Correct formulation of lattice dynamics of molecular crystals

Abstract

The authors deal with a subtle logical mistake in the standard theory of lattice dynamics for molecular crystals. Thus, usually one writes down the vibration energy up to terms quadratic in displacement of the atoms and then expresses it in terms of translation and infinitesimal rotation vectors. But since the displacement cannot be uniquely described up to quadratic terms in a rotation vector, this scheme seems to be erroneous. The correct procedure would be to start from the force and torque equations which are linear in the rotation vector. They develop this scheme to obtain the expressions for molecular force constants (in terms of atomic force constants), first retaining the coupling between internal and external vibrations and then passing over to the rigid-molecule approximation. The translational and rotational invariance conditions are also derived. However, the results are found to be identical with the existing ones derived from energy considerations. The root of this seemingly fortuitous agreement is traced to the interesting fact that up to quadratic terms in the rotation vector, the energy is indeed uniquely specified, though the displacement field is not.