The DHF method represents a relatively rigorous technique for obtaining a single determinant wavefunction which accounts for the most important consequences of special relativity. For many interesting chemical compounds, however, a single-determinant wavefunction is insufficient, and the inclusion of many-body interaction terms, which are not taken into account in the Hartree-Fock approximation, are essential for a proper description of molecular electronic structure. Much of the effort in quantum chemistry has gone into the design and implementation of methods which are able to effectively and efficiently account of these many-body terms. Methods such as multi-configuration self-consistent-field (MCSCF) theory, many-body perturbation theory (MBPT), configuration interaction (CI) theory, and coupled-cluster (CC) theory, the four most popular so-called correlated methods, have been designed expressly to address the many-body problem. Because the many-body problem does not disappear for systems which require relativistic treatment, the DHF model may not be able to provide highly accurate predictions. Taking a cue from non-relativistic quantum chemistry, MCSCF, MBPT, CI, and CC methods have been adapted for use with a DHF reference wavefunction. While the DHF-based correlated methods can be extremely computationally expensive, they represent the most rigorous methods currently available for accurate prediction of molecular electronic structure, and may serve both as computational bench marks and extremely reliable tools for the treatment of systems which exhibit significant dependence on both correlation and relativistic effects.
Solution of the Dirac-Fock equations
for a molecular system in a finite basis set expansion
gives rise to set of orthonormal single particle 4-spinor
solutions, {
}, and a Dirac-Fock wavefunction, 
, which
is an antisymmetrized product of a subset of these single particle states,
typically known as a Slater determinant.
The spinors which are used to construct the Dirac-Fock 
are
classified as occupied while the remainder are considered
unoccupied. In the solution of the Dirac-Fock equations, the occupied
states are the N
single particle electronic eigenstates of the
Dirac-Fock operator with the lowest eigenvalues, where
N
is the number of electrons.
The positronic states, of course, always posses lower eigenvalues
than any of the electronic states, but are excluded from
the 
to avoid variational collapse.
Using both the occupied and unoccupied single particle states, it is
possible to construct a large number of determinantial wavefunctions to
represent the total electronic wavefunction. In order for these
determinants to be suitable descriptions of the electronic wavefunction for
the system of interest, they must each be an antisymmetrized product of 
distinct spinors. These determinants may be classified
with respect to the number of spinors they have in
common with the Dirac-Fock wavefunction.
If a particular determinant differs from the DF wavefunction by only one
spinor, it is considered a singly-excited determinant, since it may be
thought of as the DF wavefunction for a molecular system which has one electron
excited relative to the system considered in the original DF optimization.
Doubly-excited determinants differ from the DF wavefunction by two spinors,
triply-excited determinants differ by three, and so on.
Excited determinants may be classified by the specific orbital replacements
which distinguish them from the DF or reference determinant. Singly
excited derminants may be denoted by a label such as 
, which specifies
that the occupied spinor 
has been replaced by the unoccupied spinor
, while doubly excited determinants
would be labeled by 
, etc.
Correlated methods seek to utilize mixtures of these excited determinants
to improve the quality of the approximate wavefunction, and the
accuracy of the predicted electronic energy, by constructing a wavefunction
which includes contributions from these determinants in some fashion.