Semi-universal solution of the Thomas–Fermi equation for neutral jellium spheres

The Thomas-Fermi model of electronic structures provides a universal solution for neutral atoms. As a result, the energy of a neutral atom with nuclear charge Z is given as -0.7687Z7/3. Are there other systems where such universal solutions can be found? In this work, we look for an answer to this question and discover that for neutral jellium spheres, a semi-universal solution of the Thomas-Fermi equation exists; one parameter, f = N2/3rs, determines the properties of many neutral jellium spheres with different numbers of electrons (N) and bulk density parameters (rs). This parameter is connected to an earlier approximate analytical solution given in the context of simple metal clusters. The Thomas-Fermi energy of a neutral jellium sphere is found to be a simple function of f and N, of a form similar to that for atoms. Additional examples of semi-universal solutions are given by generalizing our method to other potentials. The present work, therefore, provides a motivation to explore whether such universal forms can be found in more accurate orbital-free density functional theories.