Introduction to multiconfigurational quantum chemistry. Emmanuel Fromager

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1 Institut de Chimie, Strasbourg, France Page 1 Emmanuel Fromager Institut de Chimie de Strasbourg - Laboratoire de Chimie Quantique - Université de Strasbourg /CNRS M2 lecture, Strasbourg, France.

2 Notations Molecular orbitals: φ p (r) = µ C µp χ µ (r) φ p φ q = δ pq Non-orthogonal set of atomic orbitals (Gaussian functions): χ µ χ ν = S µν Hamiltonian in second quantization: Ĥ = p,q h pq Ê pq p,q,r,s ) pr qs (Êpq Ê rs δ qr Ê ps where h pq = [ dr φ p (r) r ] + v ne(r) φ q (r) and pr qs = dr 1 dr 2 φ p (r 1 )φ r (r 2 ) 1 r 1 r 2 φ q(r 1 )φ s (r 2 ) = (pq rs) Institut de Chimie, Strasbourg, France Page 2

3 Institut de Chimie, Strasbourg, France Page 3 Variational and non-variational approximations The exact electronic ground state Ψ 0 and its energy E 0 can be obtained two ways: E 0 = min Ψ Ψ Ĥ Ψ Ψ Ψ = Ψ 0 Ĥ Ψ 0 Ψ 0 Ψ 0 Ĥ Ψ 0 = E 0 Ψ 0 Approximate parametrized ground-state wave function: Ψ(λ 0 ) where λ 0 denotes the complete set of optimized parameters. Variational calculation Non-variational calculation Ψ(λ) Ĥ Ψ(λ) = 0 Ĥ Ψ(λ) E(λ) Ψ(λ) = 0 for λ = λ 0 λ Ψ(λ) Ψ(λ) λ=λ0 Hartree-Fock (HF) Configuration Interaction (CI) Multi-Configurational Self-Consistent Field (MCSCF) Many-Body Perturbation Theory (MBPT) Coupled Cluster (CC)

4 Institut de Chimie, Strasbourg, France Page FCI HF MP2 H 2 (1 1 Σ g +, aug cc pvqz) Energy (a.u.) Interatomic distance (a.u.)

5 Institut de Chimie, Strasbourg, France Page 5 Short-range dynamical correlation LUMO LUMO LUMO HOMO HOMO HOMO Ground-state configuration singly-excited conf. doubly-excited conf.

6 Institut de Chimie, Strasbourg, France Page 6 Static correlation LUMO HOMO LUMO HOMO LUMO HOMO

7 Institut de Chimie, Strasbourg, France Page 7 H 2 in the equilibrium geometry: Static correlation Ψ 0 = C 0 1σ α g 1σβ g +... where C 0 2 = 98% no static correlation In the dissociation limit: H A... H B and NOT H A... H+ B or H+ A... H B φ 1σg (r) = 1 ) (φ 1sA (r) + φ 1sB (r) 2 and φ 1σu (r) = 1 ) (φ 1sA (r) φ 1sB (r) 2 ( 1s α A 1sβ B + 1sα B 1sβ A + 1sα A 1sβ A + 1sα B 1sβ B ) 1σ α g 1σ β g = 1 2 ( 1s α A 1sβ B + 1sα B 1sβ A 1sα A 1sβ A 1sα B 1sβ B ) 1σ α u 1σ β u = 1 2 Ψ 0 = 1 2 ( 1σ α g 1σβ g 1σα u 1σβ u ) strong static correlation

8 Institut de Chimie, Strasbourg, France Page 8 EXERCISE: H 2 in a minimal basis (1) Show that the Hamiltonian matrix for H 2 can be written in the basis of the two single-determinant states 1σ α g 1σ β g and 1σ α u 1σ β u as follows, [Ĥ] = E g K K E u, where for i = g, u, E i = 2h ii + 1σ i 1σ i 1σ i 1σ i, h ii = 1σ i ĥ 1σ i, K = 1σ u 1σ u 1σ g 1σ g. (2) In the following, we use the minimal basis consisting of the two 1s atomic orbitals. Explain why, in the dissociation limit, E g = E u and K = 1 2 1s1s 1s1s > 0. (3) Conclude that, in the dissociation limit, the ground state is multiconfigurational and does correspond to two neutral hydrogen atoms with energy (E g K).

9 Institut de Chimie, Strasbourg, France Page FCI HF CISD 2/2 H 2 (1 1 Σ g +, aug cc pvqz) Energy (a.u.) Interatomic distance (a.u.)

10 Institut de Chimie, Strasbourg, France Page 10 Multi-Configurational Self-Consistent Field model (MCSCF) The MCSCF model consists in performing a CI calculation with a reoptimization of the orbitals Ψ(κ, C) = e ˆκ ( ξ ) C ξ det ξ where C { C ξ } ξ The MCSCF model is a multiconfigurational extension of HF which aims at describing static correlation: a limited number of determinants should be sufficient. Short-range dynamical correlation is treated afterwards (post-mcscf models) Choice of the determinants: active space H...H 2 electrons in 2 orbitals (1σ g, 1σ u ) 2/2 Be 2 electrons in 4 orbitals (2s, 2p x, 2p y, 2p z ) 2/4

11 Institut de Chimie, Strasbourg, France Page 11 Multi-Configurational Self-Consistent Field model (MCSCF) Complete Active Space (CAS) for Be: 1s 2 2s 2, 1s 2 2p 2 x, 1s2 2p 2 y, 1s2 2p 2 z, if all the determinants are included in the MCSCF calculation CASSCF if a Restricted Active Space (RAS) is used RASSCF The orbital space is now divided in three: doubly occupied molecular orbitals (inactive) φ i, φ j,... 1s active molecular orbitals φ u, φ v,... 2s, 2p x, 2p y, 2p z unoccupied molecular orbitals (virtuals) φ a, φ b,... 3s, 3p, 3d,...

12 Institut de Chimie, Strasbourg, France Page 12 a, b,... u, v,... i, j,...

13 Institut de Chimie, Strasbourg, France Page H 2 (FCI, aug cc pvqz) 1σ g 1σ u 1.6 Ψ 0 σ 2 g 1σ g +1σ u orbital occupation bond distance [units of a 0 ] Ψ ( σ 2 g σ2 u )

14 Institut de Chimie, Strasbourg, France Page 14 Multi-Configurational Self-Consistent Field model (MCSCF) EX6: In order to illustrate with H 2 the fact that active orbitals can be partially occupied, show that the active part of the density matrix A D, defined as A D vw = Ψ Êvw Ψ, where Ψ = 1 ( ) 1σ α 1 + c 2 g 1σg β c 1σu α 1σu β, equals A D = c 2 0 2c c 2 Note: In the particular case of a single determinantal wave function (c = 0) the active density matrix reduces to

15 Institut de Chimie, Strasbourg, France Page 15 Iterative optimization of the orbital rotation vector κ and the CI coefficients C i : Ψ (0) = C (0) i i normalized starting wave function i Ψ(λ) = e ˆκ Ψ(0) + ˆQ δ 1 + δ ˆQ δ convenient parametrization λ = ˆQ = 1 Ψ (0) Ψ (0), δ = i δ i i, Ψ (0) ˆQ δ = 0, Ψ(λ) Ψ(λ) = 1. κ pq. δ i. p > q MCSCF energy expression: E(λ) = Ψ(λ) Ĥ Ψ(λ) Variational optimization: E [1] λ + = and E c[1] λ + = E(λ) δ λ+ Eo[1] λ + E c[1] λ + = 0 where E o[1] λ + = E(λ) κ λ+

16 Institut de Chimie, Strasbourg, France Page 16 Newton method: E(λ) E(0) + λ T E [1] λt E [2] 0 λ E[1] λ E [1] E[2] 0 λ + = 0 E [2] 0 λ + }{{} = E[1] 0 Newton step Convergence reached when E [1] 0 = 0 ( ) EX7: Show that E o[1] 0,pq = Ψ(0) [Êpq Êqp, Ĥ] Ψ(0) and E c[1] 0 = 2 H CAS E(0) C (0) where H CAS ij = i Ĥ j and C(0) =. C (0) i. Note: E o[1] 0 = 0 is known as generalized Brillouin theorem.

17 Institut de Chimie, Strasbourg, France Page FCI HF CISD 2/2 MCSCF 2/2 H 2 (1 1 Σ g +, aug cc pvqz) Energy (a.u.) Interatomic distance (a.u.)

18 Institut de Chimie, Strasbourg, France Page 18 Multi-Reference Perturbation Theory (MRPT) Multiconfigurational extension of MP2: unperturbed wave function Φ 0 Ψ (0) unperturbed energy E (0) = 2 i ε i??? perturbers D??? zeroth-order excited energies E (0) + ε a + ε b ε i ε j??? unperturbed Hamiltonian ˆF E (0) Φ 0 Φ 0 + D D ˆF D D D??? Standard approaches are CASPT2 a and N-electron valence state PT2 (NEVPT2) b a K. Andersson, P. Å. Malmqvist, and B. O. Roos, J.Chem. Phys. 96, 1218 (1992). b C. Angeli, R. Cimiraglia, and J.P. Malrieu, J. Chem. Phys. 117, 9138 (2002).

19 Institut de Chimie, Strasbourg, France Page 19 Energy (a.u.) FCI HF CISD 2/2 MCSCF 2/2 sc NEVPT2 H 2 (1 1 Σ g +, aug cc pvqz) Interatomic distance (a.u.)

20 Institut de Chimie, Strasbourg, France Page 20 Multi-state MCSCF approach State-averaged MCSCF model: simultaneous optimization of the ground and the lowest N 1 excited states at the MCSCF level. Iterative procedure: N initial orthonormal states are built from the same set of orbitals. Ψ (0) I = i C (0) I,i i, I = 1,..., N Double-exponential parametrization: Ψ I (κ, S) = e ˆκ e Ŝ Ψ (0) I where Ŝ = N J=1 K>J S KJ ( Ψ (0) K Ψ(0) J ) Ψ(0) J Ψ(0) K and i i i = K Ψ (0) K Ψ(0) K

21 Institut de Chimie, Strasbourg, France Page 21 Multi-state MCSCF approach Gross Oliveira Kohn (GOK) variational principle for an ensemble of ground and excited states: For any set {Ψ I } I=1,N of N orthonormal states, the following inequality holds, N w I Ψ I Ĥ Ψ I I=1 N w I E I I=1 where E 1 E 2... E N are the N lowest exact eigenvalues of ordered as follows, Ĥ, and the weights are w 1 w 2... w N > 0. EXERCISE: Prove the theorem in the particular case of two states by using Theophilou s variational principle: Ψ 1 Ĥ Ψ 1 + Ψ 2 Ĥ Ψ 2 E 1 + E 2. Hint: Show that ] w 1 Ψ 1 Ĥ Ψ 1 + w 2 Ψ 2 Ĥ Ψ 2 = w 2 [ Ψ 1 Ĥ Ψ 1 + Ψ 2 Ĥ Ψ 2 + (w 1 w 2 ) Ψ 1 Ĥ Ψ 1

22 Institut de Chimie, Strasbourg, France Page 22 EXERCISE: Proof of Theophilou s variational principle for two states (1) Let = Ψ { 1 Ĥ Ψ } 1 + Ψ 2 Ĥ Ψ 2 E 1 E 2. We consider the complete basis of the exact eigenvectors ΨI of Ĥ with eigenvalues {E I} I=1,2,... I=1,2,... Both trial wavefunctions can be expanded in that basis as follows, Ψ K = I C KI Ψ I, K = 1, 2. Show that = 2 I=1(p I 1)E I + I>2 p I E I where p I = C 2 1I + C2 2I. (2) Show that = 2 I=1(1 p I )(E 2 E I ) + I>2 p I (E I E 2 ). Hint: prove first that I p I = 2. (3) Let us now decompose the two first eigenvectors (I = 1, 2) in the basis of the trial wavefunctions and the orthogonal complement: Ψ I = C 1I Ψ 1 + C 2I Ψ 2 + ˆQ 12 Ψ I where 2 ˆQ 12 = 1 Ψ K Ψ K. Explain why p I 1 when I = 1, 2 and conclude. K=1

23 Multi-state MCSCF approach State-averaged energy: E(κ, S) = N w I Ψ I (κ, S) Ĥ Ψ I(κ, S) I=1 where w I are arbitrary weights. In the so-called "equal weight" state-averaged MCSCF calculation w I = 1 N. Variational optimization: E(κ, S) κ = E(κ, S) S = 0 Note that, in contrast to the exact theory, converged individual energies (and therefore excitation energies) may vary with the weights. This is due to the orbital optimization. Short-range dynamical correlation is usually recovered within multi-reference perturbation theory (multi-state CASPT2 or NEVPT2 for example) Institut de Chimie, Strasbourg, France Page 23