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Accurate numerical integration of an electron exchange hole with a screened Coulomb interaction

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.