Internal vibrations of a molecule consisting of rigid segments. I. Non-interacting internal vibrations

For molecular crystals, a procedure is proposed for interpreting experimentally determined atomic mean square anisotropic displacement parameters (ADPs) in terms of the overall molecular vibration together with internal vibrations with the assumption that the molecule consists of a set of linked rigid segments. The internal librations (molecular torsional or bending modes) are described using the variable internal coordinates of the segmented body. In paper I of this two-part report, it is assumed as a zero-order approximation that the internal vibrations about the linkage axes between pairs of segments are uncorrelated with each other and with the overall molecular rigid-body vibrations. As a first-order approximation, the possibility that each internal vibration can be correlated with the external vibrations is also considered. An important feature of this approach is that the internal librations are required to give zero contribution to the overall momentum of the molecule at all times, so the internal coordinates must be orthogonal to the external ones. Also, each of the internal librations involves the motion of all atoms in the molecule. The resulting internal vibrational parameters are invariant to the choice of reference segment. With this procedure, the experimental ADPs obtained from crystal structure determinations involving six small molecules (sym-trinitrobenzene, adenosine, tetra-cyanoquinodimethane, benzamide, alpha-cyanoacetic acid hydrazide and N-acetyl-L-tryptophan methylamide) have been analyzed. As a consequence, vibrational corrections to the bond lengths and angles of the molecule are calculated as well as the frequencies and force constants (with e.s.d.'s) for each internal torsional or bending vibration. Compared with other models used for describing internal vibrations, there are differences in how the total ADP is partitioned between the internal and overall molecular vibrations.