Calculating the hopping rate

In figure 6.2, we have plotted the function (Kb (t))p(xo,T) both for the straight and zigzag channel for T = 15ps. For long times, as expected, this function approaches a straight line. In figure 6.3, 

Diffusion of Isobutane in Silicalite studied by Transition Path Sampling 

An explanation for this is that the a value of 3.6A we have used is somewhat too high the simulations of Bouyermaouen and Bellemans [170] report a diffusivity of 3 x 10 m /s for MD simulations using a = 3.364A, which is an order of magnitude larger than most experimental 

Apparently, the optimal a should be somewhere between 3.35A and 3.60A. Also, in our model we have assumed a rigid zeolite lattice to save CPU time, which may result in a diffusivity that is one order of magnitude too low [170,219], This suggest that one should carefully choose the model system. 

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In this section, we will briefly summarize the transition path sampling method for deterministic paths which has been developed by David Chandler and co-workers based on earlier ideas of Pratt [235]. This method is not only able to calculate the hopping rate (and therefore also the diffusion coefficient) between two stable sites (here the intersections of Silicalite). For a more complete discussion about this simulation technique, the reader is referred to refs. [227,232,234]. 

Thus, under the given conditions, the calculation of the diffusion tensor reduces to the determination of the hopping rates F(ns, ms ). 

With the estimated values of the control parameters, the rate of charge transfer can be computed. Since the charge transfer phenomena are of the diffusive type in the absence of any external potential, the diffusion coefficient, which is related to the hopping rate between pairs of the molecules, can be calculated as... 

For instance, helium spin-echo technique was used to study transport of H (or D) on Pt(lll) between 80 and 220 K at a coverage of 0.1 ML [156]. The results were consistent with hopping diffusion by means of nearest neighbor random jumps, with H atoms moving freely on the sub-nanosecond timescale. The authors conclude that quantum tunneling may make a measurable contribution to the hopping rate even at room temperature. DFT calculations were also used to address quantum effects in the low-temperature mobility of H atoms on metal surfaces [157]. 

Octamethyl pyrophosphoramide is a colorless oil, completely soluble in water, benzene, acetone, and many other common organic solvents except the paraffinic hydrocarbons. Its hydrolysis rate has not been measured, but it appears stable in the absence of alkali. In England, this systemic insecticide has been used to control aphids on hops. There it has been calculated that only a negligible quantity of the poison ultimately may find its way into the beer made from the hops. Despite calculations of this sort, the use of octamethyl pyrophosphoramide on food or fodder crops in this country is definitely not to be recommended. However, it may prove useful if properly applied to control certain insects, especially those attacking ornamental plants, such as rosebushes, and possibly on the cotton aphid and grape phylloxera. The compound has only recently been made available experimentally. 

The ion hopping rate is an apparently simple parameter with a clear physical significance. It is the number of hops per second that an ion makes, on average. As an example of the use of hopping rates, measurements on Na )3-alumina indicate that many, if not all the Na" ions can move and at rates that vary enormously with temperature, from, for example, 10 jumps per second at liquid nitrogen temperatures to 10 ° jumps per second at room temperature. Mobilities of ions may be calculated from Eqn (2.1) provided the number of carriers is known, but it is not possible to measure ion mobilities directly. 

In another model-based study Gillbro et al. [192] have used exciton-exciton annhilation to determine average EET hopping rates. In this technique the rate of exciton-exciton annhilation depends critically on the domain size and the pairwise EET rate [193, 194]. Average pairwise rates of 2 x 10 s -10 s were calculated for LHCII having EET domain sizes in the range of 300-1000 sites. 

Figure 6.5 Hopping rate for an Ag atom on Cu(100) as predicted with one dimensional harmonic transition state theory (ID HTST). The other two solid lines show the predicted rate using the DFT calculated activation energy, AE = 0.36 eV, and estimating the TST prefac tor as either 1012 or 1013 s 1. The two dashed lines show the prediction from using the ID HTST prefactor from DFT (v — 1.94 x 1012 s 1) and varying the DFT calculated activation energy by + 0.05 eV.

You have now learned about how to use DFT calculations to compute the rates of individual activated processes. This information is extremely useful, but it is still not enough to fully describe many interesting physical problems. In many situations, a system will evolve over time via many individual hops between local minima. For example, creation of catalytic clusters of metal atoms on metal oxide surfaces involves the hopping of multiple individual metal atoms on a surface. These clusters often nucleate at defects on the oxide surface, a process that is the net outcome from both hopping of atoms on the defect-free areas of the surface and in the neighborhood of defects. A characteristic of this problem is that it is the long time behavior of atoms as they move on a complicated energy surface defined by many different local minima that is of interest. 

Four distinct hopping events were considered in the calculations, which correspond to the movement of the benzene molecules between the cation and window sites of minimum energy in NaY, i.e., cation to window (C-W), C-C, W-C, and W-W. The associated rate constants of these processes were used to calculate the activation barrier to each hopping process and the Arrhenius prefactor. The MEP of benzene molecules was followed by a constrained optimization method that drags benzene from its initial site of minimum energy, through the transition state, to the final state. 

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