Alternative to the Kohn-Sham equations: The Pauli potential differential equation

A recently developed theoretical framework of performing self-consistent orbital-free (OF) density functional theory (DFT) calculations at Kohn-Sham DFT level accuracy is tested in practice. The framework is valid for spherically symmetric systems. Numerical results for the Beryllium atom are presented and compared to accurate Kohn-Sham data. These calculations make use of a differential equation that we have developed for the so called Pauli potential, a key quantity in OF-DFT. The Pauli potential differential equation and the OF Euler equation form a system of two coupled differential equations, which have to be solved simultaneously within the DFT self-consistent loop.

Figure 1

Top: Converged radial electron density $\rho$ (solid line) and Pauli potential $v_{\text{p}}$ (dashed line) of the Be atom, obtained using the Pauli potential differential equation (PPDE). Kohn-Sham (KS) results are also presented for comparison (circles and diamonds). Middle and bottom: Difference between the PPDE and the KS-DFT for $\rho$ (middle) and $v_{\text{p}}$ (bottom) using eight values for the PPDE self-consistent loop error tolerance in the range $1.0\times {10}^{-i},i=\{1,\cdots ,8\}$. Solid black curve, $i=1$; dashed colored curves, $i=2,\cdots ,7$; black circles, $i=8$.