Coriolis perturbations

Jahn, H. A. (1938), A New Coriolis Perturbation in the Methane Spectrum. I. Vibrational-Rotational Hamiltonian and Wave Functions, Proc. Roy. Soc. A 168,469. 

In symmetric top and spherical top molecules there are exact degeneracies in the unperturbed vibrational levels arising from symmetry, and Coriolis perturbations between such levels produce first order effects in the rotational structure. For symmetric top molecules the constants relating the... 

The energy level expression for states affected by the Coriolis perturbation is... 

For symmetric rotor molecules of C symmetry, the Coriolis perturbations occur between the following pairs of vibrations ( ic), a2e (flia2), ee). For the first two pairs, the perturbation increases with increasing J, while for the latter two it increases with increasing K. For molecules of C2t, symmetry, the following pairs of vibrational levels may perturb each other ( 1 2), ciib2% ( 2 iX ( 2 2)5... 

The number of spherical rotors, i.e., molecules with three equal moments of inertia, is quite limited. Methane has been the one most extensively studied, and its band contours have been analyzed by a number of workers [ ]. Coriolis perturbations have been observed to occur between the fundamentals V4 at 1306 cm and V2 at 1526 cm- ... 

The main resonance in Vg is a strong first-order a-Coriolis perturbation by Vg which strongly effects the positions of the K levels in both bands [10]. Fundamentals from a low-... 

True challenges are flows with instabilities and, in particular, the turbulent flows where Coriolis effects change the flame perturbation geometry. 

Several investigations concerned with the identification of these lines succeeded, for instance, in the case of H2O, in elucidating the rotational spectrum in excited vibrational states 356). Through comparison of wavelengths and intensities of many lines in H2O , H2 0 and DjO isotopic effects could be studied in these excited vibrational levels 357,358) Perturbations of rotational levels by Coriolis resonance which mixes different levels could be cleared up through the assignment and wavelength measurement of some DCN and HCN laser lines 359). 

Since the perturbations turn on at slightly lower Vj than the onset of changes in fine-structure parameters [5], it seems likely that the perturbations in (0, 32, 0) are due mostly to new oohx 4oobend anharmonic resonances and the rapid changes in fine-structure parameters at (0, 36,0) due mostly to new oohx = 3oobend Coriolis resonances. 

In a smaller molecule (HCP), these diagnostically important changes in vibrational resonance structure are manifest in several ways (i) the onset of rapid changes in molecular constants, especially B values and second-order vibrational fine-structure parameters associated with a doubly degenerate bending mode (ii) the abrupt onset of anharmonic and Coriolis spectroscopic perturbations and (iii) the breakup of a persistent polyad structure 15]. 

Two vibronic states which happen to occur at nearly the same energy will perturb one another if certain conditions are fulfilled. The interacting states may be derived from different electronic states, but in polyatomic spectra are often merely different vibrational states of the same electronic states. The perturbations, which are either homogeneous (AK = 0) (e.g. Fermi resonance) or heterogeneous (AK = 1) (Coriolis resonance) (Mulliken, 1937) are then analogous to the perturbations observed in infrared and Raman spectra. Such perturbations are commonplace in electronic bands where the completely unperturbed band is the exception rather than the rule. 

The vibration-rotation hamiltonian of a polyatomic molecule is more complicated than that of a diatomic molecule, both because of the increased number of co-ordinates, and because of the presence of Coriolis terms which are absent from the diatomic hamiltonian. These differences lead to many more terms in the formulae for a and x values obtained from the contact transformation, and they also lead to various kinds of vibrational and rotational resonance situations in which two or more vibrational levels are separated by so small an energy that interaction terms in the hamiltonian between these levels cannot easily be handled by perturbation theory. It is then necessary to obtain an effective hamiltonian over these two or more vibrational levels, and to use special techniques to relate the coefficients in this hamiltonian to the observed spectrum. 

As mentioned above, the terms which are responsible for the coupling of the 25+1 n state to the 25+1 states are the spin-orbit coupling and the rotational electronic Coriolis term. Thus in the second-order perturbation expression in equation (7.43), the perturbation term is... 

In order to appreciate this point more clearly, we confine our attention to the contributions to 3Qff produced by perturbations from the spin-orbit coupling 3Q0 and the electronic Coriolis mixing 30-ot- If we represent an off-diagonal matrix element of the former by (L S) and the latter by (N L), we can describe some examples of these higher order terms, as shown in table 7.1. The third-rank terms appear only in states of quartet or higher multiplicity and the fourth-rank terms in states of quintet (or higher) multiplicity. With the important exception of transition metal compounds, the vast majority of electronic states encountered in practice have triplet multiplicity or lower. 

Because the approximation described by Eq. (3.47) fails if the inversion and vibrational wave functions are strongly mixed, the Coriolis operator defined by Eq. (5.8) cannot be treated by the numerical methods described in Sections 5.1 and 5.2. Instead of the perturbation treatment described in Section 5.1, we must use a variational approach in which the energy levels are calculated as eigenvalues of an energy matrix the off-diagonal elements of this matrix are the matrix elements of the Coriolis operator ) 2,4 ... 

Aiming at a separate treatment of vibration and rotation it is obvious that we must look for such axis orientations where the Coriolis coupling term [Eq. (3.28)] either vanishes or at least can be treated as a small perturbation. With Eq. (2.48) we can generally write... 

Finally, a very useful extension of the methodology is to the inclusion of Coriolis couplings [121]. These advances render the perturbation-theoretic method very competitive with the VSCF approaches. 

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