Symmetric rotor

Linear, symmetric rotor, spherical rotor and asymmetric rotor molecules... 

For a symmetric rotor, or symmetric top as it is sometimes called, two of the principal moments of inertia are equal and the third is non-zero. If... 

A symmetric rotor must have either a C axis with n>2 (see Section 4.1.1) or an 54 axis (see Section 4.1.4). Methyl iodide has a C3 axis and benzene a Ce axis and, therefore, these are symmetric rotors whereas allene, shown in Figure 4.3(d), is also a symmetric rotor since it has an 54 axis which is the a axis allene is a prolate symmetric rotor. 

Figure 5.5 The rotational angular momentum vector P for (a) a linear molecule and (b) the prolate symmetric rotor CH3I where is the component along the a axis...

Figure 5.6 Rotational energy levels for (a) a prolate and (b) an oblate symmetric rotor...

The rotational energy levels for a prolate and an oblate symmetric rotor are shown schematically in Figure 5.6. Although these present a much more complex picture than those for a linear molecule the fact that the selection mles... 

This is the same as Equation (5.14) for a diatomic or linear polyatomic molecule and, again, the transitions show an equal spacing of 2B. The requirement that the molecule must have a permanent dipole moment applies to symmetric rotors also. 

When the effects of centrifugal distortion are included the term values of a prolate symmetric rotor are given by... 

Stark effect in diatomic, linear and symmetric rotor molecules... 

For a symmetric rotor the modification Eg to the rotational energy levels in an electric field S is larger than in a linear molecule and is given, approximately, by... 

At a simple level, the rotational transitions of near-symmetric rotors (see Equations 5.8 and 5.9) are easier to understand. For a prolate or oblate near-symmetric rotor the rotational term values are given, approximately, by... 

Examples of prolate near-symmetric rotors are the s-trans and s-cis isomers of crotonic acid, shown in Figure 5.8, the a axis straddling a chain of the heavier atoms in both species. The rotational term values for both isomers are given approximately by Equation (5.37) but, because A and B are different for each of them, their rotational transitions are not quite coincident. Figure 5.9 shows a part of a low-resolution microwave spectmm of crotonic acid in which the weaker series of lines is due to the less abundant s-cis isomer and the stronger series is due to the more abundant s-trans isomer. 

For a symmetric rotor molecule the selection rules for the rotational Raman spectmm are... 

For a symmetric rotor molecule such as methyl fluoride, a prolate symmetric rotor belonging to the C3 point group, in the zero-point level the vibrational selection mle in Equation (6.56) and the character table (Table A. 12 in Appendix A) show that only... 

The effect of the AK = 1 selection rule, compared with AK = 0 for an transition, is to spread out the sets of P, Q, and R branches with different values of K. Each Q branch consists, as usual, of closely spaced lines, so as to appear almost line-like, and the separation between adjacent Q branches is approximately 2 A — B ). Figure 6.29 shows such an example, E — A band of the prolate symmetric rotor silyl fluoride (SiH3F) where Vg is the e rocking vibration of the SiH3 group. The Q branches dominate this fairly low resolution specttum, those with AK = - -1 and —1 being on the high and low wavenumber sides, respectively. 

The selection rules are the same for oblate symmetric rotors, and parallel bands appear similar to those of a prolate symmetric rotor. However, perpendicular bands of an oblate symmetric rotor show Q branches with AK = - -1 and — 1 on the low and high wavenumber sides, respectively, since the spacing, 2 C — B ), is negative. 

Although, as in linear and symmetric rotor molecules, the term values are slightly modified by Coriolis forces in a degenerate (T2) state, the rotational selection rules... 

As is the case for diatomic molecules, rotational fine structure of electronic spectra of polyatomic molecules is very similar, in principle, to that of their infrared vibrational spectra. For linear, symmetric rotor, spherical rotor and asymmetric rotor molecules the selection mles are the same as those discussed in Sections 6.2.4.1 to 6.2.4.4. The major difference, in practice, is that, as for diatomics, there is likely to be a much larger change of geometry, and therefore of rotational constants, from one electronic state to another than from one vibrational state to another. 

Figure 9.24 shows part of the laser Stark spectrum of the bent triatomic molecule FNO obtained with a CO infrared laser operating at 1837.430 cm All the transitions shown are Stark components of the rotational line of the Ig vibrational transition, where Vj is the N-F stretching vibration. The rotational symbolism is that for a symmetric rotor (to which FNO approximates) for which q implies that AA = 0, P implies that A/ = — 1 and the numbers indicate that K" = 7 and J" = 8 (see Section 6.2.4.2). In an electric field each J level is split into (J + 1) components (see Section 5.2.3), each specified by its value of Mj. The selection mle when the radiation is polarized perpendicular to the field (as here) is AMj = 1. Eight of the resulting Stark components are shown. 

Kivelson14 has given a treatment of the distortion by the barrier forces and centrifugal effects. This has been applied phenomenologically to CH3SiH3 (a coaxial symmetric rotor) to fit the set of / = 0 to 1 transitions associated with the first few torsional states. One of the parameters involves the barrier height, which was thereby determined. 

This form of the equation is valid for linear molecules and symmetric rotors (for which I corresponds to reorientation of the symmetry axis) and can be used for nondipolar solvents. A somewhat more complicated expression would hold in the absence of axial symmetry. Maroncelli et al. estimated the value tti for AP corresponding to a charge shift of a spherical ion in a continuum model of a polar solvent of dielectric constant s and showed that it increases with increasing solvent polarity and works well when tti is significantly larger than one. 

In Table 3 are listed some experimental values related to the torsional motion for a series of symmetric rotor molecules. The given examples are chosen to illustrate the points discussed in this section. 

The structures of the neutron-rich isotopes 97Y, 98Y and 99Y reflect with special clearness the rapid change of the nuclear shape at neutron number 60. The discovery of a new isomer in 97Y has provided evidence for the shell-model character of this nucleus even at high excitation energies while 99Y shows the properties of a symmetric rotor already in the ground state. The level pattern of the intermediate isotope 98Y indicates shape coexistence. 

This fact and the results of several experimental and theoretical studies suggest that the nuclei around A = 100 change their shapes rapidly but that they have complex potential energy surfaces. In particular, these nuclei are supposed to be soft with respect to y deformations. However, recent investigations on odd-mass nuclei revealed properties of classical symmetric rotors. A good example is Y6q, the isotone of 98Sr and 100Zr,... 

The properties of 99Y suggest that this nucleus is a classical symmetric rotor. Thus, configurations can be assigned to all the bands in accordance with the predictions of the Nilsson model for A 100 and a deformation of e - 0.3. Also the mixing ratios 6 for the AI = 1 members of the bands can be accounted for in the classical picture of rotational nuclei. The half-life of the isomer at 2142 keV is obviously due to K forbiddenness. 

Y is characterized through the occurence of rotational bands all over the investigated region of energies up to 2 MeV as a symmetric rotor. 

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