Second-quantized operator products

Given two one-electron operators \(\hat{A}\) and \(\hat{B}\), quantized in a general spinor basis, we can show that their product can be written as \[\begin{align} \require{physics} \hat{A} \hat{B} &= \sum_{PQRS}^M A_{PQ} B_{RS} \hat{E}_{PQ} \hat{E}_{RS} \\ &= \sum_{PQ}^M \qty(\vb{A} \vb{B})_{PQ} \hat{E}_{PQ} + \sum_{PQRS}^M A_{PQ} B_{RS} \hat{e}_{PQRS} \thinspace . \end{align}\]

Even though the product of two one-electron operators yields a sum of one- and two-electron operators, their commutator can be calculated to reduce to: \[\begin{equation} \comm{\hat{A}}{\hat{B}} = \sum_{PQ}^M \comm{\vb{A}}{\vb{B}}_{PQ} \hat{E}_{PQ} \thinspace . \end{equation}\]

In a spin-separated spinor basis, the product of two second-quantized one-electron operators \(\hat{A}\) and \(\hat{B}\) is written as \[\begin{align} \hat{A} \hat{B} &= \sum_{\sigma \tau} \sum_{p}^{K_\sigma} \sum_{q}^{K_\tau} \qty(\vb{A} \vb{B})_{p \sigma, q \tau} \hat{a}^\dagger_{p \sigma} \hat{a}_{q \tau} + \sum_{\sigma \tau \varepsilon \kappa} \sum_{p}^{K_\sigma} \sum_{q}^{K_\tau} \sum_{r}^{K_\varepsilon} \sum_{s}^{K_\kappa} A_{p \sigma, q \tau} B_{r \varepsilon, s \kappa} \hat{a}^\dagger_{p \sigma} \hat{a}^\dagger_{r\varepsilon} \hat{a}_{q \tau} \hat{a}_{s \kappa} \end{align}\]