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

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 $3/2$.