The Asymmetric Rotor

Since asymmetric rotor molecules have three unequal moments of inertia, unless some symmetry can be found, the band contours for these tops are too complicated to permit any conclusions to be drawn concerning the structure of the molecule. 

For asymmetric rotors that have at least the symmetry associated with one of the groups C2t, D2, or Z 2fc, it is possible to recognize three types of band contours. These are called A, B, or C bands, depending on whether the change in dipole moment is along the 

Inversion doubling and intensity alternation can occur in the fine-line structure of asymmetric rotor molecules. If the fine lines of the bands were not perturbed or split, we should expect the interlinear spacing to be related to the three rotational constants of the molecule. Thus, for a C band of an asymmetric rotor in which the ratio of the two moments of inertia 7 /7 is nearly 1, the band would have the fine-line spacing 2A = 2B = 4C, where the three principal moments of inertia of the rotor are related so that A B C (7 Ip Iq) are the rotational constants. 

The presence of a band, with its distinctive shape, is often indicative of an asymmetric molecule. The band consists of a double Q-branch, separated by a maximum at the band center, and has P-and R-branches as well. If the Q-branches are weak, a series of four approximately equal maxima can be seen a P-branch, two Q-branches, and an R-branch. An example would be the band at 1247 cm for H2CO, which is also designated as a vibration for molecules of symmetry 2  

Finally, C bands can have a strong Q-branch, and P- and R-branches of moderate intensity. A band of this type occurs only for asymmetric rotors having more than three atoms. A typical C band is seen for H2CO at 1163.5 cm it is a b2 vibration for a molecule of C21 symmetry. 

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In a molecule such as the asymmetric rotor formaldehyde, shown in Figure 5.1(f), the a, b and c inertial axes, of lowest, medium and highest moments of inertia, respectively, are defined by symmetry, the a axis being the C2 axis, the b axis being in the yz plane and the c axis being perpendicular to the yz plane. Vibrational transition moments are confined to the a, b or c axis and the rotational selection mles are characteristic. We call them... 

Lars Schibbye became chief engineer of SRM in 1950 and contributed to the technical advancement of the compressor. Most. significant was the invention of the asymmetric rotor profile, which was introduced commercially by Sullair in the U.S. in 1969. The asymmetric rotor profile reduces the leakage path area and sealing line length resulting in increased efficiency. 

In Eqn. (5), the angular brackets impley averages over the asymmetric rotor wave-function as well as the vibrational wavefunction. Rz is the component of R along the space-fixed z-axis. The final step is to relate the coupling constants in Eqn. (5) to those of the monomer. In general, the expressions depend on the complexity of the monomers and on the dimer rotational state observed. For a large number of cases, a linear type dimer in a K=0 rotational state may be assumed, and Eqn. (5) may be expressed as... 

The matrixelements of this operator are given below for the most common case of the asymmetric rotor in the S representation. 

Neglecting the asymmetric rotor coupling term Aej rot), the adiabatic... 

N refers to rotational quantum number and r is the asymmetric rotor index. f(30). 

The problem was to connect the contours with the polarization properties of the transitions. Samuel Wait Jr (later Dean of Science at Rensselaer Polytechnic Institute) had joined me from RPI. He set out to compute band contours for the heaviest molecule for which such a calculation had been made. The work was at first done oil an IBM 650 by the method of iteration on the continued fraction for levels of the asymmetric rotor, and later on the IBM 704 by matrix diagonalization. 

Gwinn, W.D., Information pertaining to molecnlar stmcture, as obtained from the microwave spectra of molecules of the asymmetric rotor type. Discuss. Earaday. Soc. (19), 43, 1955. 

The microwave spectram exhibits a tunnelling sphtting due to a large amphtude internal rotation of the H2O subunit that exchanges bonded and non-bonded atoms. The internal rotor states are labelled with the asymmetric rotor quantum number for the rotational levels of free water jkak. ... 

A Side Views (3) of the Interaction of the Paired Diametrically Opposed Subunits with the Asymmetric Rotor... 

To determine the symmetry of the Si state, Christofferson et al. [7] analyzed the rotational structure of several of the bands. The appearance of the Og absorption band under high resolution is shown in the top portion of Fig. 7.7. In a molecule as large as aniline, such a contour comprises some 30,000 rotational transitions, and so there is little hope for resolving individual lines. Instead, the contour is compared with a computer simulation that calculates the asymmetric rotor energy level differences, weights the intensities of allowed rotational... 

T =the asymmetric rotor quantum number can also be expressed as K, ... 

No simple equation is presently known to describe the energy levels of the asymmetric rotor hence no simple expressions for the fine-line structure of absorption bands can be given. Several approximate equations have been suggested, but we shall not present them at this time. 

Some of the fine-line structure of the bands for H2CO can be analyzed by using the theory developed for symmetric rotors, but other structures require the asymmetric rotor theory. Generally, rotational levels identified with small values of K require consideration of the molecule as an asymmetric rotor. 

The general problem of finding the allowed energies and transition intensities of the asymmetric rotor... 

In what follows the states of the CO-H2S are labeled by KJKJC where K corresponds to the projection of /, the total angular momentum, onto the intermolecular axis, and is written E for K=<) (II forAT=l, and so on). mean the asymmetric-rotor labels for the correlated H2S free rotor states ... 

For a symmetric top, the molecule is designated a prolate or oblate rotor, depending on which inertia axis corresponds to the molecular symmetry axis. Most molecules belong to the asymmetric rotor case. 

FIGURE 5 A transition of the asymmetric rotor HDSe. Spectrai trace was recorded from ieft to right and extends approximately 170 MHz. The five major isotopic species of Se are cieariy evident. The spectrum was taken at the temperature of dry ice nevertheless, HDSe decomposes in the Stark ceii. This is evident from the decrease in iine intensity of Se compared with that of Se. 

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