Marcus Theory of Electron Transfer 2: Semiclassical Marcus Equation in Three Dimensions
This Demonstration describes the semiclassical Marcus model (also called Marcus–Levich–Jortner Theory, MLJ) in three dimensions. The rate on the axis is plotted as a function of the Gibbs free energy change and the solvent reorganization energy on the and axes. This is a linear plot; often plots are used in this application.
Nobel Prize winner Rudolph Marcus developed the theory of electron transfer . The classical Marcus equation (CME) is based on two simultaneous quadratic relations involving the driving force , the internal and external (solvent) reorganization energies and and the electronic coupling . The CME underestimates the electron transfer rate in the inverted region [2, 3]. Therefore we need the MLJ equation:
You can change the axis in the Manipulate sliders. You can also change the number of plot points, giving the graph more detail, but at the cost of slowing the computation. Use Ctrl+Return on the graph for a front view and rotate. With Shift+Return you control size. The starting point is a back view of the graph, which highlights the bell-shaped Marcus curve.
The starting values are from the so-called Closs and Miller data , which provided the first experimental proof of the inverted region. We illustrate this with the isooctane data in Snapshot 1.
We apply the semiclassical Marcus expression to the inverted region. If is much larger than the total , then inverted region effects are apparent. The enhanced rate (relative to CME) in the inverted region is related to the overlap of vibrational wavefunctions that contribute to the Franck–Condon weighted density of states (the Franck–Condon factor, FCWD). The Huang–Rhys factor is related to the vibronic coupling, which is electron-phonon coupling (represented by the symbol in the MLJ equation). The vibrational energy shows up in the spacing of the spikes in the graph at low solvent reorganization energy .
With the semiclassical equation, we can approximate the measured rates  by adjusting the vibration that mediates the electron transfer or by adjusting the electronic coupling.
Fit-values bell curve MTHF of Closs and Miller:
, (dielectric constant of the solvent)
, (refractive index of solvent)
Fit-values bell curve isooctane of Closs and Miller:
, (dielectric constant of the solvent)
, (refractive index of solvent)
The parameter is in the summation factor, determining how many transfer channels contribute to the total rate . Often only six channels are needed. Additional information on Marcus theory and electron transfer is given in [6–9]. If , the equation behaves like the classical Marcus equation.
This Demonstration displays a three-dimensional representation of the semiclassical Marcus model, also called the Marcus–Levich–Jortner theory. The starting values are from the Closs and Miller data, the first proof of the inverted region. Other aspects are defined in the interface.
The Marcus–Levich–Jortner (MLJ) theory is especially evident in the inverted region. As the value of approaches twice the total , inverted region effects start playing a more important role. Also, at lower temperatures, MLJ becomes a better approximation. But for very low temperatures, there are modified equations (see below). For very low temperature, both vibrations should be treated quantum mechanically.
As for validity of the model, the (single mode) MLJ theory can be applied: above , with the electronic coupling between and ; in polar liquids, with ; and in molecular solids, with (pertaining to both intermolecular phonons and solid matrix phonons with energies of 0.00124 to 0.0124 eV). Intramolecular frequencies can range between 300 and (0.0372 and 0.372 eV). It is also possible to apply the multi-mode MLJ equation .
Thus, in the MLJ theory the internal lambda is treated quantum mechanically. The solvent reorganization is treated classically.
In the intermediate temperature range, the most common one, the solvent (environmental) vibrational modes can be treated classically , where is an average solvent vibrational frequency. The intramolecular vibrations are quantum, that is, .
In general, a vibration can be treated as classical if ; it should be treated as quantum if . If you look at temperature effects, you have to monitor these values (given in the text above in eV) relative to (presented as output).
By adjusting the parameters, a resonance effect between the internal reorganization energy and the vibration that mediates the transfer can be observed, enhancing the rate at certain values (the sharp spikes at low solvent reorganization); it is not clear if this is an artifact of the theory. There seems to be no experimental evidence of this effect.
More information on these oscillations at low reorganization energies and are available. Papers that report oscillations in the Bell-shaped Marcus curve (resonance type effects in the Franck–Condon factor) are given [12–17].
Below 10 K, more complex models are needed [18, 19].
Snapshot 1: data based on the work of Closs and Miller (focused here on the isooctane data) at , shifting the axis to 0.1 eV as the upper limit, highlighting the region with low solvent reorganization energies
Snapshot 2: for , the semiclassical Marcus equation behaves similarly to the classical Marcus equation
Snapshot 3: in the high temperature limit, the semiclassical Marcus equation reduces to the classical Marcus equation
The results of this Demonstration were checked against the R-package  that runs in the statistical software package R  for computing and graphics.
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