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The second (unphysical) critical charge in the three-body quantum Coulomb system of a nucleus of positive charge and mass , and two electrons, predicted by Stillinger has been calculated to be equal to and for infinite and finite (proton) mass , respectively. It is shown that in both cases, the ground state energy (analytically continued beyond the first critical charge , for which the ionization energy vanishes, to ) has a square-root branch point with exponent 3/2 at in the complex -plane. Based on analytic continuation, the second, excited, spin-singlet bound state of negative hydrogen ion H is predicted to be at −0.51554 a.u. (−0.51531 a.u. for the finite proton mass ). The first critical charge is found accurately for a finite proton mass in the Lagrange mesh method, . [ABSTRACT FROM AUTHOR]

Thermodynamics of trapped charged ideal spin-1 bosons confined in a magnetic field are investigated within semi-classical approximation and truncated-summation approach. It is shown that the critical temperature increases slightly at the first, and then decreases slowly with increasing external magnetic field. Charged spin-1 Bose gases present a crossover from diamagnetism to paramagnetism as the spin factor increases. Charged spin-1 Bose gases exhibit distinct thermodynamic behaviors from the spinless case. [ABSTRACT FROM AUTHOR]