alexander a. auer - mpi cec · alexander a. auer max-planck-institute for chemical energy...
TRANSCRIPT
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Electron CorrelationAccuracy in Electronic Structure Methods
Alexander A. Auer
Max-Planck-Institute for Chemical Energy Conversion, Mulheim
2.9.2015
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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 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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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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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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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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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) >
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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 )
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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
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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
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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
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 !)
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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 !)
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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 !)
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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 !)
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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 !)
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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 !)
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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 !)
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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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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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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
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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 〉
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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 〉
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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...
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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...
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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...
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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
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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
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Hierarchy of quantum chemical methods
“Pople Diagram”
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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 !
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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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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
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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
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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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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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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using basis sets in practice
• 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
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minimal vs. optimal - electron density of CO
CO− STO-3G aug-cc-pVTZ
2s1p 5s4p3d2f
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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)
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using basis sets in practice
• 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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Hierarchy of quantum chemical methods
“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)
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