Lattice sum transformations and potential function expansions in lattice dynamics

Abstract

Mathematical techniques are presented for the treatment of both harmonic and anharmonic lattice dynamics in molecular crystals. We first give Ewald-type transformation formulas for the accurate and rapid evaluation of the slowly convergent lattice sums associated with long-range isotropic and anisotropic intermolecular force laws. The nature of these lattice sums is discussed in light of the properties of the transformed sums. We then give a systematic procedure for obtaining the Taylor series expansion of the intermolecular potential functions, in terms of the translational displacements of molecular centers of mass from their equilibrium positions. This expansion is valid for arbitrarily large orientational displacements. The need for such an expansion arises since the intermolecular potential functions are naturally expressed in molecular-pair coordinates but for many applications these potential functions must then be written in terms of lattice-fixed or space-fixed coordinates. Although the procedure involved is discussed from the viewpoint of lattice dynamics, further applications in the investigation of intermolecular dynamics of weakly bound molecular complexes and adsorption of molecules on solid surfaces are anticipated.