electron correlation - methods beyond hartree-fockelectron correlation - methods beyond hartree-fock...
TRANSCRIPT
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Electron Correlation - Methodsbeyond Hartree-Fock
how to approach chemical accuracy
Alexander A. Auer
Max-Planck-Institute for Chemical Energy Conversion, Mulheim
September 4, 2014
MMER Summerschool 2014 - SI Lecture: ElectronCorrelation
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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 Ccorrect B 3 C 44 R
MMER Summerschool 2014 - Electron Correlation 1/42
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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
MMER Summerschool 2014 - Electron Correlation 2/42
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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
MMER Summerschool 2014 - Electron Correlation 3/42
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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
MMER Summerschool 2014 - Electron Correlation 3/42
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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 !
MMER Summerschool 2014 - Electron Correlation 4/42
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Hierarchy of quantum chemical methods
“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) >
MMER Summerschool 2014 - Electron Correlation 6/42
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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) >
MMER Summerschool 2014 - Electron Correlation 6/42
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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) >
MMER Summerschool 2014 - Electron Correlation 6/42
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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) >
MMER Summerschool 2014 - Electron Correlation 6/42
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ab-initio methods : perturbation theory
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 )
MMER Summerschool 2014 - Electron Correlation 7/42
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ab-initio methods : perturbation theory
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
MMER Summerschool 2014 - Electron Correlation 8/42
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ab-initio methods : perturbation theory
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
εi + εj − εa − εb
MMER Summerschool 2014 - Electron Correlation 8/42
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ab-initio methods : perturbation theory
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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ab-initio methods : perturbation theory
Problem : convergence of perturbation series
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ab-initio methods : perturbation theory
Problem : convergence of perturbation series
MMER Summerschool 2014 - Electron Correlation 11/42
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ab-initio methods : perturbation theory
Problem : convergence of perturbation series
MMER Summerschool 2014 - Electron Correlation 12/42
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ab-initio methods : perturbation theory
EMP2 =1
4
∑ijab
|〈ij ||ab〉|2
εi + εj − εa − εb
• 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 !)
MMER Summerschool 2014 - Electron Correlation 13/42
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ab-initio methods : perturbation theory
EMP2 =1
4
∑ijab
|〈ij ||ab〉|2
εi + εj − εa − εb
• 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 !)
MMER Summerschool 2014 - Electron Correlation 13/42
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ab-initio methods : perturbation theory
EMP2 =1
4
∑ijab
|〈ij ||ab〉|2
εi + εj − εa − εb
• 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 !)
MMER Summerschool 2014 - Electron Correlation 13/42
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ab-initio methods : perturbation theory
EMP2 =1
4
∑ijab
|〈ij ||ab〉|2
εi + εj − εa − εb
• 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 !)
MMER Summerschool 2014 - Electron Correlation 13/42
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ab-initio methods : perturbation theory
EMP2 =1
4
∑ijab
|〈ij ||ab〉|2
εi + εj − εa − εb
• 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 !)
MMER Summerschool 2014 - Electron Correlation 13/42
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ab-initio methods : perturbation theory
EMP2 =1
4
∑ijab
|〈ij ||ab〉|2
εi + εj − εa − εb
• 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 !)
MMER Summerschool 2014 - Electron Correlation 13/42
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ab-initio methods : perturbation theory
EMP2 =1
4
∑ijab
|〈ij ||ab〉|2
εi + εj − εa − εb
• 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 !)
MMER Summerschool 2014 - Electron Correlation 13/42
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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
MMER Summerschool 2014 - Electron Correlation 14/42
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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
MMER Summerschool 2014 - Electron Correlation 14/42
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ab-initio methods : configuration interaction
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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
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
MMER Summerschool 2014 - Electron Correlation 16/42
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ab-initio methods : configuration interaction
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 〉
MMER Summerschool 2014 - Electron Correlation 17/42
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ab-initio methods : configuration interaction
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 〉
MMER Summerschool 2014 - Electron Correlation 17/42
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ab-initio methods : configuration interaction
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...
MMER Summerschool 2014 - Electron Correlation 18/42
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ab-initio methods : configuration interaction
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...
MMER Summerschool 2014 - Electron Correlation 18/42
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ab-initio methods : configuration interaction
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...
MMER Summerschool 2014 - Electron Correlation 18/42
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ab-initio methods : configuration interaction
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
MMER Summerschool 2014 - Electron Correlation 19/42
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ab-initio methods : configuration interaction
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
MMER Summerschool 2014 - Electron Correlation 19/42
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configuration interaction and size consistency
example : H2 molecule in minimal basis - 2 electrons, 2 orbitals :
CISD = FCI !
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configuration interaction and size consistency
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 !
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configuration interaction and size consistency
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 !
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configuration interaction and size consistency
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 !
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configuration interaction and size consistency
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 !
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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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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 !
MMER Summerschool 2014 - Electron Correlation 22/42
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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 !
MMER Summerschool 2014 - Electron Correlation 22/42
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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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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
MMER Summerschool 2014 - Electron Correlation 23/42
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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 + ...
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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 + ...
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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 + ...
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coupled cluster theory
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〉
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coupled cluster theory
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 !
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coupled cluster theory
• 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”
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benchmark studies
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benchmark studies
K. Bak, et. al J. Chem. Phys. 114, 6548 (2001).
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benchmark studies
13C-NMR chemical shifts - benchmark study on 16 small organiccompounds, deviation and standard deviation from experiment [ppm]
-15 -10 -5 0 5 10 15 20 25
HF-SCF
DFT
MP2
CCSD
CCSD(T)
+ large basis
+ vibr. corr.
A.A.Auer, J.Gauss, J.F.Stanton J. Chem. Phys., 118, 10407 (2003)
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a detective story : Cyclopropane
re (CC) re (CH)
rotational spectroscopy 1.5101 (23) 1.0742 (29)electron diffraction 1.501 (4) 1.083 (5)
theory (CCSD(T)/pVQZ) 1.5019 1.0781rot. data - vibr.corr. 1.5030 (10) 1.0786 (10)
discrepancy due to empirical force field datain analysis of rotational data !!
J. Gauss, D. Cremer and J. F. Stanton, J. Phys. Chem. A, 104, 1319 (1999)
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a detective story : Cyclopropane
re (CC) re (CH)
rotational spectroscopy 1.5101 (23) 1.0742 (29)electron diffraction 1.501 (4) 1.083 (5)theory (CCSD(T)/pVQZ) 1.5019 1.0781
rot. data - vibr.corr. 1.5030 (10) 1.0786 (10)
discrepancy due to empirical force field datain analysis of rotational data !!
J. Gauss, D. Cremer and J. F. Stanton, J. Phys. Chem. A, 104, 1319 (1999)
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a detective story : Cyclopropane
re (CC) re (CH)
rotational spectroscopy 1.5101 (23) 1.0742 (29)electron diffraction 1.501 (4) 1.083 (5)theory (CCSD(T)/pVQZ) 1.5019 1.0781rot. data - vibr.corr. 1.5030 (10) 1.0786 (10)
discrepancy due to empirical force field datain analysis of rotational data !!
J. Gauss, D. Cremer and J. F. Stanton, J. Phys. Chem. A, 104, 1319 (1999)
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a detective story : Cyclopropane
re (CC) re (CH)
rotational spectroscopy 1.5101 (23) 1.0742 (29)electron diffraction 1.501 (4) 1.083 (5)theory (CCSD(T)/pVQZ) 1.5019 1.0781rot. data - vibr.corr. 1.5030 (10) 1.0786 (10)
discrepancy due to empirical force field datain analysis of rotational data !!
J. Gauss, D. Cremer and J. F. Stanton, J. Phys. Chem. A, 104, 1319 (1999)
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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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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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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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pitfalls in electronic structure methods
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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pitfalls in electronic structure methods
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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pitfalls in electronic structure methods
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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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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hierarchy of post-HF ab-initio methods
MP2 / CC2 : noniterative / iterative N5
CCSD : iterative N2occN
4vrt (formal N6)
CC3, CCSD(T) : iterative / noniterative N3occN
4vrt (formal N7)
CCSDT : iterative N3occN
5vrt (formal N8)
CCSDTQ : iterative N4occN
6vrt (formal N10)
....
dilemma of high level ab-initio methods:high scaling of computational cost !
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hierarchy of post-HF ab-initio methods
MP2 / CC2 : noniterative / iterative N5
CCSD : iterative N2occN
4vrt (formal N6)
CC3, CCSD(T) : iterative / noniterative N3occN
4vrt (formal N7)
CCSDT : iterative N3occN
5vrt (formal N8)
CCSDTQ : iterative N4occN
6vrt (formal N10)
....
dilemma of high level ab-initio methods:high scaling of computational cost !
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CC and CI theory - computational effort
amplitude equations
〈 Ψab..ij .. |e−T HeT |Ψ0〉 = 0
example of T2 equations for CCD:
T abij ← v ab
ij − f mi tabmj − f be teaij +
1
2tabmnv
mnij + teanj v
nbie +
1
2tefij v
abef
+1
2tebij t
famnv
mnef +
1
4tefij t
abmnv
mnef −
1
2tabmi t
efnj v
mnef − tebmi t
fanjv
mnef
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Classical theory based on canonical orbitals
Tabij
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Rotation among occupied yields localized orbitals
Tγδmn
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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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