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Marcus Theory of Electron Transfer 4: Classical Marcus Equation in Three Dimensions

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This Demonstration describes the classical Marcus model in three dimensions. The rate is plotted on the axis as a function of the Gibbs free energy change and the total reorganization energy . This is a linear plot; often is plotted in this application.

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Nobel Prize winner Rudolph Marcus developed the theory of electron transfer [1]. 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 total reorganization energy is given by .The CME underestimates the electron transfer rate in the inverted region [2, 3]. The classical Marcus equation can be written as

.

You can change the axes in 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. Use Ctrl+Return on the graph for a front view. 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 [4], which provided the first experimental proof of the inverted region, characterizing the classical model.

With the classical equation, you can approximate the measured rates [5] by adjusting the parameters or the electronic coupling.

Additional information on Marcus theory and electron transfer is given in [6–9].

This Demonstration displays a three-dimensional representation of the classical Marcus model.

As for the validity of the model, the Marcus theory can be applied: above , with the electronic coupling between about 1 and 200 .

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Contributed by: René M. Williams (August 2022)
Open content licensed under CC BY-NC-SA

Snapshots

Details

Snapshot 1: increasing the solvent reorganization energy makes the curve more broad, lower and shifted to the right

Snapshot 2: note the shaper curve at 77 K. With lower , the maximum attainable rate increases within the classical Marcus model show a sharper, higher peak.

Snapshot 3: in the high temperature limit, the semiclassical Marcus equation reduces to the classical Marcus equation; note the broadening of the curve at 1000 K

The results of this Demonstration were checked against the R-package [10] that runs in the statistical software package R [11] for computing and graphics.

References

[1] R. A. Marcus, "Electron Transfer Reactions in Chemistry: Theory and Experiment (Nobel Lecture)," Angewandte Chemie International Edition, 32(8), 1993 pp. 1111–1121. doi:10.1002/anie.199311113.

[2] S. Chaudhuri, S. Hedström, D. D. Méndez-Hernández, H. P. Hendrickson, K. A. Jung, J. Ho and V. S. Batista, "Electron Transfer Assisted by Vibronic Coupling from Multiple Modes," Journal of Chemical Theory and Computation, 13(12), 2017 pp. 6000–6009. doi:10.1021/acs.jctc.7b00513.

[3] P. F. Barbara, T. J. Meyer and M. A. Ratner, "Contemporary Issues in Electron Transfer Research," Journal of Physical Chemistry, 100(31), 1996 pp. 13148–13168. doi:10.1021/jp9605663.

[4] G. L. Closs and J. R. Miller, "Intramolecular Long-Distance Electron Transfer in Organic Molecules," Science, 240(4851), 1988 pp. 440–447. doi:10.1126/science.240.4851.440.

[5] P. Hudhomme and R. M. Williams, "Energy and Electron Transfer in Photo- and Electro-active Fullerene Dyads," Handbook of Carbon Nano Materials (F. D'Souza and K. M. Kadish, eds.), Hackensack, NJ: World Scientific, 2011 pp. 545–591. doi:10.1142/9789814327824_0017.

[6] R. M. Williams. "Introduction to Electron Transfer." (Nov 11, 2021) doi:10.13140/RG.2.2.16547.30244.

[7] R. M. Williams. Photoinduced Electron Transfer—The Classical Marcus Theory [Video]. (May 17, 2022) youtu.be/YFzeMMOvhl0.

[8] R. M. Williams. Photoinduced Electron Transfer—The Semi-classical Marcus–Levich–Jortner Theory [Video]. (May 17, 2022) youtu.be/GnPIbH6nM9o.

[9] R. M. Williams. University of Amsterdam. (May 17, 2022) www.uva.nl/en/profile/w/i/r.m.williams/r.m.williams.html.

[10] J. Idé and G. Raos. "ChargeTransport: Charge Transfer Rates and Charge Carrier Mobilities." (May 17, 2022) CRAN.R-project.org/package=ChargeTransport.

[11] "What Is R?" The R Foundation. (May 17, 2022) www.r-project.org/about.html.

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