SlicedCoulombVertex

Table of Contents

1. Brief description

SlicedCoulombVertex is a set of Coulomb vertices, sliced into particle and hole index channels: \(\{ \Gamma^i_{jF}, \Gamma_i^{aF}, \Gamma_a^{iF}, \Gamma_a^{bF}\}\). The SlicedCoulombVertex is used to compute the different types of two-electron integrals, such as \(V_{ij}^{ab}=\sum_F{\Gamma^\ast}^{aF}_i\Gamma^b_{jF}\). It can be generated from the entire CoulombVertex \(\Gamma^q_{rF}\) using the SliceOperator algorithm:

- name: SliceOperator
  in:
    slicedEigenEnergies: EigenEnergies
    operator: CoulombVertex
  out:
    slicedOperator: SlicedCoulombVertex

2. Specification

The hole-particle slice \(\Gamma^i_{aF}\) is for instance found by restricting the outgoing (upstairs) states of the entire Coulomb Vertex to hole states \(i\) while restricting the incoming (downstairs) states to particle states \(a\):

\begin{equation} \Gamma^i_{aF}: \varepsilon_i < \varepsilon_\mathrm{F}<\varepsilon_a \end{equation}

The index order in cc4s is identical to the index order of the entire Coulomb Vertex. For the above example it is Gamma[Fia]. Note that all indices start with 0, thus the particle indices \(a=0\ldots N_\mathrm{v}-1\) refer to the state indices \(p=N_\mathrm{p}-N_\mathrm{v} \ldots N_\mathrm{p}-1\), where \(N_\mathrm{p}\) denotes the total number of closed-shell orbitals and \(N_\mathrm{v}\) denotes the number of virtual orbitals.

3. Literature

Created: 2022-09-19 Mon 15:00