Computational Physics - Department of Physics

50015 Many-body approaches to studies of electronic systems: Hartree-Fock theory and Density Functional Theory

15.3 Expectation value of the Hamiltonian with a given Slater determinant

We rewrite our Hamiltonian

Hˆ=−

N

∑

i= 1

1

2 ∇

2

i−

N

∑

i= 1

Z

ri+

N

∑

i<j

1

ri j,

as

Hˆ=Hˆ 0 +HˆI=

N

∑

i= 1

hˆi+

N

∑

i<j= 1

1

ri j

, (15.6)

where
hˆi=−^1
2 ∇

2

i−

Z

ri. (15.7)

The first term of Eq. (15.6),H 1 , is the sum of theNidenticalone-bodyHamiltonianshˆi. Each
individual Hamiltonianhˆicontains the kinetic energy operator of an electron and its potential
energy due to the attraction of the nucleus. The second term,H 2 , is the sum of theN(N− 1 )/ 2
two-body interactions between each pair of electrons. Let us denote the ground state energy
byE 0. According to the variational principle we have

E 0 ≤E[Φ] =

∫

Φ∗HˆΦdτ (15.8)

whereΦis a trial function which we assume to be normalized
∫
Φ∗Φdτ= 1 , (15.9)

where we have used the shorthanddτ=dr 1 dr 2 ...drN. In the Hartree-Fock method the trial
function is the Slater determinant of Eq. (15.1) which can berewritten as

Ψ(r 1 ,r 2 ,...,rN,α,β,...,ν) =√^1

N!∑P

(−)PPψα(r 1 )ψβ(r 2 )...ψν(rN) =

√

N!AΦH, (15.10)

where we have introduced the anti-symmetrization operatorA defined by the summation
over all possible permutations of two eletrons. It is definedas

A=

1

N!∑P

(−)PP, (15.11)

with the the Hartree-function given by the simple product ofall possible single-particle func-
tion (two for helium, four for beryllium and ten for neon)

ΦH(r 1 ,r 2 ,...,rN,α,β,...,ν) =ψα(r 1 )ψβ(r 2 )...ψν(rN). (15.12)

BothHˆ 0 andHˆIare invariant under electron permutations, and hence commute withA

[H 0 ,A] = [HI,A] = 0. (15.13)

Furthermore,A satisfies
A^2 =A, (15.14)

since every permutation of the Slater determinant reproduces it. The expectation value ofHˆ 0
∫
Φ∗Hˆ 0 Φdτ=N!

∫

ΦH∗AHˆ 0 AΦHdτ