Electrode potential is a general concept, in electrochemistry, which governs the charge transfer reactions such as ion insertion/extraction and reduction-oxidation at electrode/solution interfaces; the electrode potential should be appropriately included in the computational model. In this study, we consider how the standard hydrogen electrode (SHE) potential, which is the equilibrium potential of the charge transfer reaction of H⁺/H₂, is theoretically evaluated.

Methods

We employed density functional theory (DFT) calculations combined with the effective screening medium (ESM) technique¹ + the reference interaction site method (RISM);² ESM-RISM formulation³ makes it possible to simulate the electrode (+ reactive ions) and the solution based on quantum mechanics and implicit classical solvation model, respectively. ESM-RISM calculations were performed on the configuration of a vacuum/slab/solvent system as shown in Figure 1, where the DFT slab domain is on the left-hand side, and the RISM solvents, that is, H₂O, H₃O⁺, and Cl⁻ treated by the RISM equations, are on the right-hand side. Changing the chemical potential of electron, µ_e, referenced to the inner potential Φ_S at the bulk solution region, we compared the grand potentials Ω for the following reaction R:

H₃O⁺ (1M HCl aq.) + e⁻ (electrode M, M=Pt, Al) ↔︎ 1/2 H₂ (gas) + H₂O (1M HCl aq.) (R).

Results

ESM-RISM calculations revealed Ω of the left side state of reaction R and that of the right state on Pt(111) electrode. They cross at µ_e = −5.27 eV, which is equilibrium chemical potential of electron at SHE reaction (= µ_SHE). This indicates that the electrode potential vs. SHE, E(SHE), is evaluated as E(SHE) = −(µ_e − µ_SHE)/e. We also obtained the almost same value of µ_SHE on Al(111) electrode. On the other hand, Pt(111) and Al(111) is negatively and positively charged at µ_e = µ_SHE, respectively. In the presentation, we will compare the potential profile of metal/solution/vacuum region obtained from ESM-RISM and the first-principles molecular dynamics calculation using ESM.⁴ We further discuss on the difference between µ_SHE and the absolute SHE obtained by Trasatti.⁵

References

1. M. Otani, O. Sugino, Phys. Rev. B 73, 115407 (2006).
2. F. Hirata, P. J. Rossky, Chem. Phys. Lett. 83, 329, (1981).
3. S. Nishihara, M. Otani, Phys. Rev. B 96, 115429 (2017).
4. M. Otani, I. Hamada, O. Sugino, Y. Morikawa, Y. Okamoto,T. Ikeshoji, J. Phys. Soc. Jpn. 77, 024802 (2008).
5. S. Trasatti, Pure Appl. Chem. 58, 955 (1986).