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Plasmonic excitations in quantum-sized sodium nanoparticles studied by time-dependent density functional calculations

The plasmonic properties of sphere-like bcc Na nanoclusters ranging from Na$_{15}$ to Na$_{331}$ have been studied by real-time time-dependent local density approximation calculations. The optical absorption spectrum, density response function and static polarizability are evaluated. It is shown that the effect of the ionic background (ionic species and lattice) of the clusters accounts for the remaining discrepancy in the principal (surface plasmon) absorption peak energy between the experiments and previous calculations based on a jellium background model. The ionic background effect also pushes the critical cluster size where the maximum width of the principal peak occurs from Na$_{40}$ predicted by the previous jellium model calculations to Na$_{65}$. In the volume mode clusters (Na$_{27}$, Na$_{51}$, Na$_{65}$, Na$_{89}$ and Na$_{113}$) in which the density response function is dominated by an intense volume mode, a multiple absorption peak structure also appears next to the principal peak. In contrast, the surface mode clusters of greater size (Na$_{169}$, Na$_{229}$, Na$_{283}$ and Na$_{331}$) exhibit a smoother and narrower principal absorption peak because their surface plasmon energy is located well within that of the unperturbed electron-hole transitions, and their density responses already bear resemblance to that of classical Mie theory. Moreover, it is found that the volume plasmon that exist only in finite size particles, gives rise to the long absorption tail in the UV region. This volume plasmon manifests itself in the absorption spectrum even for clusters as large as Na$_{331}$ with an effective diameter of $\sim$3.0 nm.