Editing Electrode (section)

== Marcus's theory of electron transfer ==
Marcus theory is a theory originally developed by Nobel laureate [[Rudolph A. Marcus]] and explains the rate at which an electron can move from one chemical species to another,<ref name="Nobel">{{cite web|last=|first=|date=8 December 1992|title=Electron Transfer Reactions in Chemistry: Theory and Experiment|url=https://www.nobelprize.org/prizes/chemistry/1992/marcus/lecture/|access-date=2 April 2007|work=Nobelstiftung}}</ref> for this article this can be seen as 'jumping' from the electrode to a species in the solvent or vice versa.
We can represent the problem as calculating the transfer rate for the transfer of an electron from donor to an acceptor 
: D + A → D<sup>+</sup> + A<sup>−</sup>

[[File:Hush adiabatic electron transfer model parameters.png|thumb|Potential energy surface for the donor and the acceptor as]]

The potential energy of the system is a function of the translational, rotational, and vibrational coordinates of the reacting species and the molecules of the surrounding medium, collectively called the reaction coordinates. The abscissa the figure to the right represents these. From the classical electron transfer theory, the expression of the [[reaction rate constant]] (probability of reaction) can be calculated, if a non-adiabatic process and parabolic potential energy are assumed, by finding the point of intersection ({{tmath| Q_x }}). One important thing to note, and was noted by Marcus when he came up with the theory, the electron transfer must abide by the law of conservation of energy and the Frank-Condon principle. 
Doing this and then rearranging this leads to the expression of the free energy activation ({{tmath| \Delta G^{\dagger} }}) in terms of the overall free energy of the reaction ({{tmath| \Delta G^{0} }}).
<math display="block">\Delta G^{\dagger} = \frac{1}{4 \lambda} (\Delta G^{0} + \lambda)^{2} </math>

In which the <math> \lambda </math> is the reorganisation energy.
Filling this result in the classically derived [[Arrhenius equation]]
<math display="block">k = A\, \exp\left(\frac{- \Delta G^{\dagger}}{kT}\right),</math>
leads to
<math display="block">k = A\, \exp\left[{\frac {-(\Delta G^{0} + \lambda)^{2}}{4 \lambda k T}}\right] ,</math>
with ''A'' being the pre-exponential factor, which is usually experimentally determined,<ref>{{cite web |doi=10.1351/goldbook |url = https://goldbook.iupac.org/|title=The IUPAC Compendium of Chemical Terminology |year=2019 |editor1-last=Gold |editor1-first=Victor|doi-access=free }}</ref> although a semi-classical derivation provides more information as is explained below.

This classically derived result qualitatively reproduced observations of a maximum electron transfer rate under the conditions {{tmath|1= \Delta G^{\dagger} = \lambda }}.<ref>{{cite web|last=|first=|date=12 December 2020|title=Marcus Theory for Electron Transfer.|url=https://chem.libretexts.org/@go/page/107311|access-date=24 January 2021}}</ref> For a more extensive mathematical treatment one could read the paper by Newton.<ref>{{cite journal|doi=10.1021/cr00005a007|title=Quantum chemical probes of electron-transfer kinetics: The nature of donor-acceptor interactions|year=1991|last1=Newton|first1=Marshall D.|journal=Chemical Reviews|volume=91|issue=5|pages=767–792}}</ref> An interpretation of this result and what a closer look at the physical meaning of the <math>\lambda</math> one can read the paper by Marcus.<ref>{{cite journal|doi=10.1103/RevModPhys.65.599|title=Electron transfer reactions in chemistry. Theory and experiment|year=1993|last1=Marcus|first1=Rudolph A.|journal=Reviews of Modern Physics|volume=65|issue=3|pages=599–610|bibcode=1993RvMP...65..599M|url=https://resolver.caltech.edu/CaltechAUTHORS:20150414-104906889}}</ref>

The situation at hand can be more accurately described by using the displaced harmonic oscillator model, in this model [[quantum tunneling]] is allowed. This is needed in order to explain why even at near-zero [[absolute temperature]] there are still electron transfers,<ref>DeVault, D. (1984) Quantum Mechanical Tunneling in Biological Systems; Cambridge University Press: Cambridge.</ref> in contradiction with the classical theory.

Without going into too much detail on how the derivation is done, it rests on using [[Fermi's golden rule]] from time-dependent [[perturbation theory]] with the full [[Hamiltonian mechanics#Basic physical interpretation|Hamiltonian]] of the system. It is possible to look at the overlap in the wavefunctions of both the reactants and the products (the right and the left side of the chemical reaction) and therefore when their energies are the same and allow for electron transfer. As touched on before this must happen because only then conservation of energy is abided by. Skipping over a few mathematical steps the probability of electron transfer can be calculated (albeit quite difficult) using the following formula
<math display="block">w_{ET}= \frac{|J|^{2}}{\hbar^{2}}\int_{-\infty}^{+\infty}dt\, e^{-i \Delta Et / \hbar - g (t)} ,</math>
with <math> J </math> being the electronic coupling constant describing the interaction between the two states (reactants and products) and <math> g(t) </math> being the [[spectral line shape|line shape function]]. Taking the classical limit of this expression, meaning {{tmath| \hbar \omega \ll k T }}, and making some substitution an expression is obtained very similar to the classically derived formula, as expected. 
<math display="block">w_{ET} = \frac{|J|^{2}}{\hbar} \sqrt{\frac{\pi}{\lambda k T}}\exp\left[\frac {- ( \Delta E + \lambda )^{2}} {4 \lambda k T}\right]</math>

The main difference is now the pre-exponential factor has now been described by more physical parameters instead of the experimental factor {{tmath| A }}. One is once again revered to the sources as listed below for a more in-depth and rigorous mathematical derivation and interpretation.