Density-optimized radial exponents for X-ray charge-density refinement from ab initio crystal calculations

Structure factors based on periodic density-functional (DFT) calculations on 25 molecular crystals have been used to evaluate trends in refined values of the κ and κ′ expansion–contraction parameters of the Hansen–Coppens multipole formalism. As found previously and expected physically, the spherical-valence-shell κ parameters are closely related to the net atomic charges, negative atoms being expanded and vice versa. κ′ parameters, which scale the radial dependence of the non-spherical deformation functions, are remarkably consistent for particular bonding environments. Systematic trends are observed for both carbon and oxygen, but the values obtained for nitrogen show a larger variation. Average values for oxygen and carbon in different bonding environments are tabulated and can be used whenever refinement of experimental data is affected by lack of uniqueness of the charge-density parameter set. Values for nitrogen must be more finely tuned to the specific bonding environment. The relation between atomic charge and κ offers the possibility of introducing a constraint in the charge-density refinement of very large molecules, for which reduction of the size of the parameter set may be essential.