Computational Physics

54 The Hartree–Fock method

To find the probability of finding two electrons at positionsr 1 andr 2 ,we must sum
over the spin variables:

ρ(r 1 ,r 2 )=

∑

s 1 ,s 2

ρ(x 1 ,x 2 ). (4.29)

For spin-orbitals that can be written as a product of a spatial orbital and a one-particle
spin wave function, it is seen that forψkandψlhaving opposite spin, the second
term vanishes and therefore opposite spin-orbitals are still uncorrelated (the first
term of (4.28) obviously describes uncorrelated probabilities) but for equal spins,
the two terms cancel whenr 1 =r 2 , so we see that electron pairs with parallel spin
are kept apart. Every electron is surrounded by an ‘exchange hole’ [4] in which
other electrons having the same spin are hardly found. Comparing (4.28) with
the uncorrelated form (4.25), we see that exchange introduces correlation effects.
However, the term ‘correlation effects’ is usually reserved for all correlationsapart
fromexchange.
It is possible to construct Slater determinants from general spin-orbitals, i.e. not
necessarily eigenstates of the one-electron Hamiltonian. It is even possible to take
these spin-orbitals to be nonorthogonal. However, if there is overlap between two
such spin-orbitals, this drops out in constructing the Slater determinant. Therefore
we shall take the spin-orbitals from which the Slater determinant is constructed to
be orthonormal.
Single Slater determinants form a basis in the space of all antisymmetric wave
functions. In Section 4.10, we shall describe a method in which this fact is used to
take correlations into account.

4.5 Self-consistency and exchange: Hartree–Fock theory

4.5.1 The Hartree–Fock equations – physical picture

Fock extended the Hartree equation (4.11) by taking antisymmetry into account. We
first give the result which is known as the Hartree–Fock equation; the full derivation
is given in Section 4.5.2 [5, 6]:

Fψk=kψkwith (4.30)

Fψk=

[

−

1

2

∇^2 −

∑

n

Zn

|r−Rn|

]

ψk(x)+

∑N

l= 1

∫

dx′|ψl(x′)|^2

1

|r−r′|

ψk(x)

−

∑N

l= 1

∫

dx′ψl∗(x′)

1

|r−r′|

ψk(x′)ψl(x). (4.31)

The operatorFis called theFock operator. The first three terms on the right hand
side are the same as in those appearing in the Hartree equation. The fourth term is