Spherical-top molecules

For molecules that are non-linear and whose rotational wavefunctions are given in terms of the spherical or symmetric top functions D l,m,K, the dipole moment Pave can have components along any or all three of the molecule s internal coordinates (e.g., the three molecule-fixed coordinates that describe the orientation of the principal axes of the moment of inertia tensor). For a spherical top molecule, Pavel vanishes, so El transitions do not occur. 

Kluk E., Monkos K., Pasterny K., Zerda T. A means to obtain angular velocity correlation functions from angular position correlation functions of molecules in liquid. Part I. General discussion and its application to linear and spherical top molecules, Acta Physica Polonica A 56, 109-16 (1979). 

Courtney J. A., Armstrong R. L. A nuclear spin relaxation study of the spin-rotation interaction in spherical top molecules. Can. J. Phys. 50, 1252-61 (1972). 

Eagles T. E., McClung R. E. D. Rotational diffusion of spherical top molecules in liquids and gases. IV. Semiclassical theory and applications to the v3 and v4 band shapes of methane in high pressure gas mixtures, J. Chem. Phys. 61, 4070-82 (1974). 

Clodius, W. B., and Quade, C. R. (1985), Internal Coordinate Formulation for the Vibration-Rotation Energies of Polyatomic Molecules. III. Tetrahedral and Octahedral Spherical Top Molecules, /. Chem. Phys. 82, 2365. 

Sadovskii, D. A., and Zhilinskii, B. I. (1988), Qualitative Analysis of Vibration-Rotation Hamiltonians for Spherical Top Molecules, Mol. Phys. 65, 109. 

The product of these 3-j symbols is nonvanishing only under certain conditions that provide the rotational selection rules applicable to vibrational lines of symmetric and spherical top molecules. 

Higher-order classical moments have also been reported. We mention the classical expressions for the translational spectral moments M , with n = 0, 2, 4, and 6, for pairs of linear molecules given in an appendix of [204]. Spectral moments of spherical top molecules have been similarly considered [163, 205], We note that for n > 1, spectral moments show dynamic as well as static quantum correction, which become more important as the order n of the spectral moments is increased. The discussions on pp. 219, and Table 5.1, suggest that, even for the near-classical systems, quantum corrections may be substantial and can rarely be ignored. 

C. G. Gray. Theory of collision-induced absorption for spherical top molecules. J. Phys. B, 4 1661, 1971. 

Spherical-top molecules have two or more noncoincident C axes and therefore (Section 1.18) have no permanent electric dipole moment. Thus spherical tops show a particularly simple microwave spectrum, namely, no lines at all. 

The rotational relaxation of polyatomic spherical top molecules can be treated approximately on the classical rough sphere model. This has been done for homo-molecular collisions by Wang Chang and Uhlenbeck101. They find a simple expression resembling that obtained by Brout for diatomic molecules... 

The extension of the trajectory calculations to a system with any number of atoms is straightforward except for the quantization of the vibrational and rotational states of the molecules. For a molecule with three different principal moments of inertia, there does not exist a simple analytical expression for the quantized rotational energy. This is only the case for molecules with some symmetry like a spherical top molecule, where all moments of inertia are identical, and a symmetric top, where two moments of inertia are identical and different from the third. For the vibrational modes, we may use a normal coordinate analysis to determine the normal modes (see Appendix E) and quantize those as for a one-dimensional oscillator. 

Another method to prepare the initial molecular state is by using fields and the Stark effect. In this way oriented beams of notably NO and several spherical top molecules can be prepared. This has already been introduced in Section 2. In the first experiment of this kind the state selector was not used to prepare the beam, but to determine the orientation of the scattered molecules [139]. In later experiments, the... 

In symmetric top and spherical top molecules there are exact degeneracies in the unperturbed vibrational levels arising from symmetry, and Coriolis perturbations between such levels produce first order effects in the rotational structure. For symmetric top molecules the constants relating the... 

From (2.68) we see that we can add selected terms in 3/3/ to our expression for Pr in (2.51) and hence to the nuclear Hamiltonian, without altering the values of any of the physical observables. We choose these terms so that the rotational Hamiltonian has the same form as the rotational Hamiltonian of a spherical top molecule. We shall see later that with this choice for the rotational Hamiltonian, we can make use of the very powerful techniques of angular momentum theory, in particular, irreducible tensor methods, which would otherwise be denied to us. Accordingly, we modify equation (2.51) to be... 

The theory developed initially by Hubbard93 for spherical top molecules in liquids shows that collisions cause the molecule to experience changes in the magnitude and direction of its angular momentum, which then causes the fluctuations required to create magnetic fields oscillating at the Larmor frequency. For a small molecule in a liquid it can be shown that... 

The interacting u and 1/3 bands of natural silane (Lavorel et ah, 1990 Millot et ah, 1990 Lavorel et ah, 1993) and of stannane " SnH4 (Tabyaoui et ah, 1991) have been measured by high resolution infrared and stimulated Raman spectroscopy. All these results were compiled in the T.D.S. data bank (Tomsk-Dijon Spectroscopy project) and were incorporated into a comprehensive model for the rovibrational spectra of methane and other spherical top molecules (Babikov et ah, 1993 Champion et ah, 1992). 

Brownian Motion of a Rigid Body. Many molecules of interest are extended in three dimensions with three principal moments of inertia of comparable size. Gyroscopic forces complicate the discussion of Brownian motion for such bodies and a number of workers have developed convenient formal treatments without much physical novelty emerging. Steele has given a clear treatment of this problem, but obtains tractable expressions only for spherical-top molecules, which have the same moment of inertia I about all axes through the molecular mass centre, so that the Euler equations of motion fall apart in independent variables. 

A spherical-top molecule need not be geometrically spherical, so there may be three distinct principal axes for the viscous torque on the rotating molecule. Steele thus obtmns three equations of motion... 

For a spherical top molecule, the spin rotational relaxation is given as [73] ... 

D. A. Dunmur and N. E. Jessup. The influence of intermolecular interactions on the Kerr effect in gases I. Statistical theory for spherical top molecules. Molec. Phys., 57 697-711 (1979). 

D. A. Dunmur, M. R. Manterfield, and D. J. Robinson. Depolarized light scattering studies of the collision induced polarizability anisotropy of atoms and spherical top molecules. Molec. Phys., 50 573-583 (1983). 

Let us illustrate this conclusion by the example of spherical-top molecules. In the degenerate states E, T , T2, and G3/2 there is the E representation in the decomposition of the appropriate symmetric product, and hence in these cases the matrix elements are nonzero for anisotropic components of the tensors of polarizability and quadrupole moment, since E is first met in the decomposition of the spherical representation D2. Moreover, in tetrahedral systems in states of the type T and G3/2 the matrix elements of the dipole moment are nonzero, since [T2] and G3/2 contain the representation T2. 

Note that in the 2E term of a spherical-top molecule with the spin-orbital interaction taken into account, dipole transitions are allowed between the vibronic states ne) and n e ), which are nondipolar states since they originate from a nondipolar electronic term interacting with nonactive in the IR absorption E vibrations. The intensity of such transitions in accordance with Eq. (45) is proportional to dhrd2 , t e (Ogurtsov, 1984). 

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