An Analytical Method for the Approximate Determination of the Eigenfunctions and Energies of Electrons in Atoms

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Abstract
The first part of this paper shows that the reduced effective nuclear charges Zp/Z, determined without exchange by the ``self‐consistent field'' method, can be approximated by a universal function, if the distance from the nucleus is measured in the unit of length μ=0.8853a0/Z⅓, which changes from atom to atom. This universal function can also be approximated by a simple analytical expression. One of the possible applications of the above result is to use it to determine good starting potentials for calculations to be carried out by the ``self‐consistent field'' method. The universal effective nuclear charge, especially in its analytical form, also allows the determination of important atomic constants for each element of the periodic system. The second part of the paper solves the one‐electron Schrödinger equation by Rasetti's method, developing it to determine analytically the eigenfunctions and energies of electrons in the deepest s, p, d, and f energy states. The x‐ray terms obtained by this method and the experimental ones are in good agreement with each other and with those calculated by the ``self‐consistent field'' method.

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