Aspherical atom scattering factors

Substitution of the atomic density expression of Eq. (3.35) gives the aspherical atom scattering factor of atom j as... 

The introduction of standard aspherical-atom scattering factors leads to a very significant improvement in Hirshfeld s rigid bond test. The results are a beautiful confirmation of Hirshfeld s (1992) statement that an accurate set of nuclear coordinates (and thermal parameters ) and a detailed map of the electron density can be obtained, via X-ray diffraction, only jointly and simultaneously, never separately or independently . 

Koritsanszky, T., Volkov, A, Coppens, P. Aspherical-atom scattering factors from molecular wave functions. 1. Transferability and conformation dependence of atomic electron densities of peptides within the multipole formahsm, Acta Crystallogr. 58A(5) (2002) 464-472. 

The most likely cause of such discrepancy is an unsuitable atomic scattering factor. That means, some factor that affects the chemical behaviour of an atom may, for instance, not be properly accounted for in the calculated electronic structure from which scattering factors are derived. The use of oriented non-spherical atomic ground-states has been proposed [182] as a possible remedy. On this basis theoretically acceptable chemical deformation densities have been obtained. Such usage has led to the development of aspherical-atom, or multipole refinement of crystallographic structures in charge-density studies. 

According to the aspherical-atom formalism proposed by Stewart [12], the one-electron density function is represented by an expansion in terms of rigid pseudoatoms, each formed by a core-invariant part and a deformable valence part. Spherical surface harmonics (multipoles) are employed to describe the directional properties of the deformable part. Our model consisted of two monopole (three for the sulfur atom), three dipole, five quadrupole, and seven octopole functions for each non-H atom. The generalised scattering factors (GSF) for the monopoles of these species were computed from the Hartree-Fockatomic functions tabulated by Clementi [14]. 

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