Molecular rotation centrifugal distortion

All rotating molecules show the influence of molecular deformation (centrifugal distortion=c.d.) in their spectra. It is included in Eq. (5). Since there are different theoretical formulations of centrifugal distortion, a variety of centrifugal distortion constants appear in the table (column 5). For these constants the subscript V has been omitted. It is to be understood that the molecule is in the particular vibrational state Vj indicated in column 3 of the table. In most cases, v=0. 

The analyses of these LMR spectra included investigations into the effects of rotation, centrifugal distortion, and fine-structure interaction. The results led to very precise values for the respective molecular constants (see pp. 12, 15/7) [1 to 3]. 

The analysis of the rotational spectrum of an asymmetric molecule in the vibrational state ui,... vj,... v u-6 normally allows the determination of the constants listed in this table. All rotating molecules show the influence of molecular deformation (centrifugal distortion, c.d.) in their spectra. The theory of centrifugal distortion was first developed by Kivelson and Wilson [52Kiv]. The rotational Hamiltonian in cylindrical tensor form has been given by Watson [77 Wat] in terms of the angular momentum operators J, J/and as follows ... 

Values were also reported for the rotational constants, centrifugal distortion constants, and the chlorine nuclear quadrupole coupling constants of the three isotopic species F C1 02, F CF 02, and i F CF 0 0. The molecular dipole moment was found to be 1.722 0.03 D. 

Since even very low levels of theoiy can give fairly accurate geometries, rotational spectra are quite simple to address computationally, at least over low rotational quantum numbers. For higher-energy rotational levels, molecular centrifugal distortion becomes an issue, and more sophisticated solutions of Eq. (9.37) are required. 

Absorption of microwave radiation to excite molecular rotation is allowed only if the molecule has a permanent dipole moment. This restriction is less severe than it may sound, however, because centrifugal distortion can disturb the molecular symmetry enough to allow weak absorption, especially in transitions between the higher rotational states which may appear in the far IR (c. 100cm-1). Microwave spectroscopy can provide a wealth of other molecular data, mostly of interest to physical chemists rather than inorganic chemists. Because of the ways in which molecular rotation is affected by vibration, it is possible to obtain vibrational frequencies from pure rotational spectra, often more accurately than is possible by direct vibrational spectroscopy. 

It is now possible to determine precise rotational spectra for hydrogen bonded molecules of moderate size and with even very small stabilization energies. Rotational constants, centrifugal distortion constants, electric dipole moments and nuclear hyperfine interactions have been measured for a considerable number of dimers using various microwave and molecular beam techniques. 

Table 8.16. Molecular parameters for H35 Cl determined from the electric resonance spectra. All parameters are in kHz except for the electric dipole moment, n(v), which is in Debye units (D). The rotational and centrifugal distortion constants were obtained by Rank, Rao and Wiggins [102], S]2 is the spin-spin interaction constant, equal to [gig2ii2N(no/47t)/(2J + 3)(2J - 1)](R 3)vj...

It should be noted that the rotational spectroscopy of CO confined to a single vibrational level, usually the ground v = 0 level, provides only a limited amount of information about molecular structure. In the field of vibration-rotation spectroscopy, however, CO has been studied extensively and particular attention paid to the variation of the rotational and centrifugal distortion constants with vibrational quantum number. Vibrational transitions involving v up to 37 have been studied with high accuracy [78, 79, 80], and the measurements extended to other isotopic species [81] to test the conventional isotopic relationships. CO is, however, an extremely important and widespread molecule in the interstellar medium. CO distribution maps are now commonplace and with the advent of far-inffared telescopes, it is also an important... 

The microwave experiment studies rotational structure at a given vibrational level. The spectra are analyzed in terms of rotational models of various symmetries. The vibration of a diatomic molecule is, for instance, approximated by a Morse potential and the rotational frequencies are related to a molecular moment of inertia. For a rigid classical diatomic molecule the moment of inertia I = nr2 and the equilibrium bond length may be calculated from the known reduced mass and the measured moment, assuming zero centrifugal distortion. 

As 1 is a nonpolar symmetric top with symmetry, it should have no pure rotational spectrum, but it acquires a small dipole moment by partial isotopic substitution or through centrifugal distortion. In recent analyses of gas-phase data, rotational constants from earlier IR and Raman spectroscopic studies, and those for cyclopropane-1,1- /2 and for an excited state of the v, C—C stretching vibration were utilized Anharmonicity constants for the C—C and C—H bonds were determined in both works. It is the parameters, then from the equilibrium structure, that can be derived and compared from both the ED and the MW data by appropriate vibrational corrections. Variations due to different representations of molecular geometry are of the same magnitude as stated uncertainties. The parameters from experiment agree satisfactorily with the results of high-level theoretical calculations (Table 1). 

Due to the rotational structure as well as the so-called hot hand absorptions (Sec. 2.5.3), the contour of a rovibrational band depends on the temperature. Today it is possible to determine molecular constants such as moments of inertia, Coriolis coupling constants, centrifugal distortion constants, and anharmonicity coefficients by FTIR as precisely as possible in order to calculate the intensity and shape of an absorption band. In such a simulation process the temperature may be used as a parameter. The results can be compared to the experimental spectra and the temperature may be deduced by fitting the calculated to the observed bands. This is possible with IR as well as with Raman bands. A review of curve fitting procedures and their limitations has been given by Maddams (1980). 

Although centrifugal distortion is not a perturbation effect, a derivation of the form of the centrifugal distortion terms in Heff provides an excellent illustration of the Van Vleck transformation. If the vibrational eigenfunctions of the nonrotating molecular potential, V(R) rather than [V(R) + J(J + 1)H2/2/j,R2, are chosen as the vibrational basis set, then the rotational constant becomes an operator,... 

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