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.