Russell's Blog

Today, we applied the Born-Oppenheimer approximation to calculate the nuclear separation distance of the singly ionized hydrogen (H₂⁺) molecule. The trial wave function we used was the linear combination of the two atomic ground state orbitals with a normalization that takes into account the overlap between them.

Once the separation distance had been found (approximately), we calculated the energy spectrum of rotation for the hydrogen molecule. To calculate the vibrational energy spectrum, we expanded the Coulomb potential of the atom to second order. This potential is simply the harmonic oscillator problem. With the correct dimensions, the harmonic oscillator yields a good small-amplitude approximation of the vibrational energy spectrum.

It was noted that for various diatomic molecules, the rotational energy spectrum will also depend on the nuclear spin. Atoms with nuclear spin 1/2 will behave as fermions, and atoms with nuclear spin 0 will behave as bosons. Depending on the nuclear spin, symmetry may forbid certain energy levels. This is a critically important feature of matter that determines the statistical behavior of bulk quantities of the material.

In other news, we will evidently be skipping the WKB approximation so that we can spend more time on scattering.