High-order analytic translation matrix elements for real-space six-dimensional polar Fourier correlations

Analytic expressions are presented for calculating translations of high-order three-dimensional expansions of orthonormal real spherical harmonic and Gaussian-type or exponential-type radial basis functions. When used with real spherical harmonic rotation matrices, the resulting translation matrices provide a fully analytic method of calculating six-dimensional real-space rotational–translational correlations. The correlation algorithm is demonstrated by using an exhaustive search to superpose the steric density functions of a pair of similar globular proteins in a matter of seconds on a contemporary personal computer. It is proposed that the techniques described could be used to accelerate the calculation ofe.g.real-space electron density correlations in molecular replacement, docking proteins into electron microscopy density maps, and searching the Protein Data Bank for structural homologues.