Vibrational/rotational interaction

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. 

Herman, R., and Wallis, R. F. (1955), Influence of Vibration-Rotation Interaction on Line Intensities in Vibration-Rotation Bands of Diatomic Molecules, 7. Chem. Phys. 23, 637. 

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... 

Recent microwave data for the potential interstellar molecule Sis is used together with high-level coupled-cluster calculations to extract an accurate equilibrium structure. Observed rotational constants for several isotopomers have been corrected for effects of vibration-rotation interaction subsequent least-squares refinements of structural parameters provide the equilibrium structure. This combined experimental-theoretical approach yields the following parameters for this C2v molecule re(SiSi) = 2.173 0.002A and 0e(SiSiSi) = 78.1 O.2 ... 

The purpose of this report is to demonstrate the ease with which highly accurate equilibrium structures can be determined by combining laboratory microwave data with the results of ab initio calculations. In this procedure, the effects of vibration-rotation interaction are calculated and removed from the observed rotational constants, Aq, Bq and Cq. The resulting values correspond to approximate rigid-rotor constants and and are thus inversely... 

Five isotopomers of Sia were studied in Ref (20), and are labeled as follows Si- Si- Si (I) Si- Si- Si (II) Si- Si- Si (III) Si- "Si- Si (IV) Si- Si- °Si (V). Rotational constants for each (both corrected and uncorrected for vibration-rotation interaction) can be found towards the bottom of Table I. Structures obtained by various refinement procedures are collected in Table II. Two distinct fitting procedures were used. In the first, the structures were refined against all three rotational constants A, B and C while only A and C were used in the second procedure. Since truly planar nuclear configurations have only two independent moments of inertia (A = / - 4 - 7. = 0), use of B (or C) involves a redundancy if the other is included. In practice, however, vibration-rotation effects spoil the exact proportionality between rotational constants and reciprocal moments of inertia and values of A calculated from effective moments of inertia determined from the Aq, Bq and Co constants do not vanish. Hence refining effective (ro) structures against all three is not without merit. Ao is called the inertial defect and amounts to ca. 0.4 amu for all five isotopomers. After correcting by the calculated vibration-rotation interactions, the inertial defect is reduced by an order of magnitude in all cases. 

In a previous study of cyclic SiCs, a residual inertial defect of only slightly smaller magnitude was found, despite the fact that an extremely high level of calculation (surpassing that in the present study) was used to determine the vibration-rotation interaction contributions to the rotational constants. This was subsequently traced to the so-called electronic contribution, which arises from a breakdown of the assumption that the atoms can be treated as point masses at the nuclear positions. Corrections for this somewhat exotic effect were carried out in that work and reduced the inertial defect from about 0.20 to less than 0.003 amu A. However, the associated change in the rotational constants had an entirely negligible effect on the inferred structural parameters. Hence, this issue is not considered further in this work. 

The vibration-rotation interaction term makes the Hamiltonian for nuclear motion of a polyatomic molecule difficult to deal with. Frequently, this term is small compared to the other terms. We shall make the initial approximation of omitting Tvib rot. The rotational kinetic energy TTOt involves the moments of inertia of the molecule, which in turn depend on the instantaneous nuclear configuration. However, the vibrational motions are much faster than the rotational motions, so that we can make the approximation of calculating the moments of inertia averaged over the vibrational motions. 

Vibration-rotation interaction causes the rotational constants to vary with the vibrational quantum numbers [Eq. (5.72)]. Correction terms for centrifugal distortion can also be added to the energy expressions for asymmetric tops. 

Anharmonkity, Vibration-Rotation Interaction, and Centrifugal Distortion... 

ANHARMONICITY, VIBRATION-ROTATION INTERACTION, AND CENTRIFUGAL DISTORTION... 

Whereas for diatomic molecules the vibration-rotation interaction added only a small correction to the energy, for a number of polyatomic molecules the vibration-rotation interaction leads to relatively large corrections. Similarly, although the Born-Oppenheimer separation of electronic and nuclear motions holds extremely well for diatomic molecules, it occasionally breaks down for polyatomic molecules, leading to substantial interactions between electronic and nuclear motions. 

In linear molecules, the electronic-rotation interaction terms in H cause the A-type doubling of electronic states, whereas the vibration-rotation interaction terms in H cause the /-type doubling of vibrational states. In addition, the perturbation H can cause interactions between vibration-rotation levels of different electronic states. If it happens that two vibration rotation levels of different electronic states of a molecule have... 

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]. 

Energy near-resonance and favorable overlap of vibrational states are the dominant factors affecting the magnitudes of the charge-transfer cross sections in the AB + -AB systems. It was found188 that an adequate theoretical treatment of the H2+ -H2 system necessitated inclusion of the effects of vibration-rotation interaction in calculating vibrational overlaps from accurate vibrational wave functions. Charge-transfer cross sections were thus computed as a function of different vibrational and rotational levels of the incident-ion species. 

Temperature from Rotational and Vibrational Raman Scattering Effects of Vibrational-Rotational Interactions and Other Corrections... 

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. 

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