Nonrigid rotor

For a real rotating diatomic molecule, known as a nonrigid rotor, Eq. (D) becomes... 

Experiment Theory, rigid rotor Theory, nonrigid rotor ... 

For a diatomic molecule, the energy levels of the nonrigid rotor are given, to the order of perturbation considered here, by... 

Analysis of nonrigid rotors is by no means routine. The amount of perturbation depends on the size of the matrix element connecting the states and the frequency separation between the states, which is a function of the rotational constants and vibrational potential function. The Stark effect is very useful for verification of assignments. The resonance may be treated by using the following Hamiltonian... 

Except for floppy molecules, thermal contributions at room temperature can be quite accurately evaluated using the familiar rigid rotor-harmonic oscillator (RRHO) approach. If data at high temperatures are required, this approach is no longer sufficient, and an anharmonic force field and analysis, combined with a procedure for obtaining the rotation-vibration partition function therefrom, are required. Two practical procedures have been proposed. The first one, due to Martin and co-workers " is based on asymptotic expansions for the nonrigid rotor partition function inside an explicit loop over vibration. It yields excellent results in the medium temperature range but suffers from vibrational level series collapse above 2000 K or more. A representative application (to FNO and CINO) is found in Ref. 42. 

Fig.l. a) Experimental fs DFWM spectrum of formic acid vapor at 10 mbar. b) Fitted simulation of the spectrum based on a nonrigid asymmetric rotor model. 

The rotationally resolved spectra of the doubly hydrogen-bonded complexes (HCOOH)2 and (CH3COOH)2 have been measured for the first time by a spectroscopic technique. Rotational constants and the PT evaluated from the fitting of the fs DFWM spectra are in good agreement with results from ab initio calculations. The values of the CD constants have been obtained from the analysis based on a new general nonrigid asymmetric rotor approach. 

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. 

In some instances, a quantitative understanding of anharmonic effects may be required to acheive a priori accuracies of better than a factor of two. Procedures for incorporating one-dimensional corrections, particularly for hindered rotors are well developed and commonly employed. Increased quantum chemical and computational capabilities should now allow for studies of the fully coupled nonrigid anharmonic state densities and/or partition functions via direct Monte Carlo sampling. Such accurate state density studies are a necessary prerequisite to furthering our understanding of the accuracy limits of both quantum chemical estimates and of RRKM theory itself. 

In Sect. 3 the Wilson-Howard operator is discussed as an example of application. From this it appears that the Eckart conditions39)can be inferred from arguments which are easily extended to Sayvetz conditions40 of any type. The general derivation of Hamiltonians of nonrigid molecules can then be presented in Sect. 4, and an effective semirigid rotor Hamiltonian is formed by a Van Vleck transformation. Finally Sect. 5 gives a complete example of a calculation on a specific molecule, C3. 

Here nonrigidity will be considered under the perturbation treatment as well by excluding terms in the large amplitude coordinates and momenta from the zeroth order Hamiltonian. The resulting effective semirigid rotor Hamiltonians are therefore operators confined to the separate eigenspaces of a zeroth order Hamiltonian with terms of the small amplitude motion only [Eq. (4.36)]. 

At this introductory stage we can carry the comparison with the treatment of ordinary molecules further. In the first approximation these are described by the rigid rotor — harmonic oscillator model. In the next approximation an improvement is achieved by using effective operators with properties as described above. Similarly we may expect that the semirigid rotor — harmonic oscillator model for nonrigid molecules may be improved by introducing effective operators of the form,... 

Biirgi H-B, Houshell WD, Nachbar RB Jr, Mislow K (1983) Conformational dynamics of propane, di-tert-butylmethane, and bis(9-triptycyl)methane. An analysis of the symmetry of two threefold rotors on a rigid frame in terms of nonrigid molecular structure and energy hypersurfaces. J Am Chem Soc 105 1427-1438... 

Spherical rotor, 361 band types, 364 degeneracy, 362 on<>rgy, nonrigid, 309 rigid, 362... 

In the previous section the rigid-rotor approximation has been applied, while in the following we account for the nonrigidity of the molecules, which means that the nuclear positions are no longer fixed at their equilibrium values. We first address the effect of the rotation itself on the energy levels (centrifugal distortion), and then the effect of molecular vibrations on the spectroscopic parameters is presented. 

The rigid-rotor treatment discussed in the previous section accounts for the general features of the rotational spectrum. These gross features are modified somewhat when the effects of nonrigidity, nuclear coupling, and so forth are taken into account. In this section, the effects of centrifugal distortion are considered. 

For diatomic and linear rotors, distortion terms of the type DP and HP are added to the rigid-rotor Hamilitonian. Considering only the major P effect and noting (/, M P J, M) = h J J + Vp-, we have for the rotational energy of a nonrigid diatomic or linear molecule... 

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