First-Principles Free-Energy Barriers for Photoelectrochemical Surface Reactions: Proton Abstraction at ${\mathrm{TiO}}_2(110)$

We explicitly calculate the free-energy barrier for the initial proton abstraction in the water splitting reaction at rutile ${\mathrm{TiO}}_2(110)$ through ab initio molecular dynamics. Combining solid-state embedding, an energy based reaction coordinate and state-of-the-art free-energy reconstruction techniques renders the calculation tractable at the hybrid density-functional theory level. The obtained free-energy barrier of approximately 0.2 eV, depending slightly on the orientation of the first acceptor water molecule, suggests a hindered reaction on the pristine rutile surface.

Figure 1

Schematic illustration of the electrostatic QM/MM cluster embedding we use. The surface part of the QM zone (the embedded cluster) is surrounded by pseudopotentials (PPs), followed by a polarizable MM region and an inactive MM region. The embedding is capped off by a set of fitted point charges. On top of the surface are three QM water molecules followed by a hemisphere of 518 MM water molecules.

Figure 2

Simulation snap shot of the full system (other than six point charges, which are placed further away).

Figure 3

Schematic explanation of the Energy Gap reaction coordinate involving the two diabatic potential energy surfaces $E_i(\mathbf{x})$ and $E_f(\mathbf{x})$, describing the initial and final state, respectively.

Figure 4

Schematic illustration of the diabatic states employed for the Energy Gap reaction coordinate. (a) Initial state. (b) Unstable final state involving a hydronium (${\mathrm{H}}_3{\mathrm{O}}^{+}$) ion in the first solvation shell. (c) Stable final state, used in the present calculations, with the ${\mathrm{H}}_3{\mathrm{O}}^{+}$ ion in the second solvation shell.

Figure 5

Reconstructed free-energy profiles for the proton transfer reaction. The profile depends slightly on the orientation of the middle (first acceptor) water molecule: one of its hydrogen atoms pointing towards the surface (case 1, solid line) or not (case 2, dashed line). The red dashes show the free-energy derivatives obtained in the umbrella-sampling calculations. The insets show representative geometries along the reaction coordinate for case 1, which provides a lower estimate of the free-energy barrier.

Figure 6

Spin densities (0.01 a.u. contours) for representative structures of the initial state (a), the barrier (b), and the final state (c) along the proton abstraction profile.