Critical charge of a system with one electron and five or six charged centers

TL;DR

This paper investigates the stability and optimal geometrical configurations of a Coulomb system with one electron and five or six charged centers, identifying critical charges and analyzing energy singularities.

Contribution

It determines the critical charges and optimal geometries for stability of systems with five or six charged centers and analyzes the energy behavior at these critical points.

Findings

Stability domain for (5Z,e) is 0 < Z ≤ 0.350 with a dipyramid configuration.

Stability domain for (6Z,e) is 0 < Z ≤ 0.335 with an octahedron configuration.

Total energy exhibits a square-root branch point singularity at Z=Z_{cr}.

Abstract

We consider a Coulomb system of one electron and five or six infinitely massive centers of charge $Z$: $(5Z,e)$ and $(6Z,e)$. Critical charges and the possible optimal geometrical configurations are found. It is shown that the domain of stability for $(5Z,e)$ is $0 < Z \leq Z_{cr}^{(5Z,e)}=0.350$ with the optimal geometrical configuration given by a dipyramid (equilateral triangle base) circumscribed in a prolate spheroid. For $(6Z,e)$ the stability is $0 < Z \leq Z_{cr}^{(6Z,e)}=0.335$ with the optimal geometrical configuration given by an octahedron (square base), circumscribed in an oblate spheroid. For both systems we obtain an indication that total energy at $Z=Z_{cr}$ has a square-root branch point singularity with exponent $3/2$.