# Nonlocal and Size-Dependent Dielectric Function for Plasmonic Nanoparticles
## 4 Conclusions  
In conclusion, we presented a simplified theoretical model to describe the nonlocal and size effects of nanoparticles. Our implementation is based on the hydrodynamic Drude model and two size-corrective terms derived from the size-dependent dielectric function. We have demonstrated the compatible effectiveness and the accuracy of our theoretical model through comparisons to analytical results of the EELS and the optical responses for gold spherical dimers. These analytical results demonstrated a number of effects that result from nonlocality and finite-size effects in the dielectric response, including analogous quantum confinement effects that decrease the electromagnetic field enhancement and their corresponding weight relations of the contributions of surface and bulk vibrational modes reflected by EELS, blueshifts of surface plasmon resonances and main energy loss peaks. From these results, we can state that the nonlocal effects are of great significance as they obviously determine the blueshifts of surface plasmon resonances for the gold dimers with radii smaller than approximately 35 nm and the electromagnetic field enhancement for the gold dimers with radii ranging from 20 to 50 nm. In addition, blending the adequate size-dependent effects and nonlocal effects can more accurately reflect reductions in the main energy loss peaks and the electromagnetic  
Figure 7: Maximum field enhancement as a function of separation for gold dimer with \(R=50\) nm, and for different dielectric models as defined in text (and depicted in Figure 2).  
field enhancement due to quantum confinement effects for the gold dimers with radii smaller than approximately 20 nm, as well as the weight relations of the contributions of surface and bulk vibrational modes for the gold dimers with radii ranging from 1 to 50 nm; for the gold dimers with radii larger than 50 nm, the nonlocal effects in the EELS and optical responses decrease as the radii increase further, and the local effects almost dominate the EELS and optical responses as radii up to 100 nm.  
Our NLSD model used here accounts for the plasmonic nanoparticles behavior combining local, nonlocal and size-dependent effects, and reconciles the discrepancy between classical electromagnetic theory and quantum mechanical theory for plasmonics and nanophotonics in broader energy and size ranges. Importantly, although we have focused on gold spherical dimers, our approach can be extended to motivate new, and more precise experimental studies on other metallic nanostructures with complex configurations, particularly those where quantum-size effects and strongly interacting effects are likely to play a large role. In the future, we expect that our approach will make possible a deep exploration of the dissipative quantum electromagnetics, where couple loss or dissipation in a quantum system to a bath of oscillators maintained a strong connection with the classical electromagnetic system [47].  
Conceptualization, K.-J.H.; Formal analysis, K.-J.H.; Funding acquisition, Z.-C.B.; Methodology, K.-J.H.; Supervision, S.-J.Q.; Writing--original draft, K.-J.H.; Writing--review and editing, S.-J.Q., Z.-P.Z., Z.D. and Z.-C.B.  
This research was funded by the National Natural Science Foundation of China (Grant No. 61741505) and the Guizhou Province Science and Technology Projects (No. QKHZ [2017]2887).  
The authors declare no conflict of interest.