Solution of the Thomas-Fermi-Dirac Equation

Abstract

THE fundamental equation of the Thomas–Fermi–Dirac statistical atom model1 is the so-called Thomas–Fermi–Dirac equation where ψ″ denotes the second derivative of ψ by x, β represents the constant β = 31/3/(321/3π2/3Z2/3) and Z the atomic number. This equation was solved by Umeda2 for all neutral atoms with the boundary conditions derived by Brillouin3: where x0 corresponds to the border of the atom. The second and the third boundary conditions are equivalent to the minimization of the energy of any single electron in the statistical atom. This requirement and the boundary conditions resulting from it are, however—as has been already mentioned by Jensen4—not in agreement with the statistical point of view, according to which the total energy of the statistical atom has to be a minimum, which is not equivalent to the minimization of the energy of the single electrons.