Quantum mechanical trapping

Finally, quantum mechanical trapping at the resonance energy can be verified using a time-delay analysis on the quantum S-matrix. In Fig. 8, the average time delay for the J = 0 partial wave of the F + HD — HF + D reaction, defined using Eq. (22), is plotted versus collision energy. A clear... 

In an alternative model, quantum-mechanical tunneling of the electron is invoked from trap to trap without reference to the quasi-free state. The electron, held in the trap by a potential barrier, may leak through it if a state of matching... 

Because of their importance as basic primary centers, we will now discuss the optical bands associated with the F centers in alkali halide crystals. The simplest approximation is to consider the F center - that is, an electron trapped in a vacancy (see Figure 6.12) - as an electron confined inside a rigid cubic box of dimension 2a, where a is the anion-cation distance (the Cr -Na+ distance in NaCl). Solving for the energy levels of such an electron is a common problem in quantum mechanics. The energy levels are given by... 

At the end of this section, it is worthwhile to point out that resonances in quantum mechanics are intimately related to the existence of trapped classical trajectories. The smaller the classical forces between r and 7 on one hand and R on the other, the longer is the lifetime and vice versa. In this sense it might be helpful for understanding the complex quantum dynamics by imagining the trajectories of a classical billiard ball moving on multidimensional potential-energy surfaces (see, e.g., Chapter 5 of Ref. 4). 

The Fe111/11 case is particularly simple. For electron transfer reactions in general, several normal modes may contribute to the trapping of the exchanging electron at a particular site. In addition, intramolecular vibrational modes are of relatively high frequency, 200-4000 cm-1, and at room temperature the classical approximation is not valid since only the v = 0 level is appreciably populated. In order to treat the problem more generally, it is necessary to turn to the quantum mechanical results in a later section. 

The most striking application of electron transfer theory has been to the direct calculation of electron transfer rate constants for a series of metal complex couples.36 37 46 The results of several such calculations taken from ref. 37b are summarized in Table 2. The calculations were made based on intemuclear separations appropriate to the reactants in close contact except for the second entry for Fe(H20)j3+/2+, where at r = 5.25 A there is significant interpenetratidn of the inner coordination spheres. The Ke values are based on ab initio calculations of the extent of electronic coupling. k includes the total contributions to electron transfer from solvent and the trapping vibrations using the dielectric continuum result for A0. the quantum mechanical result for intramolecular vibrations, and known bond distance changes from measurements in the solid state or in solution. 

As discussed above, the 5 -G is the preferred cleavage site in a GG trap, and in the GGG trap it is either the 5 -G or the middle G depending on the nucleobases neighboring the trap (for references see Davis et al. 2000 for quantum-mechanical calculations see Voityuk et al. 2000). Hole transfer always proceeds to the closest G, and only subsequently relaxation of the hole occurs which leads to the localization of the hole within the G tract (Davis et al. 2000). In triple-stranded DNA, the G of the third strand of the CG G serves as an effective hole trap (Doh-no et al. 2002 see also Kan and Schuster 1999a). 

As the name suggests, shape-type resonances result from the shape of the potential at hand. But, what attributes must a potential have in order to trap the particle for a finite time and thus form a metastable state The wave nature of particles in quantum mechanics provides two typical ways for a... 

In the time-dependent picture, resonances show up as repeated recurrences of the evolving wavepacket. Resonances and recurrences reveal, in different ways, the same dynamical effect, namely the temporary excitation of internal motion within the complex. In the context of classical mechanics, the existence of quantum mechanical resonances is synonymous with trapped trajectories performing complicated Lissajou-type motion before they finally dissociate. The larger the lifetime, the more frequently the wavepacket recurs to its starting position, and the narrower are the resonances. 

The quantum mechanical wavepacket closely follows the main classical route. It slides down the steep slope, traverses the well region, and travels toward infinity. A small portion of the wavepacket, however, stays behind and gives rise to a small-amplitude recurrence after about 40-50 fs. Fourier transformation of the autocorrelation function yields a broad background, which represents the direct part of the dissociation, and the superimposed undulations, which are ultimately caused by the temporarily trapped trajectories (Weide, Kiihl, and Schinke 1989). A purely classical description describes the background very well (see Figure 5.4), but naturally fails to reproduce the undulations, which have an inherently quantum mechanical origin. 

As in the cases discussed in Section 8.1 an unstable periodic orbit, illustrated by the broken line in Figure 8.9, guides the indirect classical trajectories and likewise the temporarily trapped part of the quantum mechanical wavepacket. It essentially represents large-amplitude bending motion, that is strongly coupled, however, to the stretching coordinate... 

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