Configuration Interaction in Quantum Chemistry

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

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

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

4. Molecular orbital theory

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

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

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

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

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

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

11. Electron correlations − Introduction to Configuration Interaction −

12. Definition of “electron correlations”in Quantum Chemistry Restricted HF Static correlation is dominant. Numerically Exact Dynamical correlation is dominant. Fig. Potetntial energy curves of H2 molecule. 6-31G** basis set. [Szabo, Ostlund, “Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory”, Dover] • 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.

13. Dynamical correlations: lack of Coulomb hole • Slater det. : Products of one-electron function →Independent particle model • Possibility of finding two electrons at : H2–like molecule case

14. － Introducing dynamical correlations via configuration interaction • Interacting a doubly excited configuration • Chemical intuition: Changing the orbital picture →

15. - Left-right correlation • in olefin compounds - x = + x = No correlations included = Configuration interaction

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

17. Static correlations: improper electronic structure Ionic configuration: 2 e on A Covalent config.: 2 e at each A and B Ionic configuration: 2 e on B Covalent config.: 2 e at each A and B • 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.

18. Introducing static correlations via configuration interaction A A A A B B B B • Interacting a doubly excited configuration • Some particular change the weights of covalent and ionic configurations.

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

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)

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. Variational determination of the wave function coefficients • CI energy functional • Lagrange multiplier method • Constraint: Normalization condition • Variation of Lagrangian • Eigenvalue equation

23. H2O H2O H2O H2O Percentage (%) R ~ large H2O H2O H2O H2O Number of water molecules Fig. Correlation energy per water molecule as a percentage of the Full-CI correlation energy (%) . The cc-pVDZ basis sets were used. 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. • kα+kβ electrons in nα+nβ orbitals → • Porphyrin: nα=nβ =384 , kα=kβ =152 → ~10221 determinants Full-CI CISD

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

25. No interaction Far away 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.

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(N6), similar to CI method.

27. SDTQ56 SDTQ SDT SDTQ5 SD Hierarchy in CI and CC methods and numerical performance Excitation order in wf. • 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. ~kcal/mol “Chemical accuracy” CI法 Error from Full-CI(hartree) CC法 Fig. Error from Full-CI energy. H2O molecule with cc-pVDZ basis sets.[1] 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. cc-pVDZ cc-pVTZ cc-pVQZ Statistics: Bond length HF • Comparison with the experimental data (normal distribution [1]) • H2, HF, H2O, HOF, H2O2, HNC, NH3, … (30 molecules) • “CCSD(T)” : Perturbative Triple correction to CCSD energy MP2 CCSD CCSD(T) CISD [1]“Molecular Electronic Structure Theory”, Helgaker, Jorgensen, Olsen, Wiley, 2000. Error/pm=0.01Å Error/pm=0.01Å Error/pm=0.01Å

29. Statistics: Atomization energy • Normal distribution • F2, 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. Statistics: reaction enthalpy • Normal distribution • CO+H2→CH2O HNC→HCN H2O+F2→HOF+HF N2+3H2→2NH3 etc. (20reactions) • 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

32. Beyond single-configuration description B A + B A A B • 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.

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

34. MCSCF applications to potential energy surfaces 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. • CI guarantees qualitative description whole potential surfaces • From equilibrium structure to bond-dissociation limit • From ground state to excited states

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 MCSCF • MRCC (Multi-Reference Coupled-Cluster) • One of the most accurate treatment for the electron correlations.

36. Theory for Excited States

37. Excited states: definition Hamiltonian orthogonality Orthogonality • 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

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

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

40. Excitation operators and coefficients: 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. [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 GBT from CC equation [1]H. Nakatsuji, Chem. Phys. Lett., 59(2), 362-364 (1978); 67(2,3), 329-333 (1979); 334-342 (1979).

42. SAC-CI(SD-R)compared with Full-CI Accurate solution at Single and Double approximation→Applicable to molecules

43. Summary

44. CIS, CISD, SAC-CI (SD-R) are compared

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

46. Practical aspect in CI-related methods Fragment based approximated methods (divide & conquer, FMO, etc.) were excluded. Nact: Number of active orbitals , MxEX: The maximum order of excitation Nact CCSD, SAC-CISD(MxEX in linear terms) ~1000 Maximum number of active orbitals CCSDTQ (MxEX in linear terms) ~100 RASSCF RASPT2[1] Challenge 32 Challenge: Speed up 15 CASSCF, CASPT2[1] MxEX 10 16 2 4 Maximum number of excitations [1] P.-Å. Malmqvist, K. Pierloot, A. R. M. Shahi, C. J. Cramer, and L. Gagliardi, JCP 128, 204109 (2008).

47. End

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