Marcus Theory of Electron Transfer 5: Two-Dimensional Bell-Shaped Marcus Curves
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This Demonstration compares the two-dimensional bell-shaped Marcus curves for the classical (blue) and semiclassical approach (yellow). The main difference, in the Marcus inverted region (at large negative values of ), is simple to observe and manipulate. Note that we show log plots. The three-dimensional plots in earlier Demonstrations (see Related Links) are linear plots in the rate . Note that at very low temperatures, both models become inaccurate.[more]
This Demonstration describes the semiclassical Marcus model (also called Marcus–Levich–Jortner theory, MLJ) in two dimensions, and compares it to the classical Marcus equation (CME). The plots show log as a function of the Gibbs free energy change .
Nobel Prize winner Rudolph Marcus developed the theory of electron transfer . The 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:
where . This plot of log versus is shown in yellow.
The CME, shown in blue, is represented by:
You can change the axes with the control sliders. You can also change the number of plot points. More plot points give greater detail, but at the cost of slowing the computation.
The starting values are from the so-called Closs and Miller data , which provided the first experimental proof of the inverted region.
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, which contribute to the Franck–Condon weighted density of states. 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 temperature or at low solvent reorganization energy .
We can approximate the measured rates  with these equations by adjusting the input data.
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 in the summation factor determines 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 [6–11]. If , the equation behaves like the CME.
The 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. MLJ becomes a better approximation at lower temperatures. But for very low temperatures, there are modified equations (see below). For very low temperatures, 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 if , where is an average solvent vibrational frequency. The intramolecular vibrations are quantum, that is, .
In general, a vibration can be treated as classical if , and as quantum if . You must monitor these values as temperature is varied.
By adjusting the parameters, a resonance effect between the internal reorganization energy and the vibration 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, in the absence of experimental evidence.
More information on these oscillations at low reorganization energies and can be found in the references. Oscillations in the bell-shaped Marcus curve (resonance type effects in the Franck–Condon factor) are discussed in [12–17].
Below 10 K, more complex models are needed [18, 19].[less]
Contributed by: René M. Williams (January 2023)
Open content licensed under CC BY-NC-SA
Snapshot 1: data based on the work of Closs and Miller (focused here on the iso-octane data) at , with low solvent reorganization energies
Snapshot 2: for , the semiclassical Marcus equation behaves similarly to the CME
Snapshot 3: in the high temperature limit (1000 K), the semiclassical Marcus equation reduces to the CME
Snapshot 4: the very low temperature graph shows the effects of the summation factor: bell curves are added and shifted with increase of
Input data based on the work of Closs and Miller (focused here on the iso-octane data) at .
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