The plasmonic response of nanostructures is fundamentally governed by the interaction between electromagnetic fields and free electrons in the metal. Classical electrodynamics, based on the local Drude model, provides a reasonable description of plasmonic behavior in bulk materials and large nanostructures. However, as the size of the nanostructure approaches the nanometer scale, nonlocal and nonclassical effects become significant, necessitating corrections to the classical framework. These corrections arise due to electron-electron interactions, surface effects, and quantum confinement, which alter the optical response of plasmonic systems.

Nonlocal effects, described by the hydrodynamic model, account for the spatial dispersion of the electron density. In the local Drude model, the dielectric function depends only on frequency, assuming an instantaneous response of electrons to the electric field. However, in nanoscale systems, the finite wavelength of electron oscillations introduces spatial variations in the induced charge density. The hydrodynamic model incorporates this by introducing a nonlocal term proportional to the gradient of the electron density. The nonlocal dielectric function includes a wavevector-dependent correction, leading to a smearing of the induced charge density at the metal surface. This results in a blueshift of plasmon resonances and a damping of higher-order modes, particularly in structures with sharp features or gaps smaller than a few nanometers. The strength of nonlocal effects is characterized by the Thomas-Fermi screening length, typically on the order of a few angstroms for noble metals.

Nonclassical corrections, on the other hand, arise due to the quantum mechanical nature of electrons at interfaces. The Feibelman d-parameters provide a phenomenological framework to describe these effects by introducing complex-valued surface response functions. These parameters account for the centroid of the induced charge and current densities relative to the nominal surface position. The d-parameters modify the boundary conditions for Maxwell's equations, leading to shifts in plasmon resonance frequencies and changes in the near-field distribution. Unlike the hydrodynamic model, which treats the bulk response nonlocally, the d-parameters focus on the surface contribution, making them particularly relevant for thin films and core-shell structures. The real part of the d-parameters describes the phase shift of the surface response, while the imaginary part accounts for additional damping due to electron scattering at the interface.

Size-dependent damping is a critical aspect of plasmonic response in nanostructures. In bulk metals, damping is dominated by electron-phonon scattering and defect scattering. However, as the size of the nanostructure decreases, surface scattering becomes increasingly important. The damping rate increases inversely with the particle size due to the reduced mean free path of electrons. This effect is often described by a modified Drude model that includes a size-dependent damping term. Additionally, Landau damping, which arises from the decay of plasmons into electron-hole pairs, becomes significant in small nanostructures. Surface-enhanced Landau damping occurs when the plasmon frequency matches the energy difference between discrete electronic states near the Fermi level. This process is particularly pronounced in ultrasmall nanoparticles and metallic clusters, where quantum confinement leads to discrete electronic levels.

The interplay between nonlocal and nonclassical effects complicates the interpretation of plasmonic response in nanostructures. For example, in a metallic nanosphere, nonlocal effects dominate at sizes below 10 nm, leading to a blueshift of the plasmon resonance. In contrast, the Feibelman d-parameters become more relevant for thin films or layered structures, where the surface contribution outweighs the bulk response. In some cases, both corrections must be considered simultaneously, as they can either counteract or reinforce each other depending on the geometry and material properties.

Theoretical studies have shown that nonlocal and nonclassical corrections can significantly alter the optical properties of plasmonic nanostructures. For instance, in dimer configurations with subnanometer gaps, nonlocal effects suppress the field enhancement predicted by classical electrodynamics. Similarly, the d-parameters modify the near-field distribution in thin films, affecting the coupling between plasmonic modes and external emitters. These corrections are essential for accurate modeling of nanoscale light-matter interactions, particularly in applications such as sensing, nonlinear optics, and quantum plasmonics.

The hydrodynamic model and Feibelman d-parameters provide complementary approaches to address the limitations of classical electrodynamics. While the hydrodynamic model extends the bulk dielectric function to include spatial dispersion, the d-parameters focus on the surface response. Both frameworks have been validated against more rigorous quantum mechanical calculations, such as time-dependent density functional theory, confirming their utility for practical applications. However, their accuracy depends on the choice of parameters, such as the electron density and the effective mass in the hydrodynamic model, or the d-parameters themselves, which must be derived from first-principles calculations or experimental data.

In summary, the plasmonic response of nanostructures requires corrections beyond the classical Drude model to account for nonlocal and nonclassical effects. The hydrodynamic model addresses spatial dispersion in the bulk, while the Feibelman d-parameters describe quantum mechanical surface effects. Size-dependent damping and Landau damping further modify the optical response, particularly in ultrasmall nanostructures. These corrections are crucial for understanding and predicting the behavior of plasmonic systems at the nanoscale, enabling the design of advanced nanophotonic devices with tailored optical properties. Future developments in theoretical and computational methods will continue to refine these models, providing deeper insights into the quantum nature of plasmonic phenomena.