# Nonlocal and Size-Dependent Dielectric Function for Plasmonic Nanoparticles
## 2 Methods and Theoretical Framework  
Figure 1 displays the fundamental system under study; a dimer consists of two closely spaced freestanding metallic nanospheres of radius \(R\), and a certain distance \(d\) apart. The system is excited by an electron beam with high kinetic energy up to 200 keV passing by or penetrating through at the positions as indicated in the figure, and illuminated from the top by a plane wave polarized along the dimer axis, respectively. For this study, the metal is assumed to be gold, and the experimental and theoretical values of permittivity for contrast taken from [PERSON] and [PERSON] [33] and [PERSON] et al. [34], respectively. We chose the data models above for comparison for the following reasons. First, of all, in general, ab initio density functional theory (DFT) approaches need to deal with pseudopotentials, and include quantum effects that completely change the spectral distribution and the field enhancements of the plasmon resonant modes supported by a given nanostructure; they are computationally demanding and are currently limited to small-sized nanoparticles with a few thousand electrons (e.g., valid restriction of 12.3 nm drawn from Ref. [35]). Second, the application of classic local experimental dielectric data for small-sized nanoparticles ranging between 2 and 20 nm becomes questionable as described in the Introduction. These factors will be discussed in detail below combined with calculation results. All of the numerical simulations are implemented via the general public license toolbox MNPBEM [36; 37], which is based on a boundary element method (BEM) approach. The dielectric function of noble metal nanoparticles (e.g., Cu, Au, and Ag) is well described by using the hydrodynamical Drude model limit by three components [38; 39; 40]:  
\[\epsilon_{\text{Nonlocal}}(\mathbf{k},\omega)=\epsilon_{\infty}+\epsilon_{ \text{inter}}(\omega)+\epsilon_{\text{intra}}(\mathbf{k},\omega), \tag{1}\]  
where the \(\mathbf{k}\) dependence in \(\epsilon_{\text{Nonlocal}}(\mathbf{k},\omega)\) is necessary to describe the spatially nonlocal response of structures with features less than 10 nm [41; 42]. \(\epsilon_{\infty}\) is the value as \(\omega\rightarrow\infty\). \(\epsilon_{\text{inter}}(\omega)\) representsthe contribution from interband electron transitions and can be physically described in the form of a Lorentz oscillator model,  
\[\varepsilon_{\text{inter}}(\omega)=\sum_{j}\frac{\Delta\varepsilon_{\text{LJ}} \omega_{\text{LJ}}^{2}}{\omega_{\text{LJ}}^{2}-\omega(\omega+i2\delta_{\text{LJ} })}. \tag{2}\]  
In this work, we take index \(j=2\) labeling the individual d-band to sp-band electron transitions since there are two interband transitions in Au at optical frequencies [33]. \(\varepsilon_{\text{intra}}(\textbf{k},\omega)\) is responsible for the intraband effects which are usually referred to as free-electron effects, where both the plasmonic optical response of metals and nonlocal effects are contained and can be described by  
\[\varepsilon_{\text{intra}}(\textbf{k},\omega)=-\frac{\omega_{\text{D}}^{2}}{ \omega\left(\omega+i\gamma\right)-\beta^{2}\textbf{k}^{2}}, \tag{3}\]  
where \(\omega_{\text{D}}\) is the plasma frequency, \(\gamma\) is the collision frequency, and \(\beta\) is the hydrodynamical parameter measuring the degree of nonlocality. For the three-dimensional situation \(\beta^{2}=3v_{\text{F}}/5\), where \(v_{\text{F}}=1.39\times 10^{6}\) m/s is the Fermi velocity for the Au nanoparticles. Then, all parameters in Equations (2) and (3) can be estimated by fitting Equation (1) to the experimentally determined dielectric data of bulk Au in Ref. [33]. Here, we use all the parameters taken from Ref. [39], \(\omega_{\text{D}}=8.812\) eV, \(\gamma=0.0752\) eV, \(\varepsilon_{\infty}=3.559\), \(\Delta\varepsilon_{\text{L1}}=2.912\), \(\omega_{\text{LJ}}=4.693\) eV, \(\delta_{\text{L1}}=1.541\) eV, \(\Delta\varepsilon_{\text{L2}}=1.272\), \(\omega_{\text{LJ}}=3.112\) eV, \(\delta_{\text{L1}}=0.525\) eV.