The Hartree-Fock approximation underlies the most commonly used ...

arises from the antisymmetrization of the wave function. It results from the exchange
correlation. The energy of two electrons in orbitals ψ1 and ψ2 is
if their spin is antiparallel, but
E(↑↓) = h11 + h22 + J12
E(↑↑) = h11 + h22 + J12 − K12
if their spin is parallel. The energy is lower when the spin is parallel (K12 > 0) because
the antisymmetrization prevents the electrons from being at the same location.
In summary: Given a determinantal wave function, the energy can be obtained in
the following way:
(1) each electron in spatial orbital ψi contributes hii to the energy,
(2) each unique pair of electrons contributes Jij (irrespective of spin),
(3) each pair of electons with parallel spin contributes −Kij.
Restricted Hartree-Fock equation
Using the above expression for the energy, the Hartree-Fock equation becomes:
f(1)ψj(1) = ǫjψj(1)
where the Fock operator can now be expressed as:
f(1) = h(1) +
N/2
a
2Ja(1) − Ka(1)
and the restricted Coulomb and exchange operators are:
Ja(1) = dr2ψ ∗ a(r2) 1
ψa(r2)
and
Ka(1)ψi(1) =
r12
dr2ψ ∗ 1
a (r2)
r12
The total energy of the system can be written as:
N/2
E = 2
a
N/2
= 2
a
(a|h|a) +
haa +
a
N/2
a
b
40
N/2
b
ψi(r2) ψa(r1) .
2(aa|bb) − (ab|ba)
2Jab − Kab