Quasiclassical analysis of the spectra of two groups of central potentials

A method is suggested for analyzing the spectra of central attractive potentials either with Coulomb singularity (intra-atomic potentials) or finite at zero point (potentials in spherical clusters and nuclei). It is shown that, if the orbital degeneracy is removed, then $\varepsilon_{nl}-\varepsilon_{n0}\cong a_{\varepsilon_{n0}}(l+1/2)^2$ for small $l$ in the shell $n$. In atoms and ions, the coefficient $a_\varepsilon$ is nonnegative, so that the energy in the $n$ shell increases with $l$. The validity of this formula for the inner electrons is illustrated by calculating the spectrum of the mercury atom. In cluster potentials, the opposite situation, as a rule, occurs: the larger $l$, the lower the corresponding level ($a_\varepsilon<0$). However, in the soft potentials of small clusters, spectral regions with different signs of $a_\varepsilon$ coexist and the orbitally degenerate level exists in the spectral region where $a_\varepsilon=0$. Aluminum clusters Al$_N$ are taken as an example to find out how the position of the region with the degenerate level varies with varying cluster size $N$, and it is found that this region is «pushed out» to higher energies with an increase in $N$. In this connection, the presence of multiply ionized Al$_N$ clusters of the corresponding size in a low-temperature aluminum plasma is discussed.