medium [@ToscanoEtal2015]. Here in Eq.  and below, we write $n$ for the linearized density ${n}_{1}$ and similarly we will from now on simply write ${\bf v}$ for the linearized velocity ${\bf v}_{1}$. As a result, we obtain the linearized hydrodynamic equation [@RazaEtal2015JPCM; @Schmittetal2016] $$\label{eq:hds} 
m_e\partial_t{\mathbf{v}}=-e{\mathbf{E}}-m_e\gamma{\mathbf{v}}-m_e\beta^2\frac{1}{n_0}\nabla n,$$ as well as the linearized continuity relation $$\label{eq:nv}
\partial_t n = -n_0 \nabla\cdot {\mathbf{v}}.$$ Inserting Eqs.  (linearized as ${\mathbf{J}}=-en_0{\mathbf{v}}$) and into , and taking the time-derivative $\partial_t$, we obtain $$\label{eq:hdT} 
\partial_{tt} {\mathbf{J}}+\gamma\partial_t{\mathbf{J}}-\beta^2\nabla(\nabla\cdot{\mathbf{J}})-\omega_p^2\varepsilon_0\partial_t{\mathbf{E}}=0,$$ where $\omega_p$ is the plasma frequency with $\omega_p^2=n_0e^2/(m_e\varepsilon_0)$. By Fourier transformation we replace $\partial_t$ with $-{\mathrm{i}}\omega$, where ${\mathrm{i}}$ is the imaginary unit and $\omega$ is the angular frequency, and obtain the frequency-domain relation between polarization current density and the electric field within the hydrodynamic model as $$\label{eq:hdFre}
\omega(\omega+{\mathrm{i}}\gamma){\mathbf{J}}+\beta^2\nabla(\nabla\cdot{\mathbf{J}})={\mathrm{i}}\omega\omega^2_p\varepsilon_0{\mathbf{E}}.$$ This equation describes electron-field interaction within the plasmonic nanostructure $\Omega_S$. We will neglect spill-out of electrons outside the classical geometric surface of the structure, which for our purposes is a good assumption for noble metals such as silver and gold [@ToscanoEtal2015]. Mathematically, this is arranged by imposing a hard-wall condition on the boundary $\partial\Omega_S$, namely ${\mathbf{n}}\cdot{\mathbf{J}}=0$ on $\partial\Omega_S$ [@Jewsbury:1981a; @Yan:2013a].

General nonlocal optical response model
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We also briefly present the mathematical derivation of 