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The standard hydrodynamic Drude model with hard-wall boundary conditions cangive accurate quantitative predictions for the optical response of noble-metalnanoparticles. However, it is less accurate for other metallic nanosystems,where surface effects due to electron density spill-out in free space cannot beneglected. Here we address the fundamental question whether the description ofsurface effects in plasmonics necessarily requires a fully quantum-mechanicalapproach, such as time-dependent density-functional theory (TD-DFT), that goesbeyond an effective Drude-type model. We present a more general formulation ofthe hydrodynamic model for the inhomogeneous electron gas, which additionallyincludes gradients of the electron density in the energy functional. In doingso, we arrive at a Self-Consistent Hydrodynamic Model (SC-HDM), where spill-outemerges naturally. We find a redshift for the optical response of Na nanowires,and a blueshift for Ag nanowires, which are both in quantitative agreement withexperiments and more advanced quantum methods. The SC-HDM gives accurateresults with modest computational effort, and can be applied to arbitrarynanoplasmonic systems of much larger sizes than accessible with TD-DFT methods.Moreover, while the latter typically neglect retardation effects due totime-varying magnetic fields, our SC-HDM takes retardation fully into account.

Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics