# Generalized Unsöld Theorem for Hydrogenic Functions

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According to Unsöld's theorem (see the Demonstration "Unsöld's Theorem"), the sum over all states for an -subshell of hydrogen-like orbitals reduces to a spherically symmetrical function: . For a pure Coulomb potential with any nuclear charge , the different states for a given are also degenerate. The author has derived a generalization of Unsöld's theorem, an explicit form for the sum over both and for hydrogenic orbitals, namely, , where is a Whittaker function that can alternatively be written as . We can define a radial distribution function (RDF) for a completely filled -shell by . This is normalized according to , reflecting the orbital degeneracy of the energy level . In this Demonstration the function is plotted for selected values of (1 to 10) and (1 to 25).

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Contributed by: S. M. Blinder (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Snapshot 1: RDF for the 1 orbital

Snapshot 2: note undulations due to the individual -subshells

Snapshot 3: for large , the correspondence-principle limit is approached

References:

S. M. Blinder, "Generalized Unsöld Theorem and Radial Distribution Function for Hydrogenic Orbitals," *Journal of Mathematical Chemistry*, 14(1), 1993 pp. 319–324.

L. S. Bartell, "On the Limiting Radial Distribution Function for Hydrogenic Orbitals," *Journal of Mathematical Chemistry*, 19(3), 1996 pp. 401–403.

## Permanent Citation