  We investigate the process by which a water molecule diffuses on the surface
of an Al(111) electrode under constant bias voltage by first-principles density
functional theory. To understand the diffusion path of the water on the
Al(111), we calculated the minimum energy path (MEP) determined by the nudged
elastic band method in combination with constant electron chemical potential
(constant-$\mu_{\rm e}$) methods. The simulation shows that the MEP of the
water molecule, its adsorption site, and the activation barrier strongly depend
on the applied bias voltage. This strong dependence of the water diffusion
process on the bias voltage is in good agreement with the result of a previous
scanning tunneling microscopy (STM) experiment. The agreement between the
theoretical and experimental results implies that accurate treatment of bias
voltage plays a significant role in understanding the interaction between the
electric field and the surface of the material. Comparative studies of the
diffusion process with the constant total number of electrons
(constant-$N_\mathrm{e}$) scheme show that the absence of strong interaction
between the molecular dipole and the electric field leads to a different
understanding of how water diffuses on a metal surface. The proposed
constant-$\mu_{\rm e}$ scheme is a realistic tool for the simulation of
reactions under bias voltage not only using STM but also at the electrochemical
interface.
