Second-quantized Hamiltonians

Expressed in a general spinor basis, the second-quantized Hamiltonian can be written as: \[\begin{equation} \require{physics} \hat{\mathcal{H}} = \sum_{PQ}^M h_{PQ} \hat{E}_{PQ} + \frac{1}{2} \sum_{PQRS}^M g_{PQRS} \hat{e}_{PQRS} \thinspace , \end{equation}\] where \(\hat{E}_{PQ}\) is a one-electron excitation operator and \(\hat{e}_{PQRS}\) is a two-electron excitation operator.

In a spin-separated spinor basis, it takes the following form: \[\begin{equation} \hat{\mathcal{H}} = \sum_{\sigma} \sum_{pq}^{K_\sigma} h_{p\sigma, q\sigma} \hat{E}^\sigma_{pq} + \frac{1}{2} \sum_{\sigma \tau} \sum_{pq}^{K_\sigma} \sum_{rs}^{K_\tau} (p\sigma q\sigma | r\tau s\tau) \hat{e}^{\sigma \sigma \tau \tau}_{pqrs} \thinspace . \end{equation}\] because of the nature of the interactions appearing in the molecular Hamiltonian.