The Vibrational Spectra of Molecules and Complex Ions in Crystals. I. General Theory

Abstract

It is shown that if a crystal is approximated as a harmonic oscillator, only a very small number of vibrational modes may be active in the infra-red or Raman spectra. These modes are all totally symmetric with respect to translation. They must also have proper symmetry with respect to the factor group of the space group which we have called the unit cell group. The active frequencies may be calculated from a single unit cell. It is shown further that the number and symmetry type of vibrational modes may be correctly calculated from the local symmetry (site group) of the center of gravity of each kind of molecule. In the harmonic approximation the spectrum should consist of very sharp lines. The wide bands obtained experimentally are probably due to combination lines with lattice vibrations and indicate anharmonic coupling between molecular and lattice modes. Such combinations may cause apparent violations of selection rules for internal vibrations. The symmetry basis for obtaining selection rules in this case is given. The importance of obtaining spectra over a wide temperature range and particularly at very low temperatures is emphasized. In cases where all internal modes have a frequency great compared to the frequency of lattice modes the problem is examined from the point of view of the Born-Oppenheimer approximation. It is shown that for some symmetries perturbations linear in the lattice displacement may occur. In these cases considerable splitting of degenerate molecular vibration modes is possible. Groups for which first-order splitting is not expected are listed. Symmetry species of other groups are classified into those which may and which may not be split in this approximation, and the lattice modes responsible for splitting are classified as rotations or translations and by symmetry type.