Schrödinger fluid dynamics of many-electron systems in a time-dependent density-functional framework

For an N-electron system, a connection is explored between density-functional theory and quantum fluid dynamics, through a dynamical extension of the former. First, we prove the Hohenberg–Kohn theorem for a time-dependent harmonic perturbation under conditions which guarantee the existence of the corresponding steady (or quasiperiodic) states of the system. The corresponding one-particle time-dependent Schrödinger equation is then variationally derived starting from a fluid-dynamical Lagrangian density. The subsequent fluid-dynamical interpretation preserves the ’’particle’’ description of the system in the sense that the N-electron fluid has N components each of which is an independent-particle Schrödinger fluid characterized by a density function ρj and an irrotational velocity field uj, j = 1,⋅⋅⋅,N. However, the mean velocity u of the fluid is not irrotational, in general. The force densities and the stress tensor occurring in the Navier–Stokes equation are physically interpreted. The present work is another step towards the interpretation of physicochemical phenomena in three-dimensional space.