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WIREs Comput Mol Sci
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Elementary electron transfer reactions: from basic concepts to recent computational advances

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Abstract The basic chemicophysical concepts and the most recent developments in the dynamics of the elementary electron transfer reactions are reviewed, posing particular attention to discrete state approaches, which combine use of a few experimental data with reliable ab initio calculations of the equilibrium nuclear configurations and normal coordinates of vibration of the redox partners. WIREs Comput Mol Sci 2013, 3:542–559. doi: 10.1002/wcms.1147 This article is categorized under: Structure and Mechanism > Molecular Structures

The role of nuclear motion in electron transfer (ET). Curve R and P represent schematic profiles of the change of Gibbs free energy with the nuclear coordinate for ET reactants and products. In the classical Marcus theory the passage from R to P can occur at xC, where R and P cross each other. λ is the reorganization energy, i.e., the energy the system has to spend to reach the equilibrium position of one state, but staying in the other state. ΔG0 and indicates the ET free energy change and the free energy activation energy. In a one‐dimensional system with equal curvatures for the curves R and P, the free energy activation energy can be easily related to λ and ΔG0 by noting that, since GR = Kx2/2 − ΔG0 and GP = K(xxP)2/2, xC = xP/2 + ΔG0/KxP, so that .

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Franck–Condon integrals (absolute values) for electron transfer from bacteriopheophytin (BPh) to ubiquinone (QA) as a function of the energy difference between the ground vibronic state of BPh–QA and the final state of BPh–Q. Twenty vibrational states were allowed to change quantum number, with a maximum change of three quanta. (Reproduced with permission from Ref . Copyright 2007, Springer‐Verlag.)
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Displaced modes of bacteriochlorophyll anion (blue) and bacteriopheophytin (red) upon relieving and accepting an electron. Displacements are in dimensionless units.

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Total density of states versus energy for some of the redox pairs present in bacterial photosynthetic reaction centers: bacteriochlorophyll/pheophytin, pheophytin/ubiquinone, and ubiquinone/ubiquinone. The curves have been calculated by using the vibrational frequencies obtained at DFT/B3LYP level for the isolated molecules in the gas phase and the Beyer–Swinehart algorithm. The curves indicate how the density of the manifold of vibrational states of the electron transfer (ET) products change as the exothermicity of the ET reactions increases. The experimental ΔE of the three processes are also reported., The case of the pair ubiquinone/ubiquinone anion is intriguing for the huge difference with respect to the other two long‐range ET processes, which would suggest that different mechanisms must probably be invoked.

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Computed (lines) and observed (squares and circles) temperature dependence of ET rates for the redox pair BPh/QA of bacterial photosynthetic reaction centers. Circles and squares refer to measurements at 545 nm and 665 nm, respectively. Computed curves: DE = 5400 cm−1(dashed line), DE = 6200 cm−1(dotted line), DE = 5830 cm−1(full line). Experimental values are taken from Ref . Theoretical rates have been obtained by evaluating the two parameters ΔE and |HBA|2 by a least squares fit of the experimental data, which yielded 5830 and 10.0 cm−1, respectively. The other two curves have been obtained by setting ΔE to 5400 and 6200 cm−1, and evaluating the electronic coupling term by least squares, which yielded |HBA|2 = 9.9 and 10.1 cm−1, respectively. (Reprinted with permission from Ref . Copyright 2011, Royal Society of Chemistry.)
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Thermally averaged Franck–Condon (FC) weighted density of states for electron transfer from B to QA as a function of the electronic energy difference between initial and final states. The FC‐weighted density of states has been computed by the generating function method, including the whole sets of normal coordinates of the two redox pairs, without posing any limit to normal modes excitation numbers.

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Probability energy distributions for removing an electron from the donor and for releasing an electron to the acceptor, in the framework of the Franck–Condon principle. EA is the energy required to adiabatically extract an electron from donor, and EB that gained for the injection of an electron to the acceptor. Because the potential energy profiles corresponding to the neutral and charged (initial and final) states are displaced each other, the maximum probability for removing an electron occurs at EA + λA, and that for releasing an electron to the acceptor at EB − λB.

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