Net electron flow across a geometric cross-section

3 Net electron flow across a geometric cross-section 

We next accumulate the above probability current across a cross-section. Let jx xo,y,z,t) be a flux at time t passing across an x-y plane that is set at a point a = xo in the Cartesian coordinates. Define the following function 

This integral means the total flux that passes through the surface of section, 5 at X = rco at time t. Likewise, one can define a similar quantity like jy (x, j/0, z,t) and Sy yo, t). The numerical integral of Eq. (7.39) is approximated by the sum 

We further define a net flow accumulated during a given time-interval in such a way that 

we have observed a clear and qualitative (topological) directionality of the net electron flow, which is schematically summarized in red arrows in the bottom panel of Fig. 7.21. Interestingly, electrons flow from the singlebond site to the double-bond counterpart in each monomer (08— 06 and 03— 01) as two FAM are coming closer to each other. This seems rather counter intuitive at first sight. On the other hand, only slight electron flow is observed in the proton-bridging site in the same direction as the motion of double proton transfer. 

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