Ab Initio Free Energy Surfaces for Coupled Ion-Electron Transfer

Abstract

The Marcus theory of electron transfer assumes that diabatic energy gaps are sampled from a single ensemble. This assumption can break down in spatially anisotropic environments, such as Faradaic reactions at electrochemical interfaces, where distinct solvent ensembles arise along a collective variable describing the anisotropy. Treating this collective variable as an additional reaction coordinate linearly independent from the Marcus reaction coordinate, we develop a formalism that enables calculation of the resulting Coupled Ion-Electron Transfer (CIET) free-energy surface directly from constrained ab initio trajectories. Applied to CO2 redox on a gold electrode, this method reveals strong coupling to the anisotropy, predicting significantly different activation barriers compared to either coordinate alone.

Figures (2)

• Figure 1: Snapshots of the model aqueous CO$_2$–Au interface constrained to (left) the oxidized and (right) the reduced state. The circled CO$_2$ is linear in the oxidized state and bent in the reduced state. Au, C, O, H, and K are shown in gold, blue, red, white, and purple, respectively. The $z$-coordinate is taken normal to the electrode surface in the direction of the bulk metal. For the snapshot shown, a harmonic restraint is used to fix $\tilde{\xi}\approx0.44.$ Similarly, the cDFT constraints fix $\tilde{q}\approx0,1$ in the left and right panels respectively. (b) Contour plot, and (c) perspective visualization of the 2D ground-state diabatic free energy surface with thermodynamic driving force $\Delta G_{OR}^\text{ex}=0$. (d) Same as (c) except setting $\Delta G_{OR}^\text{ex}=$1 eV, which shifts the reduced diabat relative to the oxidized. The basins at points (0,0) and (1,1)---labeled "Ox" and "Red" in the contour plot respectively---are signatures of a CIET-like mechanism.
• Figure 2: (a) Contour plot of the electronic coupling $H_{OR}$ for CO$_2$ redox across the 2D surface. (b) Contour plot of 2D ground-state adiabatic free energy surface setting $\Delta G^\text{ex}_{OR}=-0.2$ eV, which corresponds to $\Delta G^\text{ex}_{\text{eff}}=-0.5$ eV, characteristic of the reduction direction.