Forward hopping motion

Both our QM and QM/MM simulations show that hydroxide migration occurs in step-wise hops followed by long resting periods. The average hopping rates (counting all proton transfer events, not only the forward-hopping motions) are 1.67 and 0.48 ps for QM/MM and QM simulations respectively. 

The deviations between the MIGRATION curve resulting from correlated forward-backward hopping motions of the ions and the experimental spectrum indicate that additional movements of charged particles or groups do contribute to the conductivity in the dispersive regime. The total spectrum displayed by the symbols in Fig. 12, can, however, be described very well if an additional contribution to the conductivity is considered (red line in Fig. 12). The red line in Fig. 12 was obtained by the following equation ... 

The stochastic motion of particles in condensed matter is the fundamental concept that underlies diffusion. We will therefore discuss its basic ideas in some depth. The classical approach to Brownian motion aims at calculating the number of ways in which a particle arrives at a distinct point m steps from the origin while performing a sequence of z° random steps in total. Consider a linear motion in which the probability of forward and backward hopping is equal (= 1/2). The probability for any sequence is thus (1/2). Point m can be reached by z° + m)/2 forward plus (z° m)/2 backward steps. The number of distinct sequences to arrive at m is therefore... 

Fig. 10. (a) Energy barrier for hopping transport illustrating activation energy for forward motion and activation energy for reverse motion, (b) Series of energy... 

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