Perspective on quantum electrochemistry. A simple method for measuring the electron transfer rate constant

Quantum rate dynamics and charge screening at the nanoscale level

“…Finally, the thermal broadening of the energy E = e 2 /C q states is taken into account, using the grand canonical ensemble, [3][4][5] by correlating C q = dq/dV with f, where dq = Àedn and dV= Àdm/ e, leading to C q = Ne 2 (df/dm) by noting that n = Nf. df/dV is obtained as the derivative of f with respect to V such as that df/ dV = bf (1 À f ) and hence C q can be written as function of f such as 2-5…”

“…This is a correlation that was ignored for at least 50 years, despite the understanding of the concepts of k and G since the 1960s. The correlation between these quantities was identified simply as [2][3][4][5]…”

“…Note that G is equal to G 0 for the ideal transmission of a single electron across an ideal quantum channel, for which N is assumed to be unity, corresponding to situations in which P N n¼1 T n ðmÞ is also equal to unity. The term g r in eqn (1) refers to the degeneracy associated with two possible oxidation and reduction states for a particular scenario of electrochemical reactions, 4 in which g r is identified as 2. For cases where there are no electrochemical reactions, g r must be assumed to be 1 in eqn (1).…”

“…1a, which explains the assembly of a redox-active monolayer onto a gold electrode. It is important to note that each redox moiety of the monolayer is modelled as a quantum point contact, 3,4 following the scheme of eqn (1). Each of these quantum point contacts has individual quantum resistors (h/e 2 ) and capacitors (c q ), which as illustrated in Fig.…”

“…Because of the latter situation, in which one of the DOS is metallic and the other is a nanoscale structure, any accessible DOS of the nanoscale structure suitably coupled to the conductive electrode can be effectively measured. It is worth emphasizing that the DOS associated with C q , as measured in a diffusionless ET situation, 3,4 is not related to the Marcus-Gerischer DOS, 9 which is theoretically described for a diffusiondependent ET context.…”