Kraitchman’s equations

To obtain molecular dimensions the moments of inertia of isotopic species must be determined. These (most conveniently) are sufficient in their natural abundance. Of particular importance here is 13C, but other atoms such as 2H, lsO and 15N require isotopic enrichment. Application of Kraitchman s equations (53MI20400) then yields atomic coordinates to a few thousandths of an Angstrom and bond angles to an accuracy of better than... 

The basic equations of the -method will be presented later within the framework of the more general r -fit problem. A rigid mass point model, which is strictly true only for the equilibrium configuration, is assumed. The application of Kraitch-man s equations (see below) to localize an atomic position requires (1) the principal planar moments (or equivalent inertial parameters) of the parent or reference molecule with known total mass, and (2) the principal planar moments of the isotopomer in which this one atom has been isotopically substituted (with known mass difference). The equations give the squared Cartesian coordinates of the substituted atom in the PAS of the parent. After extracting the root, the correct relative sign of a coordinate usually follows from inspection or from other considerations. The number, identity, and positions of nonsubstituted atoms do not enter the problem at all. To determine a complete molecular structure, each (non-equivalent) atomic position must have been substituted separately at least once, the MRR spectra of the respective isotopomers must all have been evaluated, and as many separate applications of Kraitchman s equations must be carried out. 

Since the early investigations of Costain, the restructure has been preferred over the restructure because of its better consistency and because small partial structures can be determined from a very limited SDS, neglecting the rest of the molecule. However, the limitation of Kraitchman s equations to a substitution set of singly substituted (or symmetrically disubstituted) isotopomers made it desirable to have a method which could also include isotopomers that do not satisfy these restrictive requirements, and which could also handle the problem of small coordinates in a more balanced fashion than the r -method. The rs-fit method, described in the preceding chapter, offers greater flexibility than Kraitchman s equations, but can, in principle, not locate unsubstituted atoms. 

Calculating an effective or r0-structure introduced, for the first time, least-squares fitting into the determination of molecular structure from MRR spectra. Unpublished computer programs for this purpose must have been in use by various groups for a long time. A more widely used program with many options, STRFTQ, has been coded by Schwendeman [6], It uses internal coordinates and fits inertial or planar moments, but also isotopic differences of moments. When the differences Pg exp(.v) - Pg exp( 1) are fitted, the structure is called a pseudo-Kraitchman or p-Kr structure, because these differences are the quantities that also dominate Kraitchman s equations. Therefore, the p-Kr structure should approach the restructure. By the criterion adhered to in the present paper (see preceding section), the p-Kr structure is a r0-type structure. 

As discussed in the preceding section, the r5-method (Kraitchman s equations or the rs-fit) is practically insensitive to a (small) change of a planar moment component, as represented by the rovib contributions, if the change is common to both the parent and the isotopomer s, and if the rotation by the diagonalizing matrix T(s) (Eq. 41) is small. Therefore, it usually does not matter whether experimental ground state planar moments F exp(s) or, after the T s have been obtained from a preceding... 

The basic principles of the rm-method can be exemplified most clearly by the case of the diatomic molecule, a = 1,2, where, by Kraitchman s equation in the simple form for linear molecules, the equilibrium and the substitution coordinates with respect to the PAS of the parent, xea and x, respectively, are given by,... 

More complete discussions of Kraitchman s equations can be found in standard treatises [16]. In addition, analogous equations are available for multiple isotopic substitution of equivalent atoms, e.g., H2O to D2O [17]. 

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. 

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 ... 

The principal application of the Kraitchman equations [Eq. (9)1 is for the determination of the atomic coordinates, at, bSi and cs. From a study of the rotational spectrum of the parent and of a species with single isotopic substitution the coordinates of the substituted atom may be determined. These coordinates are referred to as substitution coordinates or rs coordinates. Each new species yields new coordinates, and since all of the coordinates are in the same coordinate system, the calculation of substitution or rs bond distances and bond angles is a simple process. Costain,s demonstrated that there are definite advantages to the use of the Kraitchman equations to obtain molecular parameters. These advantages are sufficient to make the use of Kraitchman s equations the preferred method of structure determination from ground-state rotational constants. 

TABLE 4 Comparison of the c Coordinates of the Cl Atom and the Out-of-Plane C atom of 2-Chloropropane Determined by Kraitchman s Equations or by Assuming = 0 and Lmfljct = 0 ... 

Tj Effective distance derived from rotational constants via Kraitchman s equations for a sequence of isotopic substitutions. 

In the scheme the coordinates of an atom located far from a principal inertial plane can be determined accurately, whereas those of an atom located close to an inertial plane are poorly defined, irrespective of the atomic mass, hi the latter case the relative signs of the coordinates are difficult to determine, because Kraitchman s equations give only the absolute values. For small coordinates, doubly-substituted... 

Costain s contribution was to show that by using effective moments of inertia for the vibrational ground state in Kraitchman s equations, instead of the equilibrium moments which are strictly required, useful structural parameters are obtained. The chief characteristic of the rs parameters is that they are to a high degree of accuracy independent of the isotopic species used to determine them. Thus one of the main objections to the Vq structure , its variability according to the isotopic species used to calculate it, is removed in the rs structure. Apart from this, its chief advantage is that it is very easy to calculate. 

The second, and more important, objection to the r structure is that its physical significance is difiScult to assess. This is because the rs structure is defined only in terms of the operations needed to calculate it, Le. through equations of the type (23) (Kraitchman s equations). The effective moments of inertia which in the r method are inserted into Kraitchman s equations involve averages of a complicated kind over the vibrational motions of the molecule, with both harmonic and anharmonic terms playing a part. Only for the diatomic molecule is the situation clear-cut here it is easy to show that ... 

With more complicated molecules, additional isotopic data are needed. For a linear molecule such as XYZ, the moments of inertia for two molecular species and the expression from Table n give two equations to be solved for the two bond lengths. Alternately, the coordinates of, for example, the X-atom zx, that is, the distance from the center of mass, can be evaluated from Kraitchman s equation (see Section VIII.D) using isotopic data from X YZ. Subsequently, this coordinate can be used in the moment-of-inertia and first-moment equations for the XYZ species. 

Substitution structures involve the use of Kraitchman s equations, which provide the position of an atom in a molecule utilizing the changes in moments of inertia from isotopic substitution. One isotopic form is selected as the parent molecule, and Kraitchman s equations give coordinates of the isotopically substituted atom in the center-of-mass principal axis system of the parent molecule. For diatomic or linear molecules, BCraitchman s equation has the form... 

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