Non-resonant case

The energy denominators in SOS-PT contain one or more factors of the general form (iT i.f / + to) where is the excitation frequency from the ground vibronic state to K,k>, l), - is the population decay rate for K,k>, and m is an optical frequency or a sum of Hie optical frequencies that characterize the particular process. Resonant, or near-resonant processes occur when or 0. Otherwise, the process is non-resonant and we neglect The resonant case is discussed in Section 1, while the remainder of this review focuses on the non-resonant case. 

As in the BK procedure, the electrical properties and the potential energy surface may be expanded as a Taylor series in the normal coordinates. Orders of perturbation theory are defined in the same way as for the non-resonant case. Electrical property terms that are quadratic, cubic,. .. in the normal coordinates are taken to be first-order, second-order,. .. terms in the potential energy function that are... 

In the general non-resonant case there are three contributions to the observed ROA spectra aG, the isotropic ROA invariant stemming from the electric dipole-magnetic dipole optical activity tensor, which is also responsible for the anisotropic invariant y, and the anisotropic invariant due to the quadru-pole transition tensor. The anisotropic invariants are also often written as = P(G ) and = P(A). ... 

It is convenient to carry out some further calculations with regard to whether u) is commensurable to 27t or not. The non-resonant case where (jj/2n is irrational is pretty simple because only trivial resonances occur there. Therefore, a polynomial transformation brings the map to the form... 

Formula (10.4.20) is similar to the formula (10.4.14) for the non-resonant case and the only difference is that in. the case of a weak resonance only a finite number of the Lyapunov values Li,..., Lp is defined (for example, only L is defined when N = b). If at least one of these Lyapunov values is non-zero, then Theorem 10.3 holds i.e. depending on the sign of the first non-zero Lyapunov value the fixed point is either a stable complex focus or an unstable complex focus (a complex saddle-focus in the multi-dimensional case). 

Naively one could think that scattering on weak barriers cannot possibly yield a sharp peak in G(eo, T). Indeed, the transmission probability as a function of e does not have any peak at e = eo, in contrast to the case of resonant tunneling. At high T, G(eo,T) is a weakly oscillating (with a period A) function of eo- The only difference with the non-interacting case is an enhanced amplitude of the oscillations. In fact, however, the interaction-induced vanishing of the transmission probability at ep for T = 0 does lead to a narrow Lorentzian peak of G(eo,T), provided that T is low enough and the barriers are not too asymmetric. 

In conclusion, we note that the pausing" of the -trajectory indicating an optimum scale factor prompts the statement of a generalized virial theorem. Since general techniques to ascertain resonances in the non-selfadjoint case are so much more difficult in comparison to the usual (Hermitean) case it is obvious that every bit of complementary information counts. This statement becomes no less important when realizing that the complex part of a resonance eigenvalue in many situations is order of magnitudes smaller in size compared to the real part. 

Fig. 15.8. Schematic one-dimensional illustration of electronic predissociation. The photon is assumed to excite simultaneously both excited states, leading to a structureless absorption spectrum for state 1 and a discrete spectrum for state 2, provided there is no coupling between these states. The resultant is a broad spectrum with sharp superimposed spikes. However, if state 2 is coupled to the dissociative state, the discrete absorption lines turn into resonances with lineshapes that depend on the strength of the coupling between the two excited electronic states. Two examples are schematically drawn on the right-hand side (weak and strong coupling). Due to interference between the non-resonant and the resonant contributions to the spectrum the resonance lineshapes can have a more complicated appearance than shown here (Lefebvre-Brion and Field 1986 ch.6). In the first case, the autocorrelation function S(t) shows a long sequence of recurrences, while in the second case only a single recurrence with small amplitude is developed. The diffuseness of the resonances or vibrational structures is a direct measure of the electronic coupling strength.

The seed is selectively and sequentially pumped with a light source at two wavelengths and the non-resonant fluorescence is observed in each case. The ratio of the two fluorescence signals is related to the temperature. The method was first demonstrated by Haraguchi, et al. (27), who measured temperatures in a variety of flames and whose work has been extended by... 

For RIKES with circular pump polarization and for IRS, the background interference is surpressed. In the RIKES case, the non-resonant part of x drops out according to Kleinman symmetry 3). In IRS only the imaginary part of x(3) contributes, but all of the probe intensity is admitted to the detector. 

There is indirect evidence from hydrogen maser studies [221] that the reactions H + H2, HD, D2 (v = 1) show a preference for resonant exchange reactions in the case of H + H2 (v = 1) and H + HD(i> = 1) and for non-resonant exchange for H + D2 (v = 1) in accord with theoretical calculations [222]. With recent experimental developments, particularly UV lasers, it can be expected that spectroscopic methods will be applied to measuring energy disposal for these reactions. 

As an example of principle number 4 above, consider a locally excited large panel on which resonant bending waves account for most of the vibratory response. However, assume that these resonant waves have a wavelength shorter than that of free waves in the surrounding air. In such a case the resonant waves are poorly coupled to the air, and radiate very little sound. What radiation there is can be dominated by non-resonant forced motion around the drive point (and at other discontinuities). As a result, applied damping can reduce the resonant response, but not the forced motion and the radiation of sound. 

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