Symmetric top approximation

The inclusion of overall molecular rotation into the state sums may be carried using the symmetric top approximation in which two of the moments of inertia are set equal to their average. The key assumption necessary for this treatment is that the rotational constants are instantaneous functions of the large amplitude x coordinate but that the rotation is otherwise separable from vibration. The symmetric top moments of inertia Ii(x) and /j(t) are obtained from a principle axis analysis at the geometry (x,q = 0) which is presumed to be the minimum of the well V(x,q) holding X fixed. The rotational energy levels are given by... 

The inclusion of overall rotation into the reactive problem proceeds analogously to the formalism introduced for the bound problem. A principle axis analysis of the complex with fixed values of (x,s) produces rotational constants that explicitly depend on (x,s), Ii(x,s) where i=l-3. For a symmetric top approximation to the rotational energy levels, we get the rotational-vibrational cumulative state density at the TS of... 

In the following chapter we will present the transients obtained. The transients are analyzed according to the preceding chapter. In Table 1 the molecular constants obtained from fitting are summarized. Note, that the second rotational constant C can not be determined directly. When using high intensity laser beams additional transients appear that can be related to C-type transients. From their position, an approximate value ( +/- 0.1 GHz) can be obtained that is used in the simulation. It was set to 12 GHz in the simulation for cyclopropane and to 6.5 GHz for the cyclobutane simulations. This has only an effect on the thermal population of the sample as the term (C-A)K2 of the well known term equation for symmetric top cancels when calculating the Raman transitions. 

This is an allowed transition that should give rise to perpendicular bands (the molecule being assumed to be an approximately symmetric top in both lower and upper states). Further, it should lead to a small increase of apex angle and probably to a small increase of S-0 length ... 

Wherever this is applicable in rotational fine structure analysis, the customary approach involves analyzing the spectra in terms of a near prolate or oblate top. Deviations from this approximation decrease with increasing quantum number K. For a more extensive discussion, for example of the subband structure in symmetric top spectra, the reader is referred to Allen and Cross (1963). 

In a real molecule of course, the PES cannot be changed at will. Nevertheless, mixings are well known in spectroscopy and can be observed in quantities like rotational constants, intensities or any other characteristic feature of an absorption or emission spectrum. The external parameter is usually one of the rotational quantum numbers, J or K. The energy of a rovibrational state is approximately given by an expression for the symmetric top molecule, i.e.. 

To calculate Iy(t) from Eq. (3.36) it is very convenient to make several approximations. The first has been mentioned already the molecule is taken to be an approximate symmetric top. Thus, the rotational energy of rotational level JK in the manifold of the zero-order vibrational state y> is58... 

As the molecules become more complex, so the terms that influence the spectrum and the line intensities increase also although the general form of the relationships does not change radically. For a symmetric-top molecule, one can formulate an approximate expression, adapted from Gordy and Cook (ref 3, p. 209) ... 

This is a fluorine-bonded structure with an approximately 2° out-of-plane distortion of the BF3 in the complex, and a B...F-H bond angle of 108°. The complex is a symmetric top due only to rapid vibrational averaging of the HF orientation. 

Nucleic acids > ca. 10 bp long are not spherically symmetric. To a good approximation they are equivalent to circular cylinders with a hydrodynamic diameter of 20-23 A for DNA (33-35) and 25 A for RNA (35). The correlation function for such symmetric top molecules consist of three exponentials, whose arguments are combinations only of the correlation time for end over end tumbling (tl) and for rotation about the principal symmetiy axis (ts). Thus for anisotropic motion, two independent correlation times are needed to describe the rotational diffusion. The spectral density function also depends on the angle (0) the interproton vector makes with the principal axis. J(0), and hence the cross-relaxation rate constant, varies as a function of this angle according to (.16) ... 

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