Orbital-Free Density Functional Theory for Periodic Solids: Construction of the Pauli Potential

The practical success of density functional theory (DFT) is largely credited to the Kohn-Sham approach, which enables the exact calculation of the noninteracting electron kinetic energy via an auxiliary noninteracting system. Yet, the realization of DFT's full potential awaits the discovery of a direct link between the electron density, <i>n</i>, and the noninteracting kinetic energy, <i>T</i>_<i>S</i>[<i>n</i>]. In this work, we address two key challenges toward this objective. First, we introduce a new algorithm for directly solving the constrained minimization problem yielding <i>T</i>_<i>S</i>[<i>n</i>] for periodic densities─a class of densities that, in spite of its central importance for materials science, has received limited attention in the literature. Second, we present a numerical procedure that allows us to calculate the functional derivative of <i>T</i>_<i>S</i>[<i>n</i>] with respect to the density at a constant electron number, also known as the Kohn-Sham potential <i>V</i>_<i>S</i>[<i>n</i>](<b>r</b>). Lastly, the algorithm is augmented with a subroutine that computes the "derivative discontinuity", i.e., the spatially uniform jump in <i>V</i>_<i>S</i>[<i>n</i>](<b>r</b>) which occurs upon increasing or decreasing the total number of electrons. This feature allows us to distinguish between "insulating" and "conducting" densities for noninteracting electrons. The code integrates key methodological innovations such as the use of an adaptive basis set ("equidensity orbitals") for wave function expansion and the QR decomposition to accelerate the implementation of the orthogonality constraint. Notably, we derive a closed-form expression for the Pauli potential in one dimension, expressed solely in terms of the input density without relying on Kohn-Sham eigenvalues and eigenfunctions. We validate this method on one-dimensional periodic densities, achieving results within "chemical accuracy".