Simple, Analytic, Atomic Wave Functions

Abstract

A method is presented for the determination, with relatively small effort, of analytic one-electron wave functions for an electron in an approximately self-consistent screened Coulomb field. The method enables one to avoid direct numerical solution of the Schr\"odinger equation. Instead, for each state one has to adjust two parameters to make a certain expression for the screening function reasonably approximate that which follows from the Thomas-Fermi ion for the atom in question. When this is done, one has automatically a simple, analytic expression for the eigenfunction and the corresponding energy eigenvalue for that state. In particular, pairs of parameters have been obtained for states of several alkali atoms, including all of the normally filled states of sodium, the $6s$ state of potassium and most of the important excited states of cesium. The analytic wave functions thus obtained are simple enough so that the expectation values of many quantities can be easily calculated by analytic methods, a few of which are illustrated. Thus, the normalization constants are calculated for three of the states of cesium. Moreover, calculations for the fine structure and hyperfine structure energy splittings of cesium are carried out as a test of the goodness of the wave functions. Good agreement is obtained with experiment and with previous calculations.