Slater's Rules

Slater's rules describe how to compute matrix elements of the Hamiltonian between two N-electron Slater determinants. Before the rules can be used, the two Slater determinants in question 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 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:

Szabo and Ostlund describe these rules in section 2.3.3 of their book (pp. 68-74). The rules are derived explicitly in section 2.3.4 (pp. 74-81).