Equivalent isotopomers

Ethylene was one of the first systems subjected to detailed vibrational analysis using HOCM modified to account for lattice anharmonicity. Agreement with experiment is excellent (Fig. 5.5). The differences in the VPIE s of the equivalent isotopomers cis- trans-, and gem-dideuteroethylene (Fig. 5.6) are of considerable interest since they neatly demonstrate the close connection between molecular structure and isotope chemistry. The IE s are mainly a consequence of hindered rotation in the liquid (moments of inertia for cis-, trans-, and gem-C2D2H2 are slightly... 

Fig. 5.6 Vapor pressure differences between equivalent isotopic isomers of dideuteroethylene. The three mercury levels shown (left to right) measure the vapor pressures of ordinary ethylene C2D4, /ram-ethylene-1,2dj, and ei.v-cthylene- l dj. The vapor pressure of gem-ethylene-1,ld2 is very similar to the cis-isotopomer and is not shown in this photograph (Bigeleisen, J., Science 147, 463 (1965))...

Commonly encountered cubic equations of state are classical, and, of themselves, cannot rationalize IE s on PVT properties. Even so, the physical properties of iso-topomers are nearly the same, and it is likely in some sense they are in corresponding state when their reduced thermodynamic variables are the same that is the point explored in this chapter. By assuming that isotopomers are described by EOS s of identical form, the calculation of PVT isotope effects (i.e. the contribution of quantization) is reduced to a knowledge of critical property IE s (or for an extended EOS, to critical property IE s plus the acentric factor IE). One finds molar density IE s to be well described in terms of the critical property IE s alone (even though proper description of the parent molar densities themselves is impossible without the use of the acentric factor or equivalent), but rationalization of VPIE s requires the introduction of an IE on the acentric factor. 

The method described above can be applied to isotopomer pairs for which critical property IE data exists or can be estimated. Calculated values of ln(p7p) are insensitive to IE s on the acentric factor, Aoo/oo (equivalently Aa/a). The VPIE, on the other hand, is strongly dependent on Aoo/oo. For 3He/4He and H2/D2 critical property IE data are complete and MpIE and VPIE are available across the entire liquid range, are one to two orders of magnitude larger, and known to better precision than for other pairs (save perhaps H2O/D2O). For heavier pairs critical property IE data are usually incomplete or uncertain, and often data on MpIE and VPIE exist only over a limited temperature range. 

For di- and triethylene lithium the use of isotopic mixtures enabled the authors to show that the Li(C2H4)(C2D4) and Li(C2H4)2(C2D4) species have only one isotopomer and therefore two and three equivalent ligand molecules (Scheme 4). 

Once again (cf equation 205), the behavior of tetrabutylammonium fluoride (TBAF) is exceptional. Thus, the treatment of (E)-1 -decenyl(phenyl)iodonium tetrafluoroborate with ten equivalents of TBAF in acetonitrile leads exclusively to 1-decyne (equation 212)146. Furthermore, the reactions of the a- and /1-deuterio isotopomers of the decenyliodonium salt with one equivalent of TBAF in dichloromethane are clearly consistent with alkyne formation via the a-elimination-alkylidenecarbene pathway (equations 213 and 214)146. 

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. 

The expression is easily coded, since T<0>, Eq. 41, and the r 1 are known. It simplifies for substitution on a principal plane or axis and for symmetrically equivalent multiple substitution, because several of the elements of the matrix T will then vanish. It is clear from Eq. 45 that the derivatives are nonvanishing only for those atoms a that have actually been substitued in the particular isotopomer s. Therefore, the Jacobian matrix X generated from these derivatives is, in general, a sparse matrix. 

Although the r/E-fit and the p-Kr r, (-rM) fit are not equivalent (the former determines three more variables), it could be shown [55] that the molecular structures determined by the r/e-fit and the r -fit are strictly identical, including the covariance matrix. This is due to the specific form of the Jacobian matrix X of the coupled least-squares problem r/ , which permits a decomposition by a non-singular transformation into a smaller least-squares problem rM plus a subsequent direct calculation of the constant rovib contributions Eg. The r -part of the problem alone determines the molecular structure which must then be used (including the covariance matrix of the structural parameters) for the calculation of the contributions Eg. When rotational constants of new isotopomers are to be predicted from the structure determined, the r/E-method performs much better than the r -method due to the presence of the additional rovib parameters . ... 

However, we must also consider the minor isotopomers containing two Hg atoms, [ Hg- Hg-Hg]2+ and [ Hg-Hg- Hg]2+. The latter is not detected because it contains two equivalent nuclei whose resonances will be coincident with those of [ Hg-Hg-Hg]2+. In contrast, the nonequivalent spins in the remaining isotopomer, [ Hg- Hg-Hg]2+, are expected to couple strongly with each other. The direct Hg- Hg coupling (139 600 1000 Hz ) is large compared to the difference in chemical shifts ( 45 000 Hz), so an AB spin system is formed. The lines at —1400 and -1550 8 are the intense inner transitions of the AB spin system, while the two outer transitions are well outside the spectral width and are not observed. This is believed to be the first observation of direct Hg- Hg coupling. 

The geometry and the force constants of a molecule determine the isotopic shifts of the vibrational frequencies. However, the products of the vibrational frequencies of two different isotopomers within the same irreducible representation are related by the Teller-Redlich product rule. For two symmetry-equivalent vibrations the following expression is obtained ... 

The carbon skeleton of isoprenoids is derived from the branched C5 skeleton of isoprene. Isopentenyl diphosphate and dimethylallyl diphosphate represent the biologic equivalents of isoprene. From research on cholesterol biosynthesis in liver tissues and on ergosterol in yeast, mevalonate was accepted as the universal precursor of isoprenoids. However this assertion is inaccurate. Incorporation of labeled acetate and glucose isotopomers into bacterial isoprenoids and into diterpenes of ginkgo embryos indicated fortuitously the existence of an alternative mevalonate-independent route. Its full elucidation required experiments using and H-labeled precursors followed by extensive nuclear magnetic resonance analyses as well as a combination of biochemical and molecular biology methods. These additional studies revealed a complete set of novel unsuspected enzymes. 

The spectra of such metals in clusters, decoupled from all other nuclei, appear as the superposition of the different subspectra arising from the different isotopomers. The situation is slightly different for elements with one isotope such as Pt and those with two or more active isotopes such as ii7/ii9sn. In the latter case non-equivalent spin systems are obtained in highly synunetric molecules, enabling a complete determination of the NMR parameters in such molecules. 

The CAS Registry numbers for the carbonyl halides, and their principal isotopomers, are given in Table A3.1. Synonyms and foreign language equivalents for phosgene are given in Table A3.2, and for carbonyl difluoride in Table A3.3. 

Berry and Harmony [7]. What is required is that the values for those isotopomers of the SDS involving H D substitutions be corrected by an amount that is equivalent to a translation of the D atom by a distance of Sr along die C-D axis as shown in Fig. 4 for the linear and for any general molecule. In the linear molecule case, for the substitution of H by D (as in HCN, say), die correction takes the form... 

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... 

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