Although conceptually simple, the air-water interface displays rich behavior. Different definitions of the electrostatic potential, each relevant for distinct experimental scenarios, lead to widely varying surface potential magnitudes and even different signs. Based on quantum-chemical density-functional molecular dynamics simulations, all relevant surface potentials are evaluated and compared. The spatially averaged surface potential, accessible to electron holography, is dominated by the trace of the water molecular quadrupole moment and amounts to more than + 4 V inside the water phase, very different from results obtained with force-field water models. The surface potential inside a cavity is much smaller, less than 200 mV in magnitude, and depends specifically on the cavity radius. This is the electrochemical surface potential relevant for ion transfer reactions and ion surface adsorption. Charge transfer between water molecules leads to pronounced surface potentials as well. However, when probing electrophoresis by explicitly applying a lateral electric field, the zeta potential turns out to be zero. Thus, charge transfer between water molecules does not translate to a non-zero electrophoretic mobility at the pristine vapor-liquid water interface.
