Slater's Rules

Whether we perform a full CI  or only a limited CI, we must be able to express $\hat{H}$ 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 $H_{ij} = \langle \Phi_i \vert \hat{H} \vert \Phi_j \rangle$ 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

$$\langle i \vert \hat{h} \vert j \rangle = \int \phi_i^{*}({\bf r}_1) \hat{h}({\bf r}_1) \phi_j({\bf r}_1) d{\bf r}_1$$ (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

$$\langle ij \vert \vert kl \rangle = \langle ij \vert kl \rangle - \langle ij \vert lk \rangle$$ (tex2html_deferredtex2html_deferred2.tex2html_deferred12ij | | kl = ij | kl - ij | lk )

where

$$\langle ij \vert kl \rangle = \int \phi_i^{*}({\bf r}_1) \ph... ...2}} \phi_k({\bf r}_1) \phi_l({\bf r}_2) d{\bf r}_1 d{\bf r}_2$$ (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 $\langle \Phi_1 \vert \hat{H} \vert \Phi_2 \rangle$, where we have

$$\vert \Phi_1 \rangle = \vert abcd \rangle$$ (tex2html_deferredtex2html_deferred2.tex2html_deferred14| _1 = | abcd )

$$\vert \Phi_2 \rangle = \vert crds \rangle$$ (tex2html_deferredtex2html_deferred2.tex2html_deferred15| _2 = | crds )

then we must first interchange columns of $\vert \Phi_1 \rangle$ or $\vert \Phi_2 \rangle$ to make the two determinants look as much alike as possible. For example, we may rearrange $\vert \Phi_2 \rangle$ as

$$\vert \Phi_2 \rangle = \vert crds \rangle = - \vert crsd \rangle = \vert srcd \rangle$$ (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

$$\langle \Phi_1 \vert \hat{H} \vert \Phi_1 \rangle = \sum_m^{... ...t m \rangle + \sum_{m>n}^{N} \langle mn \vert \vert mn \rangle$$ (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:

$${tex2html_deferred}{{tex2html_deferred}2.{tex2html_deferred}18}\vert \Phi_1 \rangle \vert \cdots mn \cdots \rangle$$ = (tex2html_deferredtex2html_deferred2.tex2html_deferred18| _1 &=& | mn )

$$\vert \Phi_2 \rangle \vert \cdots pn \cdots \rangle$$ =

$$\langle \Phi_1 \vert \hat{H} \vert \Phi_2 \rangle \langle m \vert \hat{h} \vert p \rangle + \sum_{n}^{N} \langle mn \vert \vert pn \rangle$$ =

3. Determinants that Differ by Two Spin Orbitals:

$${tex2html_deferred}{{tex2html_deferred}2.{tex2html_deferred}19}\vert \Phi_1 \rangle \vert \cdots mn \cdots \rangle$$ = (tex2html_deferredtex2html_deferred2.tex2html_deferred19| _1 &=& | mn )

$$\vert \Phi_2 \rangle \vert \cdots pq \cdots \rangle$$ =

$$\langle \Phi_1 \vert \hat{H} \vert \Phi_2 \rangle \langle mn \vert \vert pq \rangle$$ =

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

$${tex2html_deferred}{{tex2html_deferred}2.{tex2html_deferred}20}\vert \Phi_1 \rangle \vert \cdots mno \cdots \rangle$$ = (tex2html_deferredtex2html_deferred2.tex2html_deferred20| _1 &=& | mno )

$$\vert \Phi_2 \rangle \vert \cdots pqr \cdots \rangle$$ =

$$\langle \Phi_1 \vert \hat{H} \vert \Phi_2 \rangle$$ = 0

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