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Quantum hydrodynamic theory (QHT) can describe some of the characteristicfeatures 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 fullyquantum mechanical approaches, as exemplified by time-dependent densityfunctional theory using a jellium model. However, QHT involves kinetic energyfunctionals, the practical implementation of which typically inducessignificant numerical instabilities, particularly in nonlinear opticalphenomena. To mitigate this problem, we develop a numerical solution to QHTthat is quite stable, even in a nonlinear regime. The key to our approach is torewrite the dynamical equation of QHT using the effective Schr\"odingerequation. We apply the new method to the linear and nonlinear responses of ametallic nanoparticle and compare the results with fully quantum mechanicalcalculations. The results demonstrate the numerical stability of our method, aswell as the reliability and limitations of QHT.

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