# SlicedCoulombVertex

## 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 $$a$$:

$$\Gamma^i_{aF}: \varepsilon_i < \varepsilon_\mathrm{F}<\varepsilon_a$$

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: 2023-03-16 Thu 10:59