Vibration-rotation interaction constants

The approximate symmetry of the band is due to the fact that Bi — Bq, that is, the vibration-rotation interaction constant (Equation 5.25) is small. If we assume that B = Bq = B and neglect centrifugal distortion the wavenumbers of the i -branch transitions, v[i (J)], are given by... 

If B can be obtained for at least two vibrational levels, say Bq and Bi, then B and the vibration-rotation interaction constant a can be obtained from Equation (5.25). Values for and a, together with other constants, are given for H CI in Table 6.2. 

From the following wavenumbers of the P and R branches of the 1-0 infrared vibrational band of H Cl obtain values for the rotational constants Bq, Bi and B, the band centre coq, the vibration-rotation interaction constant a and the intemuclear distance r. Given that the band centre of the 2-0 band is at 4128.6 cm determine cOg and, using this value, the force constant k. 

By obtaining values for B in various vibrational states within the ground electronic state (usually from an emission spectrum) or an excited electronic state (usually from an absorption spectrum) the vibration-rotation interaction constant a and, more importantly, B may be obtained, from Equation (7.92), for that electronic state. From B the value of for that state easily follows. 

In Equation 12.8 Be is the rotational constant, Be = h/(8jt2I), (I is the moment of inertia), coe is the vibrational frequency, 27T(oe = (k/ix)1, (k the vibrational force constant and x the reduced mass), re the equilibrium bond length (isotope independent to reasonable approximation), and ae is the vibration-rotation interaction constant... 

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]. 

A secondary motive is our general desire to verify and extend our understanding of vibration-rotation interactions in molecular spectra, and particularly to interpret data on different isotopic species in a consistent manner. Consider, for example, a constants (which measure the dependence of the rotational constant B on the vibrational quantum numbers vr) determined experimentally for several isotopic species of the same molecule. It is clear that these constants are not all independent, since they are related to the potential function which is common to all isotopic species. However, the consistency of the data and of our theoretical formulae can only be tested through a complete anharmonic force field calculation (there are at this time no known relationships between the a values analogous to the Teller-Redlich product rule). Similar comments apply to many other vibration-rotation interaction constants. 

More Complicated Molecules.—Calculations have been reported on a number of more complicated molecules, as indicated in Table 4. The work on BFa and SOa, and on NH3 and NF3, is of particular interest since these are the simplest symmetric top structures for which the calculation is practical, and for which there exist sufficient spectroscopic data to make it worthwhile for symmetric top molecules there are extra observable vibration-rotation interaction constants associated with the vibrational degeneracy that provide further information on the force field (see Table 3). Formaldehyde and ethylene, and their simple halogen derivatives, and also methane, are obvious candidates for further work. 

P. G. Szalay, J. Gauss, and J. F. Stanton, Theor. Chim. Acta, 100, 5 (1998). Analytic UHF-CCSD(T) Second Derivatives Implementation and Application to the Calculation of the Vibration-Rotation Interaction Constants of NCO and NCS. 

As pointed out by Gwinu and Gaylord19), the solutions of the eigenvalue problems associated with a vibration-rotation problem do not and must not depend on the choice of the rotating axis system as long as an adequate Hamiltonian is used. What do depend on the axis system used are the numerical values of elements of the inertial tensor or vibration-rotation interaction constants determined from analysing the data. 

The observed vibration frequencies of a molecule depend on two features of the molecular structure the masses and equilibrium geometry of the molecule and the potential eneigy surface, or force field, governing displacements from equilibrium. These are described as kinetic and potential effects, respectively for a polyatomic molecule the form and the frequency of each of the 3N—6 normal vibrations depend on the two effects in a complicated way. The object of a force field calculation is to separate these effects. More specifically, if the kinetic parameters are known and the vibration frequencies are observed spectroscopically, the object is to deduce the potential eneigy surface. A major difficulty in this calculation is that the observed frequencies are often insufficient to determine uniquely the form of the potential energy surface, and it is necessary to use data on the frequency shifts observed in isotopically substituted molecules or data on vibration/rotation interaction constants observed in high resolution spectra in order to obtain a unique solution. 

The relationship between the potential function K(R) and the observable spectroscopic parameters is summarized in Figure 2. The harmonic vibration frequencies are obtained as the eigenvalues of a secular determinant involving the quadratic force constants and the atomic masses and molecular geometry (the F and G matrices of Wilson s well-known formalism) by a calculational procedure discussed in detail by Wilson, Decius, and Cross.1 The eigenvectors determine the normal coordinates Q in terms of which the kinetic and quadratic potential energy terms are both diagonal (R = LQ). The various anharmonidty constants and vibration/rotation interaction constants are obtained in terms of the... 

Further information on the force field comes from the observation of vibration/rotation interaction constants. These are obtained from infrared and Raman spectra at high resolution and from microwave spectra observed in vibrationally excited states. They are of the following types ... 

Many interatomic distances and angles calculated by ah initio techniques have been reported in the recent literature. However, it should be emphasized that all data in this table are obtained from direct experimental measurements. In a few cases, ab initio calculations of vibration-rotation interaction constants have been combined with the primary experimental measurements to derive r values in the table. 

In favorable cases the analysis of the rotational spectrum of asymmetric molecules in the vibrational state Uj t j ajN 6 eillows the determination of the constants listed in this table. The vibration-rotation interaction constants must be determined by the analysis of at least two vibrational states of the same normal yibration. 

Harmonic force fields and harmonic frequencies of PHg were obtained by SCF MO [8,13,17] and CEPA calculations [8] for the transfer of force fields of other penta-coordinated molecules to PHg, see [26, 27]. A predicted IR gas-phase spectrum with rotational fine structure (at 300 K) is displayed in [17]. Anharmonicity constants and vibration-rotation interaction constants were derived from an anharmonic force field. From a scaling procedure, making use of harmonic and anharmonic force fields of PHg and PH3 (as reference molecule) the following wave numbers (in cm ) of the PHg fundamentals were predicted [18] (v = stretching, 6 = deformation, s = symmetric, as = antisymmetric, op = out-of-plane) ... 

The Dunham coefficients Yy are related to the spectroscopical parameters as follows 7io = cOe to the fundamental vibrational frequency, Y20 = cOeXe to the anharmonicity constant, Y02 = D to the centrifugal distortion constant, Yn = oie to the vibrational-rotational interaction constant, and Ym = / to the rotational constant. These coefficients can be expressed in terms of different derivatives of U R) at the equilibrium point, r=Re. The derivatives can be either calculated analytically or by using numerical differentiation applied to the PEC points. The numerical differentiation of the total energy of the system, Ecasccsd, point by point is the simplest way to obtain the parameters. In our works we have used the standard five-point numerical differentiation formula. In the comparison of the calculated values with the experimental results we utilize the experimental PECs obtained with the Rydberg-Klein-Rees (RKR) approach [58-60] and with the inverted perturbation approach (IPA) [61,62]. The IPA is method originally intended to improve the RKR potentials. 

Coriohs resonances. A simultaneous upper state analysis taking into account the Coriohs resonances is performed and the vibration-rotation interaction constants are determined [OOHeg]. 

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