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

TL;DR

This paper presents highly accurate calculations of quadrupole transition probabilities in the hydrogen molecular ion's bound rotational-vibrational spectrum using the Lagrange-mesh method, covering multiple vibrational and rotational states.

Contribution

It introduces a precise variational approach with the Lagrange-mesh method to compute energies and quadrupole transition probabilities in H₂⁺ with high accuracy across many states.

Findings

Energies accurate to 13 digits for ground states

Transition probabilities calculated with six significant figures

Coverage of states up to vibrational level 4 and rotational momentum 40

Abstract

The three-body Schr\"odinger 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.