Quadruple barrier

In Ref [26] three limiting cases were considered, i.e. a single barrier (Fig. 6.10), a double barrier (Fig. 6.17) and a quadruple-barrier reaction pathway (Fig. 6.18). The first process does not involve any intermediate. The second process consists essentially of consecutive double proton transfer steps, where each step involves a single barrier. There are two possibilities, either protons 1, 2 are transferred first, followed by protons 3, 4, or vice versa, proceeding via the zwitterionic intermediates 1100 or 0011. It is again assumed that the intermediates can be treated as separate species, i.e. that there are no delocalized states involving different potential wells. This assumption will be realized when the barriers are large. Each reaction step is then characterized by an individual rate constant. The process con-... 

Figure6.18 Degenerate quadruple-barrier intermediates 0101 and 1010 are not further...

In the quadruple-barrier case one needs to distinguish whether dissociation/ neutralization or propagation are the rate-limiting steps. Furthermore, a parameter a is needed describing the ratio of the forward reaction rate constants of the anion and the cation propagation, i.e. 

Figure 7.3 Distribution of the first 15 complex poles in the fourth quadrant of the k plane of the quadruple barrier resonant tunneling system with parameters given in the text.

Figure 7.4 Plot of the transmission coefficient T[E) vs E for a quadruple barrier system with parameters as discussed in the text, around the first three-resonance miniband. The exact numerical calculation (full line) is indistinguishable from the resonance expansion using the three resonant poles (dashed line). The dotted line represents the calculation without the interference resonant terms.

Figure 7.5 Plot of the transmission coefficient 7(E) vs E for the quadruple barrier system with parameters as discussed in the text as a function of different number of poles as indicated in the inset. The considered energy range, using Eq. (101), extends up to five times the height of the barriers. The region of many overlapping resonances requires of much more poles to reproduce the exact numerical calculation.

Figure 7.13 Plot of Breathing mode of the probability density I (x,t)p, along the internal region of the quadruple barrier system when the initial state is placed on the central well. One sees that the internal dynamics consists of a quasi-periodical processes involving spatial oscillations leading to the reconstruction of the initial state. At each oscillation there is a leakage through the ends of the open system.

In the first expression, the summation is extended to the quadruples of sequentially bonded atoms ABCD and the energy constants Vn are specific for the quadruples of the types of the atoms involved. The torsion angle is that between the planes ABC and BCD. The n = 1 term describes a rotation which is periodic by 360°, the n = 2 term is periodic by 180°, the n = 3 term is periodic by 120° and so on. The Vn constants determine the contribution of atoms A and D to the barrier of rotation around the... 

Figure 6.17 Degenerate two-barrier quadruple hydron transfer involving a single zwitterionic intermediate. Two hydrons lose zero-point energy in the configuration corresponding to the top ofthe barrier. Isotopic fractionation 4>a can occurforthe dissociation. Reproduced with permission from Ref [26].

Figure 6.19 Simulated Arrhenius diagrams of a degenerate quadruple hydron transfer. Arrhenius laws are assumed for the HHHH-transfer. (a) Single-barrier case, (b) Doublebarrier case with 1. (c) Double-barrier case with

Figure 7.12 shows a plot of the behavior of the survival probability for a quadruple characterized by parameters typical of semiconductor heterostructures [61] barrier heights Vo = 200 meV, barrier widths, bo = 4.0 nm, well widths, Wo = 5.0 nm. The effective mass is m = 0.067 with the electron mass. Here the first triplet of complex poles [k ] and resonant states m (x) of the problem suffice to describe the behavior of the survival probability, namely Eq. (141) with M = 3. The initial state V (x,0) is taken by simplicity as a square infinite box state. One sees clearly the oscillating nonexponential behavior of the survival probability in this system. 

Figure 7.12 Plot of the survival probability 5(t) as a function of time for the quadruple tunneling barrier system discussed in the text that exhibits decaying nonexponential Rabi oscillations. The inset shows S(t) in a wider time interval that includes the exponential-postexponential transition at long times.

The nature of the metal-metal bonding has been studied by variable temperature NMR spectroscopy and identical activation barriers were found for all complexes. This observation indicates that rotation about the metal-metal bond is the predominant process. The activation energy for this rotational process (10.1 0.5 kcal mol ) is a measure of the 6 bond strength and confirms the existence of the Mo-Mo quadruple bond. 

A second purpose of the present work is to assess the performance of the explicitly correlated coupled-cluster model CCSD(F12) that we have recently implemented in the TuR-BOMOLE program package [68, 69]. This model has the potential to yield electronic molecular energies at the level of coupled-cluster theory with single and double excitations (CCSD [37, 70]) at the limit of a complete one-particle basis set. In conjunction with corrections for higher excitations (connected triples and connected quadruples) it should be possible to compute the barrier height for the above reaction with an accuracy of about 1-2 kJ mol that is, with an error of about 0.5-1.0%. 

Because the most important quadruple excitations, doubles from double excitations within the active space, are already included at the CAS-CISD level, adding the Davidson correction has a negligible effect on the calculated rotational barrier for allyl. In benzyl, the effect is larger, amounting to 0.5 kcal/mol. This may be due, at least in part, to the use of a (5/5) instead of a (7/7) active space for the CAS-CISD calculation on benzyl. 

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