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).


L. S. Bartell has derived the classical analog of , which, in accordance with Bohr's correspondence principle, approaches the quantum result in the limit . The checkbox produces a red plot of the classical function.

The generalized Unsöld theorem has found several theoretical applications, including derivation of the canonical Coulomb partition function, density-functional computations, supersymmetry, and study of high- Rydberg states of atoms.


Contributed by: S. M. Blinder (March 2011)
Open content licensed under CC BY-NC-SA



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


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.

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