Quantizing two-electron operators in general spinor bases

A two-electron operator in coordinate representation \[\begin{equation} \require{physics} g^c(\vb{r}_1, \ldots, \vb{r}_N) = \frac{1}{2} \sum_{i \neq j}^N g^c(\vb{r}_i, \vb{r}_j) \thinspace , \end{equation}\] is represented in second quantization by \[\begin{equation} \hat{g} = \frac{1}{2} \int \int \dd{\vb{r}}_1 \dd{\vb{r}}_2 \hat{\phi}^\dagger(\vb{r}_1) \hat{\phi}^\dagger(\vb{r}_2) \thinspace g^c(\vb{r}_1, \vb{r}_2) \thinspace \hat{\phi}(\vb{r}_2) \hat{\phi}(\vb{r}_1) \thinspace , \end{equation}\] where \(\hat{\psi}(\vb{r})\) is a field operator. Recognizing the two-electron excitation operator \(\hat{e}_{PQRS}\), the two-electron second-quantized operator can be conveniently written as: \[\begin{equation} \hat{g} = \frac{1}{2} \sum_{PQRS}^M g_{PQRS} \thinspace \hat{e}_{PQRS} \end{equation}\] with the integrals \[\begin{equation} g_{PQRS} = \int \int \dd{\vb{r}_1} \dd{\vb{r}_2} \phi_P^\dagger(\vb{r}_1) \phi_R^\dagger(\vb{r}_2) \thinspace g^c(\vb{r}_1, \vb{r}_2) \thinspace \phi_S(\vb{r}_2) \phi_Q(\vb{r}_1) \thinspace . \end{equation}\]

Using antisymmetrized integrals, we can equivalently write the second-quantized two-electron operator as: \[\begin{equation} \hat{g} = \frac{1}{4} \sum_{PQRS}^M \tilde{g}_{PQRS} \thinspace \hat{e}_{PQRS} \thinspace . \end{equation}\]

The operations that appear in the integral are such that the resulting integral can be calculated analogously the one-electron case as: \[\begin{equation} g_{PQRS} = \sum_{\sigma \tau \rho \pi} g^{\sigma \tau \rho \pi}_{P \sigma Q \tau R \rho S \pi} \thinspace , \end{equation}\] which is a quadruple summation over the components of every spinor, and where \(g^{\sigma \tau \rho \pi}_{P \sigma Q \tau R \rho S \pi}\) represents the integral over one underlying scalar two-electron operator \(g^{c, \sigma \tau \rho \pi}(\vb{r}_i, \vb{r}_j)\) that appears in the tensor operator \(g^c(\vb{r}_i, \vb{r}_j)\). For non-relativistic purposes, the only two-electron operator is the electron-electron Coulomb repulsion operator, whose underlying scalar components are given by: \[\begin{equation} g^{c, \sigma \tau \rho \pi}(\vb{r}_i, \vb{r}_j) = \delta_{\sigma \tau} \delta_{\rho \pi} \thinspace \frac{1}{\norm{\vb{r}_i - \vb{r}_j}} \thinspace , \end{equation}\] such that the Coulomb repulsion integrals can then be calculated as \[\begin{equation} g_{PQRS} = \sum_{\sigma \tau} (P \sigma Q \sigma | R \tau S \tau) \thinspace , \end{equation}\] which, due to the specific tensor nature of the Coulomb operator, corresponds to an interaction between two Pauli distributions at \(\vb{r}_1\) and \(\vb{r}_2\): \[\begin{equation} g_{PQRS} = \int \int \dd{\vb{r}_1} \dd{\vb{r}_2} \phi_P^\dagger(\vb{r}_1) \phi_Q(\vb{r}_1) \thinspace \frac{1}{ \norm{\vb{r}_1 - \vb{r}_2} } \thinspace \phi_R^\dagger(\vb{r}_2) \phi_S(\vb{r}_2) \thinspace . \end{equation}\] We can think of the vertical bar in the notation \(( \cdot | \cdot )\) as a way to represent the scalar operator \[\begin{equation} \frac{1}{\norm{\vb{r}_i - \vb{r}_j}} \thinspace , \end{equation}\] which means that the total symbol \(( \cdot | \cdot )\) represents a way to write down a (scalar) Coulomb interaction energy9 between the distributions on the left- and right-hand sides.

Using the expansion of the components of the spinor in terms of the underlying scalar bases, the matrix elements of a two-electron operator (i.e. the two-electron integrals) can be calculated as: \[\begin{equation} g_{PQRS} = \sum_{\sigma \tau} \sum_{\mu \nu}^{K_\sigma} \sum_{\rho \lambda}^{K_\tau} C^{\sigma *}_{\mu P} C^{\sigma}_{\nu Q} C^{\tau *}_{\rho R} C^{\tau}_{\lambda S} (\mu \sigma \nu \sigma | \rho \tau \lambda \tau) \thinspace , \end{equation}\] in which \(\vb{C}^{\sigma}\) is the coefficient matrix for the \(\sigma\)-component and \((\mu \sigma \nu \sigma | \rho \tau \lambda \tau)\) is a Coulomb integral over scalar basis functions: \[\begin{equation} (\mu \sigma \nu \sigma | \rho \tau \lambda \tau) = \int \int \dd{\vb{r}_1} \dd{\vb{r}_2} \chi^{\sigma *}_\mu(\vb{r}_1) \chi^{\sigma}_{\nu}(\vb{r}_1) \frac{1}{\norm{\vb{r}_1 - \vb{r}_2}} \chi^{\tau *}_{\rho}(\vb{r}_2) \chi^{\tau}_{\lambda}(\vb{r}_2) \thinspace . \end{equation}\]

  1. We should remark that \(g_{PQRS}\) is a measure of the Coulomb interaction energy, which should not be confused with any Coulomb integrals or exchange integrals, which merely result from the single-Slater determinant picture in a mean-field theory.↩︎