# Nonlocal and Size-Dependent Dielectric Function for Plasmonic Nanoparticles
## 3 Results and Discussion
demonstrated that, while they can be completely neglected for Au dimers of all size when the contribution of the bulk vibrational mode is dominant (Figure 2a), and, for Au dimers of a large radius, when the contribution of the surface vibrational mode is dominant (Figure 2b), they still have a great influence on dimers of a small radius, just as the results have been demonstrated for isolated particles (Figure 3). Second, we should notice that the classic local experimental dielectric data used in this paper was obtained from reflection and transmission measurements on vacuum-evaporated thin films at room temperature [33]; the application of such local continuum electrodynamics for small-sized nanoparticles becomes questionable. This can be attributed to two effects. First of all, their imaginary part of the interband contribution due to size-dependent effects is not negligible in the spectral region of surface plasmon polariton resonance [14; 15]. Second, nonlocal effects due to the scattering of the conduction electrons arise as the nanoparticle size is decreased, relative to metal films [39; 40]. By contrast, either the EELS results or the optical properties that are going to be demonstrated in the following have shown that outcomes from our NLSD model can fit the DFT model better than other models for small-sized nanoparticles. These agreements benefit from the first term in the NLSD model, namely, the Nonlocal model of Equation (1), which have the advantage of describing the dynamical optical response of structures that are too large to treat using quantum mechanics but too small for classical electromagnetic theories to be valid [39; 40]. It should be noted that a consistent energy loss with peak near 1.25 \(\,\mathrm{eV}\) when the radius \(R=100\) nm (as indicated by the inset gray arrow in Figures 2a and 4(upper)), which is similar to the main plasmon resonances, will be presented below in Figure 6a for the extinction cross sections excited by optical plane wave, from approximately 955 nm. In order to gain physical insight into the origin of these common resonances in both situations of excitations, the surface and bulk configurations that form the total loss probabilities where \(R=100\) nm are plotted in Figure 4, respectively. The results shown in Figure 4 indicate that the total loss probabilities near 1.25 eV for any dielectric model are all dominated by surface loss probabilities. In addition, it is interesting that we found that the total energy loss probabilities shown in Figure 2b are also dominated by the surface loss probabilities, since all the bulk loss probabilities are almost zero when the electron beam located in the middle of the gap (not shown). We attribute this surface energy loss dominating effect to the contribution of the surface vibrational mode. One should note that, at least for now, the main energy loss peaks as depicted in Figure 2a are the coupling contribution of the surface and the bulk vibrational modes. However, it can be seen that these surface energy loss peaks near 1.25 eV as shown in Figure 4(upper) do not show up in  
Figure 4: (Upper panel) Surface and (lower panel) bulk loss probabilities for gold spherical dimers of radius \(R=100\) nm, where the other simulation configurations remain the same as defined in Figure 2a. Figure 2b where the contribution of the surface vibrational mode is dominant. In addition, according to our surface loss probability calculation results (not shown, similar to the form of Figure 4(upper)), they also do not show up when \(R=50\) nm where the other simulation configurations remain the same as defined in Figure 2a. However, the reducing contribution of the surface vibrational mode with nanoparticles of the small radius can be distinctly revealed by the NLSD model. For example, Figure 5 plots the simulated EELS probability maps of our dimer system with four different dielectric models for the 1.25 eV resonances. Our model captures the main aspects of the nonlocal electrodynamics for small-sized nanoparticles.