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The optimum scale factor for the Slater free-electron exchange approximation for atoms has been obtained by minimizing the total electronic energy. In general, the scale factor lies between the Slater and the Gaspar-Kohn-Sham limit of 1 and 23, respectively. The variation that occurs as the different shells are populated is qualitatively understandable in terms of the inhomogeneity of the over-all charge distribution.
E.A. Kmetko (Thu,) studied this question.