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An accurate description of the optical response of subwavelength metallic particles and nanogap structures is a key problem of plasmonics. Quantum hydrodynamic theory (QHT) has emerged as a powerful method to calculate the optical response of metallic nanoparticles (NPs) since it takes into account nonlocality and spill-out effects. Nevertheless, the absorption spectra of metallic NPs obtained with conventional QHT, i.e., incorporating Thomas-Fermi (TF) and von Weizsäcker (vW) kinetic energy (KE) contributions, can be affected by several spurious resonances at energies higher than the main localized surface plasmon (LSP). These peaks are not present in reference time-dependent density-functional-theory (TD-DFT) spectra, where, instead, only a broad shoulder exists. Moreover, we show here that these peaks incorrectly reduce the LSP peak intensity and have a strong dependence on the simulation domain size so that a proper calculation of QHT absorption spectra can be problematic. In this article, we introduce a more general QHT method accounting for KE contributions depending on the Laplacian of the electronic density ($q$), thus, beyond the gradient-only dependence of the TFvW functional. We show that employing a KE functional with a term proportional to $q^2$ results in an absorption spectrum free of spurious peaks, with LSP resonance of correct intensity and numerically stable Bennett state. Finally, we present a novel Laplacian-level KE energy functional that is very accurate for the description of the optical properties of NPs with different sizes as well as for dimers.