Calculation of the vibrational frequencies of polyatomic molecules, including those of crystals

Abstract

The calculation of the vibrational frequencies of polyatomic molecules in the harmonic approximation is discussed with reference to the use of computers. To avoid the problem of redundancies in describing the kinetic energy of the system, the secular equation is derived using mass-weighted cartesian displacements. Methods for the solution of this equation are examined with particular reference to the use of symmetrised mass-weighted Cartesian displacements in which vibrations are separated from translations and rotations. The approach is extended to an examination of the optically active vibrations of crystals when only Bravais cell atoms (apart from adjacent cell atoms required to define internal co-ordinates) need be considered.

A program is described which, from an input of masses and Cartesian co-ordinates of the atoms, together with an appropriate force-constant matrix and internal co-ordinate definitions, calculates frequencies, and eigenvectors. The program is applied to (crystalline) molybdenum trioxide.