The Slater-Condon rules
In order to calculate expectation values over one- and two-electron operators, we can use the Slater-Condon rules in second quantization.
The bane of the Slater-Condon rules (T. Helgaker, Jørgensen, and Olsen 2000) are the matrix elements \(\require{physics} \matrixel{\vb{k}_2}{ \hat{E}_{PQ} }{\vb{k}_1}\) and \(\matrixel{\vb{k}_2}{ \hat{e}_{PQRS} }{\vb{k}_1}\), with \(\ket{\vb{k}_1}\) and \(\ket{\vb{k}_2}\) differing in zero, one, or two electron excitations:
- If the bra and ket have equal occupations, we can calculate that \[\begin{equation} \ev{ \hat{E}_{PQ} }{\vb{k}} = \delta_{PQ} k_P % \thinspace , \end{equation}\] such that the corresponding matrix element of a one-electron operator becomes \[\begin{equation} \matrixel{\vb{k}}{\hat{f}}{\vb{k}} % = \sum_P^M k_P f_{PP} % \thinspace . \end{equation}\]
We furthermore have that \[\begin{equation} \ev{ \hat{e}_{PQRS} }{\vb{k}} % = (k_P k_R - \delta_{PR} k_P) % (\delta_{PQ} \delta_{RS} - \delta_{PS} \delta_{RQ}) % \thinspace , \end{equation}\] such that the corresponding matrix element of a two-electron operator becomes \[\begin{equation} \matrixel{\vb{k}}{\hat{g}}{\vb{k}} % = \frac{1}{2} \sum_{PQ}^M k_P k_Q (g_{PPQQ} - g_{PQQP}) % \thinspace . \end{equation}\]
- Given two ONVs that differ in one excitation, for example \[\begin{align} & \ket{\vb{k}_1} = % \ket{k_1, \ldots, 0_W, \ldots, 1_X, \ldots, k_M} % \label{eq:ONV_k1_1_excitation} \\ & \ket{\vb{k}_2} = % \ket{k_1, \ldots, 1_W, \ldots, 0_X, \ldots, k_M} % \label{eq:ONV_k2_1_excitation} \thinspace , \end{align}\] with \(W < X\) (if not, a simple relabeling suffices), we can calculate \[\begin{equation} \matrixel{\vb{k}_2}{ \hat{E}_{PQ} }{\vb{k}_1} % = \delta_{QX} \delta_{PW} % \Gamma_W^{\vb{k}_2} \Gamma_X^{\vb{k}_1} % \thinspace , \end{equation}\] such that we find for the one-electron operator: \[\begin{equation} \matrixel{\vb{k}_2}{\hat{f}}{\vb{k}_1} % = \Gamma^{\vb{k}_2}_W \Gamma^{\vb{k}_1}_X f_{WX} % \thinspace . \end{equation}\]
We also have that \[\begin{equation} \begin{split} \matrixel{\vb{k}_2}{ \hat{e}_{PQRS} }{\vb{k}_1} = % \Big[ & k_{1, R} ( % \delta_{QX} \delta_{PW} \delta_{RS} % - \delta_{SX} \delta_{PW} \delta_{QR} % ) \\ & + k_{2, P} ( % \delta_{SX} \delta_{RW} \delta_{QP} % - \delta_{QX} \delta_{RW} \delta_{PS} % ) % \Big] % \Gamma^{\vb{k}_2}_W \Gamma^{\vb{k}_1}_X % \thinspace , \end{split} \end{equation}\] such that \[\begin{equation} \matrixel{\vb{k}_2}{\hat{g}}{\vb{k}_1} % = \frac{1/2} \Gamma^{\vb{k}_2}_W \Gamma^{\vb{k}_1}_X % \sum_P^M k_{1, P} (g_{WXPP} - g_{WPPX} + g_{PPWX} - g_{PXWP}) % \thinspace , \end{equation}\] in which it doesn’t matter (for both formulas) which occupation numbers (the ones in \(\vb{k}_1\) or \(\vb{k}_2\)) are considered, since the corresponding term would vanish if they are different. The occupation number factors merely make sure that the corresponding orbital is occupied.
- Obviously, given two ONVs that differ in two excitations, for example \[\begin{align} & \ket{\vb{k}_1} = % \ket{ % k_1, \ldots, 0_W, \ldots, 0_X, \ldots, % 1_Y, \ldots, 1_Z, \ldots, k_M% } \\ & \ket{\vb{k}_2} = % \ket{ % k_1, \ldots, 1_W, \ldots, 1_X, \ldots, % 0_Y, \ldots, 0_Z, \ldots, k_M % } % \thinspace , \end{align}\] with \(W < X\) and \(Y < Z\) (if otherwise, a simple relabeling suffices), any matrix element for a one-electron operator vanishes.
For the two-electron case, we have that \[\begin{equation} \begin{split} \matrixel{\vb{k}_2}{ \hat{e}_{PQRS} }{\vb{k}_1} = % \Big[ & \delta_{SY} \delta_{QZ} \delta_{RW} \delta_{PX} % + \delta_{SZ} \delta_{QY} \delta_{PW} \delta_{RX} \\ & - \delta_{SY} \delta_{QZ} \delta_{PW} \delta_{RX} % - \delta_{SZ} \delta_{QY} \delta_{RW} \delta_{PX} % \Big ] % \Gamma^{\vb{k}_2}_W \Gamma^{\vb{k}_2}_X % \Gamma^{\vb{k}_1}_Y \Gamma^{\vb{k}_1}_Z % \thinspace , \end{split} \end{equation}\] such that we have \[\begin{equation} \matrixel{\vb{k}_2}{\hat{g}}{\vb{k}_1} % = \Gamma^{\vb{k}_2}_W \Gamma^{\vb{k}_2}_X % \Gamma^{\vb{k}_1}_Y \Gamma^{\vb{k}_1}_Z % (g_{WYXZ} - g_{WZXY}) % \thinspace . \end{equation}\]
- Matrix elements of one- or two-electron operators between ONVs that differ in 3 or more excitations are zero.