H.-B. Bürgi

Dynamics of molecules in crystals from multi- temperature anisotropic displacement parameters. I. Theory

A new model for analysing the temperature evolution of anisotropic displacement parameters (ADPs) is presented. It allows for a separation of temperature-dependent from temperature-independent contributions to ADPs and provides a fairly detailed description of the temperature-dependent large-amplitude molecular motions in crystals in terms of correlated atomic displacements and associated effective vibrational frequencies. It can detect disorder in the crystal structure, systematic error in the diffraction data and the effects of non-spherical electron-density distributions on ADPs in X-ray data. The analysis requires diffraction data measured at multiple temperatures.

Dynamics of molecules in crystals from multi-temperature anisotropic displacement parameters. II. Application to benzene (C6D6) and urea [OC(NH)2]

The temperature evolution of atomic anisotropic displacement parameters (ADPs) of perdeuterobenzene and of urea in the temperature range between 12 and 123 K is investigated in terms of the model presented in paper I. For the benzene molecule, the temperature-dependent contributions to the ADPs are well described by three molecular librations and three molecular translations. For the urea molecule, the analysis revealed a low-frequency high-amplitude normal mode ( approximately 64 cm(-1)), which combines out-of-plane deformations of the NH(2) groups with molecular libration. The pyramidalization motion allows the hydrogen-bonding pattern to be retained quite well, whereas this pattern is heavily distorted in the higher-frequency molecular librations. The results presented for urea go a step beyond those obtainable in a conventional rigid-body or segmented-rigid-body analysis because they show how correlations of atomic displacements in molecular crystals can be determined from the temperature evolution of ADPs. For both molecules, the analysis reveals temperature-independent contributions to the ADPs accounting for the high-frequency internal vibrations. It is the first time that such contributions have been extracted directly from single-crystal diffraction data for light atoms like hydrogen and deuterium as well as for heavier atoms like carbon, nitrogen and oxygen. These contributions agree well with those calculated from independent spectroscopic information.

Anharmonicity in anisotropic displacement parameters

A quasi-harmonic molecular-mean-field model for analyzing anharmonic temperature evolution of anisotropic displacement parameters is described. Anharmonic effects are taken into account through a Gruneisen-type temperature dependence of effective vibrational frequencies. The method is applied to neutron and X-ray diffraction data of hexamethylenetetramine measured between 15 and 298 K. The resulting Gruneisen parameters and other characteristics of molecular motion in the solid state agree well with those obtained from independent vibrational data. The analysis also suggests errors in the ADPs due to insufficient extinction corrections in the diffraction data.

Structure of C60: partial orientational order in the room-temperature modification of C60

Using published synchrotron X-ray data, the room-temperature scattering density distribution of pure C 60 has been parametrized in terms of a combination of eight oriented symmetry-related images of the molecule, and of a freely spinning molecule. Corresponding populations are 61 and 39%. The oriented part of the model is obtained, in good approximation, by imposing m3m symmetry on the energetically more favourable major orientation in the low-temperature structure of C 60

Single-Crystal X-Ray Diffraction Study of the Room Temperature Structure and Orientational Disorder of C70

The room temperature structure of the hexagonal family of modifications of pure C70 has been studied by single-crystal X-ray diffraction. The experimental evidence indicates rotational symmetry of the molecular electron density and supports the previously published assumptions that the molecules are freely rotating tops arranged in a hexagonal close packing, space group P63/mmc. The tops librate about axes perpendicular to the rotation axis with a r.m.s. amplitude of 7.33(3)°. The distances between the atomic planes perpendicular to the molecular axis agree well with earlier work.

Orientational disorder as a function of temperature in the clathrate structure of hydroquinone and C60

In the compound [C(6)H(6)O(2)](3)C(60), hydroquinone (C(6)H(6)O(2)) forms a three-dimensional hydrogen-bonded network enclosing roughly spherical cages with point symmetry 3;m and a diameter of 13.2 A at 100 K. Although C(60) fits tightly into these cages, it shows threefold orientational disorder, the molecular site symmetry being 2/m. The disorder has been studied with single-crystal Mo Kalpha X-ray data at four temperatures, 100, 200, 293 and 373 K. In the refinement, C(60) was restrained to the icosahedral molecular symmetry m3;5; and to rigid-body translational and librational displacements including third- and fourth-order cumulants to account for curvilinear atomic movements, R(|F|) = 3.2-4.7%. At 100 K, bond lengths in C(60) refine to the expected values 1.450 (1) and 1.388 (1) A. The ratio of these values increases with increasing temperature, but the radius of the molecule remains constant at 3. 537 (2) A. The r.m.s. libration amplitudes of C(60) are relatively small (5.5 degrees at 373 K) and the probability function of orientations of C(60) inside the cage shows large values only at the refined positions, indicating that the energy barrier of reorientation is large. Refinement of an ordered twinned structure was unsuccessful; the orientations of neighboring C(60) appear to be uncorrelated.