Function uw12::three_el::indirect_3el_fock_matrix

Function Documentation

linalg::Mat uw12::three_el::indirect_3el_fock_matrix(const integrals::Integrals &W, const integrals::Integrals &V, const ri::ABSProjectors &abs_projectors, size_t sigma)

Calculates the indirect fock matrix contribution

The indirect fock matrix contribution

\[F_{\alpha\beta}^{\sigma} = \frac{d E_c^{3el,-}}{d D_{\alpha\beta}^{\sigma}} \]
has three contributions from the three occupied orbital indices \(i,j,k\). These are given by:
\[\begin{align*}f_{\alpha\beta}^{\sigma} (i) = \sum_{jk} \sum_{\mu'\nu'} (jk|w_{12}|\alpha \mu') [S^{-1}]_{\mu'\nu'} (\nu' j| r_{12}^{-1}|k\beta) \newline f_{\alpha\beta}^{\sigma} (j) = \sum_{ik} \sum_{\mu'\nu'} (\alpha k|w_{12}|i \mu') [S^{-1}]_{\mu'\nu'} (\nu' \beta| r_{12}^{-1}|ki) \newline f_{\alpha\beta}^{\sigma} (k) = \sum_{ij} \sum_{\mu'\nu'} (j\beta|w_{12}|i \mu') [S^{-1}]_{\mu'\nu'} (\nu' j| r_{12}^{-1}|\alpha i) \end{align*}\]
Contributions are parallelised over the occupied indices.

Parameters

  • W – Integrals for \(w_{12}\)

  • V – Integrals for \(r_{12}^{-1}\)

  • abs_projectors – RI projectors \(S^{-1}\)

  • sigma – Spin index

Returns

Unsymmetrised indirect fock contribution for single spin channel