On optimization procedures based upon the atomic thomas-fermi energy

For the total atomic Thomas-Fermi (TF) energy many expressions in terms of the kinetic and potential energy contributions can be given. Thirty of these expressions exhibit either a maximum or a minimum if some variational approximation to the TF function is used. Three of these expressions, to note E, G, and J (see text) have been used in an optimization procedure, in which four two-parameter and two three-parameter approximate TF functions have been considered. One-parameter functions cannot be optimized, as the one parameter must be fixed to ensure proper normalization. It is found that optimization of E and G give reasonable and similar results, whereas the results of optimization of J are generally not very impressive. Where possible, expectation values of the type 〈rⁿ〉 (with n = ?1, 1, 2, and 3) have been calculated from ten approximate TF functions. A new estimate of the exact atomic TF energy, as well as of the derivative of the TF function at the origin, has been obtained.