Atomic eigenfunctions of the D-C Hamiltonian may not be achieved
analytically as were the hydrogenic solutions of the single-particle Dirac
equation. Instead, it is necessary, as it was in the non-relativistic
theory, to appeal to approximate models such as the Dirac-Hartree-Fock (DHF)
method. In a manner analogous to Hartree-Fock theory, DHF begins with the
assumption that an n-electron atomic wavefunction, 
, can be
represented by an antisymmetrized product of n single particle functions,
{
}. In contrast to the HF method and in accordance with the
4-component nature of the single particle operators, 
, the
single particle functions 
are 4-component spinors. If we make the
approximation that each electron experiences a central potential only, then
the single particle functions may be separated into radial, angular, and
spin parts analogous to the hydrogenic eigenfunctions. Anticipating the
existence of electronic and positronic solutions, we may propose two
separate forms for 
, analogous to the approximations of the
hydrogenic solutions presented above:
where 
are two component spinors dependent upon the spin and
angular coordinates and the quantum numbers l, j and
, and are given by the 
eigenfunctions 
presented for the hydrogenic case in Section 2.1.3.
The form of the radial functions has, again, been chosen such that the
resultant wave equations and electronic energy expression are simplified.
The total wavefunction formed by these one particle functions may be
expressed as
where 
is the antisymetrizer operator.
If this form of the atomic wavefunction is assumed, and the Dirac-Coulomb
Hamiltonian is utilized, then the electronic energy is given by
In order to obtain solutions of the form ( 2.59), it is necessary to
derive a Fock-type operator, which, like its non-relativistic counterpart,
has employed the central field approximation in order to define an averaged
radial potential operator, 
to replace the exchange and Coulomb
operators. This potential will still be dependent upon the form of the
single particle functions, {
}, and must be solved for iteratively,
in a self-consistent manner. Such a potential is presented by Grant in his
extensive discussion of relativistic atomic structure[5]. The
resultant Fock operators[6] may be derived by requiring that the
form of the occupied single particle spinors are variationally
optimized with respect to the electronic energy. In the variational
construction of the Dirac-Fock operator, it is important to be sure that the
occupied 
represent purely electronic solutions. If
positronic states are occupied, instead, the single particle spinors will
collapse into the negative energy continuum.
In the case that the atomic DHF wavefunction is to be represented on a
numerical grid, some relatively straightforward boundary conditions may be
applied in the occupation selection procedure to be sure that the final
wavefunction is not contaminated by positronic solutions.
This method of
solution has been implemented in a variety of widely available programs,
most notably the program due to Desclaux[6].
