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.