Junya HASEGAWA Fukui Institute for Fundamental Chemistry Kyoto University 1 Prof M Kotani 19061993 2 Contents Molecular Orbital MO Theory Electron Correlations Configuration Interaction ID: 1045423
Download Presentation The PPT/PDF document "Configuration Interaction in Quantum Che..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
1. Configuration Interaction in Quantum ChemistryJun-ya HASEGAWAFukui Institute for Fundamental ChemistryKyoto University1
2. Prof. M. Kotani (1906-1993)2
3. ContentsMolecular Orbital (MO) TheoryElectron CorrelationsConfiguration Interaction (CI) & Coupled-Cluster (CC) methodsMulti-Configuration Self-Consistent Field (MCSCF) methodTheory for Excited StatesApplications to photo-functional proteins3
4. Molecular orbital theory4
5. Electronic Schrödinger equationElectronic Schrödinger eq. w/ Born-Oppenheimer approx. Electronic Hamiltonian operator (non-relativistic)Potential energy Wave functionThe most important issue in electronic structure theory 5
6. Many-electronwave functionOrbital approximation: product of one-electron orbitalsThe Pauli anti-symmetry principle Slater determinant Anti-symmetrized orbital productsOne-electron orbitals are the basic variables in MO theory6
7. One-electron orbitalsLinear combination of atom-centered Gaussian functions.Primitive Gaussian function 7
8. Variational determination of the MO coefficientsEnergy functionalLagrange multiplier method8
9. Hartree-Fock equationVariation of MO coefficientsHartree-Fock equationA unitary transformation that diagonalizes the multiplier matrixCanonical Hartree-Fock equation 9→Eigenvalue equation Eigenvalue: Multiplier (orbital energy) Eigenvector: MO coefficients
10. Restricted Hartree-Fock (RHF) equationSpin in MO theory: (a)spin orbital formulation → spatial orbital rep. ) (b) Restricted (c) UnrestrictedRestricted Hartree-Fock (RHF) equation for a closed shell (CS) systemRHF wf is an eigenfunction of spin operators: a proper relation 10
11. Electron correlations− Introduction to Configuration Interaction −11
12. Electron correlations defined as a difference from Full-CI energy Two classes of electron correlations Dynamical correlationsLack of Coulomb hole Static (non-dynamical) correlationsBond dissociation, Excited statesNear degeneracyNo explicit separation between dynamical and static correlations. Definition of “electron correlations” in Quantum ChemistryRestricted HFNumerically ExactFig. Potetntial energy curves of H2 molecule. 6-31G** basis set. [Szabo, Ostlund, “Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory”, Dover]Static correlation is dominant.Dynamical correlation is dominant.
13. Slater det. : Products of one-electron function →Independent particle modelPossibility of finding two electrons at : H2–like molecule case Dynamical correlations: lack of Coulomb hole
14. Interacting a doubly excited configuration Chemical intuition: Changing the orbital picture→Introducing dynamical correlations via configuration interaction－
15. 15Left-right correlation in olefin compounds Configuration interaction- x=+ x=-=No correlationsincluded
16. 16Angular correlationOne-step higher angular momentum - x=+ x=Configuration interaction-=No correlationsincluded
17. 2-electron system in a dissociating homonuclear diatomic molecule Changing orbital picture into a local basis: Each configuration has a fixed weight of 25 %.No independent variable that determines the weight for each configuration when the bond-length stretches. Static correlations: improper electronic structureIonic configuration: 2 e on AIonic configuration: 2 e on BCovalent config.: 2 e at each A and BCovalent config.: 2 e at each A and B
18. Interacting a doubly excited configuration Some particular change the weights of covalent and ionic configurations. Introducing static correlations via configuration interactionABABABAB
19. Configuration Interaction (CI) and Coupled-Cluster (CC) wave functions19
20. Some notationsNotationsOccupied orbital indices: i, j, k, ….Unoccupied orbital indices: a, b, c, …..Creation operator: Annihilation operator: Spin-averaged excitation operatorSpin-adapted operator (singlet)Reference configuration: Hartree-Fock determinantExcited configurationCorrect spin multiplicity (Eigenfunction of operators)20
21. 21Configuration Interaction (CI) wave function: a general formCI expansion: Linear combination of excited configurations Full-CI gives exact solutions within the basis sets used.CI Singles (CIS)CI Singles and Doubles (CISD)CI Singles, Doubles, and Triples (CISDT)Full configuration interaction (Full CI)∙∙∙∙
22. 22Variational determination of the wave function coefficientsCI energy functionalLagrange multiplier methodConstraint: Normalization condition Variation of LagrangianEigenvalue equation
23. 23Availability of CI methodA straightforward approach to the correlation problem starting from MO theoryNot only for the ground state but for the excited statesAccuracy is systematically improved by increasing the excitation order up to Full-CI (exact solution)Energy is not size-extensive except for CIS and Full-CIDifficulty in applying large systemsFull-CI: number of configurations rapidly increases with the size of the system.kα + kβ electrons in nα + nβ orbitals →Porphyrin: nα = nβ =384 , kα =kβ =152 → ~10221 determinantsFig. Correlation energy per water molecule as a percentage of the Full-CI correlation energy (%) . The cc-pVDZ basis sets were used.Number of water moleculesPercentage (%)H2OH2OH2OH2OH2OH2OH2OH2OR ~ largeCISDFull-CI
24. 24Coupled-Cluster (CC) wave functionCI wf: a linear expansion CC wf: an exponential expansionSingle excitationsDouble excitationsTriple excitationsCC Singles (CCS)CI Singles and Doubles (CCSD)CC Singles, Doubles, and Triples (CCSDT)∙∙∙∙Linear terms =CINon-linear terms
25. 25Why exponential?Size-extensiveNon interacting two molecules A and BSuper-molecular calculation ↔ CI case A part of higher-order excitations described effectively by products of lower-order excitations.Dynamical correlations is two body and short range. Far awayNo interaction
26. 26Solving CC equationsSchrödinger eq. with the CC w.f.CC energy: Project on HF determinantCoefficients: Project on excited configurations (CCSD case)Non-linear equations. Number of variable is the same as CI method.Number of operation count in CCSD is O(N6), similar to CI method.
27. 27Hierarchy in CI and CC methods and numerical performanceRapid convergence in the CC energy to Full-CI energy when the excitation order increases.Higher-order effect was included via the non-linear terms.In a non-equilibrium structure, the convergence becomes worse than that in the equilibrium structure.Conventional CC method is for molecules in equilibrium structure.SDSDTSDTQSDTQ5SDTQ56Excitation order in wf.Error from Full-CI (hartree)CI法CC法Fig. Error from Full-CI energy. H2O molecule with cc-pVDZ basis sets.[1] ~kcal/mol“Chemical accuracy”Table. Error from Full-CI energy. H2O at equilibrium structure (Rref) and OH bonds elongated twice (2Rref)). cc-pVDZ sets were used.[1][1]“Molecular Electronic Structure Theory”, Helgaker, Jorgensen, Olsen, Wiley, 2000.
28. 28Statistics: Bond lengthComparison with the experimental data (normal distribution [1])H2, HF, H2O, HOF, H2O2, HNC, NH3, … (30 molecules)“CCSD(T)” : Perturbative Triple correction to CCSD energycc-pVDZcc-pVTZcc-pVQZHFMP2CCSDCCSD(T)CISDError/pm=0.01ÅError/pm=0.01ÅError/pm=0.01Å[1]“Molecular Electronic Structure Theory”, Helgaker, Jorgensen, Olsen, Wiley, 2000.
29. 29Statistics: Atomization energyNormal distributionF2, H2, HF, H2O, HOF, H2O2, HNC, NH3, etc (total 20 molecules) Error from experimental value in kJ/mol (200 kJ/mol=48.0 kcal/mol)[1]“Molecular Electronic Structure Theory”, Helgaker, Jorgensen, Olsen, Wiley, 2000.
30. 30Statistics: reaction enthalpyNormal distributionCO+H2→CH2O HNC→HCN H2O+F2→HOF+HF N2+3H2→2NH3 etc. (20 reactions)Increasing accuracy in both theory and basis functions, calculated data approach to the experimental values. Error from experimental data in kJ/mol (80 kJ/mol=19.0 kcal/mol)[1]“Molecular Electronic Structure Theory”, Helgaker, Jorgensen, Olsen, Wiley, 2000.
31. Multi-Configurational Self-Consistent Field method31
32. Single-configuration descriptionApplicable to molecules in the ground state at near equilibrium structureHartree-Fock methodMulti-configuration descriptionBond-dissociation, excited state, ….Quasi-degeneracy → Linear combination of configurations to describe STATIC correlationsMulti-Configuration Self-Configuration Field (MCSCF) w.f. Complete Active Space SCF (CASSCF) methodCI part = Full-CI: all possible electronic configurations are involved.Beyond single-configuration description32ABABAB+
33. Trial MCSCF wave function is parameterized by Orbital rotation: unitary transformation CI correction vectorMCSCF energy expanded up to second-order MCSCF method: a second-order optimizaton33
34. MCSCF applications to potential energy surfacesCI guarantees qualitative description whole potential surfacesFrom equilibrium structure to bond-dissociation limitFrom ground state to excited states34Soboloewski, A. L. and Domcke, W. “Efficient Excited-State Deactivation in Organic Chromophores and Biologically Relevant Molecules: Role of Electron and Proton Transfer Processes”, In “Conical Intersections”, pp. 51-82, Eds. Domcke, Yarkony, Koppel, Singapore, World Scientific, 2011.
35. Dynamical correlations on top of MCSCF w.f.MCSCF handles only static correlations.CAS-CI active space is at most 14 elec. in 14 orb. → For main configurations. → Lack of dynamical correlations. CASPT2 (2nd-order Perturbation Theory for CASSCF)Coefficients are determined by the 1st order eq. Energy is corrected at the 2nd order eq. ← MP2 for MCSCFMRCC (Multi-Reference Coupled-Cluster)One of the most accurate treatment for the electron correlations.35
36. Theory for Excited States36
37. Excited states: definitionExcited states as EigenstatesMathematical conditions for excited statesOrthogonalityHamiltonian orthogonalityCI is a method for excited statesCI eigenequationHamiltonian matrix is diagonalized.Eigenvector is orthogonal each other37Hamiltonian orthogonalityOrthogonality
38. Excited states for the Hartree-Fock (HF) ground stateFrom the HF stationary condition to Brillouin theoremParameterized Hartree-Fock state as a trial stateUnitary transformation for the orbital rotation HF energy expanded up to the second orderStationary condition38
39. Excited states for the Hartree-Fock (HF) ground stateCI Singles is an excited-state w. f. for HF ground stateBrillouin theorem: Single excitation is Hamiltonian orthogonal to HF stateCIS wave functionHamiltonian orthogonality & orthogonality→ CIS satisfies the correct relationship with the HF ground stateCI Singles and Doubles (CISD) does not provide a proper excited-state for HF ground state39
40. Excited states for Coupled-Cluster (CC) ground state [1]CC wave function (or symmetry-adapted cluster (SAC) w. f.)CC w.f. into Schrödinger eq. Differentiate the CC Schrödinger eq. Generalized Brillouin theorem (GBT) → Structure of excited-state w. f. Excitation operators and coefficients:[1]H. Nakatsuji, Chem. Phys. Lett., 59(2), 362-364 (1978); 67(2,3), 329-333 (1979); 334-342 (1979).
41. Symmetry-adapted cluster-Configuration Interaction (SAC-CI)[1]A basis function for excited statesOrthogonalityHamiltonian orthogonality → SAC-CI wave function[1]H. Nakatsuji, Chem. Phys. Lett., 59(2), 362-364 (1978); 67(2,3), 329-333 (1979); 334-342 (1979).GBT from CC equation
42. SAC-CI(SD-R)compared with Full-CIAccurate solution at Single and Double approximation→Applicable to molecules
43. Summary43
44. CIS, CISD, SAC-CI (SD-R) are comparedHF/CISCISDSAC/SAC-CI (SD-R)Ground state Wave functionHF determinantUp to DoublesCCSD level Electron correlationsNoYesYes Size-extensivityYesNoYesExcited state Wave functionSingle excitationsSingles and doublesSingles, doubles, effective higher excitations Electron correlationsNoNot enough. Near Full-CI result. Size-extensivityYesNoYes (Numerically)Applicable targetsQualitative description for singly excited statesNo. Excitation energy is overestimatedQuantitative description for singly excited statesNumber of operation ((N: # of basis function)O(N4)O(N6)O(N6)
45. Hierarchical view of CI-related methods45Dynamical correlations Non-EQExcited statesApplicabilityto structuresEQEQ: EquilibriumGS: Ground statesEX: Excited statesGSEXCorrIPHartree-FockMP2CCCISCIS(D), CC2SAC-CIFull-CIMRCCCASPT2MCSCFPerturbation 2nd orderCC levelUncorrelatedIP: Independent Particle modelCorr: Correlated modelStatic correlations
46. Practical aspect in CI-related methods46[1] P.-Å. Malmqvist, K. Pierloot, A. R. M. Shahi, C. J. Cramer, and L. Gagliardi, JCP 128, 204109 (2008).Fragment based approximated methods (divide & conquer, FMO, etc.) were excluded.Nact: Number of active orbitals , MxEX: The maximum order of excitation NactMxEXCCSD, SAC-CISD(MxEX in linear terms)2 4 ~1000 CASSCF, CASPT2[1]16153210~100CCSDTQ (MxEX in linear terms)RASSCFRASPT2[1]Maximum number of excitationsMaximum number of active orbitalsChallengeChallenge: Speed up
47. End47
48. Some important conditions for an electronic wave functionThe Pauli anti-symmetry principle Size-extensivity Cusp conditions Spin-symmetry adapted (for the non-relativistic Hamiltonian op.) 48CoordinatesE