Introduction

The problem of simplifying the computation of the normal modes of vibrations of molecules and solids has been presented, over the past century, as a classic application of symmetry. It has been extensively discussed in a plethora of books on applications of group theoretical techniques. The dynamical problem of surfaces has been a relative latecomer.

A major contribution of group theoretical techniques to the dynamics of condensed matter systems has been to simplify the secular problem for the determination of normal mode eigenfrequencies and eigenvectors in the harmonic approximation. The secular matrix is found to be reducible, i.e. “block-diagonalizable”, with respect to the Irreps of the symmetry group of the system's Hamiltonian.

For the sake of pedagogy, and in order to prepare the way for tackling the dynamics of the more complex condensed matter systems, we consider first the simpler dynamics of molecules.

Dynamical properties of molecules

The application of group theoretical techniques to study the dynamical properties of molecules involves the determination of symmetrized normal modes prior to the computation of the eigenfrequencies and eigenvectors. A typical example of such an approach has been presented in Chapter 6, to motivate the concept of projection operators. In that example, we were able to obtain the symmetry-adapted translation, rotation, and vibrational vectors describing the dynamics of water molecules. Here, we expand on this approach and extend it to enable the computation of corresponding eigenfrequencies and eigenvectors.