Constructing and representing exchange-correlation holes through artificial neural networks.

One strategy to construct approximations to the exchange-correlation (XC) energy E of Kohn-Sham density functional theory relies on physical constraints satisfied by the XC hole ρ(r, u). In the XC hole, the reference charge is located at r and u is the electron-electron separation. With mathematical intuition, a given set of physical constraints can be expressed in a formula, yielding an approximation to ρ(r, u) and the corresponding E. Here, we adapt machine learning algorithms to partially automate the construction of X and XC holes. While machine learning usually relies on finding patterns in datasets and does not require physical insight, we focus entirely on the latter and develop a tool (ExMachina), consisting of the basic equations and their implementation, for the machine generation of approximations. To illustrate ExMachina, we apply it to calculate various model holes and show how to go beyond existing approximations.