Anharmonic MotionvsChemical Bonding: on the Interpretation of Electron Densities Determined by X-ray Diffraction

Abstract

Anharmonic and electron-density ref'mements 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 anharrnonic 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 K2PtC16 at room temperature and at 100K, and Si at room temperature. Results for KzPtC16 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.

Electron Density in YTiO3

“…Comparison with the room-temperature maps shows that peaks around Y are sharper at low temperature. As this is not necessarily inconsistent with expectations based on anharmonicity (Restori & Schwarzenbach, 1996), no firm conclusions can be drawn. (7) 1.00 (9) --1.02 (9) --0.49 (9)…”

Neutron and X-ray Diffraction Study of the Thermal Motion in K2PtCl6 as a Function of Temperature

“…The observed lattice constants a are given in Table 1 and in the literature (Takazawa et aL, 1990;Restori & Schwarzenbach, 1993, 1996. They may also be calculated from the distances d(Pt--K) = a(3)1/2/4 in Table 3.…”

“…The results of the refinements on the neutron data sets will be compared with those obtained for five X-ray data sets whose resolutions are considerably larger than those of the neutron data sets. Three extensive and precise X-ray data sets were measured at 293, 100 (Restori & Schwarzenbach, 1993, 1996 (6) 3.6439 (6) 3.4602 (4) 4.2348 (4) 2.3136 (5) 3.2719 (7) 3.6242 (7) 3.4503 (5) 4.2230 (6) 2.3097 (10) 3.2664 (14) 3.6305 (14) 3.4509 (2) 4.2235 (2) 2.3133 (8) 3.2715 (11) 3.6179 (11) 3.4468 (11) 4.2188 (13) 2.3164 (8) 3.2758 (11) 3.5768 (11) 3.4280 (1) 4.1964 (1) 2.3152 (4) 3.2742 (6) 3.5750 (6) 3.4262 (1) 4.1942 (1) 2.3144 (5) 3.2730 (7) 3.5668 (7) 3.4215 (7) 4.1885 (9) 2.3148 (8) 3.2736 (11) 3.5698 (11) 3.4233 (3) 4.1907 (4) 2.3158 (10) 3.2751 (14) 3.5594 (14) 3.4187 (1) 4.1852 (1) 2.3147 (8) 3.2735 (12) 3.5540 (12) 3.4152 (1) 4.1810 (1) 2.3155 (7) 3.2746 (11) 3.5483 (11) 3.4128 (11) We only note that anharmonic terms of K and C1 at 293 and 100 K assume significant values. The 8 K data shown in Table 1 have not been reported previously.…”

“…They were collected at Aarhus (Denmark) on a Huber four-circle diffractometer equipped with a displex helium cooling device. A precise absorption correction was carried out as in Restori & Schwarzenbach (1993, 1996. Structure refinements including a correction for secondary extinction (Becker & Coppens, 1974) were performed with the program LSYEXP (Restori & Schwarzenbach, 1992).…”

“…The charge density determined from X-ray diffraction has been reported to show features near Pt typical for a d 8 ion in an octahedral ligand field (Takazawa et aL, 1990). In contrast, accurate X-ray studies at room temperature (Restori & Schwarzenbach, 1993) and 100 K (Restori & Schwarzenbach, 1996) indicated that nonspherical features near K and C1 can be attributed exclusively to anharmonic thermal motion rather than to bonding. Results for Pt also indicate the presence of anharmonic motion, but are more uncertain since the relevant features are very close to the atom center and may be influenced by the uncertainty of the dispersion corrections and the limited resolution of the data.…”