Helium- and Lithium-like ionic sequences: Critical charges

In non-relativistic quantum mechanics we study the Coulomb systems of infinitely massive center of charge Z and two-three electrons: $(Z,e,e)$ and $(Z,e,e,e)$. It is shown that in both cases the total energy curve in $Z$ is smooth, without any visible irregularities. Thus, for both systems the physical integer charges $Z=1,2,...$ do not play a distinguished role as would be associated with charge quantization. By definition, a critical charge $Z_{cr}$ is a charge which separates a domain of the existence of bound states from a domain of unbound ones (continuum). For both systems the critical charges are found, $Z_{cr,2e}=0.91085$ and $Z_{cr,3e}=2.009$, respectively. Based on numerical analysis, the Puiseux expansion in fractional powers of $(Z-Z_{cr})$ is constructed for both systems. Our results indicate the existence of a square-root branch point singularity at $Z_{cr}$ with exponent 3/2. A connection between the critical charge and the radius of convergence of 1/Z-expansion is briefly discussed.Comment: 10 pages, LaTeX, typos corrected, Fig.1 added, a Note Added with calculated critical charge for $2^1S$ state for $(Z,e,e)$ system, $Z_{cr,2e}^{(2^1S)}\ =\ 1.02

A note about the ground state of the H3+{\rm H}_3^+ hydrogen molecular ion

Three simple $7-, (7+3)-, 10-$parametric trial functions for the ${\rm H}_3^+$ molecular ion are presented. Each of them provides subsequently the most accurate approximation for the Born-Oppenheimer ground state energy among several-parametric trial functions. These trial functions are chosen following a criterion of physical adequacy and includes the electronic correlation in the exponential form $\sim\exp{(\gamma r_{12})}$, where $\gamma$ is a variational parameter. The Born-Oppenheimer energy is found to be $E=-1.340 34, -1.340 73, -1.341 59$\,a.u., respectively, for optimal equilateral triangular configuration of protons with the equilibrium interproton distance $R=1.65$\,a.u. The variational energy agrees in three significant digits (s.d.) with most accurate results available at present as well as for major expectation values.Comment: 12 pages, 1 figure, 3 table

Charged Hydrogenic, Helium and Helium-Hydrogenic Molecular Chains in a Strong Magnetic Field

A non-relativistic classification of charged molecular hydrogenic, helium and mixed helium-hydrogenic chains with one or two electrons which can exist in a strong magnetic field $B \lesssim 10^{16}$G is given. It is shown that for both $1e-2e$ cases at the strongest studied magnetic fields the longest hydrogenic chain contains at most five protons indicating to the existence of the $\rm{H}_5^{4+}$ and $\rm{H}_5^{3+}$ ions, respectively. In the case of the helium chains the longest chains can exist at the strongest studied magnetic fields with three and four \al-particles for $1e-2e$ cases, respectively. For mixed helium-hydrogenic chains the number of heavy centers can reach five for highest magnetic fields studied. In general, for a fixed magnetic field two-electron chains are more bound than one-electron ones.Comment: 32 pages, 2 figures, 9 table

The HeH+HeH^+ molecular ion in a magnetic field

A detailed study of the low-lying electronic states {}^1\Si,{}^3\Si,{}^3\Pi,{}^3\De of the $\rm{HeH}^+$ molecular ion in parallel to a magnetic field configuration (when \al-particle and proton are situated on the same magnetic line) is carried out for $B=0-4.414\times 10^{13}$ G in the Born-Oppenheimer approximation. The variational method is employed using a physically adequate trial function. It is shown that the parallel configuration is stable with respect to small deviations for \Si-states. The quantum numbers of the ground state depend on the magnetic field strength. The ground state evolves from the spin-singlet {}^1\Si state for small magnetic fields $B\lesssim 0.5$ a.u. to the spin-triplet {}^3\Si unbound state for intermediate fields and to the spin-triplet strongly bound $^3\Pi$ state for $B \gtrsim 15$ a.u. When the $\rm{HeH}^+$ molecular ion exists, it is stable with respect to a dissociation.Comment: 13 pages, 5 figures, 4 table