Studies of Atomic Self-Consistent Fields. I. Calculation of Slater Functions

Abstract

A refined technique is described for approximating the numerically given radial part of atomic wave functions associated with self-consistent fields with exchange by means of Slater's analytical functions obtained by replacing each exponential in a hydrogen-like wave function by the sum of one, two, three, or more exponentials. Exponents and coefficients of these exponentials are calculated for the $3p$-function of ${\mathrm{Cl}}^{-}$, corresponding to an accuracy of 0.0015 for the normalized radial part, and, with slightly less accuracy, for all the functions of two closed-shell ions, ${\mathrm{F}}^{-}$ (without exchange) and ${\mathrm{Na}}^{+}$, and for some neutral first-row atoms, $\mathrm{C}\left({}_{}{}^1{}_{}{}^{}D\right)$, $\mathrm{N}\left({}_{}{}^2{}_{}{}^{}P\right)$, and $\mathrm{O}\left({}_{}{}^1{}_{}{}^{}S\right)$. The interpolation problem is discussed, and a new interpolation rule for the coefficients is stated, which gives excellent agreement (0.001) in the examples chosen, namely the $1s$-functions of the He-like ions and the $2p$-functions of ${\mathrm{Na}}^{+}$, ${\mathrm{Mg}}^{+2\mathrm{}}$, and ${\mathrm{Si}}^{+4\mathrm{}}$.