Complementary Variational Principles in the Thomas-Fermi Theory

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Abstract
The normalization of the electron density associated with approximate variational solutions of the Thomas-Fermi (TF) equation for neutral atoms is discussed, and the applicability of a procedure of Anderson, Arthurs, and Robinson for obtaining the closest approximation is investigated. It is concluded that using single-parameter trial functions, the normalization requirement for the electron density cannot be strictly satisfied when the numerical value of the parameter is determined from the minimization of the difference between the complementary upper and lower bounds to the exact TF energy of a neutral atom. It is also concluded that, using multiple-parameter trial functions, this procedure is applicable since the numerical values of all but one of the parameters can be found from the minimum between the complementary bounds without violating the normalization requirement.

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