Vanishing of the atomic form factor derivatives in non-spherical structural refinement – a key approximation scrutinized in the case of Hirshfeld atom refinement

Abstract

When calculating derivatives of structure factors, there is one particular term (the derivatives of the atomic form factors) that will always be zero in the case of tabulated spherical atomic form factors. What happens if the form factors are non-spherical? The assumption that this particular term is very close to zero is generally made in non-spherical refinements (for example, implementations of Hirshfeld atom refinement or transferable aspherical atom models), unless the form factors are refinable parameters (for example multipole modelling). To evaluate this general approximation for one specific method, a numerical differentiation was implemented within the NoSpherA2 framework to calculate the derivatives of the structure factors in a Hirshfeld atom refinement directly as accurately as possible, thus bypassing the approximation altogether. Comparing wR 2 factors and atomic parameters, along with their uncertainties from the approximate and numerically differentiating refinements, it turns out that the impact of this approximation on the final crystallographic model is indeed negligible.

X-ray constrained wavefunctions based on Hirshfeld atoms. I. Method and review

“…Besides, asphericity is also characterized in Cartesian coordinates. For peculiarities of refinements using aspherical atomic form factors see also Midgley et al (2021).…”

“…The latter integrals may need to be performed numerically (Jayatilaka & Dittrich, 2008). Midgley et al (2021) have presented a very similar set of equations, in fractional coordinates, and without the use of symmetry operators to generate all the atoms in the unit cell from an asymmetric unit. Note that the above equations correct some minor errors that are present in Jayatilaka (2012).…”

Synthesis and Characterization of Phenylalanine Amides Active against Mycobacterium abscessus and Other Mycobacteria

“…Solvent water in the crystal structure of 14 (C)· 1.5 H 2 O is not shown (suffix a denotes symmetry-related atoms). The structure of 14 (C) was refined with aspherical atomic scattering factors using NoSpherA2. , …”

Influence of modelling disorder on Hirshfeld atom refinement results of an organo-gold(I) compound

“…Any of these wavefunctions were read by the NoSpherA2 (Kleemiss et al, 2021) software; the related electron-density was partitioned into Hirshfeld atoms, whose Fourier transforms are the non-spherical scattering factors, which were then tabulated in a .tsc file and handed over to olex2.refine (Bourhis et al, 2015) for the least-squares refinement. More details on the procedure are discussed by Midgley et al (2021).…”