# CoulombIntegrals

## 1.Brief description

$V^{pq}_{sr} = \sum_{G} {\Gamma^\ast}^{pG}_s \Gamma^q_{rG}$

CoulombIntegrals $$V^{pq}_{sr}$$ are computed from the CoulombVertex, which can be respresented in an arbitrary basis set depending on the employed interface (Hummel, Tsatsoulis, and Grüneis 2017).

CoulombIntegrals are computed from the CoulombVertex using the VertexCoulombIntegrals algorithm using; for example, the following yaml input file.

- name: VertexCoulombIntegrals
in:
slicedCoulombVertex: CoulombVertex
out:
coulombIntegrals: CoulombIntegrals


We note that the VertexCoulombIntegrals algorithm of Cc4s provides a recipe for the computation of a set of CoulombIntegrals $$\{V^{ab}_{ij}, V^{ai}_{bj}, V^{ab}_{cd} ... \}$$ . The respective numerical evaluation of these integrals is performed only if the numerical values of the tensors are needed by an algorithm.

## 2.Literature

Hummel, Felix, Theodoros Tsatsoulis, and Andreas Grüneis. 2017. “Low Rank Factorization of the Coulomb Integrals for Periodic Coupled Cluster Theory.” The Journal of Chemical Physics 146 (12): 124105. http://doi.org/10.1063/1.4977994.

Created: 2022-09-19 Mon 15:00