# The UV Plasmonic Behavior of Rhodium Tetrahedrons--A Numerical Analysis
## 3 Results and Discussion
### Rh Tetraledra: Effect of Rounding Edges  
As shown in the previous section, the optical response strongly depends on the shape of the NP. Therefore, we now consider how the LSPR for an isolated perfectly shaped TD is modified when some deformations are introduced. Specifically, here, we will consider the effect of rounding the corners and edges of the TD. This type of deformation can arise during the chemical synthesis or from the extended use of the TDs as catalysts. Specifically, we have studied the effect of rounding the edges and corners by varying their curvature radius \(R_{c}\) from 0 nm (perfect TD) to 2 nm. This study has been done starting from a perfect TD with side length \(L=20\) nm illuminated under normal incidence with a monochromatic plane wave under and \(\mathbf{E_{x}}\) polarization (see Figure 1a). Figure 4 shows the effect of increasing \(R_{c}\) on the absorption cross-section (a) \(C_{abs}\) and (b,c) the local field enhancement \(|\mathbf{E}|^{2}\). The local field enhancement has been evaluated spectrally by (b) averaging its value over the NPs close surface (\(\langle|\mathbf{E}|^{2}\rangle\)) to see at which photon energy this magnitude is maximized, and (c) locally on the surface of the TD to identify the regions were the enhancement is being produced (hot-spots). These maps have been recorded for the photon energy of maximum average enhancement \(\langle|\mathbf{E}|^{2}\rangle\). When increasing the value of \(R_{c}\), two trends can be distinguished in the \(C_{abs}\) spectra--a blue-shift and a decrease of the resonance peak. This effect can be understood attending to two different effects: a decrease in the volume of the TD as \(R_{c}\) increases, and the reduction of the amplitude of charge oscillation due to the de-localization of charges at the edges and corners when rounded. A similar behavior has \(\langle|\mathbf{E}|^{2}\rangle\), whose spectra are also blue-shifted and quenched when increasing \(R_{c}\). This is related to weaker and more de-localized hot-spots as seen in Figure 4c. This de-localization of the hot-spots with increasing \(R_{c}\) is related with the de-localization of charge at edges and tips, something that can be explained attending to the _lightning rod effect_--sharp corners and edges concentrate the dipolar fields in particles [24]. It is worth mentioning that the near-field resonances (peaks appearing in \(\langle|\mathbf{E}|^{2}\rangle\) spectra) are red-shifted with respect to those in the far-field (peaks appearing in \(C_{abs}\) spectra), a phenomenon attributed to the different contribution of the evanescent and propagating components to the scattered field [25]. In this case, the red-shift is \(\approx\)0.3 eV. Figure 4: **(a)** Absorption cross-sections (\(C_{abs}\)) of edge length \(L\) = 20 nm Rh NCswith rounded edges and corners with a curvature radii \(R_{c}\) varying from 0 (perfect TD) to 2 nm; **(b)** local field enhancement (\(|\mathbf{E}|^{2}\)) averaged over the TD’s surface; **(c)** local field enhancement in logarithmic scale (\(|\mathbf{E}|^{2}\)) over the surface of the different TD when illuminated with the corresponding resonant wavelength. To better illustrate the aforementioned de-localization of the charge at edges and tips, we have studied the surface charge density \(\sigma\) at the edges and flat faces of the TDs. The red/blue regions represent positive/negative charge densities. Since we associate high values of \(\sigma\) with a high chemical reactivity, these calculations allow us to identify the most reactive regions of the NP's surface and to qualitatively quantify the loss of reactivity when increasing \(R_{c}\). Figure 5 shows the normalized absolute value of the averaged surface charge density \(\langle|\sigma|\rangle\) on (a) the corners and edges, and (b) on the flat faces of the TD, respectively. It is clear from these plots that most of the charge is concentrated on the edges and corners, and that this density decreases with increasing \(R_{c}\). These results are consistent with the lightning rod effect [24].