Selection Rules and the Protonic Spectrum of Molecules

Recently, we have reported variational solutions to Schrödinger's equation for C${\mathrm{H}}_4$, N${\mathrm{H}}_3$, ${\mathrm{H}}_2$O, and HF, using a Hamiltonian which included the kinetic-energy operators of the protons. The results of these calculations implied the existence of protonic spectra similar to the electronic spectra. We show here that the selection rules which apply to electrons also apply to the protons. Furthermore, we find a two-particle-transition operator which allows an electron and a proton or two protons to be simultaneously excited with intensities proportional to the square of $\frac{k{m}_p}{M}$, where $k$ is the wave number of the light, $m_p$ is the mass of the proton, and $M$ is the total mass of the molecule. For completeness, the effects of the radiation field on the coordinates of the c. m. are given also.