The raman signal of a hindered rotor
We present a method for calculation of Raman modes of the quantum solid phase I solid hydrogen and deuterium. We use the mean-field assumption that the quantised excitations are localized on one molecule. This is done by explicit solution of the time-dependent Schroedinger equation in an angle-dependent potential, and direct calculation of the polarisation. We show that in the free-rotor limit, the H$_2$ and D$_2$ frequencies differ by a factor of 2, which evolves toward $\sqrt{2}$ as the modes acquire librational character due to stronger interactions. The ratio overshoots $\sqrt{2}$ if anharmonic terms weaken the harmonic potential. We also use density functional theory and molecular dynamics to calculate the E$_{2_g}$ optical phonon frequency and the Raman linewidths. The molecular dynamics shows that the molecules are not free rotors except at very low pressure and high temperature, and become like oscillators as phase II is approached. We fit the interaction strengths to experimental frequencies, but good agreement for intensities requires us to also include strong preferred-orientation and stimulated Raman effects between S$_0$(1) and S$_0$(0) contributions. The experimental Raman spectrum for phase II cannot be reproduced, showing that the mean-field assumption is invalid in that case.