Isotopic anharmonicity

The reason that does not change with isotopic substitution is that it refers to the bond length at the minimum of the potential energy curve (see Figure 1.13), and this curve, whether it refers to the harmonic oscillator approximation (Section 1.3.6) or an anharmonic oscillator (to be discussed in Section 6.1.3.2), does not change with isotopic substitution. Flowever, the vibrational energy levels within the potential energy curve, and therefore tq, are affected by isotopic substitution this is illustrated by the mass-dependence of the vibration frequency demonstrated by Equation (1.68). 

It is instructive to calculate the anharmonic correction to the zero point energy contribution to fractionation factors for isotope exchange equilibria involving hydrogen and deuterium. For example consider the exchange... 

Isotope effects on anharmonic corrections to ZPE drop off rapidly with mass and are usually neglected. The ideas presented above obviously carry over to exchange equilibria involving polyatomic molecules. Unfortunately, however, there are very few polyatomics on which spectroscopic vibrational analysis has been carried in enough detail to furnish spectroscopic values for Go and o)exe. For that reason anharmonic corrections to ZPE s of polyatomics have been generally ignored, but see Section 5.6.3.2 for a discussion of an exception also theoretical (quantum package) calculations of anharmonic constants are now practical (see above), and in the future one can expect more attention to anharmonic corrections of ZPE s. 

Wolfsberg, M. Correction to the effect of anharmonicity on isotopic exchange equilibria in Isotope Effects in Chemical Processes, in Spindel, W., ed. Adv. Chem. Ser. 89, 185 (1969). 

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... 

Both and 0 are commercially available, and isotopic substitution has been used in metal carbonyl spectroscopy in two totally different ways. In studies that attempt to correct for the effects of anharmonicity (see II.7. below), sub-... 

Keywords Anharmonic effects Displacive phase transition Isotope effects KDP-type ferroelectrics Order-disorder phase transition... 

Within the harmonic approximation the choice of a system of internal coordinates is irrelevant provided they are independent and that a complete potential function is considered ). For example, the vibrations of HjO can be analysed in terms of valence coordinates (r, >2, or interatomic coordinates (r, r, 3) and any difference in the accuracy to which observed energy levels are fitted (considering all the isotopic species H2O, HDO and D2O) will be due to the neglect of anharmonic terms. If one makes the approximation of a diagonal force field so that one is comparing the two potentials... 

To obtain the anharmonic terms in the potential, on the other hand, the choice of coordinates is important 130,131). The reason is that the anharmonic terms can only be obtained from a perturbation expansion on the harmonic results, and the convergence of this expansion differs considerably from one set of coordinates to another. In addition it is usually necessary to assume that some of the anharmonic interaction terms are zero and this is true only for certain classes of internal coordinates. For example, one can define an angle bend in HjO either by a rectilinear displacement of the hydrogen atoms or by a curvilinear displacement. At the harmonic level there is no difference between the two, but one can see that a rectilinear displacement introduces some stretching of the OH bonds whereas the curvilinear displacement does not. The curvilinear coordinate follows more closely the bottom of the potential well (Fig. 12) than the linear displacement and this manifests itself in rather small cubic stretch-bend interaction constants whereas these constants are larger for rectilinear coordinates. A final and important point about the choice of curvilinear coordinates is that they are geometrically defined (i.e. independent of nuclear masses) so that the resulting force constants do not depend on isotopic species. At the anharmonic level this is not true for rectilinear coordinates as it has been shown that the imposition of the Eckart conditions, that the internal coordinates shall introduce no overall translation or rotation of the body, forces them to have a small isotopic dependence 132). 

Fig. 8. An overview of the zero order energy levels of the doubly isotopically tagged helix. The isotope shifts are Sa (13C=I80) and Sb (l3C=l80) while Aa and Ab are diagonal anharmonicities. The shaded areas represent the helix one- and two-exciton bands that become perturbed by the isotopes. The solid (bra evolution) and dashed (ket evolution) arrows represent one of the Liouville paths contributing to the echo.

FTIR reporter probe, 300 carbon dioxide as a, 303 anharmonicity and, 305 bending frequency and, 306 electrolytic coupling and, 307 intermolecular forces and, 305, 346 intrinsic frequency of, 306 l80C,60 isotopic, 307 peroxide related to, 303 polarizability and, 307 windows and, 301... 

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