Intemuclear distances thermal average

During the last decades, a large body of structural information has been derived from gas-electron diffraction studies. The corresponding results are nearly exclusively reported in the literature in terms of r distances, or the equivalent thermal average intemuclear distances, which are denoted r. The r distances are defined by the relation, r = r — If. Alternative methods for interpreting gas-electron diffraction data are possible, for example, in terms of -geometries5, but they are currently too complex to apply in routine stmctural analyses, because they require detailed information on the molecular potential energy surface which is not usually available. 

Equations (5.7)—(5.9) define distances between average nuclear positions. A different type of average is obtained when the intemuclear distances, not the positions, are averaged. The meaning of the subtle shift in language is clear when the mathematical relation is considered. Thermal average intemuclear distances, or r -parameters, are related to re in the following way ... 

The fa intemuclear distance in the expression of molecular intensities (vide supra) is an effective parameter without rigorous physical meaning. It is, however, related in a good approximation to the thermal average... 

Crystallographic studies provide two grades of intemuclear distances R, between nuclei at special positions (frequently with a precision 0.005 to 0.002 A) determined by geometric coefficients times the unit cell parameters, and between two nuclei which are both (or at least one) on general positions (typically 10 times less precise). It should be noted that R values with 4 or 5 decimals 48) usually are derived from micro-wave rotational spectra, especially of gaseous diatomic molecules, giving the time average of R 2. The thermal vibrations at room temperature frequently have an amplitude of 0.05 to 0.1 A which would only be decreased by a factor around 2 or 3 at the absolute zero. 

The resolution of intemuclear distances is limited by thermal vibration to only about 0.1 A and cannot be improved experimentally except in special cases, such as PF3, when nonbonded distances allow a resolution. Closely spaced, inequivalent distances are measured only as their weighted average values, their differences remaining very uncertain. 

Bond angle for indices, see r. For example, 9, 9 and represent angles defined by a set of three nuclear positions, equilibrium, average (ground vibrational state) and average (thermal equilibrium), respectively, and 9 by a set of three intemuclear distances r. Some electron diffraction papers report distances as and angles as 0 or 9 ... 

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