On the Charge Density Refinement of Odd-Order Multipoles Invariant under Crystal Point Group Symmetry

Charge density studies utilise a multipolar expansion of the atomic density (and the associated atomic scattering factor) in order to model asphericity. Contributions of the individual multipoles to the atomic density are then refined as multipole population coefficients. Refinement of these coefficients pertaining to odd-order multipoles that are invariant under the crystal point-group symmetry is often problematic, with ill-defined values and correlations plaguing the convergence to a good model. These difficulties have been discussed in generic terms in the literature, but never explicitly analysed in detail. In this communication, we show that the charge density multipolar atomic scattering factor can be partitioned in three contributions that differ in their behaviour under the point group symmetry of the crystal. This partitioning rationalises and predicts the conditions that give rise to ill-conditioning of the charge density refinement of these multipoles.