Internal molecular vibrations from crystal diffraction data by quasinormal mode analysis

The established procedure for analyzing molecular vibrations in terms of normal modes has been adapted so that experimental anisotropic thermal parameters can be used to study low-frequency internal vibrations of simple molecules in crystals. This involves quasinormal modes, which are linear combinations of selected low-frequency internal modes such as the torsional librations about individual bonds. Higher-frequency modes are neglected, since their contribution to the atomic mean-square displacements should be small. The force constants for selected low- frequency internal modes, together with the tensor components (T,L,S) that describe the overall molecular vibration, become the variables in an iterative least-squares refinement in which the observations are the atomic U_ij values. As a result, the concerted motion of the atoms for each quasinormal mode is defined and also its vibrational frequency. Corrections to bond lengths and angles due to internal vibrations can be calculated. In tests involving two different lipid crystal structures, the internal motions were introduced as torsions about two or three bonds occurring near the junction of an extended hydrocarbon chain with a relatively rigid massive atomic grouping. Compared with the simple rigid-body model, there were highly significant improvements in agreement between experimental and calculated U_ij values. Force constants for torsion about three C-S bonds were also in agreement [26 (5), 23 (6) and 22 (6) J mol^-1 deg^-2]. In one of the crystal structures (determined at 123 K), the six C-C bonds of a paraffin chain have average lengths 1.526 (2) Å before correction, 1.527 (3) Å after correction for simple rigid-body libration and 1.536 (4) Å after corrections including the quasinormal vibrations. The latter agrees with the electron diffraction value 1.542 (4) Å for n-hexadecane.