Anharmonicity in anisotropic displacement parameters

Abstract

A quasi-harmonic molecular-mean-®eld model for analyzing anharmonic temperature evolution of anisotropic displacement parameters is described. Anharmonic effects are taken into account through a Gru È neisen-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 Gru È neisen 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 ADP's due to insuf®cient extinction corrections in the diffraction data.

Dipole–Dipole Coupling Constant for a Directly Bonded CH Pair—A Carbon-13 Relaxation Study

“…The effective, vibrationally averaged DCC is analogous to the interaction-strength constant describing the dipolar interaction in solid state (20). The crystallographic data (44)(45)(46), the ab initio calculations (46), and the vibrational spectra of HMTA in solid and in solution (47)(48)(49) are all consistent with this picture. The lowest vibrational frequency in HMTA corresponds to a wave number of about 450 cm −1 (49).…”

Molecular Motions in Crystalline Anthracene and Naphthalene from Multitemperature Diffraction Data

“…3.5. This value is of the right order of magnitude, but somewhat higher than found previously for hexamethylenetetramine (2.3) [15]. It seems likely that part of the increase is due to the hydrocarbon nature of anthracene and naphthalene, and part to the limited temperature and resolution ranges covered by the respective diffraction data.…”

“…Here, Einsteinss idea is applied to molecular coordinates. It turns out to be a useful approximation, not only for librations and deformations that tend to show relatively little intermolecular coupling (low dispersion in the Brillouin zone), but also for the much more strongly coupled molecular translations (acoustic phonons) [14] [15].…”

“…Such a model must 1) account for temperature-dependent and -independent components of ADPs as well as for softening of the crystal field due to thermal expansion (anharmonicity), 2) be independent of force fields of unknown quality, 3) produce quantities, which may be compared to experimental information from other than diffraction experiments, and 4) be reasonably simple to use. A scheme fulfilling some of these conditions has recently been reported [11], implemented in a computer program [12] [13] and tested on benzene, urea [14], and hexamethylenetetramine [15]. Qualitative aspects of the scheme are summarized here; some algebraic details are given in the Experimental (see Models of Motion).…”

“…is a polynomial expansion of the unit-cell volume V in T, and V min is the unit-cell volume at the lowest available temperature [15]. 4) The temp.…”