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 non-interacting 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, $n$, and the non-interacting kinetic energy, $T_{S}[n]$. In this work, we address two key challenges towards this objective. First, we introduce a new algorithm for directly solving the constrained minimization problem yielding $T_{S}[n]$ 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 $T_{S}[n]$ with respect to the density at constant electron number, also known as the Kohn-Sham potential $V_{S}[n](\rv)$. Lastly, the algorithm is augmented with a subroutine that computes the ``derivative discontinuity", i.e., the spatially uniform jump in $V_{S}[n](\rv)$ which occurs upon increasing or decreasing the total number of electrons. This feature allows us to distinguish between ``insulating" and ``conducting" densities for non interacting 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".