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

• • Contributed by Jack Simons
• Professor Emeritus and Henry Eyring Scientist (Chemistry) at University of Utah

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.