Rigid Rotators: Molecular Rotational Spectra

Abstract

For a rotating molecule the angular momentum, associated with the rotation of this “nearly rigid” body can be expressed in terms of Euler angles and their partial derivatives. Hence, this may be a good first example. Consider the simplest case: a diatomic molecule, e.g., the HCl molecule with one hydrogen and one Chlorine nucleus and 1 + 17 electrons. The full 20-body problem is extremely complicated, but at very low energies no excitations associated with the electron degrees of freedom will come into play. The electron cloud binds the two atomic nuclei into a nearly rigid structure. The position of the diatomic molecule in 3-D space can be described by a radial coordinate, r, giving the distance between the H and Cl nuclei, and two angles, θ, and φ, giving the orientation in space of the molecule axis, or H-Cl line. The wave function can be written as Ψ(r, θ, φ) = R(r)Y _lm (θ, φ). The electron cloud gives rise to a potential, V(r), with a deep (nearly parabolic) well with a minimum ar r = r _e , where this is the equilibrium distance between the two atomic nuclei. The radial problem is associated with the vibrational motion of the molecule, a nearly harmonic oscillator motion to good first approximation. The energy associated with this vibration, ħω₀is approximately 30 times that associated with the lower rotational excitations.