# Nonlocal and Size-Dependent Dielectric Function for Plasmonic Nanoparticles
## 1 Introduction  
Classical local electrodynamics in nanostructures under the assumption that the polarization characteristics at a given point are locally related to the external electromagnetic excitation, which has been thoroughly explored from isolated nanoparticles and dimers to other complex structures [1; 2; 3; 4; 5]. On the other hand, the nonlocal response of a metal nanostructure to an external disturbance stimulus determines many important plasmonic properties and has attracted much theoretical interest [6; 7; 8; 9]. In recent years, the [PERSON] group [10] have studied plasmons in metal nanoparticles by using a popular hydrodynamic model which incorporates nonlocal effects by introducing a pressure term that inhibits squashing electrons into small plasmonic nanoparticles. Similarly, the work of [PERSON] et al. [11] included nonlocality in the optical response of metallic nanoparticles by introducing quantum corrections obtained from first principles calculations, also using the hydrodynamic model. [PERSON] et al. [12; 13] have demonstrated a Maxwell-hydrodynamic model for studying the charge, energy and angular momentum conservation laws of high-order harmonics due to nonlocal and nonlinear interactions between electromagnetic waves and metallic metamaterials in the terahertz regime. Meanwhile, parallel to the above remarkable progress of nonlocal properties, there has been a growing interest in exploring the quantum properties of surface plasmon and the prospects of plasma devices based on reliable operation at the quantum level. However, theoretical investigation of plasmonic resonances for nanostructure sizes ranging between 2 and 20 nm remains a challenge. This length scale is too small for classical electromagnetic theories to be valid but too large for full-fledged quantum simulations to be feasible. Therefore, some of the studies are committedto provide a unified theoretical framework that breaks wide discrepancy between classical electromagnetic theory and quantum mechanical theory for plasmonics and nanophotonics, and ultimately control plasmonic responses in nanostructures. Crucial to this endeavor is to understand finite-size effects in nanostructures since the control of nanoparticle size represents one of the most effective approaches to harness the tremendous potentials of plasmonics and nanophotonics. Recent studies have shown a theoretical method with a suitable modification of the bulk complex dielectric function for free and bound electrons within metallic nanoparticles [14; 15; 16; 17; 18; 19], for the determination of plasma frequency, damping constants, and size distributions of noble metal nanoparticles, such as gold, silver and copper; in addition, the corresponding optical characterization deduced by [PERSON]'s theory has been tested to fit full UV-Vis experimental extinction spectra. The above theoretical framework follows the idea of [PERSON] and [PERSON] [20], who considered that the augmentation of the damping constant in the Drude model results from additional collisions of free electrons with the particle boundary, as well as the idea of [PERSON] et al. [21], who took into account the interband transitions of bound electrons for the complex dielectric function. Consequently, the metal dielectric function can be regarded as the sum of the free (intraband transitions) and bound (interband transitions) electron contributions; then, a size-dependent (SD) dielectric function (Equation (4)) based on a stepwise modification of the Drude model is put forward. Under this theoretical framework, the significant finite-size effects of optical properties for metallic nanoparticles can be expected to be observed below a certain limited size range (e.g., valid restriction of 20 nm drawn from Refs. [14; 15]). A developed microscopic approach based on accurate inclusion of the Lorentz friction for the plasmon damping within the random phase approximation (RPA) model [22; 23; 24] has also been performed in recent years.