Next: The Variational Theorem Up: Fundamental Concepts Previous: The Correlation Energy

## Slater's Rules

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]), 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

 (tex2html_deferredtex2html_deferred2.tex2html_deferred11i | h | j = _i^*(r_1) h(r_1) _j(r_1) dr_1 )

and the two-electron integrals are written as

 (tex2html_deferredtex2html_deferred2.tex2html_deferred12ij | | kl = ij | kl - ij | lk )

where

 (tex2html_deferredtex2html_deferred2.tex2html_deferred13ij | kl = _i^*(r_1) _j^*(r_2) 1r_12 _k(r_1) _l(r_2) dr_1 dr_2 )

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

 (tex2html_deferredtex2html_deferred2.tex2html_deferred14| _1 = | abcd )

 (tex2html_deferredtex2html_deferred2.tex2html_deferred15| _2 = | crds )

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

 (tex2html_deferredtex2html_deferred2.tex2html_deferred16| _2 = | crds = - | crsd = | srcd )

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

 (tex2html_deferredtex2html_deferred2.tex2html_deferred17_1 | H | _1 = _m^N m | h | m + _m>n^N mn | | mn )

2. Determinants that Differ by One Spin Orbital:

 = (tex2html_deferredtex2html_deferred2.tex2html_deferred18| _1 &=& | mn ) = =

3. Determinants that Differ by Two Spin Orbitals:

 = (tex2html_deferredtex2html_deferred2.tex2html_deferred19| _1 &=& | mn ) = =

4. Determinants that differ by More than Two Spin Orbitals:

 = (tex2html_deferredtex2html_deferred2.tex2html_deferred20| _1 &=& | mno ) = = 0

The derivation of these rules can be found in Szabo and Ostlund [1], section 2.3.4 (pp. 74-81).

Next: The Variational Theorem Up: Fundamental Concepts Previous: The Correlation Energy
C. David Sherrill
2000-04-18