Analysis of Thomas-Fermi Energy Functional Under Two Potentials

Abstract

In this paper, by constructing Free-energy Functionals, the Thomas-Fermi theory has been extended to include the non-zero temperature effects in many-particle systems. Using the Sobolev-Lieb and the HÅ‘lder inequalities, the constructed Free-energy Functionals were put into a form from which an extended Thomas-Fermi equation was derived. Hitherto, in this work, the two states Ψ₀(r) and Ψ₁(r), corresponding respectively to the square root of two densities r₀(r) and r₁(r), had been used to construct the new free-energy Functionals F[r₀(r)] and F[r₁(r)]. The states Ψ₀(r) and Ψ₁(r) were required to be mutually orthogonal, and the functional F[r₀(r)]  was considered as the ground state functional while F[r₁(r)] was the excited state functional with temperature 0.  From the functionals, the electron density matrix was derived and finally the total energy was computed for many-electrons system under the influence of the Coulomb and Yukawa Potentials. Various results were obtained and discussed.