# Nonlocal and Size-Dependent Dielectric Function for Plasmonic Nanoparticles
## 3 Results and Discussion
However, nevertheless, either our NLSD model or the RPA approach included with the Lorentz friction induced corrections as discussed in Section 1 has observed anomalous shift of plasmonic resonance for nanoparticles with radii significantly exceeding the previously suggested limiting 5 nm by [PERSON]'s theory. Figure 6b renders the field enhancement \(\frac{|\mathrm{E}|}{|\mathrm{E_{0}}|}\) at the middle point of the gap as a function of \(R\) for gold spherical dimers and their corresponding plasmonic resonances as shown in Figure 6a. Quite different from the extinction spectral profiles predicted, Figure 6b shows approximate tendency between the NLSD model and the DFT model. This means that the impacts of nonlocality and quantum confinement can be compatible, and both reduce the enhancement of fields. The quantum confinement effects are important when the mean free path of valence electrons is comparable to the particle size, and the mean free path of electrons in gold at room temperature is below 50 nm. These play a large role in interpreting the discrepancy in predicted field enhancements between the NLSD model and  
Figure 6: (**a**) maximum plasmonic resonances of extinction cross-sections that correspond to different radius for gold spherical dimers with constant inter-particle gap \(d=1\) nm, and for different dielectric models as defined in text; (**b**) maximum field enhancement at the middle point of the inter-particle gap (the red circle depicted in Figure 1) as a function of \(R\) for gold spherical dimers and their corresponding plasmonic resonances as shown in (**a**). experimental dielectric data would gradually narrow when the radius is greater than 50 nm, as shown in Figure 6b. Similar behavior occurs for the prediction of the DFT model, except that the radius which stands out for such change is approximately 65 nm. We attribute this difference to the inclusion of nonlocal and finite-size effects in the NLSD model, which lead to an increase in the contribution of the bulk vibrational mode and to reduce the contribution of the surface vibrational mode when the radius becomes less than 50 nm and gradually decreases, as the EELS results (Figure 5) discussed above. Both effects are not exclusive of small particles as the inset shown in Figure 6b. In addition, they are also important for small inter-particle gaps since the strongly interacting between metallic geometries encountered in plasmonics. For example, as one can see from the maximum field enhancement as a function of separation for gold dimer with \(R=50\) nm shown in Figure 7, the predictions of the NLSD model and the DFT model are closely similar, and the gaps in predicted field enhancements between them and the SD model and the experimental dielectric data would gradually narrow or even vanish as the inter-particle gap \(d\) increases. As we can see both in Figures 6b and 7, between NLSD and nonlocal models, there is also a slight difference in their outcomes with a radius of less than approximately 65 nm (Figure 6b), or with a gap of less than approximately 0.5 nm (Figure 7). These are entirely caused by the size-dependent effects due to the size-corrective terms of the NLSD model, which also lead to a decreasing trend in the outcomes of the NLSD model toward the DFT model, particularly for dimers with small radii as the inset shown in Figure 6b. Furthermore, we also should note that the maximum field enhancement shown in Figure 7 may be not very accurate when the inter-particle gap \(d\) decreases below 0.5 nm, since the NLSD model may not be capable of calculating the quantum tunneling effects under such subnanometer gaps, as mentioned in Section 1.