Studies of Atomic Self-Consistent Fields. Analytic Wave Functions for the Argon-Like Ions and for the First Row of the Transition Metals

Abstract

An improved method for calculating analytic approximations to the self-consistent-field (SCF) functions is developed. The numerically given function is divided by a polynomial having the same nodes as the function, and the monotonically decreasing quotient is then expressed as a sum of exponentials. An exact method for expanding an arbitrary function into exponentials is discussed, but the results turn out to be extremely sensitive even to rounding-off errors. For practical purposes a method of successive approximations is used instead; it works inwards from the tail towards the origin and, in each step, two exponentials are simultaneously determined. Applications are carried out to the argon-like ions (${\mathrm{Cl}}^{-}$, A, ${\mathrm{K}}^{+}$, ${\mathrm{Ca}}^{+2\mathrm{}}$) and to the copper ion. By expressing a SCF function $f_{\mathrm{nl}}\mathrm{}\left(r\right)\mathrm{}$ in terms of ($n+1\mathrm{}$) exponentials, the error may be brought below the order of magnitude 0.001-0.002. By using the data available for ${\mathrm{Ca}}^{+}$ and ${\mathrm{Cu}}^{+}$, analytic wave functions have finally been interpolated for the first row of the transition metals (Sc, Ti, V, Cr, Mn, Fe, Co, Ni); the degree of accuracy is discussed.