Computational Physics

58 The Hartree–Fock method

We now define the operators

Jk(x)ψ(x)=

∫

ψk∗(x′)

1

r 12

ψk(x′)ψ(x)dx′ and (4.39a)

Kk(x)ψ(x)=

∫

ψk∗(x′)

1

r 12

ψ(x′)ψk(x)dx′ (4.39b)

and furthermore
J=

∑

k

Jk; K=

∑

k

Kk. (4.40)

Jis called theCoulomboperator andKtheexchangeoperator as it can be obtained
from the Coulomb operator by interchanging the two rightmost spin-orbitals. In
terms of these operators, we can write the energy as

E=

∑

k

〈

ψk

∣

∣

∣∣h+^1

2

(J−K)

∣

∣

∣∣ψk

〉

. (4.41)

This is the energy-functional for a Slater determinant. We determine the minimum
of this functional as a function of the spin-orbitalsψk, and the spin-orbitals for
which this minimum is assumed give us the many-electron ground state. Notice
however that the variation in the spin-orbitalsψkis not completely arbitrary, but
should respect the orthonormality relation:

〈ψk|ψl〉=δkl. (4.42)

This implies that we have a minimisation problem with constraints, which can be
solved using the Lagrange multiplier theorem. Note that there are onlyN(N+ 1 )/ 2
independent constraints as〈ψk|ψl〉=〈ψl|ψk〉∗. Using the Lagrange multipliers

(^) klfor the constraints(4.42), we have
δE−

∑

kl

(^) kl[〈δψk|ψl〉−〈ψk|δψl〉] = 0 (4.43)
with
δE=

∑

k

〈δψk|h|ψk〉+complex conj.

+

1

2

∑

kl

(〈δψkψl|g|ψkψl〉+〈ψlδψk|g|ψlψk〉

−〈δψkψl|g|ψlψk〉−〈ψlδψk|g|ψkψl〉)+complex conj.

=

∑

k

〈δψk|h|ψk〉+complex conj.

+

∑

kl

(〈δψkψl|g|ψkψl〉−〈δψkψl|g|ψlψk〉)+complex conj. (4.44)