Whether we perform a full CI
or only a limited CI, we must be able to
express 
in matrix form so that we can diagonalize it and obtain
the eigenvectors and eigenvalues of interest. In this section we discuss
Slater's rules (or the Slater-Condon rules
[9, 10, 11]),
which allow us to express matrix elements 
in terms of one- and two-electron
integrals. At the moment, we will express these results in terms
spin-orbitals using physicist's notation. The one-electron integrals are
written as
and the two-electron integrals are written as
where
Before Slater's rules can be used, the two Slater determinants must be
arranged in maximum coincidence. Remember that switching columns in
a determinant introduces a minus sign. For instance, to calculate
, where we have
then we must first interchange columns of 
or
to make the two determinants look as much alike as possible.
For example, we may rearrange as
After the determinants are in maximum coincidence, we see how many spin
orbitals they differ by, and we then use the following rules:
1. Identical Determinants: If the determinants are identical, then
2. Determinants that Differ by One Spin Orbital:
3. Determinants that Differ by Two Spin Orbitals:
4. Determinants that differ by More than Two Spin Orbitals:
The derivation of these rules can be found in Szabo and Ostlund [1], section 2.3.4 (pp. 74-81).