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We evaluate the electron emission current density from jellium metallicsurfaces in the Thomas-Fermi-von-Weizs\"acker approximation. We implement theweighted density approximation (WDA) for description of the exchange andcorrelation energy of interacting electrons. We find the emission mechanismexhibits crossover from quantum to classical over-barrier escape of electronsfrom the surface. Well below a surface-specific threshold field strength $E_d$,electron tunneling is mainly affected by the change of the metal workfunction,and is less sensitive to the detailed shape of the surface barrier. Inparticular, since the position of the image charge plane for electrons leavingthe surface does not precisely coincide with the centroid of the inducedscreening charge at the surface in response to the applied electric field, wefind an effective increase in the metal workfunction by $\Delta W \sim 0.01$eV, which decreases the dark current.

Field emission from metal surfaces in the Thomas-Fermi-von-Weizs\"acker model