Symmetric rotor rigid

Before discussing the spherical rotor, it is appropriate to focus on the rotational energy levels for symmetric rotors (see also Tab. 4.3-2). For the rigid prolate top, the rotational energy is given by... 

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... 

Anisotropic and Rapid Internal Motions. The cross-relaxation rate constants depend not only on the intemuclear separation but also the correlation time. Even for a spherically symmetric rotating body, each cross-relaxation rate constant depends on two parameters. However, for a rigid spherically symmetric rotor, there is a single unique correlation time, that can be determined by relaxation methods on X nuclei, by cross-relaxation between protons that have a Imown fixed separation or by non-NMR methods based on rotational diffusion. 

Evidcintly the rotational structure of the band of a spherical rigid rotor is similar i,o that of a perpendicular (X) band of a lineal rigid rotor. The two types of bands of a. symmetric rotor are more complex. The paralhd ( ) typo of Itand, for which AK = 0, will, however, bo similar in structure to that of a spherical lotor or a perpendicular (X) band in a linear rotor. For a perpendicular (X) band, however, the position of the Q branch is dependent both upon the initial value of K in the transition and upon the sign of AK, as can be seen from the following expressions ... 

For a symmetric rotor, in the present approximation, only the z component of in, the vibrational angular momentum, needs to be considered. The problem may be treated as a perturbation employing zero-order wave functions which are products of rigid rotor and harmonic oscillator functions. When the molecule is in a state such that vka + Vkb — 1, where Qka and Qw> are degenerate, it is necessary to solve the secular determinant... 

The term values of the rigid symmetric rotor are given by... 

A small lesson in quantum chemistry reveals that rotational spectra in non-relativistic quantum mechanics is very simple yet very informative as regards molecular types and their transitions, heat capacities etc. Eor e.g. a symmetric rotor one obtains trivially in the rigid case... 

The three rotational constants completely determine the energy level scheme of a rigid asymmetric top. This scheme may be considerably more complex than that of a symmetric rotor, especially if k is close to zero. Like it has already been shown for linear and symmetric-top molecttles, the mrmber of parameters reqttired to theoretically reproduce accurately measirred rotational spectra may increase considerably if effects of rotation-vibration interaction, spin coupling, or internal rotation must be taken into account. Details of practical importance will be considered below. 

Sixfold barriers to internal rotation occur in molecules such as toluene andp-fluoro-toluene whose molecular frame has C2v symmetry about the rotor axis. The simplest spectroscopic model of internal methyl rotation assumes a rigid, threefold symmetric methyl rotor attached to a rigid molecular frame with the C2 axis coincident with the rotor top axis.25 The effective one-dimensional sixfold torsional potential takes the traditional form,... 

Less symmetric molecules require a considerably more complicated treatment, but in the end their spectral transitions arc functions of their three moments of inertia (see Section 10.3.5). From a computational standpoint, then, prediction of rotational spectral lines depends only on the moments of inertia, and hence only on the molecular geometry. Thus, any method which provides good geometries will permit an accurate prediction of rotational spectra within the regime where tlie rigid-rotor approximation is valid. 

The former feature is demonstrated by a part of the fs DFWM spectrum of benzene as depicted in Fig. 3. The data displayed is an extension to the published spectra in Ref. [5]. The experimental trace in Fig. 3a shows regions around the J-type recurrences at a total time delay of ca. 1.5 ns. In Fig. 3b a simulated spectrum is given, computed on the basis of a symmetric oblate rotor with the rotational constant B" = 5689 MHz and the CDs Dj- 1.1 kHz and Djk = -1.4 kHz. For comparison in Fig. 3c the same recurrences are calculated with all CDs set to zero. It can be seen that the CDs cause a strong modulation, splitting and time shift in the recurrences. Even recurrences are differently affected than odd ones. One can conclude that high temperatures do not prevent the occurrence of rotational recurrences and thus, the application of RCS. On the contrary, they enable the determination of CDs by analysis of spectral features at long time delay and hence, reflect the non-rigidity of molecules. 

In sections 3 and 4, we have considered non-rigid molecules with a solid reference frame. In order to achieve the comparison between the NRG s and the Longuet-Higgins and Altmann s groups, let us consider some very symmetric systems, such as linear molecules with equivalent rotors, in which the reference frame is only a rotational axis, or centro-symmetric molecules with permutational rearrangements around a central point, in which the reference frame is a single atom. 

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... 

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... 

Just as we corrected the expressions for the rigid rotor to allow for the centrifugal effect and an interaction with the vibration, we also must adjust the expression for the harmonic oscillator to account for the anharmonicity in the oscillation. The potential energy surface for the molecule is not symmetrical (Fig. 25.2). The parabola (dotted figure) represents the potential energy of the harmonic oscillator. The correct potential energy is shown by the full lines the vibration is anharmonic. The vibrational energy levels for such a system can be approximated by a series ... 

Fi om Table XVI-3, several statements about the rotational fine structure in the Raman effect may be made immediately. The polarized, totally symmetric band of a linear rigid rotor will resemble a perpendicular (.L) infrared band of the same type of rotor, except that the line spacing is twice as great ( AJ1 = 2 instead of jA./j = 1). The degenerate Raman band, on the other hand, will more nearly resemble a parallel (j ) infrared... 

For the experiments, the apparatus took the form of a rigid rotor supported in a single rigidly mounted bearing. This simple symmetrical arrangement overcomes many of the difficulties associated with a two bearing system where it is necessary to have two identical bearings. 

This suggests that we can determine littie more about the structure of symmetric tops other than a single rotational constant, even though symmetric tops have two distinct rotational constants. If molecules acted like perfect rigid rotors, this would be the case. But they re not perfect, and that does allow us to obtain additional information from a real spectrum. We will get to this in the next section. 

Early calculations with this formulation (Vesovic Wakeham 1993) suggest that the explicit terms connected with the inelasticity of the pair potential are small, so that it may provide a simpler route to the prediction of the thermal conductivity of mixtures if the result for atom-rigid-rotor systems is more general. The development of this alternative formulation and of methods for the evaluation of effective cross sections for realistic non-spherically symmetric potentials are active lines of research. 

Transitions between the rotational states of a polyatomic molecules can produce a microwave spectrum. We will not discuss the details of the microwave spectra of polyatomic molecules, but make some elementary comments. As with diatomic molecules, we apply the rigid-rotor approximation, assuming that a rotating polyatomic molecule is locked in its equilibrium conformation. Any molecule in its equilibrium conformation must belong to one of four classes linear molecules, spherical top molecules, symmetric top molecules, and asymmetric top molecules. 

The rotor assembly is modelled as a rigid shaft, symmetrically supported by two identical bearings. Each bearing supports a mass M, a static load Wo and a synchronous excitation due to an unbalance, characterised by its eccentricity et. Fig. 2. 

This is not at all like the rigid rotor operator given above. Here the operator is divergent whenever two nuclear moments of inertia are the same. It is thus quite impossible to describe a symmetric top molecule in this formulation. It seemed to pose such a severe problem that Eckart observed in the abstract of his paper [Eckart, 1934] that ... 

Here, the nonrandom excitation of C2H4F is described by the dynamics of the F - - C2H4 bimolecular reaction. To simulate chemical activation, proper initial conditions must be chosen for the reactants and for their relative properties. The procedure for choosing initial conditions for the reactant s relative properties is given below in the discussion of bimolecular reactions. The quasi-classical method may be used to select initial conditions for molecular reactants. The energy for a symmetric-top polyatomic molecule in a specific vibrational-rotational state may be approximated by the harmonic oscilla-tor/rigid rotor model... 

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