Chapter 7 - Diatomic molecules

Abstract

The Schrödinger equation for the hydrogen molecular ion is an eigenvalue equation with two independent variables. A MATLAB program is given for solving the eigenvalue problem of the hydrogen molecular ion using spline collocation. Our MATLAB program is described in detail for a smaller grid but it is found that the spline collocation method produces results accurate to seven significant figures with a larger grid having 192 grid points for each variable. The Hartree-Fock equations for diatomic molecules has the same single-electron terms that occur for the hydrogen molecular ion and additional terms corresponding to the direct and exchange interactions of an electron with the other electrons in the molecule. Numerical results are given for the OH molecule which is responsible for the powerful airglow spectra produced in the higher levels of our atmosphere and may be observed in emission and absorption spectra of exoplanets orbiting distant stars. We then consider the possibility of detecting life on exoplanets. For life to be remotely visible from space, we restrict our attention to rocky planets with water on their surfaces. This is called the habitable zone around a distant star. Very many planets that satisfy these criterion have already been observed, and efforts continue to be made to find definitive signs of life. The Earth has a long history in which it has passed through ages in which its surface features and the composition of its atmosphere was quite different. The same could well be true of other habitable planets. Our nearest star – Proxima Centauri – is a red dwarf that is two hundred million years older than our Sun with a planet in its habitable zone.