Rovibrator rigid

The more familiar rigid rovibrator has the dynamical symmetry associated with chain (II). The Hamiltonian up to quadratic terms is... 

One may go one step further and determine thermal rigidity factors for capture of dipoles in individual rovibrational states i, that is, derive /rigid,/(T). 

In order to make a rigorous comparison between calculated and experimental shielding tensors, it is necessary to correct the calculated shielding for rovibrational averaging (28-33). Typically, ab initio calculations are carried out on an isolated rigid... 

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... 

Finally, in physical situations characterized by potential energy functions intermediate between purely rigid and nonrigid rovibrators, one should consider more complex algebraic treatments in which both U(3) and 0(4) invariant operators are included. Consequently, the Hamiltonian operator can no longer be diagonal in the chosen algebraic basis (related to either one or the other of the two dynamical symmetries). However, matrix elements for any operator of interest have already been explicitly computed in analytical form [35]. 

By starting from more complex operators, such as (4.104), (4.105), or (4.106), one obtains approximate solutions expressed as power series in the rovibrational quantum numbers. (For a complete discussion, see Ref. 11.) The rotational factor, corresponds to a rigid-rotor model. 

Calculations of the rovibrational component of statistical sums in the rigid rotator-harmonic oscillator approximation with inclusion of additional corrections for the anharmonicity of oscillations require data on a number of molecular constants cOe, ai, Dg, and Bg (or Rg) (Handbook, 1962 Gurvich et al., 1978-1984). 

The spectrum of the Watson Hamiltonian gives the rovibrational energy levels of the molecule under consideration. The resolution proceeds through the application of perturbation theory (Rayleigh-Schrodinger perturbation theory), which allows the partitioning of the Watson Hamiltonian into the rigid-rotor harmonic oscillator Hamiltonian H and a perturbation... 

Explicit expressions for the prior distributions require the expressions for the energies of the rovibrational states. For most diatomics, it is sufficient to use the rigid rotor approximation Eyj = E + Byj(j + 1), where By is the rotational constant in the vibrational manifold v. It is however necessary to recognize the anharmonicity of the vibrational levels because near the energy cutoff, the ttans-lational density of states oc(E - E v, ) varies quite rapidly. 

The most important molecular constants are Vq, the energy of the vibrational levels above the rotationless ground state and the rotational constants By and B. The centrifugal distortion constants, D and d, H and h, are orders of magnitude smaller than B and account for the deviations from the rigid rotor energy levels. The constants Ware necessary to reproduce the measured spectrum since the Vi and Zvz states are in Fermi resonance. The constants take into account the Coriolis interaction between the Vi and Zv + v. For the latter state no transitions have been observed and its constants have been obtained from the perturbation induced on the levels of Vj. The RMS in Table 1 is the standard deviation normalized to a single rovibrational transition of the calculated and measured spectrum. This quantity should be very close to the precision of the wave-number measurement if the theoretical model used for the analysis is satisfactory. 

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