# 11.2: The Slater-Condon Rules Give Expressions for the Operator Matrix Elements Among the CSFs

- Page ID
- 63316

To form the \(H_{K,L}\) matrix, one uses the so-called **Slater-Condon rules** which express all non-vanishing determinental matrix elements involving either one- or two- electron operators (one-electron operators are additive and appear as

\[ F = \sum\limits_i f(i); \]

two-electron operators are pairwise additive and appear as

\[ G = \sum\limits_{ij}g(i,j). \]

Because the CSFs are simple linear combinations of determinants with coefficients determined by space and spin symmetry, the \(H_{I,J}\) matrix in terms of determinants can be used to generate the \(H_{K,L}\) matrix over CSFs.

The Slater-Condon rules give the matrix elements between two determinants

\[ |>=|\phi_1\phi_2\phi_3... \Phi_N| \]

and

\[ |'>=|\phi_1' \phi_2' \phi_3' ...\phi_N'| \]

for **any** quantum mechanical operator that is a sum of one- and two- electron operators (F + G). It expresses these matrix elements in terms of one-and two-electron integrals involving the spin-orbitals that appear in | > and | '> and the operators f and g.

As a first step in applying these rules, one must examine | > and | '> and determine by how many (if any) spin-orbitals | > and | '> differ. In so doing, one may have to reorder the spin-orbitals in one of the determinants to achieve maximal coincidence with those in the other determinant; it is essential to keep track of the number of permutations (\(N_p\)) that one makes in achieving maximal coincidence. The results of the Slater-Condon rules given below are then multiplied by \((-1)^{N_p}\) to obtain the matrix elements between the original | > and | '>. The final result does not depend on whether one chooses to permute | > or | '>.

Once maximal coincidence has been achieved, the Slater-Condon (SC) rules provide the following prescriptions for evaluating the matrix elements of any operator F + G containing a one-electron part \(F = \sum\limits_i f(i)\) and a two-electron part \(G = \sum\limits_{ij} g(i,j)\) (the Hamiltonian is, of course, a specific example of such an operator; the electric dipole operator \(\sum\limits_i e\textbf{r}_i\) and the electronic kinetic energy \( \frac{-\hbar^2}{2m_e}\sum\limits_i \nabla_i^2 \) are examples of one-electron operators (for which one takes g = 0); the electron-electron coulomb interaction \( \sum\limits_{i>j} \frac{e^2}{r_{ij}} \) is a two-electron operator (for which one takes f = 0)):

The Slater–Condon rules express integrals of one- and two-body operators over wavefunctions constructed as Slater determinants of orthonormal orbitals in terms of the individual orbitals. In doing so, the original integrals involving N-electron wavefunctions are reduced to sums over integrals involving at most two molecular orbitals, or in other words, the original 3N dimensional integral is expressed in terms of many three- and six-dimensional integrals.

## Contributors and Attributions

Jack Simons (Henry Eyring Scientist and Professor of Chemistry, U. Utah) Telluride Schools on Theoretical Chemistry and Jeff A. Nichols (Oak Ridge National Laboratory)