# Nonlocal and Size-Dependent Dielectric Function for Plasmonic Nanoparticles
## 2 Methods and Theoretical Framework
Note that the above dielectric function calculation model neglects quantum-mechanical exchange and correlation effects, but the significant nonlocal effects from the reduced mean free path of the conduction electrons due to electron-interface scattering in small metal structures can be taken into account by using a modified collision frequency \(\gamma^{\prime}=\gamma+4 Av_{\text{F}}V/S\) in Equation (3) for the three-dimensional problems [43]. In principle, conclusions from the previews studies that adopt such modified collision frequency have shown that the Nonlocal model of Equation (1) can model the spatially nonlocal dielectric response of arbitrarily shaped and sized structures [39; 40]. However, although the volume of the structure \(V\) and surface area \(S\) in the modified collision frequency implies the influence of size factor, only the impact of free electrons was considered to be modified by size. In addition, correctly selecting the value of dimensionless parameter \(A\) remains a challenging since its complex details are of discrepancy in different theoretical dielectric models (consult Section 1 in Ref. [44] and its references for a detailed discussion). Therefore, in order to overcome this difficulty in a classical mean free path approach, a more reasonable alternative way for describing size-dependent dielectric functions is needed by considering the dominated conditions and a scope of the contributions of intraband transitions (free electrons) and interband transitions (bound electrons). Herein, according to  
Figure 1: Schematic illustration of the system under study: the interaction of the electron beam passes by or penetrates through a dimer consisted with two Au nanospheres with radius R and gap d, as well as the plane wave polarized along the dimer axis. The inset shows the exact located positions (or impact parameters) of electron beams: the red ring represents the middle of the gap and the blue cross represents the centre in one of the nanospheres. previous studies about suitable modification of the bulk complex dielectric function for free and bound electrons within metallic nanoparticles [14; 15; 16], the full size-corrected dielectric function can be written as the sum of three terms:  
\[\varepsilon_{\text{SD}}\left(\omega,R\right)=\varepsilon_{\text{bulk}}\left( \omega\right)+\Delta\varepsilon_{\text{free}}\left(\omega,R\right)+\Delta \varepsilon_{\text{bound}}\left(\omega,R\right), \tag{4}\]  
where \(\varepsilon_{\text{bulk}}\left(\omega\right)\) corresponding to the experimentally measured bulk values from UV-visible to NIR-FIR, and the last two terms are size-corrective contributions for free and bound electrons, respectively:  
\[\Delta\varepsilon_{\text{free}}\left(\omega,R\right) =-\omega_{\text{D}}^{2}\sum_{n=1}^{\infty}\left(-1\right)^{n} \frac{\left(i\omega C\
u_{\text{F}}/R\right)^{n}}{\left(\omega^{2}+i\omega \gamma_{\text{free}}\right)^{n+1}}, \tag{5}\] \[\Delta\varepsilon_{\text{bound}}\left(\omega,R\right) =-K_{b}e^{R/R_{0}}\int_{\omega_{g}}^{\infty}\frac{\sqrt{x-\omega_{g }}}{x}\left[1-F(x,T)\right]\] (6) \[\times\frac{dx}{x^{2}-\omega^{2}+\gamma_{\text{bound}}-2i\omega \gamma_{\text{bound}}},\]  
where \(\hbar\omega_{g}\) is the gap energy; \(F(x,T)\) is the Fermi distribution function of conduction electron of energy \(\hbar x\) and temperature \(T\) with Fermi energy \(E_{\text{F}}\), \(\gamma_{\text{free}}=\gamma\), \(\gamma_{\text{bound}}\) stands for the damping constant in the band to band transition; \(K_{b}\) is a proportionality factor with units of \(s^{-3/2}\); \(C\) is a constant that depends on the material. For noble metals, it has a value of about 0.8 that has been theoretically justified from first principles calculations [45]; \(R_{0}\) is a reference radius value that represents the range for which the density of states can be considered to reach the value of the bulk. For Au, the following values determined in Refs.