Two-electron excitation operators

Taking into account the compound effect of two annihilations and two subsequent creations, we can define \(\hat{e}_{PQRS}\) as a two-electron excitation operator: \[\begin{equation} \hat{e}_{PQRS} = \hat{a}^\dagger_P \hat{a}^\dagger_R \hat{a}_S \hat{a}_Q \thinspace . \end{equation}\] Using the elementary anticommutation relations for fermions, we can alternatively write: \[\begin{equation} \hat{e}_{PQRS} = \hat{E}_{PQ} \hat{E}_{RS} - \delta_{QR} \hat{E}_{PS} \thinspace . \end{equation}\]

Due to the nature of the two-electron excitation operators, they possess the following permutational symmetry: \[\begin{equation} \hat{e}_{PQRS} = -\hat{e}_{RQPS} = \hat{e}_{RSPQ} = -\hat{e}_{PSRQ} \end{equation}\] and the Hermitian adjoint of a two-electron excitation operator is \[\begin{equation} \hat{e}^\dagger_{PQRS} = \hat{e}_{QPSR} \thinspace . \end{equation}\]