Kraitchman relations

An improvement on the rg structure is the substitution structure, or structure. This is obtained using the so-called Kraitchman equations, which give the coordinates of an atom, which has been isotopically substituted, in relation to the principal inertial axes of the molecule before substitution. The substitution structure is also approximate but is nearer to the equilibrium structure than is the zero-point structure. 

In mechanics, the inertial properties of a rotating rigid body are fully described by its inertial moment tensor I. We can simplify the subsequent equations if we employ in place of I the closely related planar moment tensor P, apparently first used by Kraitchman [4], At any stage of the calculations, however, an equivalent equation could be given which involves I instead of P. The principal planar moments P (g = x, y, z) are the three eigenvalues of the planar moment tensor P and the principal inertial moments Ig the eigenvalues of the inertial moment tensor I. Pg, Ig, and the rotational constants Bg = f/Ig are equivalent inertial parameters of the problem investigated (/conversion factor). 

In Watson s treatment, these small or imaginaiy coordinates are not dropped, set equal to zero, or recomputed by some other method (such as COM relation) in fact, the values from Kraitchman s equations (Eq. (31) for the linear molecule) provide the essential mass-dependent vibration-rotation contributions to the moments of inertia. 

According to Rudolph [33], structures determined by fitting structural parameters directly to the experimentally determined rotational constants Bq, the moments of inertia Iq or the planar moments Pq of a series of isotopomers are called rg structures. If the covariances of the observables are transformed correctly, the structures resulting fi-om any of these fits are identical. However, if covariances are not transformed, these structures are not identical nor equivalent (for examples, see [34]). Therefore, one should always specify what kind of covariance matrix is used with a particular fit. Pseudo-Kraitchman (p-Kr) or pseudo-r fits are those in which the structures are fit to differences of rotational constants, moments of inertia or planar moments. The structures called and again are equivalent to each other if the covariances are transformed correctly. On the other hand, the structure is theoretically different fi om the and structures because of a lack of a linear relation between the AB and A/. In practice, structures are very close to other p-Kr structures. The rig structure is obtained when the isotopic moments Iq are fitted to structural parameters and three e parameters. It is... 

The second common type of operationally defined structure is the so-called substitution or rt structure.10 The structural parameter is said to be an rs parameter whenever it has been obtained from Cartesian coordinates calculated from changes in moments of inertia that occur on isotopic substitution at the atoms involved by using Kraitchman s equations.9 In contrast to r0 structures, rs structures are very nearly isotopically consistent. Nonetheless, isotope effects can cause difficulties as discussed by Schwendeman. Watson12 has recently shown that to first-order in perturbation theory a moment of inertia calculated entirely from substitution coordinates is approximately the average of the effective and equilibrium moments of inertia. However, this relation does not extend to the structural parameters themselves, except for a diatomic molecule or a very few special cases of polyatomics. In fact, one drawback of rs structures is their lack of a well-defined relation to other types of structural parameters in spite of the well-defined way in which they are determined. It is occasionally stated in the literature that r, parameters approximate re parameters, but this cannot be true in general. For example, for a linear molecule Watson12 has shown that to first order ... 

An important advance in structure determination was made by Kraitchman in 1953.14 He showed that the coordinates of an atom that was isotopically substituted in a molecular species—referred to as the parent species-are related to the principal second moments by expressions like the following ... 

It is nowadays quite unusual to need to relate a molecular geometry to its inertial constants from first principles. When this is essential, procedures using the Kraitchman isotopic substitution equations, described in the reference texts, tend to give best results. However, extensive material describing structures obtained by electron diffraction as well... 

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