APIG PSEs

As the seniority numbers of a fully paired Slater determinant \(\require{physics} \ket{\Phi_0}\) and a single pair excited Slater determinant \(\ket{\Phi_i^a}\) is zero, \[\begin{align} & \hat{\Omega} \ket{\Phi_0} = 0 \\ & \hat{\Omega} \ket{\Phi_i^a} = 0 \thinspace , \end{align}\] and a general APIG wave function also has seniority zero (cfr. equation \(\eqref{eq:seniority_zero_APIG}\)), we can use the DOCI Hamiltonian \(\eqref{eq:h_doci}\) in a pSE (projected Schrödinger equation) approach. Using a pSE approach, we require the projected Schrödinger equation to be fulfilled for the subspace of a reference determinant \(\ket{\Phi_0}\) and its double excitations \(\set{\ket{\Phi_i^a}}\): \[\begin{align} & \matrixel{\Phi_0}{\hat{\mathcal{H}}_\text{DOCI}}{\text{APIG}} = E \braket{\Phi_0}{\text{APIG}} \\ & \matrixel{\Phi_i^a}{\hat{\mathcal{H}}_\text{DOCI}}{\text{APIG}} = E \braket{\Phi_i^a}{\text{APIG}} \thinspace . \end{align}\] Using \(\eqref{eq:h_doci}\) on the bra, we find after some rearrangements: \[\begin{equation} \label{eq:PSE_APIG} \begin{split} & g_{ai} \braket{\Phi_0}{\text{APIG}}^2 - g_{ia} \braket{\Phi_i^a}{\text{APIG}}^2 \\ & + \qty[ 2 (h_{aa} - h_{ii}) + (g_{aa} - g_{ii}) + \sum_{j \neq i}^{N_P} (\gamma_{aj} + \gamma_{ja} - \gamma_{ij} - \gamma_{ji} ) ] \braket{\Phi_0}{\text{APIG}} \braket{\Phi_i^a}{\text{APIG}} \\ & + \sum_{j \neq i}^{N_P} \Big(g_{ji} \braket{\Phi_0}{\text{APIG}} - g_{ja} \braket{\Phi_i^a}{\text{APIG}} \Big) \braket{\Phi_j^a}{\text{APIG}} \\ & + \sum_{b = N_P + 1, b \neq a}^K \Big( g_{ab} \braket{\Phi_0}{\text{APIG}} - g_{ib} \braket{\Phi_i^a}{\text{APIG}} \Big) \braket{\Phi_i^b}{\text{APIG}} \\ & + \sum_{j \neq i}^{N_P} \sum_{b = N_P + 1, b \neq a}^K g_{jb} \Big( \braket{\Phi_0}{\text{APIG}} \braket{\Phi_{ij}^{ab}}{\text{APIG}} - \braket{\Phi_i^a}{\text{APIG}} \braket{\Phi_j^b}{\text{APIG}} \Big) = 0 \thinspace . \end{split} \end{equation}\] For real orbitals, we have some minor simplifications: \[\begin{equation} \label{eq:PSE_APIG_real} \begin{split} & g_{ai} (\braket{\Phi_0}{\text{APIG}}^2 - \braket{\Phi_i^a}{\text{APIG}}^2) \\ & + \qty[ 2 (h_{aa} - h_{ii}) + (g_{aa} - g_{ii}) + 2 \sum_{j \neq i}^{N_P} (\gamma_{aj} - \gamma_{ij}) ] \braket{\Phi_0}{\text{APIG}} \braket{\Phi_i^a}{\text{APIG}} \\ & + \sum_{j \neq i}^{N_P} \Big(g_{ji} \braket{\Phi_0}{\text{APIG}} - g_{ja} \braket{\Phi_i^a}{\text{APIG}} \Big) \braket{\Phi_j^a}{\text{APIG}} \\ & + \sum_{b = N_P + 1, b \neq a}^K \Big( g_{ab} \braket{\Phi_0}{\text{APIG}} - g_{ib} \braket{\Phi_i^a}{\text{APIG}} \Big) \braket{\Phi_i^b}{\text{APIG}} \\ & + \sum_{j \neq i}^{N_P} \sum_{b = N_P + 1, b \neq a}^K g_{jb} \Big( \braket{\Phi_0}{\text{APIG}} \braket{\Phi_{ij}^{ab}}{\text{APIG}} - \braket{\Phi_i^a}{\text{APIG}} \braket{\Phi_j^b}{\text{APIG}} \Big) = 0 \thinspace . \end{split} \end{equation}\]

Unfortunately, by using this projection set, we have only solved \(N_P \times (K - N_P)\) equations, but there are \(N_P \times K\) geminal coefficients to be determined. We will later enlarge our projection set by including all singly-excited determinants as well.

Anyways, once the geminal coefficients are determined, we can calculate the energy (for real orbitals) as \[\begin{equation} E = \sum_i^{N_P} \qty( h_{ii} + \sum_j^{N_P} \gamma_{ij} + \sum_{a = N_P + 1}^K g_{ia} \frac{\braket{\Phi_i^a}{\text{APIG}}}{\braket{\Phi_0}{\text{APIG}}} ) \thinspace . \end{equation}\]