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

(tex2html_deferredtex2html_deferred4.tex2html_deferred18_I | H | _J = ¯_I | H_0 | ¯_J ) |

where and are identical to and , respectively, except that the core orbitals have been deleted from and , and has been replaced by defined by

(tex2html_deferredtex2html_deferred4.tex2html_deferred19H_0 = E_c + _i=1^N-N_c h_c(i) + _i>j^N-N_c 1r_ij ) |

where

(tex2html_deferredtex2html_deferred4.tex2html_deferred20E_c = 2 _i^n_c h_ii + _ij^n_c { 2 (ii|jj) - (ij|ji) } ) |

Finally, is the one-electron Hamiltonian operator for electron

(tex2html_deferredtex2html_deferred4.tex2html_deferred21h_c(i) = h(i) + _j=1^n_c { 2 J_j(i) - K_j(i) } ) |

with and representing the standard Coulomb and exchange operators, respectively. Note that, although we have written the frozen core energy

(tex2html_deferredtex2html_deferred4.tex2html_deferred22P^c_ = _i^n_c C_^i C_^i ) |

where is the contribution of atomic orbital to molecular orbital

(tex2html_deferredtex2html_deferred4.tex2html_deferred23h^c_ = h_ + 2 _ (| ) P^c_ - _ (| ) P^c_ ) |

and the frozen core operator in molecular orbitals

tex2html_{d}eferredtex2html_{d}eferred4.tex2html_{d}eferred24E_{c} |
= | (tex2html_deferredtex2html_deferred4.tex2html_deferred24E_c &=& _ P^c_ ( h_ + h^c_ ) ) | |

= | Tr(P^{c} h) + Tr(P^{c} h^{c}) |

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,

(tex2html_deferredtex2html_deferred4.tex2html_deferred25 E^(2) = _a>b _r>s |ab | | rs |^2 _a + _b - _r - _s ) |

where is the orbital energy (eigenvalue) of orbital

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).