  The numerical implementation of an exchange-correlation functional is not
always an accurate reproduction of its theoretical specification. For example,
density functionals for exchange and correlation can be defined by an exchange
or correlation hole function that is integrated with the Coulomb interaction to
form an energy. This construction can be used to modify a density functional
for use with any electron-electron interaction. Its most prominent use is in
the Heyd-Scuseria-Ernzerhof (HSE) functional that generalizes the
Perdew-Burke-Ernzerhof (PBE) model of exchange to a screened Coulomb
interaction with an error function form. However, we find non-negligible
numerical errors in the standard implementation of the HSE exchange hole
integration. We formulate and implement a new method for evaluating the
exchange hole integration that is simple, accurate, and efficient. Its
numerical errors are bounded and minimized by applying basic elements of
approximation theory.
