# Coulomb operator

The Coulomb operator, named after Charles-Augustin de Coulomb, is a quantum mechanical operator used in the field of quantum chemistry. Specifically, it is a term found in the Fock operator. It is defined as:[1]

${\displaystyle {\widehat {J}}_{j}(1)f_{i}(1)=f_{i}(1)\int {\left|\varphi _{j}(2)\right|}^{2}{\frac {1}{r_{12}}}\,dr_{2}}$

where

${\displaystyle {\widehat {J}}_{j}(1)}$ is the one-electron Coulomb operator defining the repulsion resulting from electron j,

${\displaystyle f_{i}(1)}$ is the one-electron wavefunction of the ${\displaystyle i^{th}}$ electron being acted upon by the Coulomb operator,

${\displaystyle \varphi _{j}(1)}$ is the one-electron wavefunction of the ${\displaystyle j^{th}}$ electron,

${\displaystyle r_{ij}}$ is the distance between electrons ${\displaystyle (i)}$ and ${\displaystyle (j)}$.