Section 4.1.2.4. Molecular crystals

The full Born–von Kármán treatment becomes excessively cumbersome when applied to most molecular crystals. For example, for naphthalene with two molecules or 36 atoms in the primitive cell, the dynamical matrix has dimensions 108 × 108. Moreover, the physical picture of molecules or of groups of atoms, vibrating in certain modes as quasi-rigid units, is lost in the full treatment.

To simplify the setting up of the dynamical matrix, it is assumed that the molecules vibrate as rigid units in the crystal with each molecule possessing three translational and three rotational (librational) degrees of freedom. The motion of these rigid groups as a whole is described by the external modes of motion, whereas the internal modes arise from distortions within an individual group. The frequencies of these internal modes, which are largely determined by the strong intramolecular forces, are unaffected by the phase of the oscillation between neighbouring cells: the modes are taken, therefore, to be equivalent to those of the free molecule. The remaining external modes are calculated by applying the Born–von Kármán procedure to the crystal treated as an assembly of rigid molecules.

The dynamical matrix D(q) now has dimensions , where is the number of molecules in the primitive cell: for naphthalene, D is reduced to 12 × 12. The elements of D can be expressed in the same form as equation (4.1.2.5) for an atomic system. , refer to molecules which are L cells apart and the indices , () label the six components of translation and rotation. in equation (4.1.2.5) is replaced by where represents the 3 × 3 molecular-mass matrix for and the 3 × 3 moment-of-inertia matrix referred to the principal axes of inertia for . The 6 × 6 force-torque constant matrices are derived by taking the second derivative of the potential energy of the crystal with respect to the coordinates of translation and rotation.