Resonance wave functions

Figure 5. Selected resonance wave functions for HCO. The left-hand panel shows (ft, r) cuts for fixed angle y and the right-hand panel illustrates the (ft, y) behavior for fixed value of r. The distances are given in ao and the angle is given in degrees. Plotted is the modulus square of the wave function. Except for the lowest panel, all wave functions ate easily assigned by the quantum numbers t) (H-CO stretch), in (C-O stretch), and in (H-C-O bend).

Figure 12. Angular dependence of the (0, 7, 0) resonance wave function of HCO. (Reprinted, with permission of the American Institute of Physics, from Ref. 32.)...

Figure 17. Plots of resonant wave functions for HO2 as functions of R and 7 for two energy regimes, E 0.15 eV and E 0.25 eV. The O2 coordinate r is fixed. Shown is the logarithm of the modulus square of the total wave function i . The right-hand side in each column depicts the corresponding rotational state distributions at these energies. The precise energies (in eV) are from top to bottom 0.151, 0.152, 0.154, 0.156, 0.158, and 0.252, 0.254, 0.257, 0.259, 0.263, respectively. The coordinate R ranges from lao to 5ao- (Reprinted, with permission of the American Institute of Physics, from Ref. 37.)...

Another advantage of the quantum calculations is that they provide a rigorous test of approximate methods for calculating dissociation rates, namely classical trajectories and statistical models. Two commonly used statistical theories are the Rice-Ramsperger-Kassel-Marcus (RRKM) theory and the statistical adiabatic channel model (SACM). The first one is thoroughly discussed in Chapter 2, while the second one is briefly reviewed in the Introduction. Moreover, the quantum mechanical approach is indispensable in analyzing the reaction mechanisms. A resonance state is characterized not only by its position, width and the distribution of product states, but also by an individual wave function. Analysis of the nodal structure of resonance wave functions gives direct access to the mechanisms of state- and mode-selectivity. 

Figure 4 Illustration of resonances in a one-dimensional square-well potential. The two lowest solid lines are bound-state wave functions, whereas the upper two solid lines illustrate resonance wave functions. The dashed curve represents a non-resonant scattering state . Shown is the modulus square of x the scaling is different for the different wave functions. In the numerical example in the text, Vi = 8 and V2 = 12.

Figure 5 (a) Wave function for resonance state (5,1,3) of HOCl. (b) An example of an unassignable resonance wave function for N02- These two molecules will be discussed in more detail in 5.2 and 6.2, respectively. 

In this chapter, we discussed the principle quantum mechanical effects inherent to the dynamics of unimolecular dissociation. The starting point of our analysis is the concept of discrete metastable states (resonances) in the dissociation continuum, introduced in Sect. 2 and then amply illustrated in Sects. 5 and 6. Resonances allow one to treat the spectroscopic and kinetic aspects of unimolecular dissociation on equal grounds — they are spectroscopically measurable states and, at the same time, the states in which a molecule can be temporally trapped so that it can be stabilized in collisions with bath particles. The main property of quantum state-resolved unimolecular dissociation is that the lifetimes and hence the dissociation rates strongly fluctuate from state to state — they are intimately related to the shape of the resonance wave functions in the potential well. These fluctuations are universal in that they are observed in mode-specific, statistical state-specific and mixed systems. Thus, the classical notion of an energy dependent reaction rate is not strictly valid in quantum mechanics Molecules activated with equal amounts of energy but in different resonance states can decay with drastically different rates. 

Levine (1988) has found that the ABA resonances studied by Manz and coworkers (Bisseling et al., 1985, 1987) can be fit by Eq. (8.17) with v = 1.8. These ABA resonances include a large number of mode-specific states and the ABA system is certainly not statistical state specific. The inferences to be made, in light of this result, is that the ability to fit a collection of resonance widths to Eq. (8.17) does not prove the system is statistical-state-specific. As discussed above, the evidence for statistical state specificity is the absence of any patterns in the positions of the resonances in the spectrum so that all the resonance states are intrinsically unassignable. This will be the case when the expansion coefficients, for the resonance wave functions i j , are Gaussian random variables for any zero-order basis set (Polik et al., 1990b). 

The above mapping of for a particular resonance onto product states requires that the resonance be isolated and assignable. However, in the case of classical chaotic motion, the resonance wave function becomes highly irregular and unassignable, so that the above mapping scheme breaks down. The dissociation of NO2 appears to fall into this latter category (Reisler et al., 1994). 

The complex stabilization method of Junker (7), although it was introduced in a different way, gives practically the same computational prescription as the CESE method, as far as the way of using complex coordinates is considered. Another approach of this type, resembling the CESE method as well as the complex stabilization method, is the saddle-point complex-rotation technique of Chung and Davis (29). These methods provide cleair physical insight into the resonance wave function. They differ in the way the localized paxt of the wave function is expanded in basis sets and how it is optimized. 

The CESE method is applicable directly. Nicolaides and Themelis (49) considering the asymptotic behaviour of the LoSurdo-Stark resonance states, showed that the wave function is regul2irizable by the complex rotation transformation and that the complex energy of the resonance can be obtained as a solution of the CESE. In the case of a magnetic field, the asymptotic behaviour of the resonance wave function is governed by the atom potential, hence the results for field-free atoms (26) remain valid. 

R.M. More, E. Gerjuoy, Properties of resonance wave functions, Phys. Rev. 7 (1973) 1288. G. Garcia-Galderon, R. Peierls, Resonant states and their uses, NucLPhys. A 265 (1976) 443. 

J. Bang, F.A. Gareev, M.H. Gitzzatkulov, S.A. Goncharov, Expansion of continuum functions on resonance wave functions and amplitudes, Nucl. Phys. A 309 (1978) 381. 

Figure 3.9 shows the resonance-mediated reaction mechanism. The HF(v = 3)-H VAP on the new PES is very peculiar with a deeper vibrational adiabatic well close to the reaction barrier and a shallow van der Waals (vdW) well. The ID wave function for the ground resonance state in Fig. 3.9 shows that this state is mainly trapped in the inner deeper well of the HF(v = 3)-H VAP with a considerable vdW character, whereas the excited resonance wave function is mainly a vdW resonance. Because of the vdW characters, these two resonance states could likely be accessed via overtone pumping from the HF(v = 0)-H vdW well. 

In addition to the resonance enhancement factor, there is the so-called dynamic enhancement factor, which is connected with the ratio v/D. For a crude estimate of this ratio, one can expand the compound resonance wave function wave functions (e.g. one-particle wave functions) (/>, which are admixed to compound resonances by strong interactions,... 

Here [Pg.86]

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