# CoulombVertex

## Table of Contents

## 1. Brief description

The `CoulombVertex`

\(\Gamma^q_{rF}\)
is a resolution-of-identiy decomposition of the
electron repulsion integrals of the form

where \({\Gamma^\ast}^{pF}_s = \overline{\Gamma^s_{pF}}\) denotes the
conjugate transpose.
For periodic systems it is computed from the Fourier transform of the
co-density of an orbital pair \({\psi^\ast}^q(\mathrm{r})\) and
\(\psi_r(\mathrm{r})\), and the square-root of the Coulomb potential
\(\sqrt{4\pi/\mathrm{G}^2}\), as specified below.
The index order of \(\Gamma^q_{rF}\) in a `Cc4s`

tensor is \((F,q,r)\).

The `CoulombVertex`

needs to be provided by an interfaced electronic
structure theory package
and can be read into `Cc4s`

using the Read algorithm.

- name: Read in: fileName: "CoulombVertex.yaml" out: tensor: CoulombVertex

## 2. Specification

For periodic systems the `CoulombVertex`

\(\Gamma^q_{rF}\) is computed from
the Fourier transform \(\tilde\Gamma^q_{rG}\) of the
right vertex of the Coulomb interaction

where \(\mathbf{G}\) denotes the momentum vector associated to the momentum index \(G\) and where the indices \(q\) and \(r\) refer to the outgoing and incoming states at the right interaction vertex, respectively. Note, that the square root of the numerical integration weight \(w_G\) and of the Coulomb kernel \(4\pi/\mathbf{G}^2\) are contained in the above definition, such that the desired resolution-of-identity factorization of of the electron repulsion integrals holds

\begin{equation} V^{pq}_{sr} = \sum_G {\tilde\Gamma^\ast}^{pG}_s \tilde\Gamma^q_{rG}. \end{equation}
To reduce the size of the employed `CoulombVertex`

while minimizing the
error of the above factorization, the
singular value decomposition of the Fourier transformed
Coulomb interaction vertex
\(\tilde\Gamma^q_{rG} = U^F_G \Sigma^F_F {W^\ast}^q_{rF}\) is
considered.
The `CoulombVertex`

used by `Cc4s`

is finally defined using only
the \(N_\mathrm{F}\) largest singular values \(\Sigma^F_F\) and their respective
left singular vectors \(U^F_G\)

The singular value index \(F\) is called `AuxiliaryField`

index in `Cc4s`

,
as indicated by `type`

field in the `dimensions`

section of the
`CoulombVertex`

tensor specification.
The coefficients \(U_F^G\) are needed by the
`FiniteSizeCorrection`

algorithm
and they are contained in the object
`CoulombVertexSingularVectors`

.
For more details, see (Hummel, Tsatsoulis, and Grüneis 2017).

An example `CoulombVertex.yaml`

file is given below

version: 100 type: Tensor scalarType: Complex64 dimensions: - length: 356 type: AuxiliaryField - length: 96 type: State - length: 96 type: State elements: type: IeeeBinaryFile unit: 0.1917011272153577 # = sqrt(Eh/eV) metaData: halfGrid: 1

## 3. Literature

*The Journal of Chemical Physics*146 (12): 124105. http://doi.org/10.1063/1.4977994.