Approximate Analytical Solutions of the Thomas–Fermi–Dirac and Thomas–Fermi–Dirac–Gombás Equations

Abstract

Approximate analytical solutions of the Thomas–Fermi–Dirac (TFD) and Thomas–Fermi–Dirac–Gombás (TFDG) differential equations are obtained for neutral and positively charged atoms. The approximate solutions are of the form ψ = φ0 + η and Ψ = φ0 + η̃, respectively, where φ0 is a universal approximate analytical solution of the Thomas–Fermi (TF) differential equation for neutral atoms and η and η̃ are correction functions. The function φ0 has been obtained previously, by making use of a variational principle (the equivalent of the TF equation), and the functions η and η̃ are determined by solving second-order inhomogeneous linear differential equations with the boundary and subsidiary (normalization) conditions that are required in the TFD and TFDG models, respectively. The correction functions η and η̃ depend on the atomic or ionic radius, which, in turn, depends on the atomic number Z and electron number N of the system considered. The approximate solutions ψ and Ψ are tested by calculating the first ionization energies of Ar, Kr, and Xe. The values obtained are about the same as those calculated from the exact numerical solution of the TFD and TFDG equations. In three Appendices approaches for improved approximate analytical solutions of the TFD and TFDG equations are outlined. Appendix A proposes an iterative procedure for obtaining the correction functions η and η̃ in successively more refined approximations, which is expected to be of importance for multiply charged ions or for neutral atoms of low atomic number. Appendix B establishes a variational principle (which to the author's knowledge has not been formulated before) for obtaining approximate analytical solutions of the TFD and TFDG equations, or for achieving further refinements in the approximate solutions ψ and Ψ that are derived in this paper. Appendix C describes Jensen's modification of the TFD and TFDG models, which, with the technique developed in this paper, enables one to obtain approximate analytical solutions for negative ions.