The surface potential of the water liquid-vapor interface is studied by molecular dynamics using the TIP4P model. The surface potential predicted by this empirical model is -(130 +/- 50) mV. This value for the surface potential is of reasonable magnitude but of opposite sign to the expectations derived from laboratory experiments. The electrostatic potential displays a nonmonotonic variation with depth into the liquid. This nonmonotonic variation is explained on the basis of the nondipolar charge distribution of the H2O molecule and the observation that the more probable molecular orientations in the interfacial region place the molecular symmetry axis near the plane of the interface. It is shown that minor changes in the assumed molecular charge distribution can bring the computed surface potential into agreement with experimental expectations without qualitatively altering the nonmonotonic variation of the electrostatic potential through the interfacial region. Computed quantum mechanical descriptions of the electron distribution of the isolated H2O molecule are not compatible with the surface structure predicted by the TIP4P model and the experimental expectation that the surface potential of the water liquid-vapor interface is small, roughly of the of order of 10-100 mV. The surface potential is sensitive to details in the large distance wings of the molecular electron distribution. It is hypothesized that the surface environment qualitatively alters the wings of the distribution from the result obtained by a superposition of the isolated molecule electron densities.