A Local-Scaling Time-Dependent Current Density Functional Theory

Abstract

The formally exact expression of the exchange-correlation potential in the local-scaling density functional theory (LSDFT) was derived recently by the author, and it is equivalent to that of Holas and March. The basic equation of the LS-DFT was also derived, and it is equivalent to the differential virial theorem (DVT). The LS-DFT has been extended to the local-scaling current density functional theory (LS-CDFT). This has been compared with the formulation of Holas and March. In this note, the LS-CDFT is extended to the timedependent system; i.e., the local-scaling time-dependent current density functional theory (LS-TDCDFT) is formulated. As a straightforward extension of ref. 3, the density and the paramagnetic current density jp are adopted as the basic variables, which is the same procedure used by Colwell et al. It is explicitly shown that the time-dependent exchange-correlation scalar potential vxc and the vector one Axc are same functionals of firstand second-order density matrices as those of the time-independent case, while the total effective potentials are not so. The equation of motion of jp provides the DVT for the time-dependent system in the magnetic field. The time-dependent term of the DVT is equal to that of the nonmagnetic time-dependent system. Throughout the entire formulation, the notation of the tensor product is suppressed. Here, an N-electron system in an external scalar potential vextðr; tÞ, an external vector potential Aextðr; tÞ, and an interaction potential uðr; r0Þ 1⁄4 =jr r0j is considered, where is the coupling constant treated as a variable parameter. In a time-independent theory, the quantity to be minimized is the energy. But in the time-dependent case, the action S plays the role: