Effect of Anharmonic Libron Interactions on the Single-Libron Spectrum of SolidH2andD2

The effects of interactions between the elementary excitations (librons) in the orientationally ordered phase of solid ${\mathrm{H}}_2$ and ${\mathrm{D}}_2$ are studied using diagrammatic perturbation theory. This formulation leads naturally to the construction of a renormalized dynamical matrix which includes all anharmonic effects. The cubic anharmonic interactions are by far the dominant ones, and we have calculated the energy shifts of each of the zero-wave-vector libron modes self-consistently to lowest order in the expansion parameter $\frac{1}{z}$, where $z=12\mathrm{}$ is the number of nearest neighbors. We find the libron energies (in units of the electrostatic quadrupole-quadrupole coupling constant $\Gamma$) to be 11.29 (13.66), 14.07 (17.72), and 19.55 (29.04) with the corresponding harmonic values in parentheses. In contrast to the harmonic theory, these anharmonic results provide a striking fit to the observed Raman spectrum of solid ${\mathrm{H}}_2$ and ${\mathrm{D}}_2$ with reasonable values of $\Gamma$, e.g., $\Gamma =0.59\mathrm{}$ ${\mathrm{cm}}^{-1\mathrm{}}$ for ${\mathrm{H}}_2$ and $\Gamma =0.83\mathrm{}$ ${\mathrm{cm}}^{-1\mathrm{}}$ for ${\mathrm{D}}_2$. We develop an expression for the Raman intensities in terms of the single-libron spectral weight function. The group-theoretical simplifications in our calculations are discussed in detail in the appendices. The cubic anharmonicity is shown in the accompanying paper to lead to a two-libron spectrum which explains the appearance of "extra" high-energy lines in the Raman spectrum of solid hydrogen. These effects are shown to be included in the renormalized dynamical matrix in the present approximation. Sum rules for the Raman intensities are derived and are used to check the calculations.