Kraitchman equations

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. 

Let T(s) be the orthogonal transformation that diagonalizes nI1](s) with eigenvalues II w(s)- Kraitchman equates these eigenvalues to the experimental principal planar moments Pgu 1 exp(s) of the isotopomer s ... 

The other two principal planar moments and expressions for b and c are given by cyclic permutations of a, b, c. Equations of the form of Eq. (11) are known as the Kraitchman equations. Special cases arise for molecules with various symmetry elements. For substitution of an atom in a linear molecule, the magnitude of the a coordinate of the substituted atom is... 

For polyatomic molecules, even the simplest linear ones, there is no proof that the rg structure should be closer to equilibrium, but it has nevertheless been assumed to be the case. Because the Kraitchman equations rely most heavily on AI values upon isotopic substitution, it has seemed reasonable to suppose that even an incomplete cancellation of vibration-rotation terms would lead to a structure that was closer to equilibrium than the ro structures. For this reason, and because rg structures are simply not possible, the rg method has been used extensively to obtain structures for polyatomic molecules. Unfortunately, although the early work of Costain... 

The b coordinates of all three non-hydrogen atoms were imaginary when the Kraitchman equations were used to locate the position of the nuclei in isothiocyanic acid, HNCS [218]. To determine a reasonable structure for this planar molecule, the authors used the first moment condition for the b coordinates and the product of inertia condition. The second moment condition was used also but with the substitution moment of inertia 4 instead of the ground state moment /q. The authors argued that this procedure gives a more reliable structure instead of a hybrid, particularly since Watson... 

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. 

Instead of fitting the moments of inertia by means of the Kraitchman equations, the moments of inertia or the planar second moments may be fit directly by adjusting the bond distances and bond angles to give the best fit in the least squares sense. This may be done in a variety of ways, as indicated above. To obtain the values in the column headed p-Kr (rs) the parameters were adjusted to give a best fit to differences between the moments of inertia of the isotopically substituted species and the corresponding moments of the parent species. The structure labeled p-Kr (P1 s) is a best fit to differences in planar second moments. The numbers in the column labeled rQ (Ps) resulted from a direct fit of all... 

Two special procedures employed in structure calculations will be mentioned here. One is the double substitution method introduced by Pierce26 and used for the determination of the coordinates of atoms near a principal inertial plane, which is the situation for which the Kraitchman equations become unreliable. In Pierce s method rotational constants must be available for four isotopic species, two of which are... 

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 original rs-method is based on equations presented by Kraitchman [4], He had noticed that the numerically dominant part of the inertial moment tensor for an isotopomer in which (exactly) one atom has been substituted, can be replaced by the known inertial moments of the parent molecule, and that the remainder of the tensor then depends only on the Cartesian coordinates of the substituted atom. Equating the roots of the secular equation for this hybrid tensor to the experimental inertial moments of the isotopomer, Kraitchman obtained his famous equations which, quite in contrast to any r0-type method, always aim at the determination of the Cartesian coordinates of just this one substituted atom. The rest of the molecule is irrelevant, it does not influence the result and need not be known. Costain has checked, for several small molecules, the invariance of -determined bond lengths when employing different pairs of parent molecule and isotopomer [7]. He found it superior to the comparable r0-data and recommended to prefer -structures for... 

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). 

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. 

Typke has introduced the rs-fit method [7] where Kraitchman s basic principles are retained. A system of equations is set up for all available isotopomers of a parent (not necessarily singly substituted) and is solved by least-squares methods for the Cartesian coordinates (referred to the PAS of the parent) of all atomic positions that have been substituted on at least one of the isotopomers The positions of unsubstituted atoms need not be known and cannot be determined. The method is presented here with two recent improvements true derivatives are used for the Jacobian matrix X, and the problem of the observations and theircovariances, which is rather elaborate, is fully worked out. The equations are always given for the general asymmetric rotor, noting that simplifications occur in more symmetric situations, e.g. for linear molecules, which could nonetheless be treated within the framework presented. 

Equations 36 and 37 are the basis for Typke s rs-fit method [7], By Eq. 36, the eigenvalues 77y[Pg.81]

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]. 

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