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

Abstract

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 $\mathrm{R}:\phantom{\rule{0.28em}{0ex}}1/2\phantom{\rule{0.16em}{0ex}}{\mathrm{H}}_{2}(\mathrm{gas})+{\mathrm{H}}_{2}\mathrm{O}\phantom{\rule{0.16em}{0ex}}(1\phantom{\rule{0.16em}{0ex}}\mathrm{M}\phantom{\rule{0.16em}{0ex}}\mathrm{HCl}\phantom{\rule{0.16em}{0ex}}\mathrm{aq}.)\ensuremath{\leftrightarrow}{\mathrm{H}}_{3}{\mathrm{O}}^{+}(1\phantom{\rule{0.16em}{0ex}}\mathrm{M}\phantom{\rule{0.16em}{0ex}}\mathrm{HCl}\phantom{\rule{0.16em}{0ex}}\mathrm{aq}.)+{e}^{\ensuremath{-}}$ (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 ${\mathrm{\ensuremath{\Phi}}}_{\mathrm{S}}$ at the bulk solution region, which is the reference potential for the electrode potential, that is, the chemical potential of electrons (${\ensuremath{\mu}}_{\mathrm{e}}$). Grand potentials of the left and right sides in reaction R at the equilibrium point derive the corresponding SHE potential of ${\ensuremath{\mu}}_{\mathrm{e}}^{\mathrm{SHE}}\phantom{\rule{0.16em}{0ex}}=\phantom{\rule{0.16em}{0ex}}\ensuremath{-}5.27\phantom{\rule{0.16em}{0ex}}\mathrm{eV}$ vs ${\mathrm{\ensuremath{\Phi}}}_{\mathrm{S}}$ 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 ${\ensuremath{\mu}}_{\mathrm{e}}^{\mathrm{SHE}}=\ensuremath{-}5.22\phantom{\rule{0.16em}{0ex}}\mathrm{eV}$ vs ${\mathrm{\ensuremath{\Phi}}}_{\mathrm{S}}$, 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 ${\mathrm{\ensuremath{\Phi}}}_{\mathrm{S}}$.