# Nonlocal and Size-Dependent Dielectric Function for Plasmonic Nanoparticles
## 3 Results and Discussion  
The first example of the compatible effectiveness produced by our NLSD approach is shown in Figure 2 for comparisons of EELS calculated with different dielectric models and gold spherical dimers of different sizes, where electron beam penetrates the centre in one of the nanospheres (Figure 2a) and passes by the middle of the gap (Figure 2b) as shown in Figure 1, and the gap \(d\) fixed on 1 nm. Figure 2a shows that the agreement of the total energy loss probabilities between the NLSD approach(black solid line) and the theoretical dielectric data set (green dashed line), which is obtained from density functional theory (DFT) calculations by [PERSON] et al. [34], is remarkable for radii up to 20 nm. Moreover, for the NLSD approach, the evolves of the size-dependent response as the radius increases are observably more outstanding than the full size-dependent dielectric model (blue dotted line, Equation (4)), and the total energy loss probabilities become almost consistent with those calculated by using the local, frequency-dependent and experimental complex bulk dielectric values (red dash-dotted line) as radii up to 100 nm. In addition, blueshifts of the main energy loss peaks become more apparent as the radius increases and it is accompanied by plasmon broadening. These nonlocal effects are known as the availability of the additional loss channels [7], and can have an effect in the higher energy region when compared to classical local electrodynamics because of the interplay between \(\omega\) and \(\mathbf{k}\) in Equation (3). All of the above agreements between the model predictions support that our approach provides a straightforward theoretical framework to describe metal behavior combining local, nonlocal and size-dependent effects in broader energy and size ranges simultaneously. Moreover, according to the various developed theories of the SD model as mentioned in Section 1, the significant finite-size effects on the optical properties of nanoparticles can only be expected to be observed below a certain limited size range of about 20 nm. Our model captures the main aspects of the finite-size effects for small-sized nanoparticles. For radii below 20 nm, Figure 2b shows a significant difference of the total (and surface) energy loss probabilities between the NLSD approach and nonlocal dielectric model (thinner gray solid curves) of Equation (1). \(A=0.1\) was used in the modified collision frequency \(\gamma^{\prime}=\gamma+4 Av_{\mathrm{F}}V/S\) when using the Nonlocal model [39; 40]. In addition, unlike the outcomes between the SD model and the experimental dielectric data, which show remarkable differences in all size range, the total (and surface) energy loss probabilities calculated by using the NLSD model are almost identical with the Nonlocal model for radii larger than 20 nm this is why the forecasted results of Nonlocal model are not shown in Figure 2b where the radii larger than 20 nm. Similarly, for clarity, Figure 2a does not show the outcomes from the Nonlocal model because they are identical with those from the NLSD model in all size ranges. In consideration of the size-corrective terms from SD and NLSD model being the same, these results might appear unexpected at the first view. The nonlocal effects themselves are also size-dependent. This can be seen from Figure 2b where radii are below 20 nm; besides capturing the blueshifts, the main energy loss peaks of the Nonlocal model are obviously less than those by using experimental dielectric data, but in a low-energy region (e.g., 0.65-2.4 eV), the outcomes of the Nonlocal model are higher than those by using experimental dielectric data, which have the same changing tendency as the results of SD and NLSD models. However, the NLSD model is a more suitable implementation for revealing the contribution of the surface vibrational mode with nanoparticles of the small radius.