Thermal motion correction

In very precise crystal structure analyses, where the objective is to compare the geometry of the molecule in the crystal with that calculated by ab-initio quantum theory, thermal motion corrections are applied to give at rest values. The vibrations of rigid molecules in crystals consist of translational and librational modes... 

Deformation density maps have been used to examine the effects of hydrogen bonding on the electron distribution in molecules. In this method, the deformation density (or electrostatic potential) measured experimentally for the hydrogen-bonded molecule in the crystal is compared with that calculated theoretically for the isolated molecule. Since both the experiment and theory are concerned with small differences between large quantities, very high precision is necessary in both. In the case of the experiment, this requires very accurate diffraction intensity measurements at low temperature with good thermal motion corrections. In the case of theory, it requires a high level of ab-initio molecular orbital approximation, as discussed in Chapter 4. 

Hg(CN)2 in the solid state has a structure (I42d neutron diffraction), completely different from that of Cd(CN)2 Almost-linear molecules (r(Hg—C) 201.9, r(C—N) 116.0pm (corrected for thermal motion) a(C—Hg—C) 175.0°) are arranged such that four secondary bonds N" Hg (274.2 pm) yield the often-occurring 2 + 4 coordination around Hg.103 Analysis of the 199Hg MAS NMR spectrum of Hg(CN)2 has yielded the chemical shift and shielding tensor parameters.104... 

The last molecule considered in this parameterization study was 1,5,9,13-tetraazacyclohexadecane (7). A comparison between MM2 and X-ray results (see structure 7)5 reveals good fit between theory and experiment (the X-ray C—C bond lengths are shorter than the MM2 corresponding ones, partly since the data were collected at room temperature with no corrections for thermal motion). 

The conventional approach used to describe the response of a molecule to a static electric field is either to perform pure electronic BO calculations or to perform calculations where the BO values are corrected for vibrational and rotational (thermal) motion of the nuclei. The vibrationally corrected polarizabilities usually do an excellent job of correcting the errors inherent in the pure electronic BO values. Bishop has written several excellent reviews on this topic [78-80]. 

There are four independent molecules in the crystal of205 at 138 K two of them show rotational disorder about their central bonds3 8. Another phase formed on cooling consists of twinned crystals, which cannot be used for a structure determination318. Approximate Z)3h sysmmetry and bond lengths of about 1.60 and 1.53 A (corrected for the large thermal motion) in the ordered molecules at 138 K correspond to those in the gas phase. 

The object of a crystal-structure determination is to ascertain the position of all of the atoms in the unit cell, or translational building block, of a presumed completely ordered three-dimensional structure. In some cases, additional quantities of physical interest, e.g.. the amplitudes of thermal motion, may also be derived from the experiment. The processes involved in such crystal-structure determinations may he divided conveniently into (I) collection of the data. (2) solution of the phase relations among the scattered x-rays (phase problem)—determination of a correct trial structure, and (3) refinement of this structure. 

If Bj were purely a measure of thermal motion at atom j (and assuming that occupancies are correct), then in the simplest case of purely harmonic thermal motion of equal magnitude in all directions (called isotropic vibration), B, is related to the magnitude of vibration as follows ... 

The inter nuclear distance at which the energy minimum occurs defines the bond length. This is more correctly known as the equilibrium bond length, because thermal motion causes the two atoms to vibrate about this distance. In general, the stronger the bond, the smaller will be the bond length. 

In neutron diffraction structure analyses, the correction for thermal motion is much more complex for hydrogen atoms than for nonbydrogen atoms. It has long been recognized that vibrational thermal motion causes an apparent reduction in interatomic distances relative to those for the atoms at rest, and the methods for thermal motion analysis and geometry correction are well developed [190—192]. 

Fig. 3.2. Thermal ellipsoids (at 99% probability) for 1,2,4-triazole by neutron diffraction at 15 K illustrating the relative thermal motion of hydrogen and nonhydrogen atoms. That of the hydrogen bonded H(l) is only slightly less than that of H(3) and H(5), and the corrections of the X-H bond lengths are +0.005 A for N-H versus +0.006 for the C-H bonds at 15 K [199]...

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