A statistical interpretation of rotation and translation functions in reciprocal space

Rotation and translation functions in reciprocal space, commonly used to find the orientation of a known molecular fragment and/or its position in the unit cell, are all identical or closely related to the function Q(R,t) = <|E_h|²|E_hp(R,t)|²>_h, where |E_h| is the observed normalized structure factor and E_hp(R,t) is the normalized structure factor for the partial structure, calculated for orientation R and position t. In this paper, this function is derived from statistical considerations without referring to a Patterson synthesis. From the centrosymmetric and noncentrosymmetric probability density functions, Q(R,t) = 1 + 2p² and Q(R,t) = 1 + p² are obtained, respectively, where p² is the fraction of atoms (scattering power) that has been placed in the correct orientation and position in the unit cell. Alternative search functions, for instance using |E_h|⁴ instead of |E_h|², are discussed.