Quadrupole transitions in the bound rotational–vibrational spectrum of the hydrogen molecular ion

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 wave functions of up to four of the lowest vibrational bound or quasibound states for 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 wave functions, 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.