Transformations into optimal parallelism in euclidean spaces (or: how to explain the shape of the electron-density distribution inside a crystal)

Let and be vectors in the finite dimensional euclidean space . We investigate the problem of how one can find a U such that ∑i Uxi,yi is maximal when U runs through the orthogonal group or the special orthogonal group, i.e., we are looking for a U such that the and the are “as parallel as possible”. For d=3 this problem arises, for instance, from the data analysis of crystallographic diffraction experiments on orientationally disordered systems: the xʹs stand for the atom positions of fragments M1 of the crystal structure, the yʹs are taken from the set M2 of maxima of the electron-density distribution reconstructed from diffraction data, and one must know the above transformations U in order to determine the M1–M2-configurations of minimum distance since they are responsible for the shape of the density distribution. It turns out that one can associate a d×d-matrix G with this problem in such a way that the relevant U are precisely those for which the trace of GU is maximal. Using this transformation we are able to provide an explicit solution by means of the singular value decomposition of G. Several further topics in connection with this problem are also discussed. In particular we investigate in which cases the optimal position is unique, and we study generalizations to nondiscrete situations and to the infinite dimensional setting.