Computational Physics

62 The Hartree–Fock method

This describes two electrons located at different nuclei, which is correct for large
nuclear separation, but this is not an eigenstate of the total spin operator. When the
nuclei are separated, the state crosses over from a restricted to an unrestricted one.
The distance at which this happens is the Coulson–Fisher point [9].
In a closed-shell system the 2Norbitals can be grouped in pairs with the same
orbital dependence but with opposite spin, thus reflecting the spin-degeneracy:

{ψ 2 k− 1 (x),ψ 2 k(x)}={φk(r)α(s),φk(r)β(s)}, k=1,...,N. (4.58)

Theφk(r)are the spatial orbitals andα(s),β(s)are the up and down spin-states
respectively. For an open-shell system, such pairing does not occur for all levels, and
to obtain accurate results, it is necessary to allow for a different orbital dependence
for each spin-orbital in most cases. Even for an open-shell system it is possible
to impose the restriction (4.58) on the spin-orbitals, neglecting the splitting of the
latter, but the results will be less accurate in that case. Calculations with the spin-
orbitals paired as in (4.58) are called restricted Hartree–Fock (RHF) and those
in which all spin-orbitals are allowed to have a different spatial dependence are
called unrestricted Hartree–Fock (UHF). UHF eigenstates are usually inconvenient
because they are not eigenstates of the total spin-operator, as can easily be verified
by combining two different orbitals with a spin-up and -down function respectively.
On the other hand, the energy is more accurate.
We shall now rewrite the Hartree–Fock equations for RHF using the special
structure of the set of spin-orbitals given in (4.58). As we have seen in the previous
section, the general form of the Fock operator is

F=h+J−K (4.59)

with

J(x)ψ(x)=

∑

l

∫

dx′ψl∗(x′)ψl(x′)

1

r 12

ψ(x);

K(x)ψ(x)=

∑

l

∫

dx′ψl∗(x′)ψ(x′)

1

r 12

ψl(x).

(4.60)

The sum overlis over all occupied Fock levels. As the Fock operator depends
explicitly on the spatial coordinate only (there is an implicit spin-dependence via
the spin-orbitals occurring in the Coulomb and exchange operators), it is possible to
eliminate the spin degrees of freedom by summing over them and find an operator
acting only on the spatial orbitalsφ(r). The uncoupled single-particle Hamiltonian
hremains the same since it contains neither explicit nor implicit spin-dependence,
and from (4.60) it is seen that the Coulomb and exchange operators, written in terms