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The Thomas-Fermi method is applied to metals, by replacing each atom by a sphere, assuming the potential to be spherically symmetrical within it, and solving the Thomas-Fermi equation subject to the boundary condition that the electronic charge within the sphere shall balance the nuclear charge, rendering it electrically neutral. Calculations are presented giving potential field, charge density, and kinetic, potential, and total energy of the metal, as function of lattice spacing. The virial theorem is verified for the energy. The total energy shows no minimum, the pressure being always positive. Calculations are also made using the Dirac method of correcting for exchange, for three atoms, Li, Na and Cu. The exchange lowers the energy, but still not quite enough to produce a minimum of energy and an equilibrium at zero pressure. The result should be useful as a first approximation in self-consistent field approximations for the structure of metals, and could be adapted to give approximate treatment for matter under very high pressure, as in stars.
Slater et al. (Mon,) studied this question.