The objective of the effective core potential method is to construct potentials which are solely dependent upon the coordinates of the valence electrons, but take into account the influence of the inert, core electrons. The all electron Hartree-Fock model for an n-electron system begins with an antisymmetrized Hartree product wavefunction of the form
where 
is the antisymmetrizer operator and {
} are the
single-particle eigenfunctions of the Fock operator
where the action of the operators 
and 
is defined by
If we now divide the orbitals into a group of 
core orbitals and 
valence orbitals, we may re-express the Fock operator as
where
In order to make the computational task of solving the Fock equations for the molecular case more affordable, it would be useful to develop an equation analogous to the Fock equation which does not contain non-local terms which depend upon the core orbitals. Such an equation would take the form
where the full coulomb and exchange terms of 
have been
replaced by an effective core potential (ECP),
, a local potential, and the nuclear charge, Z, has been
replaced by 
, where 
is the number of core electrons.
The form of this effective potential is typically derived from
numerical Hartree-Fock atomic
solutions, in accordance with the frozen core approximation.
Several methods have been suggested for obtaining
.[26]
If we consider 5.160 for a valence atomic orbital of
angular momentum l, 
, it is possible to obtain an analytic form
for 
, the local potential which will exactly reproduce
where the l subscript acknowledges the unique form of 
for
each value of valence orbital angular momentum l.
Unfortunately, this expression for 
is only valid for
, since the last term in 5.161 is
singular when 
. To circumvent this problem, the full atomic
valence orbitals, {
}, may be replaced by approximate
pseudo-orbitals, {
}, which are nodeless. Ideally we would like to
obtain a 
which will produce atomic valence pseudo-orbitals
which are as close to the original orbitals, {
}, as possible.
The use of approximate valence orbitals brings a new definition of
, since the valence orbitals now satisfy the equation
Because 
will differ from
, the ECP must represent not only the core-core and core-valence
interactions, but also the parts of the valence-valence interaction which
were lost in the switch from 
to 
. If the difference
between 
and 
is large, then the ECP, which
will contain valence-valence interaction terms for the atomic case, will
bias the potential in the molecular valence region towards the atomic case.
One popular method of forming the nodeless 
takes a linear
combination of the appropriate full electron valence orbital,
, with all of the full electron core orbitals, 
, to
give
where the parameters 
and 
are selected by minimizing the kinetic
energy of 
.
Because of the imposition of the normalization condition upon 
,
will always be less than one, which means that 
will be less
than 
outside of the core region, where the 
's will have
little to no contribution. This could potentially introduce a substantial
difference in 
and 
and thereby introduce large
valence-valence terms to the ECP.
One 
selection scheme which gets around the problem of
valence-valence contributions to the ECP
is the technique of selecting shape consistent pseudo-orbitals.
This method
requires that 
exactly match the original valence atomic orbital,
, for 
, where 
is the radius at which
experiences its outermost maximum. Inside this radius,
the wavefunction is represented by a function of the form
where N = l+2, in most cases[27]. The coefficients, 
,
are restricted by the requirement that the zeroth through third derivatives
of 
and 
are equal, and that 
is normalized.
The resultant 
are generally known as shape-consistent
pseudo-orbitals.
By solving ( 5.162) for 
we obtain
The total 
for an atom, then, is written
so that the orthonormality of the spherical harmonics, 
,
pairs
each atomic wavefunction with the proper 
term.
The radial wavefunction of the valence orbitals is required to be orthogonal
to the core orbitals of the same angular momentum as that valence orbital.
If an atomic core contains orbitals with angular momenta up to a certain
value 
then valence orbitals of a greater orbital angular momentum
than 
do not have any such orthogonality constraints on their
radial wavefunctions. 
for 
, then, will be
largely independent of l, since the only l dependent terms which should be
represented are the core-valence exchange terms, which should not be very
large. It is convenient, then, to approximate ( 5.166) as
Once the nodeless pseudo-orbitals have been selected, it is possible to simply solve 5.165 numerically. Once this has been done, a parameterized analytic form may be fit to the numerical data. One common form is given by
Once the ECP has been been modeled by some set of functions, then ( 5.160) is complete and may be implemented for a chosen set of valence orbitals.
