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Direct simulation of electron transfer using ring polymer molecular dynamics: Comparison with semiclassical instanton theory and exact quantum methods
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Figures

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FIG. 1.

Snapshots of the atomistic representation for the ET reaction, with the donor and acceptor metal ions shown in yellow, the electron ring polymer in black, and the water molecules in red and white. Typical configurations of the symmetric ET system are presented with the electron ring polymer (a) in transition between the redox sites, (b) in the reactant basin, and (c) in the product basin.

Image of FIG. 2.

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FIG. 2.

(a) FE profiles, FU), for the reactant (colored, at right) and product (left) diabatic electronic states as a function of the solvent collective variable in the atomistic representation. The various cases of the thermodynamic driving force for the ET reaction are labeled; see Tables III and VI for details. For each case, the FE profiles are vertically shifted to align the minima of the product basin. (b) The corresponding FE profiles as a function of the bead-count coordinate, F(f b). In the main panel, the profiles are vertically shifted to align the product basin; in the inset, the profiles are vertically shifted to align the reactant basin. (c) The corresponding RPMD transmission coefficients for the ET reaction, κ(t). In panels (b) and (c), the curves retain the same color scheme introduced in panel (a).

Image of FIG. 3.

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FIG. 3.

(a) ET reaction rates for the atomistic representation in the normal and activationless regimes, computed using RPMD (red) and Marcus theory (black). The various cases for the thermodynamic driving force are labeled. (b) Representative trajectories (red) from the ensemble of reactive RPMD trajectories for symmetric ET (Case I). The trajectories are plotted as a function of the ring-polymer centroid, , and the solvent collective variable, ΔU. The FE profile in these collective variables is also presented, with contour lines indicating FE increments of 10 kcal/mol. (c) Representative RPMD trajectories for activationless ET (Case IV) and the corresponding FE profile. The white arrows in panels (b) and (c) indicate the solvent reorganization mechanism for ET that is anticipated in the Marcus rate theory, and the dashed lines indicate values of ΔU at which the reactant and product diabats cross in Fig. 2(a).

Image of FIG. 4.

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FIG. 4.

(a) ET reaction rates for the atomistic representation in the inverted regime, computed using RPMD (red) and Marcus theory (black). The various cases for the thermodynamic driving force are labeled. (b) Representative trajectories (red) from the ensemble of reactive RPMD trajectories for inverted ET (Case VI). The trajectories are plotted as a function of the ring-polymer centroid, , and the solvent collective variable, ΔU. The FE profile in these collective variables is also presented, with contour lines indicating FE increments of 10 kcal/mol. The white arrows indicate the solvent reorganization mechanism for ET that is anticipated in the Marcus rate theory, and the dashed line indicates the value of ΔU at which the reactant and product diabats cross in Fig. 2(a).

Image of FIG. 5.

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FIG. 5.

(a) ET reaction rates for Model SB1, computed using RPMD (red), Marcus theory (black), and SCI theory (blue). (b) FE profiles, FU), for the reactant (colored, at right) and product (left) diabatic electronic states as a function of the solvent coordinate, s. The various cases of the thermodynamic driving force for the ET reaction are labeled; see Table IV for details. The arrow indicates the value of the solvent coordinate that maximizes , which corresponds to the dominant contribution to the SCI rate in Eq. (19).

Image of FIG. 6.

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FIG. 6.

The ET rates for Model SB1 corresponding to a Marcus-like mechanism (black) and the “direct” mechanism in Eq. (66) (red). SCI rates (blue) correspond to the kinetically favorable mechanism in all regimes. See text for details.

Image of FIG. 7.

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FIG. 7.

Normalized flux-flux autocorrelation functions C FF(t) for Model SB2, calculated using exact quantum dynamics for Cases I (black), II (blue), III (purple), and IV (red).

Tables

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Table I.

Complex times t k used to calculate the .

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Table II.

Parameters for the atomistic representation of ET.

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Table III.

Values of the asymmetry parameter ε considered in the atomistic representation and the corresponding thermodynamic driving force regimes.

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Table IV.

Values of the asymmetry parameter ε considered in the system-bath representation, the corresponding thermodynamic driving force regimes, and the electronic coupling matrix element, V 12.a

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Table V.

Parameters for the system-bath representation of ET.a

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Table VI.

ET reaction rates for the atomistic representation, obtained using RPMD and Marcus theory.a

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Table VII.

ET reaction rates for Model SB1, obtained using RPMD, Marcus theory, and SCI theory.a

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Table VIII.

ET reaction rates for a 1D asymmetric double well, obtained using SCI theory.a

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Table IX.

ET reaction rates for Model SB2, obtained using RPMD, Marcus theory, SCI theory, and exact quantum dynamics.a

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Table X.

Parameters for the left Coulombic well in the electron-ion potential energy function of Eq. (49) for Model SB1.a

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Table XI.

Parameters for the right Coulombic well in the electron-ion potential energy function of Eq. (49) for Model SB1.a

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Table XII.

Parameters for the left Coulombic well in the electron-ion potential energy function of Eq. (49) for Model SB2.a

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Table XIII.

Parameters for the right Coulombic well in the electron-ion potential energy function of Eq. (49) for Model SB2.a

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Table XIV.

The diagonal elements of the diabatic potential matrix V 11(s) and V 22(s) in Eqs. (55) and (56) for Model SB2.

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2011-08-17
2014-04-25

Abstract

The use of ring polymer molecular dynamics (RPMD) for the direct simulation of electron transfer(ET)reactiondynamics is analyzed in the context of Marcus theory, semiclassical instanton theory, and exact quantum dynamics approaches. For both fully atomistic and system-bath representations of condensed-phase ET, we demonstrate that RPMD accurately predicts both ETreaction rates and mechanisms throughout the normal and activationless regimes of the thermodynamic driving force. Analysis of the ensemble of reactive RPMD trajectories reveals the solvent reorganization mechanism for ET that is anticipated in the Marcus rate theory, and the accuracy of the RPMD rate calculation is understood in terms of its exact description of statistical fluctuations and its formal connection to semiclassical instanton theory for deep-tunneling processes. In the inverted regime of the thermodynamic driving force, neither RPMD nor a related formulation of semiclassical instanton theory capture the characteristic turnover in the reaction rate; comparison with exact quantum dynamics simulations reveals that these methods provide inadequate quantization of the real-time electronic-state dynamics in the inverted regime.

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Scitation: Direct simulation of electron transfer using ring polymer molecular dynamics: Comparison with semiclassical instanton theory and exact quantum methods
http://aip.metastore.ingenta.com/content/aip/journal/jcp/135/7/10.1063/1.3624766
10.1063/1.3624766
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