Fokker-Planck Modeling of the Electronic Localization

In order to better understand the actual ELF approach, it is worth reminding that the origin of the above / (x) relies in evaluation of the conditioned pair probability with which one electron is located at point x with the spin a once the reference electron is located at point x with the same (parallel) spin r (Becke, 1988) 

Quantum Nanochemistry—Volume II Quantum Atoms and Periodicity 

As previously shown, the Markovian treatment of the conditioned probability density given by the general Fokker-Planck path integral (5.349) with correspondences of Table 5.1 in atomic units h-m- ) 

within the potential form (5.397) the x and coordinates are separated and coupled, allowing the averages operations being performed firstly on the coordinates of the coupled x, electron, while replacing in the final result the referential electronic x influences. However, the present Markovian ELF picture is summarized by the following analytical steps (Putz, 2005)  

The result from (ii) is employed upon the specific integration rule (Parr Yang, 1989) 

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