Computational Physics

4.5 Self-consistency and exchange: Hartree–Fock theory 59

where in the second step the following symmetry property of the two-electron
matrix elements is used:

〈ψkψl|g|ψmψn〉=〈ψlψk|g|ψnψm〉. (4.45)

Note furthermore that because of the symmetry of the constraint equations, we must

have (^) kl= ∗lk.Eq. (4.44)can be rewritten as
δE=

∑

k

〈δψk|F|ψk〉+〈ψk|F|δψk〉 (4.46)

with
F=h+J−K. (4.47)

The Hermitian operatorFis the Fock operator, now formulated in terms of the
operatorsJandK. It is important to note that in this equation,JandKoccur with
the same prefactor ash, in contrast to Eq. (4.41) in which bothJandKhave a factor
1 /2 compared withh. This extra factor is caused by the presence of two spin-orbitals
in the expressions forJandKwhich yield extra terms in the derivative of the energy.
The matrix elements of the Fock operator with respect to the spin-orbitalsψkare

〈ψk|F|ψl〉=hkl+

∑

k′

[〈ψkψk′|g|ψlψk′〉−〈ψkψk′|g|ψk′ψl〉]. (4.48)

We finally arrive at the equation

〈δψk|F|ψk〉+〈ψk|F|δψk〉+

∑

l

(^) kl(〈δψk|ψl〉−〈ψl|δψk〉)= 0 (4.49)
and sinceδψis small but arbitrary, this, with (^) kl= ∗lk, leads to
Fψk=

∑

l

(^) klψl. (4.50)
The Lagrange parameters (^) klin this equation cannot be chosen freely: they must
be such that the solutionsψkform an orthonormal set. An obvious solution of the
above equation is found by taking theψkas the eigenvectors of the Fock operators
with eigenvaluesk, and (^) kl=kδkl:
Fψk=kψk. (4.51)
This equation is the same as(4.31), presented at the beginning of the previous
subsection. We can find other solutions to the general Fock equation(4.50)by
transforming the set of eigenstates{ψk}according to a unitary transformation,
defined by a (unitary) matrixU:
ψk′=

∑

l

Uklψl. (4.52)