Rigid-link constraints and rigid-body molecules

The rigid-bond condition for harmonic thermal parameters states that the difference of the mean-square displacements of atoms A and B along the covalent bond A-B is negligible. In this paper, the corresponding condition for non-bonded intramolecular distances is called a rigid link. Rigid-body motion according to the TLS formalism requires all intramolecular links to be rigid. Conversely, a cornpiece set of rigid links is not necessarily equivalent to rigid-body motion. An algorithm is presented for the determination of the maximum number Q_N of independent rigid links of an N-atom molecule. In general for site symmetry 1, Q_N = N-1 for linear and 3N-6 for planar molecules. For three-dimensional molecules, Q_N = N(N - 1)/2, N ≤ 8 and 6N-20, N ≥ 8. For particular geometries, Q_N may be smaller. For many molecules, Q_N rigid links are equivalent to rigid-body motion. Notable exceptions are most linear and planar molecules, and all molecules with six or seven atoms. Higher site symmetries reduce and often eliminate these differences between rigid links and rigid-body motion. The use of rigid-link restraints in crystallographic least squares is recommended. They provide a computationally simple means of relaxing the constraints imposed on the displacement parameters by the TLS model for any molecular site symmetry.