Nonadiabatic electron dynamics in time-dependent density-functional theory

Time-dependent density-functional theory (TDDFT) treats dynamical exchange and correlation (xc) via a single-particle potential, V_xc(r,t), defined as a nonlocal functional of the density n(r^′,t^′). The popular adiabatic local-density approximation (ALDA) for V_xc(r,t) uses only densities at the same space-time point (r,t). To go beyond the ALDA, two local approximations have been proposed based on quantum hydrodynamics and elasticity theory: (a) using the current as the basic variable (C-TDDFT) [ G. Vignale, C. A. Ullrich and S. Conti Phys. Rev. Lett. 79 4847 (1997)], (b) working in a comoving Lagrangian reference frame (L-TDDFT) [ I. V. Tokatly Phys. Rev. B 71 165105 (2005)]. In this paper we illustrate, compare, and analyze both nonadiabatic theories for simple time-dependent model densities in the linear and nonlinear regime, for a broad range of time and frequency scales. C- and L-TDDFT are identical in certain limits, but, in general, exhibit qualitative and quantitative differences in their respective treatment of elastic and dissipative electron dynamics. In situations where the electronic density rapidly undergoes large deformations, it is found that nonadiabatic effects can become significant, causing the ALDA to break down.