Symmetric top rotor

From Eqs. (3.40) and (3.35) it is obvious that the inversion—rotation wave functions i//°. (0,, X, p) of NH3 which are the eigenfunctions of the operator, , can be written as a product of the rigid-rotor symmetric top wave functions depending on the Euler angles 0,4>, x and the inversion wave functions, depending on the variable p. Integration of the Schrodinger equation... 

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... 

Rotors are classified as follows If all three principal moments of inertia are equal, we have a spherical top. If two principal moments are equal, we have a symmetric top. A prolate symmetric top has Ib = Ic and the a axis (the axis of smallest principal moment) is the symmetry axis an oblate symmetric top has la = lb and the c axis is the symmetry axis. A football (American, not British) is a prolate symmetric top a discus is an oblate symmetric top. If all three principal moments are unequal, we have an asymmetric top. Thus... 

The density of states and partition function may easily be derived from this result, which is similar to that of the symmetry axis of a symmetric top, differing only in the appearance of the reduced moment of inertia. Also, the equations for a doubly degenerate internal rotor may be obtained from those for a linear molecule by substituting /red for the moment of inertia. 

In the "nonrigid symmetric-top rotors" (such as NH ), the second-order Stark effect is observed under normal circumstances. Indeed, field strengths of the order of 1 600 000 [V/m] are required to bring the interaction into the first-order regime in this case [18]. In contrast, very weak interactions suffice to make the mixed-parity states and appropriate for the description of optically active systems. Parity-violating neutral currents have been proposed as the interaction missing from the molecular Hamiltonian [Eq.(1)] that is responsible for the existence of enantiomers [14,19]. At present, this hypothesis is still awaiting experimental verification. 

Similar methods may be used for individual rotors, for symmetric tops and for systems containing internal rotors, as long as the exact partition function... 

Fig. 2.14. The geometry of the group VI dihydride molecules when the moments of inertia about the two rotation axes are equal (as illustrated), the rotor is described as a symmetric top and has simpler rotational structure than when they are not (asymmetric top). The condition for the moments of inertia about both axes to be equal is that the bond angle should be ir/2. M is the mass of an H atom, (after J.-P. Connerade and J. Hormes [91]).

By way of introduction let us note that the depolarized spectrum Ivh(co) calculated in Section 7.5 for independent rotors consists of a superposition of Lorentzian bands all centered at zero frequency. In the simplest case of symmetric top rotors the spectrum consists of a single band with a width [q2D + 6<9] which depends only on the translational self-diffusion coefficient D and on the rotational diffusion coefficient 0. This should be compared and contrasted with the depolarized spectrum Ivh(co) of certain pure liquids (e.g., aniline, nitrobenzene, quinoline, hexafluorobenzene) shown schematically in Fig. 12.1.1. The spectrum appears to be split. This entirely novel fea-... 

Microcanonical transition-state theory (TST) assumes that all vibrational-rotational levels for the degrees of freedom orthogonal to the reaction coordinate have equal probabilities of being populated [12]. The quasi-classical normal-mode/rigid-rotor model described above may be used to choose Cartesian coordinates and momenta for these energy levels. Assuming a symmetric top system, the TS energy E is written as... 

A variety of different methods have been used to measure V, V, and (LS59, OM07) only a few of the more important will be discussed here. For asymmetric rotors, both the pure rotational spectrum and its torsion-rotation counterpart are electric dipole allowed and are affected in lowest order by the leading terms in the torsional Hamiltonian. Both types of spectra have been used extensively to determine (LS59). For symmetric tops with a single torsional degree of freedom, either the permanent electric dipole moment vanishes, as in CH CH, or the normal rotational spectrum is independent of 17 in lowest order, as in CH SiH. In... 

To determine the quantum mechanical rigid-rotor energy levels, the quantum mechanical Hamiltonian operator is formed from the classical Hamiltonian in Eq. (2.50) and the eigenvalue equation, Eq. (2.54), is solved. For a symmetric top rigid-rotor, which has two equal moments of inertia (i.e., 4 = /(, /g), the resulting energy levels are... 

A second rotational effect comes into play when rotations are strongly coupled to the vibrations, via, for instance, coriolis interactions. In that case, the projection of the principle rotational quantum number, the K quantum number in symmetric top molecules, is no longer conserved. The energy associated with this quantum number then gets mixed in with the molecule s vibrational energy, thereby increasing the density and sums of states. When this happens we say that the A -rotor is active. If the T-rotor does not couple with the vibrations, it is inactive. We first discuss what happens when a diatom dissociates and follow that with the dissociation of polyatomic molecules. 

Many nonlinear molecules can be treated as symmetric top rotors in which two of the moments of inertia are equal. The moment of inertia about the synunetiy axis is 7, while the two other moments of inertia are 7 = ly A symmetric top can be visualized as a rotating cylinder. For a given J, the cylinder can rotate in a total of 27 -I- 1 orientations, each with a different K quantum number which determines its projection along the symmetry axis. Figure 7.9 shows the case of prolate and oblate tops rotating with K J and K = 0. 

The situation with asymmetric top rotors is rather more complex because of the wide range of transition selection rules followed, although similar comments apply. The more intense absorption lines will tend to occur at higher frequencies although the relative intensities of lines may vary because of symmetry considerations. Symmetric top spectra occur in clumps centred around 2B J + 1) and so from the quantitative analysis viewpoint differ hardly at all from those of linear molecules. Asymmetric top spectra are more scattered and it is rather easier to choose an accessible and discrete line from them. 

With the potential expanded as in Eqs. (3) and (4), a convenient basis [27, 49] for the orientational motions of the molecules consists of (symmetric top) free-rotor functions molecules the latter functions are simply spherical... 

For the V3 band about 270 absorption lines were recorded between 920 and 967 cm The V3 band is an a-type band of a near-prolate asymmetric rotor, and at large Kg it should resemble the parallel band of a prolate symmetric top, i.e., AN = 0, 1, AKg ( AK) = 0. In the V3 band the symmetric top characteristics are not as obvious as in the band, however, a number of Pk and °Qk branches with N up to 28 and Rk branches with N up to 42 could be identified (for the band center, see p. 247). The assignment was supported by the results from a Fourier transform spectrum of NF2 at 890 to 980 cm The spin-rotation splitting is relatively small and unresolved in transitions with low Kg values. The asymmetry splitting is apparent in lines with low Kg and high N values, which was demonstrated with the (N=19 to 21) branch [9]. 

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