Computational Physics - Department of Physics

51015 Many-body approaches to studies of electronic systems: Hartree-Fock theory and Density Functional Theory

〈μ ν|V|ν μ〉=

∫

ψμ∗(ri)ψν∗(rj)V(ri j)ψν(ri)ψμ(ri)drirj.

Since the interaction is invariant under the interchange oftwo particles it means for example
that we have
〈μ ν|V|μ ν〉=〈ν μ|V|ν μ〉,

or in the more general case
〈μ ν|V|σ τ〉=〈ν μ|V|τ σ〉.
The direct and exchange matrix elements can be brought together if we define the anti-
symmetrized matrix element

〈μ ν|V|μ ν〉AS=〈μ ν|V|μ ν〉−〈μ ν|V|ν μ〉,

or for a general matrix element

〈μ ν|V|σ τ〉AS=〈μ ν|V|σ τ〉−〈μ ν|V|τ σ〉.

It has the symmetry property

〈μ ν|V|σ τ〉AS=−〈μ ν|V|τ σ〉AS=−〈ν μ|V|σ τ〉AS.

The antisymmetric matrix element is also hermitian, implying

〈μ ν|V|σ τ〉AS=〈σ τ|V|μ ν〉AS.

With these notations we rewrite Eq. (15.21) as

∫

Φ∗Hˆ 0 Φdτ=^1

2

A

∑

μ= 1

A

∑

ν= 1

〈μ ν|V|μ ν〉AS. (15.33)

Combining Eqs. (15.17) and (15.33) we obtain the energy functional

E[Φ] =

N

∑

μ= 1

〈μ|h|μ〉+^1

2

N

∑

μ= 1

N

∑

ν= 1

〈μ ν|V|μ ν〉AS. (15.34)

which we will use as our starting point for the Hartree-Fock calculations.
If we vary the above energy functional with respect to the basis functions|μ〉, this corre-
sponds to what was done in the previous subsection. We are however interested in defining
a new basis defined in terms of a chosen basis as defined in Eq. (15.32). We can then rewrite
the energy functional as

E[Ψ] =

N

∑

a= 1

〈a|h|a〉+

1

2

N

∑

ab

〈ab|V|ab〉AS, (15.35)

whereΨ is the new Slater determinant defined by the new basis of Eq. (15.32). Using
Eq. (15.32) we can rewrite Eq. (15.35) as

E[Ψ] =

N

∑

a= 1

∑

α β

C∗aαCaβ〈α|h|β〉+

1

2 ∑abα β γ δ∑C

∗

aαC

∗

bβCaγCbδ〈α β|V|γ δ〉AS. (15.36)

We wish now to minimize the above functional. We introduce again a set of Lagrange multi-
pliers, noting that since〈a|b〉=δa,band〈α|β〉=δα,β, the coefficientsCaγobey the relation

〈a|b〉=δa,b=∑

α β

Ca∗αCaβ〈α|β〉=∑

α

Ca∗αCaα,