Vibrational anharmonicity constant

Vibrational anharmonicity constant Vibrational coordinates Internal coordinates Normal coordinates, dimensionless Mass adjusted Vibrational force constants "eAe A,s get Ri, r 0J, etc. Qr m-i ... 

It is instructive to consider two of the formulae for the spectroscopic constants in more detail, and for this we choose — and xrs for an asymmetric top, these being respectively the coefficients of (vT + )J, the vibrational dependence of the rotational constant, and (vr + i)(fs + i), the vibrational anharmonic constant quadratic in the vibrational quantum numbers. As for diatomic molecules these two types of spectroscopic constant provide the most important source of information on cubic and quartic anharmonicity, respectively. The formulae obtained from the perturbation treatment for these two coefficients in the effective hamiltonian are as follows ... 

A similar interaction would be observed between all Fermi polyads containing sets of vibrational levels related by the selection rule A tv = 2, A tv = +1, and the hamiltonian matrix should be diagonalized for each Fermi polyad without the use of perturbation theory. If, on the other hand, the interaction (63) were smaller, or the separation between the unperturbed levels were larger, the interaction could be treated by perturbation theory it can be shown that, in second-order perturbation theory, equation (63) would contribute a term to the vibrational anharmonic constants... 

The harmonic frequencies and the anharmonic constants may be obtained from experimental vibrational spectra, although their determination becomes difficult as the size of the system increases. In Table 1.10, we have listed experimental harmonic and anharmonic contributions to the AEs. These contributions may also be obtained from electronic-structure calculations of quadratic force fields (for harmonic frequencies) and cubic and quartic force fields (for anharmonic constants). For some of the larger molecules in Table 1.11, we have used ZPVEs calculated at the CCSD(T)/cc-pVTZ level or higher, see Ref. 12. In some cases, both experimental and theoretical ZPVEs are available and agree to within 0.3 kJ/mol [12, 57],... 

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. 

In Equation 5.34 to is the harmonic frequency, v the vibrational quantum number, and xe and ye the first and second anharmonicity constants (mass dependent, co x /(coxe) = X /X = il/il, l, and i are vibrational reduced masses). The ZPE(v = 0) contribution to RPFR through first order is thus... 

It would seem to us to be extremely interesting to investigate the influence of the cation on the anharmonicity constant of internal vibrations as seen, for example, for K2Cr04 and BaCr04 above. If there is no error in the measurements for chromate, then this influence found for Ba2+ is remarkably large. We hope to measure this effect systematically. 

Once again v is the vibrational quantum number with allowed values of 0, 1, 2,..., and xc and yc are anharmonicity constants characteristic of the molecule. 

The anharmonicity constant vexe is small compared to ve, but its effect increases as v increases, and the overtones deviate more and more from simple multiples of the fundamental frequency with increasing vSee Fig. 4.7. The infrared region extends from 10 to 14,000 cm-1 (7000 A). Molecular vibrational frequencies run from 100 to 4000 cm-1, so that the fundamental and lower overtones lie in the infrared region. 

Figure 4. Theoretical dependence of rate of IVR as a function of vibrational energy of model molecule CFCl2Br with given anharmonicity constant (from Ref. 1).

Finally, we should note that a number of formulas for anharmonic constants and vibration-rotation interaction constants for symmetric-top molecules given in the spectroscopic literature are incomplete. The problem is that relatively few complete anharmonic analyses have been carried out, and the available examples are not adequate to cover all combinations of degenerate and nondegenerate modes. A detailed discussion is given by Lee and co-workers [35]. 

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