A non-Marcus model for electrostatic fluctuations in long range electron transfer

Abstract

We propose a new model for the elementary act of electron transfer between two species in solution. The central idea is that the solution in the immediate vicinity of each species may be represented by an equivalent circuit consisting of a Debye circuit shunted by a resistor. Based on this insight, we derive a new formula for the one-dimensional potential energy profile of a coupled donor–acceptor pair at finite (but large) separation d, along a charge-fluctuation reaction co-ordinate, at fixed radii of the transition states. The corresponding reorganisation energy of the reaction is also derived, and it is found to differ from that in the Marcus theory. In particular, the new model predicts that the reorganisation energy is independent of the static dielectric constant of the solution, whereas the old model predicts a strong dependence. The difference is traced to the fact that the Marcus theory omits consideration of the work required to form the charge fluctuations and focuses instead on the work required to localise the charge fluctuations. In general, the equivalent circuit approach permits many of the difficult-to-derive equations of non-equilibrium polarisation theory to be written down by inspection.

Appendix

List of symbols

ΔG ^o :

the standard free energy of the reaction

ɛ₀ :

the permittivity of free space

ɛ(0):

the relative permittivity (dielectric constant) of the solution in the low frequency limit

ɛ(∞):

the relative permittivity (dielectric constant) of the solution in the high frequency limit

R ₂ :

the resistance to reorientation of solvent molecules in the Debye circuit

R ₃ :

the resistance to ionic transport of co-ions and counter-ions across the Debye circuit

C ₂ :

the capacitance due to the polarisability of the solvent in the Debye circuit

C ₁ :

the capacitance due to the polarisability of the electrons in the Debye circuit

W ₂ :

the work of localisation of the charge

W ₃ :

the work of ionisation of the charge

W _T :

the total work of formation of the charge Q ₁ on C ₁

−e :

the charge on the electron

Q ₁ :

a charge fluctuation on one hard sphere

a _D :

the radius of the donor supermolecule in the transition state

a _A :

the radius of the acceptor supermolecule in the transition state

d :

the centre-to-centre distance between D and A

λ :

the reorganisation energy