Smoluchowski equations proton transfer

Proton transfer dynamics of photoacids to the solvent have thus, being reversible in nature, been modelled using the Debye-von Smoluchowski equation for diffusion-assisted reaction dynamics in a large body of experimental work on HPTS [84—87] and naphthols [88-92], with additional studies on the temperature dependence [93-98], and the pressure dependence [99-101], as well as the effects of special media such as reverse micelles [102] or chiral environments [103]. Moreover, results modelled with the Debye-von Smoluchowski approach have also been reported for proton acceptors triggered by optical excitation (photobases) [104, 105], and for molecular compounds with both photoacid and photobase functionalities, such as lO-hydroxycamptothecin [106] and coumarin 4 [107]. It can be expected that proton diffusion also plays a role in hydroxyquinoline compounds [108-112]. Finally, proton diffusion has been suggested in the long time dynamics of green fluorescent protein [113], where the chromophore functions as a photoacid [23,114], with an initial proton release on a 3-20 ps time scale [115,116]. 

The traditional analysis of proton transfer reactions [3, 5-7] is by a kinetic scheme e.g., eq 1, and the ensuing chemical rate equations describing the time dependence of bulk concentrations. This is insufficient for explaining the long time tail. The minimal level of complexity has to involve the spatial diffusion of the proton in the electrostatic field of the anion, as depicted by the time-dependent Smoluchowski equation [15], but with a boundary condition (the back-reaction boundary condition) which describes reversible reactions [10]. 

Pines E, Huppert D, Agmon N (1988) Geminate recombination in excited-stale proton-transfer reactions—numerical solution of the debye-smoluchowski equation with backreaction and comparison with experimental results. J Chem Phys 88 5620... 

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