# Nonlocal and Size-Dependent Dielectric Function for Plasmonic Nanoparticles
## 1 Introduction
Unlike the conventional [PERSON]'s theory that only gives reasonably intrinsic size effects for nanospheres of radii lower than 5 nm in vacuum due to neglecting the Lorentz friction losses that dominate over all attenuation channels, the specific size dependence for plasmons due to the Lorentz friction induced corrections derived upon this theoretical approach can be applied to account for the observed irregular size-effect of plasmon resonance for relatively large metallic nanoparticles (e.g., for Au nanospheres of radii 20-60 nm in vacuum drawn from Ref. [24]). This theoretical approach has also been verified to improve experimental data fitting of metallic nanospheres measured in water colloidal solutions, especially pronounced for a large size limit (e.g., of radii 10-75 nm for Au nanospheres in water colloidal solutions drawn from Ref. [22]). One should note that, although accurate accounting for the Lorentz friction is important for nanospheres of radii above 20 nm (for Au in vacuum), the major drawback of our work is that adequate corrections of the Lorentz friction are not included, since integrating them with the nonlocal and size-dependent effects within a macroscopic material dielectric function is not straightforward. Extending beyond the nonlocal and size-dependent effects we have focused on, in the following, there is a huge amount of research concerning about other \"non-classical electromagnetic effects\" by incorporating some \"quantum effects\" within classical electrodynamics. Quantum-size effects due to the confinement of electron motion in small dimensions can lead to a discrete electronic energy levels and a size-dependent conductivity [25]. Accessing plasmonic modes in the quantum tunneling regime also remains as a challenge, quantum tunneling effects become significant for subnanometer interparticle gaps and can be reflected through a quantum corrected model (QCM), which models the junction between adjacent nanoparticles by means of a local dielectric response that includes electron tunneling and tunneling resistivity at gap [26; 27]. However, such \"non-classical\" effects are of great interest and significance in the field of nano-optics, and can be incorporated into classical electrodynamical descriptions for some simple systems; directly integrating them within a general framework is not straightforward. In particular, quantum tunneling effects between metal nanoparticles will not be considered in this work. Nonlocal effects are a direct consequence of the inhomogeneity of the electron gas, and can hardly be ignored when the mean free path of an electron in the simple systems is larger than the wavelength of light [28; 29]. To a certain extent, nonlocal electrodynamics partially account for finite-size and quantum mechanical effects inherent in a nanostructure in a more rigorous way. Such as in the case of applying the hydrodynamic dielectric function to calculate the absorption coefficient from metallic nanoparticles, its size-dependent effects will show up by an effective dielectric function due to the additional boundary conditions for solving polarizabilities [30]. Although it has been argued that the hydrodynamic model yields only semiquantitative results in the linear regime, recent improvements regarding the incorporation of interband transitions and Landau damping have demonstrated that fully quantitative results may be obtained [31; 32]. In principle, as most of the modified hydrodynamic models shown in the aforementioned applications, a spatially nonlocal relationship between the material polarization and the field of plasmonics can typically be mirrored through the use of a spatially nonlocal dielectric function. Some of them take finite-size effects of the nanostructure into consideration by using a modified collision frequency. However, the corresponding changes in optical, plasmonic and dielectric responses are almost negligible, and the effective dimension of nanostructure for nonlocal electrodynamics is also being limited or obscured consequently.