Hartree-Fock Theory

Home , Hartree

Hartree-Fock Theory

ISTPC Summer School June 14th - 24th 2015 Aussois

Vincent Robert, [email protected]

UdS - UMR 7177 Laboratoire de Chimie Quantique 1. N-electron problem 2. Independent electron approximation

I Non-interacting electrons

I Orbitals (AOs and MOs)

I Electron-electron interactions

I Self-Consistent Field Method

I Slater determinant

I Hartree-Fock equations

I Roothan equations - atomic basis sets

I A featuring system : H2 3. General results : Brillouin,Koopmans’ theorems 4. Conclusions N-electron Problem

N electrons → {i} light particles M ions → {A} heavy particles

N M N M ˆ X 1 2 X 1 2 X X ZA H = − ∇i − ∇A − 2 2MA riA i=1 A=1 i=1 A=1 N N M M X X 1 X X ZAZB + + rij RAB i=1 j>i A=1 B>A mi /MA ≈ 1/2000 frozen ions : Born-Oppenheimer approximation

1 → non-independent particles ! rij

approximated solution Non-interacting electron Case “ 1 = 0” rij

N N X X Hˆ = hˆ(i) → E0 = i i=1 i=1

E(N + 1) = E(N) + N+1 convenient and simple, though approximate! Orbital Approximation “electrons in boxes”

Atomic Orbitals → Molecular Orbitals orbital : eigenfunction of a monoelectronic problem Z hˆϕ = ϕ kϕk2 = ϕ∗ϕdr = hϕ|ϕi = 1

Linear Combination of Atomic Orbitals (LCAO)

N X ψMO = ci ϕi i=1

example 1 : H¨uckel for conjugated systems {2pz } simple zeta example 2 : extended H¨uckel

{AO}i multiple zeta

ϕi = a1exp{−ζ1r} + a2exp{−ζ2r}

how do the electrons see each other ? Electron-Electron Interactions “ 1 6= 0” rij

Hartree’s assumptions :

1. instantaneous interaction → average interaction spherical for atoms

2. self-consistency {ϕi }→ Coulomb → hˆϕ˜i = i ϕ˜i →{ ϕ˜i } kϕ˜i − ϕi k → 0

3. Pauli exclusion principle “no two electrons with same spin in the same orbital”

→ spherical averaging of the Coulomb potential Self-Consistent Field Method atomic calculations

- the electrons move independently one from the other : Ψ({ri }, {θi }, {φi }) = ϕ1(r1, θ1, φ1) ··· ϕN (rN , θN , φN ) → Hartree product

- let us write as for the hydrogen atom :

P(r) m m ϕ(r, θ, φ) = r Yl (θ, φ) Yl (θ, φ) : spherical harmonics 0 (e.g. Y0 (θ, φ) ˆ=1s ) ˆ X 1 2 X Z X 1 X X H = − ∇i − + = fi + gij 2 ri rij i i j>i i j>i P - electrostatic part j>i gij : Z ∗ ∗ 1 ϕi (r1)ϕj (r2) ϕi (r1)ϕj (r2)dr1dr2 =( ii, jj): Coulomb integral r12 Spherical Averaging : Hartree’s equations

δhψ|H|ˆ ψiav = 0

- averaging the charge distribution around r1 : 2 δQ(r1) = Pj (r1)dr1

δQ(r2) δQ(r2) - energy at r2 : r if r1 > r2 and r if r1 < r2 n 1 2 o 1 R r1 R ∞ r1 Y0(r1, j) = δQ(r2)dr2 + δQ(r2) dr2 r1 0 r1 r2

- spherically averaged ith and jth charge distributions : Z ∞ Z ∞Z ∞ 1 F0(i, j) = δQ(r1)Y0(r1, j)dr1 = δQ(r1) δQ(r2) 0 0 0 rmax

2 ∗ n d li (li +1) Z o − dr 2 + r 2 − r Pi (r) = i Pi (r)

∗ X Z = Z − ri Y0(ri , j): screening eﬀect j6=i Antisymmetry of the Wave Function 2-electron system

- experimental facts : 1s2s → antisymmetric 1S and symmetric 3S - triplet/singlet spin parts : symmetric/antisymmetric MS = 0 component : α(1)β(2) ± α(2)β(1)

triplet :

Slater determinant

1 ϕi (x1) ϕj (x1) Ψ(x1, x2) = √ with xi = (ri , ωi ) 2 ϕi (x2) ϕj (x2) Note : ϕi,j = spin-orbitals

- antisymmetry:Ψ( x1, x2) = −Ψ(x2, x1)

- Pauli’s principle veriﬁed : Ψ(x, x) = 0 (identical lines !) Properties-Generalization

ϕ(xi ) = ϕ(ri )α(ωi ) or ϕ(ri )β(ωi )

- Space parts of the wave-functions : |S = 1, MS = 0i = ϕi (r1)ϕj (r2) − ϕi (r2)ϕj (r1) |S = 0, MS = 0i = ϕi (r1)ϕj (r2) + ϕi (r2)ϕj (r1)

r1 6= r2 in the triplet - Coulomb interactions larger in S = 0 than in S = 1

- Analytically, E0 − E1 = 2Kij R ∗ ∗ 1 with Kij = ϕ (r1)ϕ (r2) ϕi (r2)ϕj (r1)dr1dr2 i j r12 = hij |1/r|ji i =( ij, ji): exchange integral

→ Hund’s rule : Smax is favored E S=0 2Kij S=1 Generalization : Slater determinant Starting from 2-electron functions :

Ψ1(x1, x2) = ϕi (x1)ϕj (x2)Ψ2(x1, x2) = ϕj (x1)ϕi (x2)

1 ϕi (x1) ϕj (x1) Ψ(x1, x2) = √ : Slater determinant 2 ϕi (x2) ϕj (x2)

More generally :

ϕ (x ) ϕ (x ) ··· ϕ (x ) i 1 j 1 k 1

1 ϕi (x2) ϕj (x2) ··· ϕk (x2) Ψ({xi }) = √ . . . N! ......

ϕi (xN ) ϕj (xN ) ··· ϕk (xN )

notation : Ψ = |ϕi ϕj ··· ϕk |

{ϕj } = orthonormal molecular spin-orbital basis set Hartree-Fock Equations

E[{ϕa}] = hψ|H|ψi and hψ|ψi = 1

(i) single Slater determinant : Ψ = |ϕ1ϕ2 ··· ϕaϕb ··· ϕN |

(ii) δE = 0 under the constraints hϕa|ϕbi = δab N N X X L[{ϕa}] = E[{ϕa}] − ba(hϕa|ϕbi − δab) a=1 b=1

=⇒ Fˆ(1)ϕi (1) = i ϕi (1) : Hartree-Fock equations (∼ 1930)

Fˆ = Fock operator → monoelectronic X Fˆ(1) = hˆ(1) + Jˆa(1) − Kˆa(1) a,occupied Z ˆ ∗ −1 Ja(1) = dx2ϕa (2)r12 ϕa(2) Z ˆ ∗ −1 Ka(1)ϕi (1) = dx2ϕa (2)r12 ϕi (2) ϕa(1)

Jˆa = Coulomb, local Kˆa = exchange, non-local Coulomb, Exchange Operators, Hartree-Fock Potential

Coulomb Jˆb Z ˆ ∗ ∗ −1 ha|Jb|ai = dx1dx2ϕa (1)ϕb(2)r12 ϕb(2)ϕa(1)=( aa, bb) = Jab

exchange Kˆb Z ˆ ∗ ∗ −1 ha|Kb|ai = dx1dx2ϕa (1)ϕb(2)r12 ϕa(2)ϕb(1)=( ab, ba) = Kab

hα|βi = 0 ⇒ Kab = 0 for electrons of opposite spins

P ˆ ˆ HF potentialv ˆHF = occ Jb − Kb

X ha|vˆHF |ai = (aa, bb) − (ab, ba) b,occ → α and β electrons are not correlated Properties-Resolution - α and β electrons form two discernable sets ˆ (K operator → spin polarization) E - closed-shell systems : Restricted Hartree-Fock

N/2 X Fˆ(1) = hˆ(1) + 2Jˆa(1) − Kˆa(1) MOs

- ha|Jˆa|ai = ha|Kˆa|ai cancellation inv ˆHF : no self-interaction

- Fˆ = f ({ϕa, occ}) ⇒ iterative resolution until self-consistency : hi|F|ˆ ji = δij i

1 - along the HF procedure → vˆHF : mean ﬁeld r12 Roothaan Equations

introduction of a basis LCAO M X ϕi = cµi χµ M ∼ 1000 µ=1 two M × M matrices must be introduced : Z ∗ Sµν = dr1χµ(1)χν (1) : metric Z ∗ Fµν = dr1χµ(1)F(1)χν (1) : Fock matrix X X Fµν cνi = i Sµν cνi µ ν

FC = SC : Roothaan equations Roothaan Equations in Practice

non-orthogonal atomic orbitals

I Slater type (atomic-like) example : double dz´eta,DZ

ϕi (r) = a1exp{−ζ1r} + a2exp{−ζ2r}

I Gaussian type and contractions

I minimal basis sets Slater ← linear combination of Gaussians

X 2 ϕi (r) = ci exp{−αi r } i

I contraction : some of the coeﬃcents are frozen example : 6-31G 6 for the core 3 + 1 for the valence → split valence Unrestricted Hartree-Fock

diﬀerent α and β spatial forms

ˆ ˆ X h ˆ ˆ i X ˆ Fα(1) = h(1) + Ja,α(1) − Ka,α(1) + Ja,β(1)

Nα Nβ

σσ σσ αβ ˆ ˆ Jij , Kij and Jij : coupling between Fα and Fβ ! How to practically solve FC = SC ? † − 1 Find X such as X SX diagonal → X = S 2

F 0C 0 = C 0 with F 0 = X †FX

→ ”standard” eigen-value problem

Main drawback of UHF method : hSˆ2i 6= S (S + 1) → spin contamination issue H2 as an example

- minimal basis set : {1sa, 1sb} = {a, b} ϕ1 = ξ1a + ξ2b ϕ2 = ξ2a − ξ1b from group theory, |ξ | = |ξ | = √1 1 2 2

ϕ1 = g and ϕ2 = u

- self-consistency is automatically achieved

Fgu = hu|F|ˆ gi = 0 Fgg = hg|F|ˆ gi = g Fuu = hu|F|ˆ ui = u

u = ? EHF = f (g , u) ?

= h + 2J − K g = hgg + Jgg u uu gu gu Hartree-Fock Energy

EHF = hΨ|H|ˆ Ψi with Ψ = |aa · · · |

from a ”natural” inspection : EH2 = 2hgg +Jgg

How does this compare to 2g ? P EH2 = 2g −Jgg ⇒ EHF 6= a,occ a !

X 1 X h i E = h + (aa, bb) − (ab, ba) HF aa 2 a,occ a,b X 1 X h i = − (aa, bb) − (ab, ba) a 2 a,occ a,b

1 X h i − (aa, bb) − (ab, ba) 2 a,b avoids double counting of e-e interactions Brillouin and Koopmans Theorems

Single Excited Determinants r † |Ψai = ar aa|ΨHF i

ˆ r hΨHF |H|Ψai = 0

What do we learn from the i ?

N N+1 eA = −r eA = EHF − Er N−1 N IP = −a IP = Ea − EHF

i calculated for the N-electron system ! Pathology of the Hartree-Fock Solution

Does ΨHF = |gg¯| survive as R → ∞ ?

Since g = √1 (a+b), Ψ = 1 |a¯a| + |bb¯|+|ab¯| + |b¯a| 2 HF 2

Ψ = √1 Ψ + Ψ HF 2 ionic neutral

incompatible withΨ ≈ Ψneutral as R → ∞ !

⇒ “suppress” the ionic contributions : Ψ = λ|gg¯| − µ|uu¯|

Hartree-Fock invalidated : beyond mean ﬁeld is necessary