The inactive Fock matrix
Inspired by form of the (G)HF Fock matrix, we will define the inactive Fock matrix as \[\begin{equation} \require{physics} {}^{i} F_{PQ} = h_{PQ} + \sum_I^N ( g_{PQII} - g_{PIIQ} ) \thinspace . \end{equation}\] Using the antisymmetrized integrals \(\tilde{g}_{PQRS}\), the inactive Fock matrix can be simplified to: \[\begin{equation} {}^{i} F_{PQ} = h_{PQ} + \sum_I^N \tilde{g}_{PQII} \thinspace . \end{equation}\]
An analogous definition for the RHF inactive Fock matrix can be found in (T. Helgaker, Jørgensen, and Olsen 2000). We can then express the GHF Fock matrix in terms of the inactive Fock matrix: \[\begin{equation} {}^{i} F_{PI} = \mathscr{F}_{IP} ( \hat{\mathcal{H}} ) \thinspace , \end{equation}\] but we must keep in mind that the indices in the inactive Fock matrix yield non-zero elements over all indices, whereas the GHF Fockian matrix vanishes for a virtual first index.
Due to the symmetry of the one- and two-electron integrals of the Hamiltonian, the inactive Fock matrix is Hermitian: \[\begin{equation} {}^{i} F_{PQ}^* = {}^{i} F_{QP} \thinspace , \end{equation}\] which is a relation that does not hold for the general Fock matrix.
In RHF theory, we can also define a similar inactive Fock matrix: \[\begin{equation} \label{eq:inactive_Fock_matrix} {}^{\text{i}} F_{pq} = h_{pq} + \sum_i (2 g_{pqii} - g_{piiq}) \thinspace , \end{equation}\] which is also Hermitian: \[\begin{equation} {}^{i} F_{pq}^* = {}^{i} F_{qp} \thinspace . \end{equation}\]