Semiclassical evaluation of sums of squares of hydrogenic bound-state wavefunctions

The sum over l and m of |phisub(nlm)(r)|2, where phisub(nlm)(r) is a bound-state hydrogenic wavefunction with principal quantum number n(>>1) and angular momentum quantum numbers l and m, is evaluated approximately using the Thomas-Fermi statistical approach. This approach breaks down in the region near the nucleus and in the region near and beyond the classical edge of the nth shell at r=2n2 a0/Z, with Z the atomic number. A simple correction, proposed in the context of heavy atoms, is investigated in the region near the edge. Results are presented. The sum over n and m of |phisub(nlm)(r)|2 and the sum over l and m of |phi-tildesub(nlm)(k)|2, where phi-tildesub(nlm)(k) is the Fourier transform of phisub(nlm)(r), are also evaluated using the Thomas-Fermi method. (author)