Editing Hartree–Fock method (section)

=== Linear combination of atomic orbitals ===
{{Main|Basis set (chemistry)}}

Typically, in modern Hartree–Fock calculations, the one-electron wave functions are approximated by a [[linear combination of atomic orbitals]].  These atomic orbitals are called [[Slater-type orbital]]s.  Furthermore, it is very common for the "atomic orbitals" in use to actually be composed of a linear combination of one or more [[Gaussian orbital|Gaussian-type orbitals]], rather than Slater-type orbitals, in the interests of saving large amounts of computation time.

Various [[basis set (chemistry)|basis sets]] are used in practice, most of which are composed of Gaussian functions.  In some applications, an orthogonalization method such as the [[Gram–Schmidt process]] is performed in order to produce a set of orthogonal basis functions.  This can in principle save computational time when the computer is solving the [[Roothaan equations|Roothaan–Hall equations]] by converting the [[overlap matrix]] effectively to an [[identity matrix]]. However, in most modern computer programs for molecular Hartree–Fock calculations this procedure is not followed due to the high numerical cost of orthogonalization and the advent of more efficient, often sparse, algorithms for solving the [[generalized eigenvalue problem]], of which the [[Roothaan equations|Roothaan–Hall equations]] are an example.