Parameter-free quantum hydrodynamic theory for plasmonics: Electron density-dependent damping rate and diffusion coefficient

An accurate and efficient method to calculate the optical response of metallic structures plays a key role in plasmonics. Quantum hydrodynamic theory (QHT) provides a self-consistent and computationally efficient description of the free-electron gas, where nonlocality, spill-out effects, and Landau damping can be taken into account. In this work, we introduce a general QHT that both the quantum convection-diffusion and spill-out effects are taken into account. For sodium nanosphere of various sizes, we show that our QHT method can predict both plasmon energy and broadening, in excellent agreement with time-dependent density functional theory predictions and Kreibig formula. Both the ground and excited states are solved by using the same energy functional, with the kinetic energy described by the Thomas-Fermi (TF) and von Weizs\"{a}cker (vW) formalisms. We numerically prove that the fraction of the vW should be around $0.4$. An electron density-dependent damping rate is introduced, making the QHT a numerically stable method. We also introduce a density-dependent diffusion coefficient, which then enables a parameter-free simulation, i.e. without resorting to any input parameter such as damping rate, diffusing coefficient and the fraction of the vW. By applying our QHT method to sodium jellium cylinders of various sizes, we find that there exists a perfect linear relation between the main longitudinal localized surface plasmon resonance wavelength and the aspect radio. The width decreases with increasing aspect ratio and height. We particularly show that our QHT is robust and provides a great degree of predictability. Thus, our QHT theory represents a general, reliable, and efficient platform to study plasmonics.