1. Configuration Interaction in Quantum Chemistry Jun-ya HASEGAWA Fukui Institute for Fundamental Chemistry Kyoto University 1

2. Prof. M. Kotani (1906-1993) 2

3. Contents Molecular Orbital (MO) Theory Electron Correlations Configuration Interaction (CI) & Coupled-Cluster (CC) methods Multi-Configuration Self-Consistent Field (MCSCF) method Theory for Excited States Applications to photo-functional proteins 3

4. Molecular orbital theory 4

5. Electronic Schrödinger equation Electronic Schrödinger eq. w/ Born-Oppenheimer approx. Electronic Hamiltonian operator (non-relativistic) Potential energy – Wave function –The most important issue in electronic structure theory – 5

6. Many-electronwave function Orbital approximation: product of one-electron orbitals The Pauli anti-symmetry principle Slater determinant –Anti-symmetrized orbital products –One-electron orbitals are the basic variables in MO theory 6

7. One-electron orbitals Linear combination of atom-centered Gaussian functions. Primitive Gaussian function 7

8. Variational determination of the MO coefficients Energy functional Lagrange multiplier method 8

9. Hartree-Fock equation Variation of MO coefficients Hartree-Fock equation A unitary transformation that diagonalizes the multiplier matrix Canonical Hartree-Fock equation 9 → Eigenvalue equation Eigenvalue: Multiplier (orbital energy) Eigenvector: MO coefficients

10. Restricted Hartree-Fock (RHF) equation Spin in MO theory: (a)spin orbital formulation → spatial orbital rep. ) (b) Restricted (c) Unrestricted Restricted Hartree-Fock (RHF) equation for a closed shell (CS) system RHF 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 correlations –Lack of Coulomb hole Static (non-dynamical) correlations –Bond dissociation, Excited states –Near degeneracy No explicit separation between dynamical and static correlations. Definition of “electron correlations” in Quantum Chemistry Restricted HF Numerically Exact Fig. Potetntial energy curves of H 2 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 model Possibility of finding two electrons at : H 2 –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. 15 Left-right correlation in olefin compounds Configuration interaction - x = + x = - = No correlations included

16. 16 Angular correlation One-step higher angular momentum - x = + x = Configuration interaction - = No correlations included

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 structure Ionic configuration: 2 e on A Ionic configuration: 2 e on B Covalent 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 interaction ABABAB AB

19. Configuration Interaction (CI) and Coupled-Cluster (CC) wave functions 19

20. Some notations Notations –Occupied orbital indices: i, j, k, …. –Unoccupied orbital indices: a, b, c, ….. –Creation operator: Annihilation operator: Spin-averaged excitation operator –Spin-adapted operator (singlet) Reference configuration: Hartree-Fock determinant Excited configuration –Correct spin multiplicity (Eigenfunction of operators) 20

21. 21 Configuration Interaction (CI) wave function: a general form CI 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. 22 Variational determination of the wave function coefficients CI energy functional Lagrange multiplier method –Constraint: Normalization condition Variation of Lagrangian Eigenvalue equation

23. 23 Availability of CI method A straightforward approach to the correlation problem starting from MO theory Not only for the ground state but for the excited states Accuracy is systematically improved by increasing the excitation order up to Full-CI (exact solution) Energy is not size-extensive except for CIS and Full-CI –Difficulty in applying large systems Full-CI: number of configurations rapidly increases with the size of the system. –k α + k β electrons in n α + n β orbitals → –Porphyrin: n α = n β =384, k α =k β =152 → ~10 221 determinants Fig. Correlation energy per water molecule as a percentage of the Full-CI correlation energy (%). The cc-pVDZ basis sets were used. Number of water molecules Percentage (%) H2OH2O H2OH2O H2OH2O H2OH2O H2OH2O H2OH2O H2OH2O H2OH2O R ~ large CISD Full-CI

24. 24 Coupled-Cluster (CC) wave function CI wf: a linear expansion CC wf: an exponential expansion Single excitations Double excitations Triple excitations CC Singles (CCS) CI Singles and Doubles (CCSD) CC Singles, Doubles, and Triples (CCSDT) ∙∙∙∙ Linear terms =CI Non-linear terms

25. 25 Why exponential? Size-extensive –Non interacting two molecules A and B –Super-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 away No interaction

26. 26 Solving CC equations Schrödinger eq. with the CC w.f. CC energy: Project on HF determinant Coefficients: 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(N 6 ), similar to CI method.

27. 27 Hierarchy in CI and CC methods and numerical performance Rapid 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. SD SDT SDTQ SDTQ5 SDTQ56 Excitation order in wf. Error from Full-CI (hartree) CI 法 CC 法 Fig. Error from Full-CI energy. H 2 O molecule with cc-pVDZ basis sets.[1] ~kcal/mol “Chemical accuracy” Table. Error from Full-CI energy. H 2 O at equilibrium structure (R ref ) and OH bonds elongated twice (2R ref )). cc-pVDZ sets were used.[1] [1]“Molecular Electronic Structure Theory”, Helgaker, Jorgensen, Olsen, Wiley, 2000.

28. 28 Statistics: Bond length Comparison with the experimental data (normal distribution [1]) H 2, HF, H 2 O, HOF, H 2 O 2, HNC, NH 3, … (30 molecules) “CCSD(T)” : Perturbative Triple correction to CCSD energy cc-pVDZ cc-pVTZ cc-pVQZ HF MP2 CCSD CCSD(T) CISD Error/pm=0.01 Å [1]“Molecular Electronic Structure Theory”, Helgaker, Jorgensen, Olsen, Wiley, 2000.

29. 29 Statistics: Atomization energy Normal distribution F 2, H 2, HF, H 2 O, HOF, H 2 O 2, HNC, NH 3, 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. 30 Statistics: reaction enthalpy Normal distribution CO+H 2 →CH 2 O HNC→HCN H 2 O+F 2 →HOF+HF N 2 +3H 2 →2NH 3 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 method 31

32. Single-configuration description –Applicable to molecules in the ground state at near equilibrium structure Hartree-Fock method Multi-configuration description –Bond-dissociation, excited state, …. –Quasi-degeneracy → Linear combination of configurations to describe STATIC correlations Multi-Configuration Self-Configuration Field (MCSCF) w.f. – –Complete Active Space SCF (CASSCF) method CI part = Full-CI: all possible electronic configurations are involved. Beyond single-configuration description 32 A B A B A B +

33. Trial MCSCF wave function is parameterized by –Orbital rotation: unitary transformation –CI correction vector MCSCF energy expanded up to second-order – MCSCF method: a second-order optimizaton 33

34. MCSCF applications to potential energy surfaces CI guarantees qualitative description whole potential surfaces –From equilibrium structure to bond-dissociation limit –From ground state to excited states 34 Soboloewski, 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 1 st order eq. –Energy is corrected at the 2 nd order eq. ← MP2 for MCSCF MRCC (Multi-Reference Coupled-Cluster) –One of the most accurate treatment for the electron correlations. 35

36. Theory for Excited States 36

37. Excited states: definition Excited states as Eigenstates Mathematical conditions for excited states –Orthogonality –Hamiltonian orthogonality CI is a method for excited states –CI eigenequation –Hamiltonian matrix is diagonalized. –Eigenvector is orthogonal each other 37 Hamiltonian orthogonality Orthogonality

38. Excited states for the Hartree-Fock (HF) ground state From the HF stationary condition to Brillouin theorem –Parameterized Hartree-Fock state as a trial state –Unitary transformation for the orbital rotation –HF energy expanded up to the second order –Stationary condition 38

39. Excited states for the Hartree-Fock (HF) ground state CI Singles is an excited-state w. f. for HF ground state –Brillouin theorem: Single excitation is Hamiltonian orthogonal to HF state –CIS wave function –Hamiltonian orthogonality & orthogonality → CIS satisfies the correct relationship with the HF ground state CI Singles and Doubles (CISD) does not provide a proper excited-state for HF ground state 39

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 states –Orthogonality –Hamiltonian 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-CI Accurate solution at Single and Double approximation→Applicable to molecules

43. Summary 43

44. CIS, CISD, SAC-CI (SD-R) are compared HF/CISCISDSAC/SAC-CI (SD-R) Ground state Wave functionHF determinantUp to DoublesCCSD level Electron correlationsNoYes Size-extensivityYesNoYes Excited 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 states No. Excitation energy is overestimated Quantitative description for singly excited states Number of operation ((N: # of basis function) O(N 4 )O(N 6 )

45. Hierarchical view of CI-related methods 45 Dynamical correlations Non-EQ Excited states Applicability to structures EQ EQ: Equilibrium GS: Ground states EX: Excited states GS EX Corr IP Hartree-Fock MP2 CC CIS CIS(D), CC2 SAC-CI Full-CI MRCC CASPT2 MCSCF Perturbation 2 nd order CC level Uncorrelated IP: Independent Particle model Corr: Correlated model Static correlations

46. Practical aspect in CI-related methods 46 [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. N act : Number of active orbitals, MxEX: The maximum order of e xcitation N act MxEX CCSD, SAC-CISD(MxEX in linear terms) 2 4 ~1000 CASSCF, CASPT2[1] 16 15 32 10 ~100 CCSDTQ (MxEX in linear terms) RASSCF RASPT2[1] Maximum number of excitations Maximum number of active orbitals Challenge Challenge: Speed up

47. End 47

48. Some important conditions for an electronic wave function The Pauli anti-symmetry principle Size-extensivity Cusp conditions Spin-symmetry adapted (for the non-relativistic Hamiltonian op.) 48 Coordinates E