https://doi.org/10.1351/goldbook.RT07472
Empirical correlation found between the observed second-order rate constant, \(k_{\rm{q}}\), for an intermolecular electron-transfer reaction and the Gibbs energy of the photoinduced electron transfer process within the encounter complex (\(\Delta _{\rm{ET}}G^{\,\unicode{x26ac}}\)): \[k_{\rm{q}} = \frac{k_{\rm{d}}}{1+ \frac {k_{\rm{d}}}{K_{\rm{d}}\, \rm{Z}}\left [ \rm{exp}\left ( \frac{\Delta G^{\ddagger }}{RT} \right ) + \rm{exp}\left ( \frac{\Delta_{\rm{ET}} G^{o }}{RT} \right ) \right ]}\] with \(k_{\rm{d}}\) and \(k_{\rm{-d}}\) the rate constant for the formation and separation, respectively, of the encounter (precursor) complex, \(K_{\rm{d}} = \: ^{k_{\rm{d}}}\! /_{k_{\rm{-d}}}\), \(Z\) the universal collision frequency factor, \(R\) the gas constant, \(T\) the absolute temperature and \(\Delta G^{\ddagger}\) the activation Gibbs energy of the forward electron transfer reaction.
Note: In the original formulation of this equation the value \(\frac{k_{\rm{d}}}{K_{\rm{d}}\, \rm{Z}} = 0.25\) in acetonitrile was used.