Basic Principles and
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Slater's Rules

Slater determinants, either taken singly or in small linear combinations which are eigenfunctions of S2 called configuration state functions (CSFs), form an ``almost'' ideal trial wave function for a molecular system. Having chosen Slater determinants as trial wave functions, the next step in the variational procedure (Table 1.2) is to plug them into the Average Value Theorem and find an energy expression which can then be minimized to find the optimal LCAO-MO coefficients in the spatial part of the spin orbitals. In evaluating the Average Value Expression, it is necessary to evaluate the matrix element of the Hamiltonian between two Slater determinants, i. e. the numerator in equation (1.5). While one can just evaluate the math by brute force expansion of the determinants and simplification of all the resulting expressions, this is a tedious process, and fortunately a set of convenient rules exists to simplify it: the Slater-Condon rules[9,10,11] for determinants made from orthogonal orbitals and the Lödin rules[12] for determinants constructed from non-orthogonal orbitals. Because we will be dealing only with sets of orthogonal orbitals, only Slater's rules will be covered here. 

Before Slater's rules can be applied, however, 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  $\langle \Psi_1 \vert \hat{H} \vert \Psi_2 \rangle$, where we have

\begin{displaymath}\vert \Psi_1 \rangle = \vert abcd \rangle\end{displaymath} (23)
\begin{displaymath}\vert \Psi_2 \rangle = \vert crds \rangle,\end{displaymath} (24)


one must first interchange the columns of  $\vert \Psi_1 \rangle$ or  $\vert \Psi_2 \rangle$ to make the two determinants look as much alike as possible. For example, we may rearrange  $\vert \Psi_2 \rangle$ as

\begin{displaymath}\vert \Psi_2 \rangle = \vert crds \rangle = - \vert crsd \rangle = \vert srcd \rangle.\end{displaymath} (25)


Once the determinants are placed in maximum coincidence, the following four rules are used to reduce the matrix element into sums of one-electron integrals, denoted  $\langle m \vert \hat{h} \vert m \rangle$

1. Identical Determinants:

$\displaystyle \langle \Psi_1 \vert \hat{H} \vert \Psi_1 \rangle = \sum_m^{N} \l... ...vert \hat{h} \vert m \rangle + \sum_{m>n}^{N} \langle mn \vert \vert mn \rangle$
 (24)
$\displaystyle = \sum_m^{N} [m\vert\hat{h}\vert m] + \sum_{m>n}^{N} [mm\vert nn] - [mn\vert nm]$
 (25)

2. Determinants that Differ by One Spin Orbital:
N $\epsilon$ common occupied orbitals

$\displaystyle \vert \Psi_1 \rangle$
 =
$\displaystyle \vert \cdots mn \cdots \rangle$
 (26)
$\displaystyle \vert \Psi_2 \rangle$
 =
$\displaystyle \vert \cdots pn \cdots \rangle$
  
$\displaystyle \langle \Psi_1 \vert \hat{H} \vert \Psi_2 \rangle$
 =
$\displaystyle \langle m \vert \hat{h} \vert p \rangle + \sum_{n}^{N} \langle mn \vert \vert pn \rangle$
  
  =
$\displaystyle [m\vert\hat{h}\vert p] + \sum_{n}^{N} [mp\vert nn] - [mn\vert np]$
 (27)

3. Determinants that Differ by Two Spin Orbitals:

$\displaystyle \vert \Psi_1 \rangle$
 =
$\displaystyle \vert \cdots mn \cdots \rangle$
 (28)
$\displaystyle \vert \Psi_2 \rangle$
 =
$\displaystyle \vert \cdots pq \cdots \rangle$
  
$\displaystyle \langle \Psi_1 \vert \hat{H} \vert \Psi_2 \rangle$
 =
$\displaystyle \langle mn \vert \vert pq \rangle$
  
  =
[mp|nq] - [mq|np]
 (29)

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

$\displaystyle \vert \Psi_1 \rangle$
 =
$\displaystyle \vert \cdots mno \cdots \rangle$
 (30)
$\displaystyle \vert \Psi_2 \rangle$
 =
$\displaystyle \vert \cdots pqr \cdots \rangle$
  
$\displaystyle \langle \Psi_1 \vert \hat{H} \vert \Psi_2 \rangle$
 =
0
  

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)
This page maintained by Brian C. Hoffman
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