Vibrational rotational interactions, effects

One very effective approach for partitioning the molecular Hamiltonian so as to simplify vibration-rotation interaction effects involves the use of classical rotational sudden approximations. These are primarily applicable to high-velocity collisions, and they have been applied quite effectively to the Li" + CO2 system.In... 

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. 

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... 

These early papers, as well as most of the theoretical work on the inversion of ammonia that has been done later, have considered the problem of the solution of the Schrddinger equation for a double-minimum potential function in one dimension and the determination of the parameters of such a potential function from the inversion splittings associated with the V2 bending mode of ammonia Such an approach describes the main features of the ammonia spectrum pertaining to the V2 bending mode but it cannot be used for the interpretation of the effects of inversion on the energy levels involving other vibrational modes or vibration—rotation interactions. 

E.E.Nikitin, Effect of the vibrational-rotational interaction in vibrational relaxation of diatomic molecnles, Kinetika i Kataliz, 3, 332 (1962)... 

The simple constant-effective-mass, quartic-quadratic Hamiltonian, Eqs. (3.22), (3.27), was found quite adequate to reproduce the observed far infrared transitions, account for the rotational constant variation [via Eq. (4.2)] and faithfully reproduce the 0—1 inversion splitting derived from the vibration-rotation interaction analysis. 

The vibration-rotation interaction is the effect arising from coupling terms between angular and vibrational momenta as well as from the dependence of the rotational G-matrix elements (the /u-tensor) on the internal coordinates. The importance of this effect may to some extent be reduced provided an appropriate axis convention is used. The axis convention is the set of rules defining the orientation of the molecular axes, eg, g = x,y, z, relative to an arbitrary configuration as given by the position vectors, Ra, a. = 1, 2,... N. These rules can be expressed in three relations between the rag components, similar to the center of mass conditions(2.4). We shall refer to these relations as the axial constraints . Usually Eckart-condi-tions39 are imposed, but other possibilities may be considered. 

The theory of vibration-rotation interactions has been developed over the last 50 years by many prominent researchers. It has been presented in many texts on the subject e.g. [10], among them a rather complete summary by Aliev and Watson [11]. It is based on classical perturbation theory in the form of a sequence of contact transformations. The results relevant to the rotational constants are summarized here. The effective rotational constant about the P axis in the vibrational state characterized by the vibrational quantum numbers v=(vi. .. vjt...) with degeneracies d. .. <4...), is given by [12]... 

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. 

Inertial defect - In molecular spectroscopy, the quantityfor a molecule whose equilibrium configuration is planar, where and / are the effective principal moments of inertia. The inertial defect for a rigid planar molecule would be zero, but vibration-rotation interactions in a real molecule lead to a positive inertial defect. 

The Coriolis coupling constants in Table 11 were derived from the effect of the zero point vibration on the rotational constants through vibration-rotation interaction [20] and from an harmonic force field [21] (symmetry of vibration A, E z C3 axis). 

For nonlinear molecules, the Hamiltonian including the effects of vibration-rotation interaction is more complicated, but well understood. 

Each of the partition functions is now regarded as a product of independent translational, rotational, and vibrational partition functions—the implication being that vibration-rotation interaction is negligible, and it is then assumed that the rotations are classical and the vibrations harmonic. If the structure of each molecular species is known, the moments of inertia can be calculated, and if necessary— as may well be the case for hydrogen isotopes—a correction can be applied to account for the fact that the rotational partition function has not reached its classical value. If complete vibrational analjrses of all the molecules are also available, the vibrational partition functions can be set up, and an approximate correction for neglect of anharmonicity can also be made. Having done all this, we can calculate the isotope effect. 

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