Because of the presence of the continuum of positronic states in the DHF model, it is important to be certain that only electronic states spinors are occupied in the construction of the relativistic Fock matrix. One way of unambiguously imposing this restriction is via the method of kinetic balance. In regions where the nuclear potential is small relative to the rest mass energy, the single particle Dirac equation predicts that, for positive energy solutions, the small component is approximately related to the large component by means of the relation
This proportionality is a direct result of the form of the kinetic energy
operator, 
.
This condition should be obeyed by electronic solutions in regions far from
the nucleus. If the basis set does not reflect this relationship between
the two components, positronic states will contaminate the electronic
solutions, and a limited variational collapse will ensue.
This problem may be remedied by first selecting a large component basis, and
then constructing a small component basis by applying the 
operator to each large component. The resultant basis set,
which is said to be kineticly balanced, will be
able to fulfill the low field boundary conditions without requiring mixture
of the positive and negative energy solutions.
The action of the kinetic energy operator on the atomic basis functions has
some interesting consequences. 
acting on a large
component s-type Gaussian function gives rise to a p-type small
component Gaussian function. The action of the kinetic energy operator on a
large component p-type Gaussian gives rise to a small component
s function and a small component d function. In general, a large
component atomic basis basis function of angular momentum l gives rise to
small component atomic orbitals of angular momentum l-1 and l+1. As a
result, the small component basis can be up to almost twice as large as the
large component basis. For this reason, a relativistic basis set can
involve almost three times as many basis functions as a non-relativistic
basis of similar quality.
Once a kineticly balanced basis set has been constructed and the
scalar integrals have been evaluated, the DHF equations may be
implemented in a straightforward manner. In order to improve the efficiency
of these calculations, however, several aspects of the symmetry of the
electronic wavefunction may be exploited. In non-relativistic quantum
chemistry, the use of linear combinations of basis functions which transform
as irreducible representations of the point group of the nuclear framework's symmetry can
dramatically reduce the number of integrals which need to be solved, and
hence the size of the Fock matrix, since entire blocks of the Fock matrix
become zero by symmetry. Spin-restriction is another common method of
simplifying the evaluation of the non-relativistic HF wavefunction.
When spin-restriction is imposed, the same set of single particle functions,
, is used to describe the spatial portion of electrons with
and 
spin.
This allows for the reduction of the size of the Fock matrix, and results in
a wavefunction which is rigorously an eigenfunction of 
.
Spatial symmetry constraints and spin-restriction may be performed
simultaneously for non-relativistic methods.
The symmetry properties of the relativistic single particle spinors, on the
other hand, must be considered within the framework of double-groups.
Double groups take into account all of the spatial symmetry features which
the spatial symmetry point groups consider plus rotations by 2
.
Rotations by 2
introduce a sign flip for fermionic wavefunctions, and
so this adds an additional criterion which may be used to characterize the
symmetry of the wavefunction. This extra characteristic doubles the order
of the group, hence the appellation ``double-group symmetry''.
In order to take advantage of this type of symmetry, symmetry adapted basis
functions which transform as a particular irreducible representation of the double group of
interest are constructed from linear combinations of the original basis
functions. All of the scalar integrals are then constructed using this
symmetry adapted basis, thereby lending the same symmetry constraints to the
final DHF wavefunction.
Another consequence
of the entanglement of the spin and spatial symmetry is that it
is not possible to apply spin
restriction in the same manner as it was applied for non-relativistic
wavefunctions. Instead, the related Kramer's restriction may be imposed.
Kramer's restriction relates two spinors through the time reversal operator,
.
The time reversal operator is given by
where 
is the scalar complex conjugation operator,
and relates a pair of spinors, 
and 
, via the
relations
The degeneracy of the Kramer's pairs reduces the number of integrals and matrix elements which need to be evaluated to construct the DHF operator.