Barrier, Gaussian

The method will, however, fail badly if the Gaussian form is not a good approximation. For example, looking at the dynamics shown in Figure 2, a problem arises when a barrier causes the wavepacket to bifurcate. Under these circumstances it is necessary to use a superposition of functions. As will be seen later, this is always the case when non-adiabatic effects are present. 

While being very attractive in view of their similarity to CLTST, on closer inspection (3.61)-(3.63) reveal their deficiency at low temperatures. When P -rcc, the characteristic length Ax from (3.60b) becomes large, and the expansion (3.58) as well as the gaussian approximation for the centroid density breaks down. In the test of ref. [Voth et al. 1989b], which has displayed the success of the centroid approximation for the Eckart barrier at T> T, the low-temperature limit has not been reached, so there is no ground to trust eq. (3.62) as an estimate for kc ... 

We start with the reaction of abstraction of a hydrogen atom by a CH3 radical from molecules of different matrices (see, e.g., Le Roy et al. [1980], Pacey [1979]). These systems were the first to display the need to go beyond the one-dimensional consideration. The experimental data are presented in table 2 together with the barrier heights and widths calculated so as to fit the theoretical dependence (2.1) with a symmetric gaussian barrier. 

Effective parameters of the one-dimensional gaussian barrier for the H atom in the reactions (6.18) (see below)... 

Ab initio molecular orbital calculations (using the Gaussian 80 computer program) on the barrier to pseudorotation (for the furanose ring) of two model compounds, 2-deoxy-/ -D- /) cero-tetrofuranosylamine (781) and 2-deoxy-2-fluoro-)3-D-erythrofuranosylamine (782) were reported. Al-... 

Figure 6. Shown is the correlation between the liquid s fragility and the exponent p of the stretched exponential relaxations, as predicted by the RFOT theory, superimposed on the measured values in many liquids taken from the compilation of Bohmer et al. [50]. The dashed line assumed a simple gaussian distribution with the width mentioned in the text. The solid line takes into account the existence of the highest barrier by replacing the barrier distribution to the right of the most probable value by a narrow peak of the same area the peak is located at that most probable value. Taken from Ref. [45] with permission.

Fig. 2.14. Influence of the reverse activation energy on KER and thus, on peak shapes in metastable ion decompositions, suitable experimental setup as prerequisite. From left no or small reverse barrier causes Gaussian peak shape, whereas medium or yields flat-topped peaks and large For causes dish-shaped peaks.

These results are commonly interpreted in terms of distributions of elemental local jump processes [116]. For instance, a superposition of Debye elemental processes with a Gaussian distribution of energy barriers g(E) ... 

A textbook example for the successful application of the model of Arkhipov et al. is the work of van Woudenbergh et al. [173]. More recently, Agrawal et al. [106] compared injection limited currents and space-charge-limited currents in a copper-phthalocyanine sandwich cell with TTO and Al electrodes. An analysis of experimental data yields consistent values for the width of the DOS distribution as well as for inter-site separation [174]. These studies support the model of thermally activated injection into a Gaussian DOS distribution of hopping sites and confirm the notion that disorder facilitates injection because it lowers the injection barrier, although the transport velocity decreases with increasing disorder. 

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