Relativistic corrections to the Hartree-Fock model tend to be largest in the region immediately surrounding the nucleus. This is because of the fact that the kinetic energy of the electrons is greatest in this region. Therefore, it is reasonable to assume that the electrons in the innermost core orbitals will experience the effects of relativity much more directly than will the higher lying valence orbitals, which tend to have much less probability of existing close to the nucleus. The interaction of the valence electrons, then, with the nuclei and other electrons should be relatively well described by the non-relativistic Hamiltonian. Although the valence orbitals do not experience large direct relativistic effects, the changes in the core orbitals will have direct consequences on the nuclear shielding and orthogonality constraints which the valence orbitals experience. These are generally known as secondary effects. Since the behavior of valence electrons is the primary concern for quantum chemists, and the deeper core electrons of very heavy atoms should not change significantly in a molecular environment, an ECP method which incorporates the direct relativistic effects experienced by the core orbitals is of great practical interest.
Relativistic effective core potentials (RECP) may be obtained in a manner
directly analogous to the non-relativistic ECP's. The starting point for
RECP's is the atomic Dirac-Coulomb-Fock equations. The valence solutions,
{
}, are four-component spinors, but the large radial component
generally accounts for over 99% of the electronic density. Therefore, the
re-normalized large components are used as the starting points for the
relativistic pseudo-orbitals, 
. The methods used to derive the
relativistic pseudo-orbitals are exactly the same as the methods used in the
non-relativistic case. Once {
} have been determined, the RECP's
are obtained via a relation similar to 5.165. The DHF eigenstates,
however, are now eigenstates of the total angular momentum operator, 
, and so the resultant RECP's are no longer uniquely
defined for a particular l value. Instead, the RECP's are dependent on
l and j and are based on the relation
Since the RECP's are typically used within the framework of non-relativistic
quantum chemistry, where single particle states are eigenfunctions of
and 
rather than 
, it is necessary to construct
a 
which is only l dependent. This may be accomplished by
statistically averaging over all of the appropriate j-dependent
RECP's which are associated with a particular l value to give what are
known as l-averaged RECP's (AREP).
The total 
may be given by
completely analogous to the equation for 
.
Alternatively independent RECP's may be derived by starting with l-averaged pseudo orbitals. Such orbitals may be obtained by statistically averaging the large component of the valence spin orbitals, thereby producing l averaged atomic orbitals which are then utilized to construct nodeless pseudo-orbitals
The l-averaging may also take place in an analogous fashion after the
j-dependent pseudo-orbitals, 
, have been formed.
When RECP's are utilized to form valence LS eigenstates, the nuclear shielding and core shrinking effects are taken into account, but the third major affect of relativity on the valence orbitals, spin-orbit splitting, is not taken into account. The portion of the original RECP's which was dropped in the process of statistical averaging over l-states may be utilized to define a spin-orbit interaction term
This spin-orbit interaction operator, 
, may be
expressed as
Since this correction is based solely upon the RECP's, only one-electron integrals are required for its evaluation. This correction term was first presented in the context of CI calculations, but may be employed to estimate the extent of spin-orbit coupling of two HF wavefunctions.[28]