yielding a “completeness profile”:

solid and dashed lines for s-, p-, and d-spaces for carbon, D.P.Chong, Canadian J. Chem., 73 (1), 79, (1995)

51/55


• use smallest sets with caution : STO-3G , 3-21G

• use Pople and Ahlrichs sets for HF and DFT

• use cc-pVXZ for ab-initio methods

• never just use one basis set :two calculations for estimating basis set convergence

• pitfall : BSSE

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“Pople Diagram”

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the end ...

Thank you for your attention !

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literature

List of Books :

A. Szabo, S. Ostlund, Modern Quantum Chemistry, Dover Publications (1996)

T. Helgaker, P. Jørgensen, J. Olsen, Molecular Electronic-Structure Theory, Wiley, Chichester (2000)

B. Roos, Lecture Notes in Quantum Chemistry I + II, Springer Verlag (1994)

W. Koch,M. Holthausen, A Chemists Guide to DFT, Wiley (2001)

D. Young, Computational Chemistry, Wiley (2001)

List of Papers :

W. Klopper, K. L. Bak, P. Jørgensen, J. Olsen, T. Helgaker, J. Phys. B, 32, R103 (1999)

K. L. Bak, P. Jørgensen, J. Olsen, T. Helgaker, J. Gauss, Chem. Phys. Lett., 317, 116 (2000)

K. L. Bak, J. Gauss, P. Jørgensen, J. Olsen, T. Helgaker, J.F. Stanton, J. Chem. Phys., 114, 6548 (2001)

K. Ruud, T. Helgaker, K. L. Bak, P. Jørgensen, J. Olsen, Chem. Phys., 195, 157 (1995)

A. A. Auer, J. Gauss, J. F. Stanton, J. Chem. Phys., 118, 10407 (2003)

J. Gauss, Encyclopedia of Computational Chemistry, Wiley, New York (1998)

J. Gauss, Modern Methods and Algorithms of Quantum Chemistry, NIC Proceedings, NIC-Directors (2000)

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Documents

Alexander A. Auer - MPI CEC · Alexander A. Auer Max-Planck-Institute for Chemical Energy Conversion, Mulheim 2.9.2015. ... distinction of exchange and correlation in DFT arti cial