Computing Solvation Shell Dynamics and Energetics in Electron Transfer Reactions via Molecular Dynamics Simulations

Abstract

Marcus theory is fundamental to describing electron transfer reactions and quantifying their rates, effectively representing the energy surface associated with an electron transfer from the reactant to the product ionic state via parabolas within a reaction coordinate diagram. Here, we present an intuitive and computationally efficient generalised reaction coordinate amenable to molecular dynamics simulations. By utilising the nuclear charge of the ion, we are able to quantify in a targeted approach changes in the ion's solvation shell, thereby efficiently obtaining the free energy profile associated with the electron transfer, the transition state geometry and the evolution of the water network.

Figures (3)

• Figure 1: (a) Schematic free energy profile of Fe^2+ and Fe^3+ in water as a function of reaction coordinate. The ground states of Fe^2+ and Fe^3+ are marked by a red and a blue circle, respectively, while the transition state at which the charge transfer occurs is marked by a purple circle. $\Delta G$ represents the minimal energy needed for the charge transfer to happen, while the red dashed circle marks an high energy state in which Fe^2+ is occupying a solvation shell corresponding to the Fe^3+ ground state. (b) Schematic free energy profile as a function of the reaction coordinate, which we will obtain from the calculations. The dashed red and green circles denote the energy of the high energy configurations Fe$^{{2+}^*}$ and Fe$^{{3+}^*}$ obtained using Eq. \ref{['eq:UpSampling']}.
• Figure 2: Calculated free energy profiles for (a) Fe$^{2+}$ and Fe$^{3+}$ and (b) Na$^0$ and N$^{1+}$. The actually calculated free energies obtained from the MD-runs for different $q_{\mathrm{core}}$ values are shown as points. Error bars are omitted for clarity, since the standard error of the mean (SEM) for each point is less than 25 meV. Parabolas fitted using the ground state (i.e. minimum Fe$^{2+}$ or Fe$^{3+}$ energy) and its corresponding high energy vertical excitation state (i.e. Fe$^{{2+}*}$ or Fe$^{{3+}*}$ are shown as solid lines and clearly reveal deviations from the calculated points. Applying, e.g., positive bias $\mu_e > 0$ shifts the Fe$^{3+}$ (Na$^{1+}$) parabolas upward, as shown by the dashed orange curves in (a) and (b). The free energy barrier for electron transfer and how it changes as the electron chemical potential $\mu_e$ is varied (i.e. application of bias) is shown in (c) for Fe$^{2+} \rightleftharpoons$ Fe$^{3+}$ and in (d) for Na$^{0} \rightleftharpoons$ Na$^{1+}$. The dashed lines depict the free energy barrier variation obtained by shifting the parabolas fitted in (a) for Fe$^{2+}$ and Fe$^{3+}$ (respectively (b) for Na$^0$ and Na$^{1+}$) with respect to each other by a bias potential $\mu_e$. Connecting the MD-calculated points in (a) (respectively (b)) by linear segment and then shift the ensuing curves by $\mu_e$ with respect to each other, results in the solid lines. The squares each of the two lines separate regions where the slopes of the intersecting linear segments remain unchanged.
• Figure 3: (a) Evolution of Na-ion solvation shell as a function of the charge state of the ion ($q_{\rm core}$) with positive (H) and negative (O) charges in the solvation shell shown in blue and red, respectively. (b) The corresponding radial distribution functions (RDFs) for Na-O and Na-H as a function of the charge state $q_{\rm core}$ of the Na ion. (c) Evolvement of the coordination number of Fe to water molecules within the first solvation shell of a Fe$^q$ ion, as it charge changes from $2^+$ to $3^+$. A similar plot for Na is shown in the supplemental material SM.