Finding best-fit polyhedral rotations with geometric algebra

There are many minerals whose structure is well described as a framework of linked SiO₄ tetrahedra. Since the energy cost of stretching the Si-O bond is much greater than the cost of changing the bridging Si-O-Si bond angle, these structures may to a first approximation be analysed using the rigid-unit picture, in which the polyhedra are treated as completely rigid. In order to compare the predictions of rigid-unit theory with the results of other forms of simulation, we wish to determine how well a given set of atomic motions can be described in terms of rigid-unit motion. We present a set of techniques for finding the polyhedral rotations that most closely fit a given set of atomic motions, and for quantifying the residual distortion of the polyhedra. The formalism of geometric (Clifford) algebra proved very convenient for handling arbitrary rotations, and we use this formalism in our rotor-fitting analysis.