The Hartree-Fock approximation is an extension of the above Hartree approximation to include the permutation symmetry of the wavefunction which leads to the exchange interaction. Exchange is due to the Pauli exclusion principle, which states that the total wavefunction for the system must be antisymmetric under particle exchange. This means that when two arguments are swapped the wavefunction changes sign as follows:

where includes coordinates of position and spin. Therefore no two electrons can have the same set of quantum numbers, and electrons with the same spin cannot occupy the same state simultaneously.

Instead of using the simple product form of the wavefunction shown in Eq.(), a Slater determinant wavefunction[5, 6] which satisfies antisymmetry is used

where are the one-electron wavefunctions.

Following exactly the same method of minimising the expectation value of with respect to the one-electron wavefunctions as was used in the derivation of the Hartree equations, results in the following set of one-electron equations, the Hartree-Fock equations;

where labels the spin of particle *i*. Note the
self-interaction cancels out from the second and third terms. The
extra term in these equations, when compared to
Eq.(), is known as the exchange term and is only
non-zero when considering electrons of the same spin. The effect of
exchange on the many-body system is that electrons of like spin tend
to avoid each other. As a result of this, each electron has a ``hole''
associated with it which is known as the exchange hole (or the Fermi
hole). This is a small volume around the electron which like-spin
electrons avoid. The charge contained in the exchange hole is positive
and exactly equivalent to the absence of one-electron.

Unlike all the other terms acting on , the exchange term is a non-local integral operator and this makes the Hartree-Fock equations hard to solve in all but a few special cases.

Tue Nov 19 17:11:34 GMT 1996