Function uw12::three_el::indirect_3el_energy

Function Documentation

double uw12::three_el::indirect_3el_energy(const integrals::Integrals &W, const integrals::Integrals &V, const ri::ABSProjectors &abs_projectors)

Calculates the indirect three electron energy

The indirect three electron energy is calculated using ABS+RI as:

\[E_c^{3el, -} = \sum_{ij} Y_{ij} = \sum_{ij} \left( \sum_{AB} \tilde{t}_{AB}^{ij} X_{AB}^{ij} \right) \]
for (active) occupied indices \(i,j\), and density-fitting indices \(A,B\), and where
\[\tilde{t}_{AB}^{ij} = \sum_k (\tilde{A}| w_{12} | jk) (ki |r_{12}^{-1}|\tilde{B}) \]
for (complete) occupied indices \(k\), and
\[X_{AB}^{ij} = \sum_{\mu'\nu'} (A| w_{12}| i\mu') [S^{-1}]_{\mu'\nu'} (\nu'j| r_{12}^{-1} |B) \]
with ri indices \(\mu'\nu'\) over the complete ao+ri space.

Calculation of the energy is parallelised over indices \(i, j\).

Parameters

  • W – Integrals for \(w_{12}\)

  • V – Integrals for \(r_{12}^{-1}\)

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

Returns

Indirect three-electron energy