Restricted excitation operators

In the previous section, we have already introduced Emmanuel Fromager’s favorite \(\require{physics} \hat{E}_{pq}\)-operator. It is a singlet one-electron excitation operator: \[\begin{align} \hat{E}_{pq} % &= \hat{a}^\dagger_{p \alpha} \hat{a}_{q \alpha} % + \hat{a}^\dagger_{p \beta} \hat{a}_{q \beta} \\ % &= \hat{E}^\alpha_{pq} + % \hat{E}^\beta_{pq} \label{eq:E_pq_alpha_beta} \\ % &= \sum_\sigma % \hat{a}^\dagger_{p \sigma} \hat{a}_{q \sigma} % \label{eq:E_pq} % \thinspace . \end{align}\] We can verify the following commutation relations: \[\begin{equation} \label{eq:commutator_E_E} \comm{ % \hat{E}_{pq} % }{ % \hat{E}_{rs}% } = \delta_{rq} \hat{E}_{ps} - \delta_{ps} \hat{E}_{rq} % \thinspace , \end{equation}\] such that the singlet excitation operators can be seen as the generators of the unitary group. Furthermore, we have \[\begin{align} & \comm{ % \hat{a}^\dagger_{p \sigma} % }{\hat{E}_{qr}} = - \delta_{pr} \hat{a}^\dagger_{q \sigma} \\ % & \comm{ % \hat{a}_{p \sigma} % }{\hat{E}_{qr}} = \delta_{pq} \hat{a}_{r \sigma} % \thinspace , \end{align}\] and the ‘diagonal’ singlet one-electron excitation operator is \[\begin{equation} \label{eq:diagonal_E_pp} \hat{E}_{pp} = \hat{N}_{pp} % \thinspace . \end{equation}\] For the Hermitian adjoint of the singlet one-electron excitation operators, we have \[\begin{equation} \label{eq:E_pq_Hermitian_adjoint} \hat{E}^\dagger_{pq} = \hat{E}_{qp} % \thinspace . \end{equation}\] We will also need the following operator elsewhere: \[\begin{align} \hat{E}^-_{pq} % &= \hat{E}_{pq} - \hat{E}_{qp} \\ % &= \hat{E}_{pq} - \hat{E}^\dagger_{pq} % \thinspace , \end{align}\] which is, by the way, anti-Hermitian: \[\begin{equation} \qty(\hat{E}^-_{pq})^\dagger = - \hat{E}^-_{pq} % \thinspace , \end{equation}\] and its commutators are \[\begin{equation} \label{eq:commutator_E-} \comm{ % \hat{E}^-_{pq} % }{ % \hat{E}^-_{rs} % } % = % \qty( % \delta_{rq} \hat{E}^-_{ps} % - \delta_{ps} \hat{E}^-_{rq} % ) % - \qty( % \delta_{sq} \hat{E}^-_{pr} % - \delta_{pr} \hat{E}^-_{sq} % ) % \thinspace . \end{equation}\]

\(\hat{e}_{pqrs}\) is a singlet two-electron excitation operator, which can be written in many different ways: \[\begin{align} \hat{e}_{pqrs} % &= \sum_{\sigma \tau} % \hat{a}^\dagger_{p \sigma} \hat{a}^\dagger_{r \tau} % \hat{a}_{s \tau} \hat{a}_{q \sigma} % \label{eq:two_electron_excitation_operator} \\ % &= \hat{E}_{pq} \hat{E}_{rs} % - \delta_{qr} \hat{E}_{ps} \\ % &= \sum_{\sigma \tau} % \hat{e}^{\sigma \sigma \tau \tau}_{pqrs} \\ % &= \sum_\sigma % \hat{a}^\dagger_{p \sigma} % \hat{E}_{rs} % \hat{a}_{q \sigma} % \thinspace . \end{align}\] We can change the pair indices \(pq\) and \(rs\): \[\begin{equation} \hat{e}_{rspq} = \hat{e}_{pqrs} % \end{equation}\] and for the Hermitian adjoint of the singlet two-electron excitation operators, we have \[\begin{equation} \label{eq:e_pqrs_Hermitian_adjoint} \hat{e}^\dagger_{pqrs} = \hat{e}_{qpsr} % \thinspace , \end{equation}\] which is reminiscent of the behavior of complex conjugation of the two-electron integrals (cfr. equation \(\eqref{eq:two_electron_integrals_complex_conjugation}\)). We also have the following important commutator relation: \[\begin{equation} \comm{ % \hat{E}_{pq} % }{ % \hat{e}_{rstu} % } % = \delta_{tq} \hat{e}_{rspu} % - \delta_{pu} \hat{e}_{rstq} % + \delta_{rq} \hat{e}_{pstu} % - \delta_{ps} \hat{e}_{rqtu} % \thinspace , \end{equation}\] and we also have \[\begin{align} & \comm{ % \hat{a}^\dagger_{p \sigma} % }{ % \hat{e}_{qrst} % } % = - \delta_{pr} \hat{a}^\dagger_{q \sigma} \hat{E}_{st} % - \delta_{pt} \hat{a}^\dagger_{s \sigma} \hat{E}_{qr} \\ % & \comm{ % \hat{a}_{p \sigma} % }{ % \hat{e}_{qrst} % } % = \delta_{pq} \hat{E}_{st} \hat{a}_{r \sigma} % + \delta_{ps} \hat{E}_{qr} \hat{a}_{t \sigma} % \thinspace . \end{align}\] The ‘diagonal’ singlet two-electron excitation operator is \[\begin{equation} \hat{e}_{pppp} = 2 \hat{N}_{p \alpha} \hat{N}_{p \beta} % \thinspace , \end{equation}\] and we have the following `trace’ formulas: \[\begin{align} & \sum_{pq}^K \hat{e}_{ppqq} % = \hat{N} (\hat{N} - 1) \\ % & \sum_r^K \hat{e}_{pqrr} % = \hat{E}_{pq} (\hat{N} - 1) \\ % & \sum_{pq}^K % \hat{e}^{\sigma \sigma \tau \tau}_{ppqq} % = \hat{N}_\sigma \hat{N}_\tau % \thinspace , \end{align}\] in which \(\sigma \neq \tau\).