Electron Correlation - Methods beyond Hartree-Fock

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1 Electron Correlation - Methods beyond Hartree-Fock how to approach chemical accuracy Alexander A. Auer Max-Planck-Institute for Chemical Energy Conversion, Mülheim September 4, 2014 MMER Summerschool SI Lecture: Electron Correlation

2 Hartree-Fock theory and accuracy Hartree-Fock bond lengths [a.u.] sto-3g 4-31G 6-31G* 6-31G** exp. CH NH H 2 O HF minimum structure of C 20 - ring (R), bowl (B) or cage (C)? (energy difference in kcal/mol) HF R 20 B 78 C correct B 3 C 44 R MMER Summerschool Electron Correlation 1/42

3 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 Electron Correlation 2/42

4 Hartree-Fock theory: interaction of electrons Interaction of two electrons in HF theory : coulomb and exchange φ p(1)φ q(2) 1 φ p (1)φ q (2)dτ 1 dτ 2 r 12 = ρ p (1) 1 r 12 ρ q (2)dτ 1 dτ 2 BUT : particles only interact through their charge distribution mean field approximation - no explicit correlation of electron movement! MMER Summerschool Electron Correlation 3/42

5 Hartree-Fock theory: interaction of electrons Interaction of two electrons in HF theory : coulomb and exchange φ p(1)φ q(2) 1 φ p (1)φ q (2)dτ 1 dτ 2 r 12 = ρ p (1) 1 r 12 ρ q (2)dτ 1 dτ 2 BUT : particles only interact through their charge distribution mean field approximation - no explicit correlation of electron movement! in the HF Wavefunction two electrons will never see each other, only their charge clouds! follows from one electron orbital approximation in HF MMER Summerschool Electron Correlation 3/42

6 Electron correlation Hartree Fock : Mean field approximation no explicit correlation of electrons definition (!) of correlation energy : E corr = E exact E HF general Ansatz : improve HF method post Hartree-Fock methods often needed even for qualitative agreement with experiment! MMER Summerschool Electron Correlation 4/42

7 Pople Diagram Hierarchy of quantum chemical methods MMER Summerschool Electron Correlation 5/42

8 ab-initio methods : perturbation theory MMER Summerschool Electron Correlation 6/42

9 ab-initio methods : perturbation theory divide System into unperturbed part Ĥ0 and small perturbation Ĥ Ĥ = Ĥ0 + λ Ĥ MMER Summerschool Electron Correlation 6/42

10 ab-initio methods : perturbation theory divide System into unperturbed part Ĥ0 and small perturbation Ĥ Ĥ = Ĥ0 + λ Ĥ expand energy and wavefunction in series E = E (0) + λe (1) + λ 2 E (2) +... Ψ = Ψ (0) + λψ (1) + λ 2 Ψ (2) +... MMER Summerschool Electron Correlation 6/42

11 ab-initio methods : perturbation theory divide System into unperturbed part Ĥ0 and small perturbation Ĥ Ĥ = Ĥ0 + λ Ĥ expand energy and wavefunction in series E = E (0) + λe (1) + λ 2 E (2) +... Ψ = Ψ (0) + λψ (1) + λ 2 Ψ (2) +... sort in powers of λ to obtain energy corrections : E [0] = < Ψ (0) Ĥ0 Ψ (0) > E [1] = < Ψ (0) Ĥ Ψ (0) > E [2] = < Ψ (0) Ĥ Ψ (1) > MMER Summerschool Electron Correlation 6/42

12 ab-initio methods : perturbation theory define unperturbed and perturbed Hamiltonian for our problem : Ĥ = Ĥ 0 + Ĥ Ĥ 0 = i ˆF (i) Ĥ as difference between exact electron electron interaction and mean field interaction : Ĥ = i<j 1 r ij i (Ĵ i ˆK i ) MMER Summerschool Electron Correlation 7/42

13 ab-initio methods : perturbation theory Rayleigh-Schrödinger perturbation theory leads to : E [0] = i ɛ i E [1] = 1 ij ij 2 ij zeroth and first order give HF solution MMER Summerschool Electron Correlation 8/42

14 ab-initio methods : perturbation theory Rayleigh-Schrödinger perturbation theory leads to : E [0] = i ɛ i E [1] = 1 ij ij 2 zeroth and first order give HF solution first correction at 2nd order : Møller-Plesset PT - MP2 E [2] = 1 4 ij ab ij ij ab 2 ɛ i + ɛ j ɛ a ɛ b MMER Summerschool Electron Correlation 8/42

15 ab-initio methods : perturbation theory bond lengths [a.u.]: HF and MP2 sto-3g 4-31G 6-31G* 6-31G** exp. CH NH H 2 O HF MMER Summerschool Electron Correlation 9/42

16 ab-initio methods : perturbation theory Problem : convergence of perturbation series MMER Summerschool Electron Correlation 10/42

17 ab-initio methods : perturbation theory Problem : convergence of perturbation series MMER Summerschool Electron Correlation 11/42

18 ab-initio methods : perturbation theory Problem : convergence of perturbation series MMER Summerschool Electron Correlation 12/42

19 ab-initio methods : perturbation theory E MP2 = 1 ij ab 2 4 ɛ i + ɛ j ɛ a ɛ b ijab MP2 often improves HF results significantly MMER Summerschool Electron Correlation 13/42

20 ab-initio methods : perturbation theory E MP2 = 1 ij ab 2 4 ɛ i + ɛ j ɛ a ɛ b ijab MP2 often improves HF results significantly note: integrals also over virtual orbitals MMER Summerschool Electron Correlation 13/42

21 ab-initio methods : perturbation theory E MP2 = 1 ij ab 2 4 ɛ i + ɛ j ɛ a ɛ b ijab MP2 often improves HF results significantly note: integrals also over virtual orbitals higher order expressions lead to better correction MMER Summerschool Electron Correlation 13/42

22 ab-initio methods : perturbation theory E MP2 = 1 ij ab 2 4 ɛ i + ɛ j ɛ a ɛ b ijab 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! MMER Summerschool Electron Correlation 13/42

23 ab-initio methods : perturbation theory E MP2 = 1 ij ab 2 4 ɛ i + ɛ j ɛ a ɛ b ijab 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 N 2/3/4, MP2 N 5 ) MMER Summerschool Electron Correlation 13/42

24 ab-initio methods : perturbation theory E MP2 = 1 ij ab 2 4 ɛ i + ɛ j ɛ a ɛ b ijab 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 N 2/3/4, MP2 N 5 ) recommendation : use other methods beyond MP2 MMER Summerschool Electron Correlation 13/42

25 ab-initio methods : perturbation theory E MP2 = 1 ij ab 2 4 ɛ i + ɛ j ɛ a ɛ b ijab 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 N 2/3/4, MP2 N 5 ) recommendation : use other methods beyond MP2 standard : RI-MP2 / TZ (note: RI approximation!) MMER Summerschool Electron Correlation 13/42

26 ab-initio methods : configuration interaction exact wavefunction is a many electron function! MMER Summerschool Electron Correlation 14/42

27 ab-initio methods : configuration interaction exact wavefunction is a many electron function! idea : expand exact wavefunction in complete basis of many electron functions MMER Summerschool Electron Correlation 14/42

28 ab-initio methods : configuration interaction MMER Summerschool Electron Correlation 15/42

29 ab-initio methods : configuration interaction exact wavefunction is a many electron function! idea : expand exact wavefunction in complete basis of 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 Electron Correlation 16/42

30 ab-initio methods : configuration interaction wavefunction expression as linear CI expansion : Ψ CI = Ψ 0 + ia c a i Ψ a i ijab c ab ij Ψ ab ij +... MMER Summerschool Electron Correlation 17/42

31 ab-initio methods : configuration interaction wavefunction expression as linear CI expansion : Ψ CI = Ψ 0 + ia c a i Ψ a i ijab c ab ij Ψ ab ij +... Ψ CI = Ψ 0 + Ĉ1 Ψ 0 + Ĉ2 Ψ diagonalization of Hamiltonian matrix yields energies for ground and excited states Ψ CI Ĥ Ψ CI MMER Summerschool Electron Correlation 17/42

32 ab-initio methods : configuration interaction use all determinants that can be constructed expansion in complete basis (number of determinants grows exponentially!!) MMER Summerschool Electron Correlation 18/42

33 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) MMER Summerschool Electron Correlation 18/42

34 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 determinants CISD : singly and double substituted determinants CISDT : single, double and triple substitutions... MMER Summerschool Electron Correlation 18/42

35 ab-initio methods : configuration interaction correlation energy contributions from different levels of excitations in BeH 2 MMER Summerschool Electron Correlation 19/42

36 ab-initio methods : configuration interaction correlation energy contributions from different levels of excitations in BeH 2 excitation level E CI (DZ basis) [a.u.] total exact a.u.= kj/mol = 2.7 ev = cm 1 MMER Summerschool Electron Correlation 19/42

37 configuration interaction and size consistency example : H 2 molecule in minimal basis - 2 electrons, 2 orbitals : CISD ˆ= FCI! MMER Summerschool Electron Correlation 20/42

38 configuration interaction and size consistency example : two noninteracting hydrogen molecules (H 2 ) 2 MMER Summerschool Electron Correlation 21/42

39 configuration interaction and size consistency example : two noninteracting hydrogen molecules (H 2 ) 2 CISDTQ ˆ= FCI! MMER Summerschool Electron Correlation 21/42

40 configuration interaction and size consistency example : two noninteracting hydrogen molecules (H 2 ) 2 CISDTQ ˆ= FCI! CISD does not allow for simultaneous double excitations MMER Summerschool Electron Correlation 21/42

41 configuration interaction and size consistency example : two noninteracting hydrogen molecules (H 2 ) 2 CISDTQ ˆ= FCI! CISD does not allow for simultaneous double excitations at the CISD level of theory a single hydrogen molecule is treated more accurately than an isolated hydrogen molecule in a system of non interacting molecules! MMER Summerschool Electron Correlation 21/42

42 configuration interaction and size constancy unphysical description leads to wrong asymptotic of energy for extended systems extensivity of energy not given!! (CI is not size-extensive) MMER Summerschool Electron Correlation 22/42

43 configuration interaction and size constancy unphysical description leads to wrong asymptotic of 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 a truncated CI MMER Summerschool Electron Correlation 22/42

44 configuration interaction and size constancy unphysical description leads to wrong asymptotic of 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 a truncated CI recommendation : use CI with caution! MMER Summerschool Electron Correlation 22/42

45 improving configuration interaction MMER Summerschool Electron Correlation 23/42

46 improving configuration interaction not only include C 1, C 2 but also products of excitations C 2 1, C2 2 for FCI result! MMER Summerschool Electron Correlation 23/42

47 improving configuration interaction not only include C 1, C 2 but also products of excitations C 2 1, C2 2 for FCI result! instead of a linear ansatz, a product separable ansatz is required for size extensivity MMER Summerschool Electron Correlation 23/42

48 coupled cluster theory ansatz : instead of linear expansion, use exponential expansion! Ψ CC = e ˆT Ψ 0 MMER Summerschool Electron Correlation 24/42

49 coupled cluster theory ansatz : instead of linear expansion, use exponential expansion! Ψ CC = e ˆT Ψ 0 cluster operator ˆT analogous to CI-Operator ˆT = ˆT 1 + ˆT 2 + ˆT MMER Summerschool Electron Correlation 24/42

50 coupled cluster theory ansatz : instead of linear expansion, use exponential expansion! Ψ CC = e ˆT Ψ 0 cluster operator ˆT analogous to CI-Operator coefficients are called amplitudes: ˆT = ˆT 1 + ˆT 2 + ˆT e ˆT = 1 + ˆT + 1 2! ˆT ! ˆT MMER Summerschool Electron Correlation 24/42

51 coupled cluster theory product terms resulting from exponential expansion Ψ CC = (1 + ˆT ! ˆT ! ˆT ˆT ! ˆT ! ˆT ˆT 1 ˆT ! ˆT 1 2 ˆT ! ˆT 1 ˆT ) Ψ 0 MMER Summerschool Electron Correlation 25/42

52 coupled cluster theory Energy equations : Ψ 0 e ˆT Ĥe ˆT Ψ 0 = E Amplitude equations : Ψ ab.. ij.. e ˆT Ĥe ˆT Ψ 0 = 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! MMER Summerschool Electron Correlation 26/42

53 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 Doubles CCSD(T):Singles, Doubles + perturbative approximation to Triples CCSDT : Singles, Doubles and Triples... MMER Summerschool Electron Correlation 27/42

54 methods and basis sets Pople Diagram MMER Summerschool Electron Correlation 28/42

55 benchmark studies MMER Summerschool Electron Correlation 29/42

56 benchmark studies K. Bak, et. al J. Chem. Phys. 114, 6548 (2001). MMER Summerschool Electron Correlation 30/42

57 benchmark studies 13 C-NMR chemical shifts - benchmark study on 16 small organic compounds, deviation and standard deviation from experiment [ppm] + vibr. corr. + large basis CCSD(T) CCSD MP2 DFT HF-SCF A.A.Auer, J.Gauss, J.F.Stanton J. Chem. Phys., 118, (2003) MMER Summerschool Electron Correlation 31/42

58 a detective story : Cyclopropane r e (CC) r e (CH) rotational spectroscopy (23) (29) electron diffraction (4) (5) MMER Summerschool Electron Correlation 32/42

59 a detective story : Cyclopropane r e (CC) r e (CH) rotational spectroscopy (23) (29) electron diffraction (4) (5) theory (CCSD(T)/pVQZ) MMER Summerschool Electron Correlation 32/42

60 a detective story : Cyclopropane r e (CC) r e (CH) rotational spectroscopy (23) (29) electron diffraction (4) (5) theory (CCSD(T)/pVQZ) rot. data - vibr.corr (10) (10) MMER Summerschool Electron Correlation 32/42

61 a detective story : Cyclopropane r e (CC) r e (CH) rotational spectroscopy (23) (29) electron diffraction (4) (5) theory (CCSD(T)/pVQZ) rot. data - vibr.corr (10) (10) discrepancy due to empirical force field data in analysis of rotational data!! J. Gauss, D. Cremer and J. F. Stanton, J. Phys. Chem. A, 104, 1319 (1999) MMER Summerschool Electron Correlation 32/42

62 pitfalls in electronic structure methods up to now all methods based on orbital expansions (ground state wavefunction or density represented by a single slater determinant) MMER Summerschool Electron Correlation 33/42

63 pitfalls in electronic structure methods up to now all methods based on orbital expansions (ground state wavefunction or density represented by a single slater determinant) consider Ozone : O - O - O O = O + - O O - O + = O MMER Summerschool Electron Correlation 33/42

64 pitfalls in electronic structure methods up to now all methods based on orbital expansions (ground state wavefunction or density represented by a single slater determinant) consider Ozone : O - O - O O = O + - O O - O + = O more than one determinant needed to describe electronic structure even qualitatively! MMER Summerschool Electron Correlation 33/42

65 pitfalls in electronic structure methods HF is not a good basis to start from breakdown of PT, CI, CC etc.!! MMER Summerschool Electron Correlation 34/42

66 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-Sham equations MMER Summerschool Electron Correlation 34/42

67 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-Sham equations multireference problem! often observed : excited states, radicals, multiply charged ions, non-minimum structures, bond formation and breaking of chemical bonds... MMER Summerschool Electron Correlation 34/42

68 typical multireference case H 2 O molecule - at minimum geometry and at stretched geometry r=r e r=2r e E-E FCI W E-E FCI W RHF CISD CISDT CISDTQ CISDTQ MMER Summerschool Electron Correlation 35/42

69 multireference problems multireference treatment : optimize determinant coefficients and orbitals parameters at the same time! MMER Summerschool Electron Correlation 36/42

70 multireference problems multireference treatment : optimize determinant coefficients and orbitals parameters at the same time! artificial distinction : CI - dynamic correlation Methods : MCSCF multi configuration SCF CASSCF complete active space SCF CASPT2 CASSCF + 2nd order PT NEV-PT or NEV-CI MRCI multi reference CI MR - static correlation MMER Summerschool Electron Correlation 36/42

71 hierarchy of post-hf ab-initio methods MP2 / CC2 : noniterative / iterative N 5 CCSD : iterative NoccN 2 vrt 4 (formal N 6 ) CC3, CCSD(T) : iterative / noniterative NoccN 3 vrt 4 (formal N 7 ) CCSDT : iterative NoccN 3 vrt 5 (formal N 8 ) CCSDTQ : iterative NoccN 4 vrt 6 (formal N 10 )... MMER Summerschool Electron Correlation 37/42

72 hierarchy of post-hf ab-initio methods MP2 / CC2 : noniterative / iterative N 5 CCSD : iterative NoccN 2 vrt 4 (formal N 6 ) CC3, CCSD(T) : iterative / noniterative NoccN 3 vrt 4 (formal N 7 ) CCSDT : iterative NoccN 3 vrt 5 (formal N 8 ) CCSDTQ : iterative NoccN 4 vrt 6 (formal N 10 )... dilemma of high level ab-initio methods: high scaling of computational cost! MMER Summerschool Electron Correlation 37/42

73 CC and CI theory - computational effort amplitude equations example of T 2 equations for CCD: Ψ ab.. ij.. e ˆT Ĥe ˆT Ψ 0 = 0 T ab ij v ab ij teb ij t fa fi m tmj ab fe b tij ea tab mnvij mn + tnj ea vie nb tef ij vef ab mnvef mn tef ij tmnv ab ef mn 1 2 tab mitnj ef vef mn tmit eb nj fa vef mn MMER Summerschool Electron Correlation 38/42

74 Classical theory based on canonical orbitals T ab ij MMER Summerschool Electron Correlation 39/42

75 Rotation among occupied yields localized orbitals T γδ mn MMER Summerschool Electron Correlation 40/42

76 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, (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) MMER Summerschool Electron Correlation 41/42

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