Basis set error in finite temperature electronic structure calculations: observations, considerations, and future goals

We investigate the basis set incompleteness error (BSIE) for interacting electrons at finite temperature in the canonical ensemble. Finite temperature electronic structure calculations were conducted for the uniform electron gas (UEG), an analytical hydrogen atom, and helium in a periodic box. By exploring the components of the canonical ensemble variational free energy, we can define a correlation free energy analogous to the ground state correlation energy. As this has monotonic convergence it may help in reducing BSIE. At finite temperature, the kinetic energy dominates until the error converges exponentially for a temperature dependent basis set size M, while for larger M the potential energy error dominates showing ground-state like convergence. We also find that the entropic term in the free energy ensures that the free energy remains variational even when the kinetic energy term has not achieved exponential convergence. Comparisons are made between the analytical hydrogen atom and the helium atom in a periodic box, and it is found that the helium atom does not reproduce the analytical hydrogen atom divergence and behaves like the UEG. We close with brief observations about the basis set dependence for the momentum distribution and static structure factor for the uniform electron gas.