The use of symmetrized valence and relative motion coordinates for crystal potentials

Abstract

Symmetrized valence coordinates are linear combinations of conventional valence coordinates which display the symmetry of a set of atoms bound by the valence bonds. Relative motion coordinates are relative translations, or relative rotations, of two or more strongly bonded groups of atoms among which relatively weak forces act. They are useful for expressing interactions between molecules in molecular crystals and should be chosen, also, to reflect the symmetry of the interacting groups. Since coordinates defined by these procedures possess elements of symmetry in common with the bonding electron distributions, the force constants in the potential should be more amenable to calculation in terms of energy changes in the electronic ground state which accompany displacements of the atoms from equilibrium. It is easier to determine force constants for fitting experimental data because interaction constants coupling coordinates of unlike symmetry with regard to the crystal point group are necessarily zero. They may be small, also, for coordinates which belong to different representations of the local symmetry when this is not the same as for the crystal. Procedures are given for defining the coordinates, and for assuring that the potential energy is invariant to crystal translations and rotations. The secular equation is derived by expressing the kinetic and potential energies in terms of components of mass adjusted basis vectors which are chosen so that high and low frequency modes can be separated approximately. The necessity to remove redundancies among the coordinates in the potential is avoided.