Marcus Theory of Electron Transfer 3: Interactive Potential Energy Surfaces for Charge Separation

This Demonstration applies the classical Marcus equation to calculate and manipulate the rate of electron transfer (charge separation) by varying certain "molecular parameters". Eight molecular input parameters and four general input parameters are varied to calculate the Gibbs free energy of electron transfer , the barrier and the total reorganization energy , as well as the rate of the charge separation . The pre-exponential factor represents the rate in the absence of a barrier. You can use the sliders to observe the effects of the individual parameters on the rates and thermodynamic parameters.
The blue curve represents the initial reactant (locally excited state), while the yellow curve represents the final product (after charge separation). Also shown as vertical lines are the driving force (black), the barrier (red) and the total reorganization energy (blue). and are the redox energies of the donor and acceptor (in the same solvent with dielectric constant epsEC versus the same reference electrode); is the center-to-center distance; and are the ionic radii; is the electronic coupling; is the internal reorganization energy; the eps are dielectric constants; and is the refractive index. is the singlet state energy (or triplet state energy) of the system.
Shown at the top are the driving force, total reorganization energy, solvent reorganization energy, barrier to electron transfer and the pre-exponential factor. The starting data comes from C60[11]DMA in a benzonitrile solvent [3]. For a neutral donor and acceptor, the Coulomb correction equals 1. For a charge shift reaction, it is 0. For charged initial reactants but neutral products, it is (use the "Coulomb correction " slider). Use the axis minimum/maximum sliders to scale the graph.
This project is based on two Demonstrations developed by students (see Related Links). Those projects actually contain a minor error (for the barrier), but they inspired me to develop this project. Their modified code is used in this project and I thank these people very much!
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 energy ( and ) and the electronic coupling (). The CME underestimates the electron transfer rate in the inverted region [2, 3].
We can also use the so-called Closs and Miller data [4], which provided the first experimental proof of the inverted region. This is a charge shift reaction, so the Coulomb correction should be set to zero. We can approach the measured rates [5] by using the appropriate input parameters, like we show for C60[11]DMA in benzonitrile for charge separation. Additional information on the Marcus theory and electron transfer is available in [6–9].
See [10] for some further examples from literature data and test data.
Note that under certain "unrealistic" conditions, the solvent reorganization can become negative (which makes no physical sense). This results in complex number solutions and imaginary electron transfer rates (e.g. with low and high and , you can get ). In order to avoid Mathematica plotting problems we take the absolute value of the total reorganization energy.

SNAPSHOTS

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DETAILS

Snapshot 1: Data on charge separation for the compound C60[11]DMA in benzonitrile as solvent. Normal region.
Snapshot 2: Data on charge separation of the compound C60[3]TMPD in toluene as solvent. Optimal region.
Snapshot 3: Data on charge shift reaction of Closs and Miller in methyltetrahydrofuran as solvent. Inverted region, in the classical model.
Snapshot 4: Data on charge recombination of the compound C60[3]TMPD in toluene as solvent. Inverted region, in the classical model. Zero-zero energy adjusted to fit charge recombination.
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," (Jan 26, 2022) doi:10.13140/RG.2.2.16547.30244.
[7] R. M. Williams. Photoinduced Electron Transfer - The Classical Marcus Theory [Video]. (Jan 26, 2022) youtu.be/YFzeMMOvhl0.
[8] R. M. Williams. Photoinduced Electron Transfer - The Semi-classical Marcus-Levich-Jortner Theory [Video]. (Jan 26, 2022) youtu.be/GnPIbH6nM9o.
[9] R. M. Williams. University of Amsterdam. (Jan 21, 2022) www.uva.nl/en/profile/w/i/r.m.williams/r.m.williams.html.
[10] R. M. Williams. "22-Marcus-Mathematica-TESTDATA-Assignments." (Jan 26, 2022) doi.org/10.21942/uva.17912015.
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