Electrode potential from density functional theory calculations combined with implicit solvation theory

We present a method that makes it possible to determine an electrode potential in an electrode/electrolyte solution system. We consider the electrode potential of the standard hydrogen electrode (SHE) reaction $R:{{0.28em}{0ex}}1/2{{0.16em}{0ex}}{H}₂(gas)+{H}₂O{{0.16em}{0ex}}(1{{0.16em}{0ex}}M{{0.16em}{0ex}}HCl{{0.16em}{0ex}}aq.){↔}{H}₃{O}⁺(1{{0.16em}{0ex}}M{{0.16em}{0ex}}HCl{{0.16em}{0ex}}aq.)+{e}^{{-}}$ (electrode M), and conduct density functional theory (DFT) calculations combined with the effective screening medium (ESM) method and the reference interaction site model (RISM). The electrostatic field from a charged slab described by DFT with ESM is screened by that from the charge distribution in an electrolyte solution. This screening enables us to define the inner potential ΦS at the bulk solution region, which is the reference potential for the electrode potential, that is, the chemical potential of electrons (μₑ). Grand potentials of the left and right sides in reaction R at the equilibrium point derive the corresponding SHE potential of μₑSHE0.16em0ex=0.16em0ex-5.270.16em0exeV vs ΦS for a Pt(111) electrode. Another pathway that uses the free energy difference gives the same SHE potential; the equivalence of the electrode potentials from chemical potential and from free energy difference is validated within ESM-RISM. Even using a different electrode of Al yields the value of μₑSHE=-5.220.16em0exeV vs ΦS, which indicates that the electrode potential is independent of the electrodes. Finally, the potential energy profile in a vacuum/metal/solution/vacuum region shows that a difference between the inner and outer potentials is necessary to compare an absolute SHE potential and the SHE potential vs ΦS.