Hartree–Fock and Post-Hartree–Fock Methods Emmanuel Fromager

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Hartree–Fock and post-Hartree–Fock methods

Emmanuel Fromager

Institut de Chimie de Strasbourg - Laboratoire de Chimie Quantique - Université de Strasbourg /CNRS

M2 lecture, Strasbourg, France.

Institut de Chimie, Strasbourg, France Page 1 Hartree–Fock and post-Hartree–Fock methods

Institut de Chimie, Strasbourg, France Page 2 Hartree–Fock and post-Hartree–Fock methods

Notations

X • Molecular orbitals: φp(r) = Cµp χµ(r) hφp|φqi = δpq µ

• Non-orthogonal set of atomic orbitals (Gaussian functions): hχµ|χν i = Sµν

• Hamiltonian in second quantization:

X 1 X Hˆ = hpqEˆpq + hpr|qsi EˆpqEˆrs − δqrEˆps p,q 2 p,q,r,s

Z h 1 2 i where hpq = dr φp(r) − ∇ + vne(r) φq(r) 2 r

ZZ 1 and hpr|qsi ≡ (pq|rs) = dr1dr2 φp(r1)φr(r2) φq(r1)φs(r2) |r1 − r2|

Institut de Chimie, Strasbourg, France Page 3 Hartree–Fock and post-Hartree–Fock methods Variational and non-variational approximations

• The exact electronic ground state Ψ0 and its energy E0 can be obtained two ways:

hΨ|Hˆ |Ψi hΨ0|Hˆ |Ψ0i E0 = min = Hˆ |Ψ0i = E0|Ψ0i Ψ hΨ|Ψi hΨ0|Ψ0i

• Approximate parametrized ground-state wave function: Ψ(λ0)

where λ0 denotes the complete set of optimized parameters.

Variational calculation Non-variational calculation

Hartree-Fock (HF) Many-Body Perturbation Theory (MBPT) Conﬁguration Interaction (CI) Coupled Cluster (CC) Multi-Conﬁgurational Self-Consistent Field (MCSCF)

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Rotation in real space

θ > 0 u2⃗ u ⃗ θ u ⃗ θ 2( ) 1( ) u1(⃗ θ) = cos θ u1⃗ + sin θ u2⃗ u ⃗ θ θ u ⃗ θ u ⃗ θ 2( ) = −sin 1 + cos 2 u ⃗ 1 θ = 0

Rotation in spin-orbital space

|φ (θ)⟩ = cos θ |φ ⟩ + sin θ |φ ⟩ a† | a† θ a† θ a† 1 1 2 ≡ 1̂ (θ) vac⟩ 1̂ (θ) = cos 1̂ + sin 2̂ φ θ θ φ θ φ | 2( )⟩ = −sin | 1⟩ + cos | 2⟩ a† | a† θ a† θ a† ≡ 2̂ (θ) vac⟩ 2̂ (θ) = −sin 1̂ + cos 2̂

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Matrix representation of a rotation

θ > 0 u2⃗ u ⃗ θ u ⃗ θ 2( ) 1( ) u1(⃗ θ) = cos θ u1⃗ + sin θ u2⃗ u ⃗ θ θ u ⃗ θ u ⃗ θ 2( ) = −sin 1 + cos 2 u ⃗ 1 θ = 0

u1⃗(θ) u2⃗(θ)

cos θ −sin θ − 0 θ T U(θ) = = e K where K = = − K [sin θ cos θ ] [−θ 0]

Institut de Chimie, Strasbourg, France Page 6 Hartree–Fock and post-Hartree–Fock methods Spin-orbital rotation

• Let {ϕP }P =1,2,... denote an orthonormal basis of spin-orbitals and {ϕ˜P }P =1,2,... another orthonormal basis obtained from {ϕP }P =1,2,... by unitary transformation:

X |ϕP i −→ |ϕ˜P i = UQP |ϕQi Q=1,2,...

† T ∗ −1 • “unitary” means that U ≡ U =U where U ≡ UPQ P,Q=1,2,....

† † • U can be written as U = e−κ where κ† = −κ ← U † = e−κ = e−κ = eκ = U −1

• It can be shown that, in second quantization, the unitary transformation of basis reads

† X −κ † −κˆ † κˆ † X † aˆ ≡ e aˆ = e aˆ e =a ˆ where κˆ = κPQ aˆ aˆQ P˜ QP Q P P˜ P Q PQ

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Spin-orbital rotation

• Note that the rotation operator κˆ is anti-hermitian:

† X ∗ † X h †i † X † κˆ = κPQ aˆ aˆP = κ aˆ aˆP = − κQP aˆ aˆP = −κˆ Q QP Q Q PQ PQ PQ

• Unitary transformation for an N-electron Slater determinant:

˜ ˜ ˜ † † † −κˆ † κˆ −κˆ † κˆ −κˆ † κˆ |P1P2 ... PN i =a ˆ ˜ aˆ ˜ ... aˆ ˜ |vaci = e aˆP e e aˆP e . . . e aˆP e |vaci P1 P2 PN 1 2 N

= e−κˆ aˆ† aˆ† ... aˆ† eκˆ |vaci P1 P2 PN | {z } |vaci

−κˆ |P˜1P˜2 ... P˜N i = e |P1P2 ...PN i

Institut de Chimie, Strasbourg, France Page 8 Hartree–Fock and post-Hartree–Fock methods

Spin-restricted orbital rotation

• In a restricted formalism the same set of orbitals is used for α and β spins:

X † X X † X ˆ κˆ = κPQ aˆP aˆQ = κp,σq,σ0 aˆp,σaˆq,σ0 = κpqEpq PQ pq σσ0 | {z } pq κpqδσσ0

• Since κpq = −κqp (real algebra)

X X κˆ = κpqEˆpq − κqpEˆpq p>q p

X κˆ = κpq Eˆpq − Eˆqp p>q

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Hartree-Fock approximation

• For simplicity we consider here the particular case of a non-degenerate singlet closed-shell ground state.

• The HF method consists then in approximating the exact wave function Ψ0 by a single Slater determinant Φ0. The orbital space is thus divided in two:

doubly occupied molecular orbitals φi, φj ,... unoccupied molecular orbitals φa, φb,...

occ. Y Y † |Φ0i = aˆi,σ|vaci i σ=α,β

• The initial set of molecular orbitals is usually not optimized → the optimized HF molecular orbitals will be obtained by means of unitary transformations (orbital rotations).

Institut de Chimie, Strasbourg, France Page 10 Hartree–Fock and post-Hartree–Fock methods Hartree-Fock approximation

−κˆ X • Exponential parametrization: |Φ(κ)i = e |Φ0i with κˆ = κpq Eˆpq − Eˆqp p>q   .  .    κ = κ  denotes the column vector containing all the to-be-optimized parameters.  pq  .  . p>q

• occupied-occupied and unoccupied-unoccupied rotations can be disregarded (through ﬁrst order):

X ˆ ˆ X ˆ ˆ X ˆ ˆ κˆ = κij Eij − Eji + κai Eai − Eia + κab Eab − Eba i>j i,a a>b | {z } | {z } κˆocc. κˆunocc.   .  .  occ. unocc.   κˆ |Φ0i =κ ˆ |Φ0i = 0 → only occupied-unoccupied rotations will be optimized → κ = κ  .  ai  .  .

Institut de Chimie, Strasbourg, France Page 11 Hartree–Fock and post-Hartree–Fock methods

Hartree-Fock approximation • Hartree-Fock energy expression:

[1] ∂E(κ) • Variational optimization of κ: Eκ = = 0 + ∂κ κ+

• Iterative procedure (Newton method):

T [1] 1 T [2] [1] [1] [2] [2] [1] E(κ) ≈ E(0) + κ E0 + κ E0 κ → Eκ+ ≈ E0 + E0 κ+ = 0 → E0 κ+ = −E0 2 |{z} Newton step • Update the HF determinant: Φ0 ← Φ(κ+)

[1] • HF calculation converged when E0 = 0

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Fock matrix and canonical orbitals

[1] ˆ ˆ It can be shown [see the complements] that E0,ai = −2hΦ0|HEai|Φ0i = 0= −4fia

Brillouin theorem

X 1 where the Fock matrix elements are deﬁned as fpq = hpq + hpr|qsi − hpr|sqi Drs, rs 2

Drs = hΦ0|Eˆrs|Φ0i ← one-electron density matrix

• Generating the canonical HF orbitals:

      Docc. 0 f occ. 0 f 0occ. 0 D =   f =   −→ D0 = D, f 0 =   0 0 0 f unocc. 0 f 0unocc.

occ. occ. unocc. 0occ. 0unocc. Dij = 2δij fij = fij , fab = fab f ij = δij εi, f ab = δab εa

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• Fock operator in second-quantized form using canonical orbitals:

X X X Fˆ = fpqEˆpq = εi Eˆii + εa Eˆaa p,q i a X • Møller-Plesset partitioning of the Hamiltonian: Hˆ = Fˆ + (Hˆ − Fˆ) with Fˆ|Φ0i = 2 εi |Φ0i |{z} | {z } i | {z } (0) Hˆ0 VEˆ

• Using perturbation theory, the HF energy is recovered through ﬁrst order:

(0) (1) (0) E + E = E + hΦ0|Vˆ |Φ0i = hΦ0|Hˆ |Φ0i = EHF

• The correlation energy is deﬁned as the difference between the exact and HF energies. It is a second- and higher-order energy correction:

(2) (3) Ec = E0 − EHF = E + E + ...

• Indeed, the HF determinant is an approximation to the exact ground-state wave function:

(0) Hˆ |Φ0i = E |Φ0i + Vˆ |Φ0i 6= E0|Φ0i

Institut de Chimie, Strasbourg, France Page 14 Hartree–Fock and post-Hartree–Fock methods

Møller-Plesset perturbation theory

a a 1 hΦ |Hˆ |Φ0i=0 → no projection onto the singly-excited conﬁgurations |Φ i = √ Eˆ |Φ0i ... i i 2 ai

... but doubly-excited determinants appear when applying Vˆ to |Φ0i:

† † ˆ haˆa,σaˆi,σaˆb,σaˆj,σ|Φ0|V |Φ0i = hab|iji − hab|jii 6= 0

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X |DihD|Vˆ |Φ0i • Wave function through ﬁrst order: |Ψ0i = |Φ0i + + ... (0) (0) D E − ED 2 X hD|Vˆ |Φ0i • Energy through second order (MP2): E0 = EHF + + ... (0) (0) D E − ED

• Note that a correlated wave function cannot be packed into a single determinant. This is due to the double excitations. • When single excitations contribute at ﬁrst order to the wave function, they are not associated to −κˆ X correlation but to orbital relaxation instead: |Φ(κ)i = e |Φ0i = |Φ0i − κai Eˆai|Φ0i + ... a,i

Institut de Chimie, Strasbourg, France Page 16 Hartree–Fock and post-Hartree–Fock methods

• Note also that, for open-shell atoms, the HF calculation is based on the atomic terms (2S+1L) which consist in linear combinations of Slater determinants which all correspond to the same conﬁguration (for example 1s22s22p3 for nitrogen). The HF wave function is then multideterminantal but describes one single conﬁguration (no correlation effects).

Institut de Chimie, Strasbourg, France Page 17 Hartree–Fock and post-Hartree–Fock methods Conﬁguration Interaction method (CI) • Expansion of the wave function in the basis of determinants based on the canonical doubly-occupied and unoccupied (virtual) HF orbitals. • Those determinants are obtained when applying single, double, triple, quadruple, ... excitations to the HF determinant Φ0.

• Stationarity condition:

∂E(C) n o = 0 ⇔ HC = E(C)C where H ≡ H 0 = hdet |Hˆ |det 0 i . ∂C ξξ ξ ξ

Institut de Chimie, Strasbourg, France Page 18 Hartree–Fock and post-Hartree–Fock methods

H2 in a minimal basis

EXERCISE:

Show that the Hamiltonian matrix for H2 can be written in the basis of the two single-determinant states α β α β |1σg 1σg i and |1σu 1σu i as follows,

  Eg K [Hˆ ] =   , where KEu for i = g, u, Ei = 2hii + h1σi1σi|1σi1σii, hii = h1σi|hˆ|1σii, K = h1σu1σu|1σg1σgi.

Institut de Chimie, Strasbourg, France Page 19 Hartree–Fock and post-Hartree–Fock methods Size-consistency problem in truncated CI calculations

• Size-consistency property of a method: multiplicatively separable wave function and additively separable energy, i.e., E(1 + 2) = E(1) + E(2)

where 1 and 2 denote two non-interacting monomers( Hˆ = Hˆ1 + Hˆ2).

• Example:H2 dimer in a minimal basis set

• For the monomer I (I = 1, 2), the 1σgI and 1σuI orbitals only are considered.

† † • Ground-state HF determinant for the monomer: |Φ0(I)i =a ˆ aˆ |vaci 1σgI ,α 1σgI ,β

2 Y † † • Ground-state HF determinant for the dimer: |Φ0(1 + 2)i = aˆ aˆ |vaci 1σgI ,α 1σgI ,β I=1

EHF(1 + 2)= hΦ0(1 + 2)|Hˆ |Φ0(1 + 2)i = hΦ0(1)|Hˆ1|Φ0(1)i + hΦ0(2)|Hˆ2|Φ0(2)i

= EHF(1) + EHF(2) ←− size-consistent!

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Size-consistency problem in truncated CI calculations

• CID corresponds to FCI for the monomer:

CID |Ψ (I)i = 1 + cDˆI |Φ0(I)i

ˆ † † where DI =a ˆ aˆ aˆ1σ ,β aˆ1σ ,α ←− double excitation on monomer I 1σuI ,α 1σuI ,β gI gI

  CID 0 K CID p 2 2 H (I) − EHF(I) =   −→ E (I) = EHF(I) +∆ − ∆ + K K 2∆

CID • CID is not FCI for the dimer: |Ψ (1 + 2)i = 1 + cDˆ1 + cDˆ2 |Φ0(1 + 2)i

FCI |Ψ (1 + 2)i = 1 + cDˆ1 + cDˆ2 + c12Dˆ1Dˆ2 |Φ0(1 + 2)i

Institut de Chimie, Strasbourg, France Page 21 Hartree–Fock and post-Hartree–Fock methods

Size-consistency problem in truncated CI calculations

EX5: Show that, for the dimer,   0 KK CID   (i) the CID Hamiltonian matrix equals H (1 + 2) − EHF(1 + 2) = K 2∆0    K 0 2∆

  0 KK 0   K 2∆0 K  FCI   (ii) the FCI Hamiltonian matrix equals H (1 + 2) − EHF(1 + 2) =   K 0 2∆ K    0 KK 4∆ (iii) CID is not size-consistent since

CID p 2 2 CID CID E (1 + 2) = EHF(1 + 2) +∆ − ∆ + 2K 6= E (1) + E (2)

2 (iv) FCI is size-consistent and c12 = c

Institut de Chimie, Strasbourg, France Page 22 Hartree–Fock and post-Hartree–Fock methods Size-consistency problem in truncated CI calculations s s ! K 2 K 2 • EFCI(1 + 2) − ECID(1 + 2) =∆ 1 + 1 + 2 − 2 1 + ∆ ∆

1 K 4 =∆ − + ... 4 ∆

• FCI wave function written as a Coupled-Cluster wave function (exponential ansatz):

FCI 2 |Ψ (1 + 2)i = 1 + cDˆ1 + cDˆ2 + c Dˆ1Dˆ2 |Φ0(1 + 2)i

cDˆ cDˆ = 1 + cDˆ1 1 + cDˆ2 |Φ0(1 + 2)i = e 1 e 2 |Φ0(1 + 2)i

FCI cDˆ +cDˆ |Ψ (1 + 2)i = e 1 2 |Φ0(1 + 2)i

CCD generates quadruple excitations, by means of the exponential, as products of double excitations and thus ensures size-consistency!

Institut de Chimie, Strasbourg, France Page 23 Hartree–Fock and post-Hartree–Fock methods Coupled-Cluster model (CC)

• Note that the exponential used in CC enables to describe not only orbital rotations (single excitations) but also electron correlation.

Tˆ • Exponential ansatz in the general case: |Ψ(t)i = e |Φ0i where

X X X X X Tˆ = tS Sˆ + tDDˆ + tT Tˆ + tQQˆ + ... = tµτˆµ S D T Q µ

τˆµ|Φ0i = |µi ←− excited determinant

  .  .    t = t  ←− CC amplitudes to be optimized  µ  .  .

• Truncated and approximate CC models: CCSD, CCSDT, CCSDTQ, CCSD(T), CC2, ...

Institut de Chimie, Strasbourg, France Page 24 Hartree–Fock and post-Hartree–Fock methods

Coupled-Cluster model (CC)

• Variational optimization of the CC amplitudes not convenient:

Tˆ Tˆ • Non-variational optimization: Hˆ |Ψ(t)i = E(t)|Ψ(t)i −→ Heˆ |Φ0i = E(t)e |Φ0i

−Tˆ Tˆ • "Linked" formulation: e Heˆ |Φ0i = E(t)|Φ0i

−Tˆ Tˆ Tˆ • CC energy: E(t) = hΦ0|e Heˆ |Φ0i = hΦ0|Heˆ |Φ0i

! X 1 X 2 E(t) = hΦ0|Hˆ 1 + tDDˆ + tS Sˆ |Φ0i 2 D S

−Tˆ Tˆ • CC amplitudes: hµ|e Heˆ |Φ0i = 0 ←− easier to expand (no terms beyond fourth order!)

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Ar2 (aug−cc−pVQZ) 100

0 H) µ −100

−200

−300

Interaction energy ( Reference (Tang et al.) −400 HF MP2 CCSD(T) −500 7 8 9 10 11 12 13 14 15 Interatomic distance (a.u.)

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Complements

Institut de Chimie, Strasbourg, France Page 28 Hartree–Fock and post-Hartree–Fock methods

Hartree-Fock approximation

ˆ ˆ EXERCISE: Using the Taylor expansion of fˆ(x) = e−xA Bˆ exA about x = 0, prove the Baker-Campbell-Hausdorff (BCH) expansion:

+∞ −Aˆ Aˆ X 1 e Bˆ e = Bˆ + B,ˆ Aˆ n B,ˆ Aˆ n+1 = [ B,ˆ Aˆ n, Aˆ], B,ˆ Aˆ 1 = [B,ˆ Aˆ] n! n=1 J K J K J K J K 1 = Bˆ + [B,ˆ Aˆ] + [[B,ˆ Aˆ], Aˆ] + ... 2

• Analytical formulas for the gradient and the hessian:

1 E(κ) = E(0) + hΦ0|[ˆκ, Hˆ ]|Φ0i + hΦ0|[ˆκ, [ˆκ, Hˆ ]]|Φ0i + ... | {z } 2

X ˆ ˆ ˆ [1] ˆ ˆ ˆ κaihΦ0|[Eai − Eia, H]|Φ0i → E0,ai = hΦ0|[Eai − Eia, H]|Φ0i ai

= −2hΦ0|Hˆ Eˆai|Φ0i = 0 (Brillouin theorem)

Institut de Chimie, Strasbourg, France Page 29 Hartree–Fock and post-Hartree–Fock methods Conﬁguration Interaction method (CI)

• Iterative optimization of the CI coefﬁcients Ci:

(0) (0) X (0) |Ψ(C )i = |Ψ i = Ci |ii ←− normalized starting CI wave function i  .  . (0) ˆ   |Ψ i + Q|δi   |Ψ(δ)i = q ←− convenient parametrization δ = δi ˆ   1 + hδ|Q|δi  .  .

• CI energy expression: E(δ) = hΨ(δ)|Hˆ |Ψ(δ)i

E(0) + 2δTQHC(0) + δTQHQδ = 1 + δTQδ

(0) (0)T where Hij = hi|Hˆ |ji and Q = 1 − C C

Institut de Chimie, Strasbourg, France Page 30 Hartree–Fock and post-Hartree–Fock methods Conﬁguration Interaction method (CI)

[1] ∂E(δ) • Variational condition: E = = 0 δ+ ∂δ δ+

• Newton method:

T [1] 1 T [2] [1] [1] [2] [2] [1] E(δ) ≈ E(0) + δ E + δ E δ → E ≈ E + E δ+ = 0 → E δ+ = −E 0 0 δ+ 0 0 0 0 2 |{z}

EX: Show that the CI gradient and hessian can be expressed as Newton step

[1] (0) [2] E0 = 2 H − E(0) C and E0 = 2Q H − E(0) Q

[2] [2] (0) • Note that E0 cannot be inverted since E0 C = 0

! (0)T [2] (0) (0)T [1] • We can choose δ+ such that C δ+ = 0 −→ E0 + 2αC C δ+ = −E0 | {z } [2] where α 6= 0 G0

Institut de Chimie, Strasbourg, France Page 31 Hartree–Fock and post-Hartree–Fock methods Conﬁguration Interaction method (CI)

−1 (0) −1 H − E(0) C ∗ δ = − G[2] E[1] = −C(0) + After some algebra ... + 0 0 −1 C(0)T H − E(0) C(0)

(0)T [1] which does not depend on α since C E0 = 0

• Update of the CI vector:

−1 H − E(0) C(0) C(0) −→ C(0) + Qδ = C(0) + δ = + + −1 C(0)T H − E(0) C(0)

and then normalize. [1] (0) (0) • The CI calculation has converged when E0 = 0 −→ HC = E(0)C • This procedure is also known as Rayleigh method.

∗T. Helgaker, P. Jørgensen, and J. Olsen, Molecular Electronic- Structure Theory (Wiley, Chichester, 2004), pp. 544-545.

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