The non-rigid rotor

The total rotational energy is now the sum of the kinetic and potential energies, given by 

If we substitute for R, using equation (6.192), expand the denominator in the first term in (6.193) and neglect cubic and higher powers of Rc - Re we obtain 

We have already seen that, in quantum mechanics, the eigenvalue of J is J(J + 1) so that from (6.194) we obtain, for the non-rigid rotor, 

The rotational term values are therefore given by the power series 

The coefficients B, D, H, etc., are determined from an analysis of the experimental spectrum it is rarely necessary to go beyond the cubic term, except when very high J values are involved. The parameters D, 77, etc., are known as the centrifugal distortion corrections to the rotational kinetic energy. 

So far we have used the models of the rigid rotor, and the harmonic or anharmonic oscillator to describe the internal dynamics of the diatomic molecnle. Since the period for rotational motion is of the order 10 s, and that for vibrational motion is 10 s, the 

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In fact, the frequency ofthe torsional oscillation mode V4 is found to be more than double that ofthe ground state. The frequency ofthe torsional oscillation mode was reevaluated by Mukheijee et al [56], using a very accurate representation of the one-dimensional vibrational Hamiltonian of the non-rigid rotor in terms of a Fourier series [76-78], and other spectroscopic parameters calculated for the first time taking care of anharmonicity. A new assignment of the experimental spectrum was given. The results are displayed in Table 8. For reference purpose the vibrational frequencies of the ionic states are also listed... 

MW Double minimum potential function determined from the variation of rotational constants and from analysis of the non-rigid rotor spectra in the v = 0, 1 and 2,3 inversion states. Barrier =241 cm-1 125)... 

It is possible that the complexes benzene- -HX can be described in a similar way, but in the absence of any observed non-rigid-rotor behaviour or a vibrational satellite spectrum, it is not possible to distinguish between a strictly C6v equilibrium geometry and one of the type observed for benzene- ClF. In either case, the vibrational wavefunctions will have C6v symmetry, however. 

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. 

The thermodynamic functions of this table are analogous to those in the JANAF table for H20(g) (j ) both tables are taken from Freidman and Haar (1 ). Friedman and Haar applied their non-rigid-rotor, anharmonic-oscillator treatment (with vibrational-rotational coupling terms and low-temperature rotational corrections) to the infrared-spectra analyses of Benedict et al. (J ), and... 

The 2-3 splitting is of the order of a typical low-J rotational spacing and non-rigid-rotor spectra result. The rovibrational levels were therefore computed from the Hamiltonian of Eqs. (4.9a, b) with the help of a second-order perturbation correction used by Butcher and Costain66 for cyclopentene rather than by direct matrix... 

Figure 4.4 Dependence of NOE, nTi and nT2 upon rotational correlation time Tr, for a proton-decoupled resonance in a rigid rotor, in spectrometers with proton resonance frequencies of 250 and 500MHz. The dotted curve represents nT for a non-rigid rotor with an internal motion which rapidly halves the angular order parameter, S. The curves for other values of may be estimated by interpolation.

In a similar fashion to the description of vibrational energy levels, a simple model can be used to approximate the rotational motion. In general, that of a non-rigid rotor is used (because the atoms are able to change their relative intemuclear positions). In general, the rotation energy Ej (frequently, F(J) instead of / is used) is written as... 

The Seetion entitled The BasiC ToolS Of Quantum Mechanics treats the fundamental postulates of quantum meehanies and several applieations to exaetly soluble model problems. These problems inelude the eonventional partiele-in-a-box (in one and more dimensions), rigid-rotor, harmonie oseillator, and one-eleetron hydrogenie atomie orbitals. The eoneept of the Bom-Oppenheimer separation of eleetronie and vibration-rotation motions is introdueed here. Moreover, the vibrational and rotational energies, states, and wavefunetions of diatomie, linear polyatomie and non-linear polyatomie moleeules are diseussed here at an introduetory level. This seetion also introduees the variational method and perturbation theory as tools that are used to deal with problems that ean not be solved exaetly. 

Figure 3.57 Similar to Fig. 3.56(b), for the uncorrelated (RHF) rigid-rotor model. The Lewis (E(L>, squares, light solid line) and non-Lewis ( (NL), circles, light solid line) components of )totai) are shown for comparison. 

The Hamiltonian for a non-linear rigid rotor is quite complicated [1] and the derivation of the expressions for the sum and density of states is cumbersome. We know, however, that the partition function is given by Eq. (A.20), and it is quite easy to find the expressions for the sum and density of states that are consistent with Eq. (A.20). 

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. 

In the previous section, we considered the relatively simple non-rigid systems and we have deduce their local rNRG s for different st es of simplification of the Hamiltonian operator. Next, we shall consider molecular systems bearing a Cz rotor. The study of these more complex molecules better illustrated the advantage of the using the local NRG. 

This result was obtained in the rigid rotor approximation, i.e, with B and B 2 constant in equation (111). When non-rigidity will be introduced in the calculations, the B constants have to be developed also in Fourier expansions just as the potential. Better results may then be expected [59]. 

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