The Schroedinger equation for two electrons in the field of a charged fixed center $Z$ is solved with the Lagrange mesh method for charges close to the critical charge $Z_{cr}$. We confirm the value of the nuclear critical charge $Z_{cr}$ recently calculated in Estienne et al. {\em Phys. Rev. Lett. \bf 112}, 173001 (2014) to 11 decimal digits using an inhomogeneous (non-uniform) three-dimensional lattice of size $70 \times 70 \times 20$. We show that the ground state energy for H$^-$ is accurate to 14 decimals on the lattice $50 \times 50 \times 40$ in comparison with the highly accurate result by Nakashima-Nakatsuji, {\it J. Chem. Phys. \bf 127}, 224104 (2007).Comment: 6 pages, no figure

The three-body Schrödinger equation of the H+2 hydrogen molecular ion with Coulomb potentials is solved in perimetric coordinates using the Lagrange-mesh method. The Lagrange-mesh method is an approximate variational calculation with variational accuracy and the simplicity of a calculation on a mesh. Energies and wavefunctions of up to 4 of the lowest vibrational bound or quasibound states for the total orbital momenta from 0 to 40 are calculated. The obtained energies have an accuracy varying from about 13 digits for the lowest vibrational state to at least 9 digits for the third vibrational excited state. With the corresponding wavefunctions, a simple calculation using the associated Gauss quadrature provides accurate quadrupole transition probabilities per time unit between those states over the whole rotational bands. Extensive results are presented with six significant figures.

The second (unphysical) critical charge in the 3-body quantum Coulomb system of a nucleus of positive charge Z and mass m p , and two electrons, predicted by F Stillinger has been calculated to be equal to Z ∞ B = 0.904854 and Z mp B = 0.905138 for infinite and finite (proton) mass m p , respectively. It is shown that in both cases, the ground state energy E(Z) (analytically continued beyond the first critical charge Z c , for which the ionization energy vanishes, to ReZ < Z c ) has a square-root branch point with exponent 3/2 at Z = Z B in the complex Z-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 m p ). The first critical charge Z c is found accurately for a finite proton mass m p in the Lagrange mesh method, Z mp c = 0.911 069 724 655.

Abstract. The non-relativistic three-body Schrödinger equation of the heteronuclear molecular ion HD + is solved in perimetric coordinates using the Lagrange-mesh method. Energies and wave functions of the four lowest vibrational bound or quasibound states v = 0 − 3 are calculated for total orbital momenta from 0 to 47. Energies are given with an accuracy from about 12 digits for the lowest vibrational level to at least 9 digits for the third vibrational excited level. With a simple calculation using the corresponding wave functions, accurate dipole transition probabilities per time unit between those levels are given over the whole v = 0 − 3 rotational bands. Results are presented with six significant figures.

After the three-body molecular system H (J. Phys. B: At. Mol. Opt. Phys. 45 065101), its isotopomer, the deuterium molecular ion D is studied. The three-body Schrödinger equation is solved using the Lagrange-mesh method. Energies and wave functions for four vibrational states v = 0–3 and bound or quasibound states for total orbital momenta from 0 to 56 are calculated. The fundamental constant md = 3670.483 014 me is used. Energies are presented with an accuracy from about 13 digits for the lowest vibrational state up to 9 digits for the third vibrational excited state. Quadrupole transition probabilities per time unit between those states over the whole rotational bands were calculated. Extensive results are presented with six significant figures.