# Nonlocal and Size-Dependent Dielectric Function for Plasmonic Nanoparticles
## 3 Results and Discussion
For radius \(R=20\) nm, the surface and bulk vibrational modes by using the NLSD model are higher than the others as shown in the middle and bottom panel of Figure 5 (left column). Thus, the energy loss extended throughout the whole surface of the gold spherical dimer, whereas those by using experimental dielectric data are confined in narrow rings formed by the surface and the horizontal cross-section of the dimer, as the total EELS probabilities shown in the left-top panel in Figure 5. We found similar results for dimers with radii less than 20 nm, except that the intensities of the energy loss calculated by using the NLSD approach the DFT model as the radius decreases. For radius \(R=50\) nm, the total energy loss confined more strongly in narrow rings formed by the surface and the horizontal cross-section of the dimer for all dielectric models, and they are dominated by surface vibrational mode, as indicated by the EELS probability maps shown in Figure 5 (right column). We also found similar results for dimers with radii greater than 50 nm, except that the patterns of the total (and surface) energy loss changing from the narrow rings to crescents as the radius increases. These results mean that the nonlocal impact on the mode evolution of EELS and responses from the quantum mechanical mechanism can also be depicted by using our NLSD model. Although there are similar outcomes of mode evolution by using the SD and DFT models, they are obviously incompatible with the classic local experimental data where the size of the dimer becomes larger, such as shown in Figure 2 (e.g., \(R=100\) nm). Finally, we use our NLSD approach to explore another interesting compatible effectiveness for the optical properties of the dimer system illuminated by a plane wave as defined in Figure 1. Although the effects of quantum natural and nonlocality have different origins, their impact on the optical response of the nanostructure can be measured by its physical parameters (in our case, which means the radius and gap). Figure 6a renders the maximum plasmonic resonances of extinction cross-sections that correspond to different radius for gold spherical dimers with the constant gap \(d=1\) nm, and for different dielectric models. The comparison between predictions from our NLSD model and the  
Figure 5: Total (top panels), bulk (middle panels) and surface (bottom panels) EELS probability maps for the gold dimers with particles radii \(R=20\) nm (left column) and \(R=50\) nm (right column), and for different dielectric models as defined in the text. The inter-particle gap \(d=1\) nm. other dielectric models clarifies how nonlocality and quantum mechanical effects modify the optical properties of the system. For small radii (approximately less than 35 nm), there are significant blueshifts for the prediction of our NLSD model when compared to the predictions of SD model and experimental dielectric data, which are similar to the comparison results by using another entirely different nonlocal dielectric model of the metal from previous studies (see Figure 2 or Figure 4a in Ref. [7]), and can be accounted for the inclusion of nonlocal effects in the NLSD model, which also means that the size-dependent effects due to the size-corrective terms in the NLSD model can not affect the maximum plasmonic resonances of extinction cross-sections, since the predictions of our NLSD model and the SD model are approximately consistent with those of the Nonlocal model and experimental dielectric data, respectively, as shown in Figure 6a. One should note that these maximum plasmonic resonances of extinction cross-sections that correspond to different radius for gold spherical dimers of radii ranging between 20 and 60 nm may be more accurate if our model can be integrated with the adequate corrections of the Lorentz friction.