Approximate Variational Solution of the Thomas-Fermi Equation for Atoms

An approximate solution of the Thomas-Fermi (TF) differential equation is obtained by making use of an equivalent variational principle. The trial solution, depending on several parameters, is chosen in such a way that it satisfies the boundary conditions imposed on the TF equation together with the subsidiary condition that the electron density be normalized. The numerical values of the parameters are determined by extremalizing the variational expression with respect to the parameters. Using the approximate solution, one finds that at large distances from the nucleus, the radial electron density decreases exponentially, as required by quantum mechanics---in contrast to the original TF theory, where the above quantity decreases as the inverse fourth power of the distance from the nucleus. The approximate TF function is then used for calculating the energy necessary to remove all electrons of an atom and for calculating the interaction energies between atoms in the Firsov approximation. In the former case the improvement upon the original TF theory is found to be substantial, and in the latter the interaction energies closely approximate Abrahamson's interaction energies based on the Thomas-Fermi-Dirac model.