The Gram-Charlier and multipole expansions in accurate X-ray diffraction studies: can they be distinguished?

The Gram-Charlier temperature factor formalism has been applied to a set of accurate low-temperature data on bis(pyridine)(meso-tetraphenylporphinato)iron(II), and to a theoretical set of static structure factors on the hexaaquairon(II) ion. The refinements are compared with the multipole treatment for atomic asphericity due to chemical bonding. In a treatment of the experimental data in which only the iron atom asphericity is considered, the 'thermal motion' formalism is as efficient as the multipole formalism in accounting for the observations. It is slightly less efficient when applied to the static theoretical data, though model maps based on the two treatments are remarkably similar. A high-order Gram-Charlier refinement of the porphyrin data, followed by a multipole refinement of all data with the Gram-Charlier parameters initially fixed, and later varied, shows that simultaneous refinement of anharmonic and aspherical effects is possible, though the resulting separation may not be accurate. A combined Gram-Charlier multipole refinement on the static data, however, leads to non-significant thermal parameters. It is concluded that the statistical Gram-Charlier formalism is remarkably successful in representing bonding effects in the valence charge density if these are not specifically accounted for in the scattering formalism. Statistical anharmonic thermal motion formalisms should only be used for X-ray data analysis in combination with a formalism accounting for the effect of bonding on the atomic charge density.