Convergence stability and estimator in orbital free electronic structure calculation on a grid at finite temperature

The electronic charge density is the central quantity in density functional theory. It can be expressed as the diagonal elements of the density matrix in its real space representation, which can be computed by a recursion method based on the Trotter formula. This allows for an orbital free computation of the charge density in the Kohn Sham formalism at finite temperature whose numeric complexity increases linearly with the size of the system (that is, an order N method).No assessment of the numerical properties such an approach has been presented yet, and this paper aims to analyze its convergence properties.In particular, we wish to understand its convergence properties as a function of the number of recursion steps, to establish an error estimate in order to devise a stopping criterion, and to analyze the “locality properties” of the method which is necessary to make it an order N method. We illustrate the assessment with numerical tests performed for the free electron gas and for Helium at a density and high temperature relevant to shock physics.