Numerical scheme for nonlinear optical response of metallic nanostructure: Quantum hydrodynamic theory solved by adopting effective Schr\"odinger equation

Quantum hydrodynamic theory (QHT) can describe some of the characteristic features of quantum electron dynamics that appear in metallic nanostructures, such as spatial nonlocality, electron spill-out, and quantum tunneling. Furthermore, numerical simulations based on QHT are more efficient than fully quantum mechanical approaches, as exemplified by time-dependent density functional theory using a jellium model. However, QHT involves kinetic energy functionals, the practical implementation of which typically induces significant numerical instabilities, particularly in nonlinear optical phenomena. To mitigate this problem, we develop a numerical solution to QHT that is quite stable, even in a nonlinear regime. The key to our approach is to rewrite the dynamical equation of QHT using the effective Schr\"odinger equation. We apply the new method to the linear and nonlinear responses of a metallic nanoparticle and compare the results with fully quantum mechanical calculations. The results demonstrate the numerical stability of our method, as well as the reliability and limitations of QHT.