Tunneling Path

A Kuki, PG Wolynes. Electron tunneling paths in proteins. Science 236 1647-1652, 1987. T Ziegler. Approximate density functional theory as a practical tool m molecular energetics and dynamics. Chem Rev 91 651-667, 1991. 

In realistic systems, the separation of the modes according to their frequencies and subsequent reduction to one dimension is often impossible with the above-described methods. In this case an accurate multidimensional analysis is needed. Another case in which a multidimensional study is required and which obviously cannot be accounted for within the dissipative tunneling model is that of complex PES with several saddle points and therefore with several MEPs and tunneling paths. 

From the very simple WKB considerations it is clear that the tunneling rate is proportional to the Gamov factor exp —2j[2(F(s(0) — )] ds, where s Q) is a path in two dimensions Q= 61)62 ) connecting the initial and flnal states. The most probable tunneling path , or instanton, which renders the Gamov factor maximum, represents a compromise of two competing factors, the barrier height and its width. That is, one has to optimise the instanton path not only in time, as has been done in the previous section, but also in space. This complicates the problem so that numerical calculations are usually needed. 

A number of empirical tunneling paths have been proposed in order to simplify the two-dimensional problem. Among those are MEP [Kato et al. 1977], sudden straight line [Makri and Miller 1989], and the so-called expectation-value path [Shida et al. 1989]. The results of these papers are hard to compare because slightly different PES were used. As to the expectation-value path, it was constructed as a parametric line q(Q) on which the vibration coordinate q takes its expectation value when Q is fixed. Clearly, for the PES at hand this path coincides with MEP, since is a harmonic oscillator. 

On the other hand, it is clear that in the classical regime, T> (T i is the crossover temperature for stepwise transfer), the transition should be step-wise and occur through one of the saddle points. Therefore, there should exist another characteristic temperature. r 2> above which there exist two other two-dimensional tunneling paths with smaller action than that of the one-dimensional instanton. It is these trajectories that collapse to the saddle points atlT = T i. The existence of the second crossover temperature, 7, 2, for two-proton transfer has been noted by Dakhnovskii and Semenov [1989]. 

The bifurcational diagram (fig. 44) shows how the (Qo,li) plane breaks up into domains of different behavior of the instanton. In the Arrhenius region at T> classical transitions take place throughout both saddle points. When T < 7 2 the extremal trajectory is a one-dimensional instanton, which crosses the maximum barrier point, Q = q = 0. Domains (i) and (iii) are separated by domain (ii), where quantum two-dimensional motion occurs. The crossover temperatures, Tci and J c2> depend on AV. When AV Vq domain (ii) is narrow (Tci — 7 2), so that in the classical regime the transfer is stepwise, while the quantum motion is a two-proton concerted transfer. This is the case when the tunneling path differs from the classical one. The concerted transfer changes into the two-dimensional motion at the critical value of parameter That is, when... 

Figure 16.S A contour plot illustration of the comer cutting tunneling path...

State Proton Transfer (Section VIII). The general problem of intramolecular proton transfers includes tunneling paths (91JPC10457). The most relevant results are reported in Table VI. 

Fig. 12 The Fig. 11 intramolecular circuit can be decomposed into four tunneling paths, to apply the parallel superposition rule, and predict the transmission coefficient through Fig. 11 molecule. Molecules 1 and 2 are for the contribution of two short tunnel paths and molecules 3 and 4 for the contribution of the two longer paths through the central perylene wire... 

Because of the success of the Marcus-Coltrin tunneling path for H + H2, the same procedure was applied to other collinear three center reactions including... 

Monte Carlo Samphng of Tunneling Paths The Path Integral Instanton Method. .. 67... 

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