Time-Dependent Deformation Approximation

Abstract

One of the most important conceptual achievements of DFT is the possibility to formulate the many-body problem in a form of a closed theory that contains only a restricted set of basic variables, such as the density in the static DFT, or the density and the current in TDDFT. In classical physics, a theory of this type is known for more than two centuries. This is classical hydrodynamics. In fact, the Runge-Gross mapping theorem in TDDFT [Runge 1984] proves the existence of the exact quantum hydrodynamics. In this respect, static equilibrium DFT should be viewed as the exact quantum hydrostatics. It is indeed known that the condition of the energy minimum is equivalent to the condition for a local compensation of the external and the internal stress forces exerted on every in.nitesimal volume element of the system in equilibrium [Bartolotti 1980]. Interestingly, the equations of TDDFT in the hydrodynamic formulation can be also considered as a force balance condition, but in a local noninertial reference frame moving with the .ow. In the time-dependent case there is a local compensation of the external, inertial, and internal stress forces. This demonstrates a close similarity of static DFT and TDDFT in the co-moving frame. The above similarity was the main motivation to reconsider the formulation of TDDFT from the point of view of a local observer in the co-moving Lagrangian reference frame [Tokatly 2005a, Tokatly 2005b]. One of the goals of this work is to present the Lagrangian formulation of TDDFT in a compact and physically transparent form. Simple numerical illustrations of this approach for one-dimensional dynamics can be found in a recent paper [Ullrich 2006].