Expectation values over Fockian operators

In a general spinor basis

Given a wave function \(\require{physics} \ket{\Psi}\), we define the Fockian matrix of the operator \(\hat{O}\) as the expectation value of the Fockian operator: \[ \require{physics} \begin{equation} \mathscr{F}_{PQ}( \hat{O} ) = \ev{ \hat{\mathscr{F}}_{PQ}( \hat{O} ) }{\Psi} \thinspace . \end{equation} \] The Fockian matrix for a one-electron operator \(\hat{f}\) becomes: \[\begin{equation} \mathscr{F}_{PQ}( \hat{f} ) = \sum_{R}^M f_{QR} D_{PR} \end{equation}\] and for a two-electron operator \(\hat{g}\) (including the prefactor \(1/2\)) \[\begin{equation} \mathscr{F}_{PQ}( \hat{g} ) = \sum_{RST}^M g_{QRST} d_{PRST} \thinspace , \end{equation}\] in which \(D_{PQ}\) and \(d_{PQRS}\) are elements of the one- and two-electron density matrices. The Fockian matrix of the Hamiltonian is accordingly called the Fock matrix:

\[\begin{equation} \mathscr{F}_{PQ}( \hat{\mathcal{H}} ) = \sum_{R}^M h_{QR} D_{PR} + \sum_{RST}^M g_{QRST} d_{PRST} \end{equation}\]

The super-Fockian matrix of an operator \(\hat{O}\) is then defined as the expectation value of the super-Fockian of \(\hat{O}\): \[\begin{equation} \mathscr{G}_{PQRS}( \hat{O} ) = \ev{ \hat{\mathscr{G}}_{PQRS}( \hat{O} ) }{\Psi} \thinspace . \end{equation}\] The super-Fockian matrix for a one-electron operator \(\hat{f}\) becomes: \[\begin{equation} \mathscr{G}_{PQRS}( \hat{f} ) = \delta_{QR} \mathscr{F}_{PS}( \hat{f} ) - f_{QR} D_{PS} \end{equation}\] and for a two-electron operator \(\hat{g}\) (again, including the prefactor \(1/2\)): \[\begin{equation} \mathscr{G}_{PQRS}( \hat{g} ) = \delta_{QR} \mathscr{F}_{PS}( \hat{g} ) + \mathcal{Z}_{PQRS}( \hat{g} ) \thinspace , \end{equation}\] with the auxiliary tensor \(\mathcal{Z}_{PQRS}( \hat{g} )\): \[\begin{equation} \mathcal{Z}_{PQRS}( \hat{g} ) = \sum_{TU}^M ( g_{QTSU} d_{PTRU} - g_{QRTU} d_{PSTU} - g_{QTUR} d_{PTUS} ) \thinspace . \end{equation}\]

If the wave function \(\ket{\Psi}\) is real, the tensor \(\mathcal{Z}_{PQRS}( \hat{g} )\) has the following symmetry: \[\begin{equation} \mathcal{Z}_{PQRS} = \mathcal{Z}_{SRQP} \thinspace . \end{equation}\]

The super-Fockian matrix of the Hamiltonian is given by \[\begin{equation} \mathscr{G}_{PQRS}( \hat{\mathcal{H}} ) = \delta_{QR} \mathscr{F}_{PS}( \hat{\mathcal{H}} ) - h_{QR} D_{PS} + \mathcal{Z}_{PQRS}( \hat{g} ) \thinspace . \end{equation}\]

In a restricted spin-orbital basis

Given a wave function \(\ket{\Psi}\), we define the Fockian matrix as the expectation value of the Fockian operator: \[\begin{equation} \mathscr{F}_{pq}(\hat{O}) = \ev{ \hat{\mathscr{F}}_{pq}(\hat{O}) }{\Psi} \thinspace , \end{equation}\] The Fock matrix for a one-electron operator \(\hat{f}\) becomes: \[\begin{equation} \mathscr{F}_{pq}(\hat{f}) = \sum_{r}^K f_{qr} D_{pr} \end{equation}\] and for a two-electron operator \(\hat{g}\) \[\begin{equation} \mathscr{F}_{pq}(\hat{g}) = \sum_{rst}^K g_{qrst} d_{prst} \thinspace , \end{equation}\] in which \(D_{pq}\) and \(d_{pqrs}\) are elements of the one- and two-electron density matrices. For a sum of one- and two-electron operators, such as the Hamiltonian \(\hat{\mathcal{H}}\), we have: \[\begin{equation} \mathscr{F}_{pq}(\hat{\mathcal{H}}) = \sum_{r}^K h_{qr} D_{pr} + \sum_{rst}^K g_{qrst} d_{prst} \thinspace . \end{equation}\]

The super-Fockian matrix of the operator \(\hat{O}\) can then be calculated as \[\begin{equation} \mathscr{G}_{pqrs}(\hat{O}) = \ev{ \hat{\mathscr{G}}_{pqrs}(\hat{O}) }{\Psi} \thinspace . \end{equation}\] For a one-electron operator \(\hat{f}\), we find:: \[\begin{equation} \mathscr{G}_{pqrs}(\hat{f}) = \delta_{qr} \mathscr{F}_{ps}(\hat{f}) - f_{qr} D_{ps} \end{equation}\] and for a two-electron operator \(\hat{g}\) \[\begin{equation} \mathscr{G}_{pqrs}(\hat{g}) = \delta_{qr} \mathscr{F}_{ps}(\hat{g}) + \mathcal{Z}_{pqrs}(\hat{g}) \thinspace . \end{equation}\] For a sum of one- and two-electron operators, such as the Hamiltonian \(\hat{\mathcal{H}}\), we have: \[\begin{equation} \mathscr{G}_{pqrs}(\hat{\mathcal{H}}) = \delta_{qr} \mathscr{F}_{ps}(\hat{\mathcal{H}}) - h_{qr} D_{ps} + \mathcal{Z}_{pqrs}(\hat{g}) \thinspace , \end{equation}\] with \[\begin{equation} \mathcal{Z}_{pqrs}(\hat{g}) = \sum_{tu}^K ( g_{qtsu} d_{ptru} - g_{qrtu} d_{pstu} - g_{qtur} d_{ptus} ) \thinspace . \end{equation}\] If the two-electron operator \(\hat{g}\) is Hermitian, and the wave function \(\ket{\Psi}\) is real, the tensor \(\mathcal{Z}_{pqrs}(\hat{g})\) has the following symmetry: \[\begin{equation} \mathcal{Z}_{pqrs} = \mathcal{Z}_{srqp} \thinspace . \end{equation}\]