Studies of Atomic Self-Consistent Fields. II. Interpolation Problems

By studying the self-consistent-field (SCF) functions tabulated in the literature, it is shown that a quantity, which for a hydrogenlike function would equal the atomic number $Z$, for SCF functions will represent a form of "effective nuclear charge" $Z_{\mathrm{eff}}$, which for a series of consecutive atoms or ions appears to be an almost linear function of the atomic number, convenient for interpolation purposes. The positions and magnitudes of maxima and minima, nodes, derivatives in origin, eigenvalues, etc., are investigated by considering the corresponding effective nuclear charges. It is important that the nodes may be interpolated separately, since the logarithm of an SCF function, divided by a polynomial having the same zero points as the function itself, may be used for interpolating SCF functions as a whole. This logarithm may also be used for defining a continuously varying effective charge $Z_{\mathrm{eff}}\mathrm{}\left(r\right)\mathrm{}$, which is convenient for interpolation purposes or for estimating fields with exchange from those without exchange.