It is quite common in applications of the CI method to invoke the
frozen core
approximation, in which the lowest-lying molecular
orbitals (occupied by the inner-shell electrons)
are constrained to remain doubly-occupied in all configurations.
The frozen core for atoms lithium to neon typically consists of the 1s atomic
orbital, while that for atoms sodium to argon consists of the atomic
orbitals 1s, 2s, 2p
, 2p
and 2p
.
The frozen molecular orbitals are those which are primarily these
inner-shell atomic orbitals (or linear combinations thereof).
A justification for this approximation is that the inner-shell electrons of an atom are less sensitive to their environment than are the valence electrons. Thus the error introduced by freezing the core orbitals is nearly constant for molecules containing the same types of atoms. In fact, it is sometimes recommended that one employ the frozen core approximation as a general rule because most of the basis sets commonly used in quantum chemical calculations do not provide sufficient flexibility in the core region to accurately describe the correlation of the core electrons.
Not only does the frozen core approximation reduce the number of configurations in the CI procedure, but it also reduces the computational effort required to evaluate matrix elements between the configurations which remain. Assuming that all frozen core orbitals are doubly occupied and orthogonal to all other molecular orbitals, then it can be shown [21] that
where 
and 
are identical to
and 
, respectively, except that the core orbitals
have been deleted from 
and 
, and
has been replaced by 
defined by
where N is the number of electrons and 
is the number of core
electrons. 
is the so-called ``frozen-core energy,''
which is the expectation
value of the determinant formed from only the 
core
electrons doubly occupying the 
core orbitals
Finally, 
is
the one-electron Hamiltonian operator for electron i in the
average field produced by the 
core electrons,
with 
and 
representing the standard
Coulomb and exchange operators, respectively.
Note that, although
we have written the frozen core energy 
and frozen core
operator 
in terms of molecular orbitals, it
is not necessary to explicitly transform the one- and two-electron
integrals involving core orbitals. Assuming real orbitals,
we can define a frozen
core density matrix [22]
in atomic (or symmetry) orbitals as
where 
is the contribution of atomic orbital 
to molecular orbital i. Now the frozen core operator in
atomic orbitals becomes
and the frozen core operator in molecular orbitals 
can
be obtained simply by transforming 
. Similarly
the frozen core energy can be evaluated as
An analogous approximation is the deleted virtual approximation, whereby a few of the highest-lying virtual (unoccupied) molecular orbitals are constrained to remain unoccupied in all configurations. Since these orbitals can never be occupied, they can be removed from the CI procedure entirely because no terms involving them contribute to the CI coefficients or energy. The rationalization for this procedure is that it is unlikely that electrons will choose to partially populate high-energy orbitals in their attempt to avoid other electrons. However, this conclusion is generally true only for very high-lying virtual orbitals (such as those formed by antisymmetric combinations of symmetry orbitals in the core region). For all other virtual orbitals, such simplistic reasoning is not sufficient.
Davidson points out that those high energy SCF virtual orbitals which
result from the antisymmetric combination of the two basis functions
describing each atomic orbital in a double-
basis set (such
as the 3p-like orbital formed from the minus combination of the
larger and smaller 2p atomic orbitals on oxygen)
often make the largest contribution to the correlation energy
in Møller-Plesset (MPn) wavefunctions [23].
This can be seen from the expression for the second-order correction
to the energy in Møller-Plesset perturbation theory,
where 
is the orbital energy (eigenvalue) of orbital i.
Thus a virtual orbital r with a large energy 
will contribute
to a large energy denominator in each term of equation ( 4.25)
in which it appears. However, if orbital r lies close spatially
to one of the orbitals a or b occupied in the reference, then
this large overlap will contribute to a large two-electron integral
. This integral is squared in the numerator,
leading to a large overall contribution to the second-order energy.
For the antisymmetric combinations of the basis functions
describing the core region, this large
numerator is insufficient to overcome the even larger energy
denominator; such virtual orbitals can generally be deleted with
minimal loss in the correlation energy recovered.
Although the analysis in the preceeding paragraph is based on perturbation theory, similar conclusions can be drawn for the CI method. This is most easily verified by actual calculations, since analytical expressions for the energetic contribution of orbitals to the CI energy are not nearly as simple to obtain or interpret as is equation ( 4.25).
