Quantum Hydrodynamic Model of Density Functional Theory

In this paper, we extend the method in [5] to derive a class of quantum hydrodynamic models for the density-functional theory (DFT). The most popular implement of DFT is the Kohn-Sham equation, which transforms a many-particle interacting system into a fictitious non-interacting one-particle system. The Kohn-Sham equation is a non-linear Schrödinger equation, and the corresponding Wigner equation can be derived as an alternative implementation of DFT. We derive quantum hydrodynamic models of the Wigner equation by moment closure following [5]. The derived quantum hydrodynamic models are globally hyperbolic thus locally wellposed. The contribution of the Kohn-Sham potential is turned into a nonlinear source term of the hyperbolic moment system. This provides a new possible way to solve DFT problems.