Anharmonic motion vs chemical bonding: On the interpretation of electron densities determined by X-ray diffraction

Anharmonic and electron-density refinements against accurate X-ray diffraction data are today almost routine. However, the unambiguous identification and separation of effects due to anharmonic atomic motion and to chemical bonding is impossible with a single X-ray data set and difficult even with data measured at different temperatures, especially in heavy-atom compounds. For cubic site symmetry, analytical expressions are compared for the convolutions of: (i) the electron density of a spherical free atom with an anharmonic probability density distribution (p.d.f.); and (ii) an aspherical atom with a Gaussian p.d.f. If both the free atom and the deformation functions of the aspherical atom are represented by Gaussian-type functions, there exists for every set of anharmonic parameters an equivalent set of aspherical-atom parameters but the reverse is not necessarily true. Both models are usually suitable for parametrizing anharmonicity also in the case of real atoms and exponential-type deformation functions. Contrary to widespread belief, both models predict a qualitatively similar change of the aspherical density with decreasing temperature: the extrema move towards the atom center and their heights increase except at low temperatures. Quantitatively, however, the temperature dependence of the adjusted parameters should be different: in the case of anharmonicity, the second-, third- and fourth-order coefficients should be proportional to T, T-2 and T-3, respectively, while the population factors of the deformation functions should be independent of T. The theory is tested and verified with refinements on calculated and on measured X-ray structure amplitudes for K2PtCl6 at room temperature and at 100 K, and Si at room temperature. Results for K2PtCl6 agree well with the anharmonic model. In Si at room temperature, the two effects overlap only slightly and can be reasonably well identified; they cannot be distinguished with simulated high-temperature data.