Hartree-Fock
In computational physics, the Hartree-Fock calculation scheme is a self-consistent iterative procedure to calculate the best possible single determinant solution to the time-independent SchrÃÂödinger equation. As a consequence to this, whilst it calculates the exchange energy exactly, it does not calculate the effect of electron correlation at all. It is only applicable after the Born-Oppenheimer approximation has been made.It is often used in the same area of calculations as density functional theory, which can give approximate solutions to both exchange and correlation energies, although it is not based on a quantum mechanical solution. Indeed, it is common to use calculations that are a hybrid of the two methods - the popular B3LYP schema is one such. Additionally Hartree-Fock calculations can be used as the starting point for more sophisticated methods, such as many-body perturbation theory, or quantum Monte-Carlo calculations
The starting point for the Hartree-Fock with a set of approximate orbitals. For an atomic calculation, these are typically the orbitals for a hydrogenic atom (a atom with only one electron, but the same nuclear charge). For a molelcular, or crystaline, calculation, the initial approximate wavefunctions are typically a linear combination of atomic orbitals. This gives a collections of one electron orbitials, which due to the fermionic nature of electrons, this must be anti-symmetric, which is achieved by the use of a Slater_determinant. The anti-symmetrising was contributesd by Fock, on top of the basic procedure devised by Hartree.
Once an initial wavefunction is constructed, an electron is selected. The effect of all the other electrons is summed up, and used to generate a potential. (This is why the procedure is sometimes called a mean-field procedure.) This gives a single electron in a defined potential, for which the SchrÃÂödinger equation can be solved, giving a slightly different wavefunction for that electron. This processed is then repeated for all the other electrons, which completes one step of the procedure. The whole procedure is then repeated, until the change from one step to the next is suffciently small.
Numerical stability can be a problem with this procedure, and there are various ways of combating instability. One of the most basic, and generally applicable is called F-mixing. With F-mixing, once a single electron wavefunction is calculated, it is not used directly. Instead, some combination of that calculated wavefunction and the previous wavefunctions for that electron is used - the most common being a simple linear combination of the calculated and immediatly previous wavefunction. A clever dodge, employed by Hartree, for atomic calculations was to increase the nuclear charge, thus pulling all the electrons closer together. As the system stabilised, this was gradually reduced to the correct charge.