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Journal of the Physical Society of Japan

A Local-Scaling Time-Dependent Current Density Functional Theory

Keiji Kosaka

Published Oct 1, 2005• 1 citation
Publisher

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:

DOI
10.1143/jpsj.74.2864

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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:
This chapter discusses time-dependent current-density-functional theory (TDCDFT). It first asks under what circumstances one should choose the current density as basic variable, instead of the particle density. There are several reasons: a current-based formulation is more suitable for constructing functionals beyond the adiabatic approximation. Furthermore, the current is the natural variable to describe coupling to electromagnetic fields and vector potentials. The basic existence theorems of TDCDFT are proved, and several exact properties and the linear-response regime are discussed. The second half of the chapter deals with the so-called Vignale-Kohn functional, which is a local and frequency-dependent approximation to the exchange-correlation vector potential. The Vignale-Kohn functional is expressed in terms of viscoelastic stresses in the electron liquid. Several applications in the linear-response regime and in real time are presented. The Vignale-Kohn functional works well for describing dissipative effects in collective, plasmon-like electron dynamics.
Time-dependent quantal density-functional theory (Q-DFT) is a description of the s-system of noninteracting fermions with electronic density equivalent to that of Schr\"odinger theory, in terms of fields whose sources are quantum-mechanical expectations of Hermitian operators. The theory delineates and defines the contribution of each type of electron correlation to the local electron-interaction potential ${\ensuremath{\nu}}_{\mathrm{ee}}(\mathbf{r},t)$ of the s system. These correlations are due to the Pauli exclusion principle, Coulomb repulsion, correlation-kinetic, and correlation-current-density effects, the latter two resulting, respectively, from the difference in kinetic energy and current density between the interacting Schr\"odinger and noninteracting systems. We employ Q-DFT to prove the following sum rules and properties of the s system: (i) the components of the potential due to these correlations separately exert no net force on the system; (ii) the torque of the potential is finite and due solely to correlation-current-density effects; (iii) two sum rules involving the curl of the dynamic electron-interaction kernel defined as the functional derivative of ${\ensuremath{\nu}}_{\mathrm{ee}}(\mathbf{r},t)$ are derived and shown to depend on the frequency dependent correlation-current-density effect. Furthermore, via adiabatic coupling constant (\ensuremath{\lambda}) perturbation theory, we prove: (iv) the exchange potential ${\ensuremath{\nu}}_{x}(\mathbf{r},t)$ is the work done in a conservative field representative of Pauli correlations and lowest-order $O(\ensuremath{\lambda})$ correlation-kinetic and correlation-current-density effects; (v) the correlation potential ${\ensuremath{\nu}}_{c}(\mathbf{r},t)$ commences in $O({\ensuremath{\lambda}}^{2}),$ and, at each order, it is the work done in a conservative field representative of Coulomb correlations and correlation-kinetic and correlation-current-density effects; (vi) we derive the integral virial theorem relating ${\ensuremath{\nu}}_{\mathrm{ee}}(\mathbf{r},t)$ to the electron-interaction and correlation-kinetic energy for arbitrary coupling constant strength \ensuremath{\lambda}, and show there are no explicit correlation-current-density contributions to the energy. From this integral virial theorem we (vii) obtain the fully interacting $(\ensuremath{\lambda}=1)$ and exchange-only $(\ensuremath{\lambda}=0)$ integral virial theorems as special cases, the latter showing there is no explicit correlation-kinetic contribution to the exchange energy; and (viii) write expressions for the electron-interaction and correlation-kinetic actions for arbitrary coupling constant \ensuremath{\lambda} in terms of the corresponding fields.
A rigorous formulation of time-dependent current density functional theory (TDCDFT) on a lattice is presented. The density-to-potential mapping and the V\mathcal{V}V-representability problems are reduced to a solution of a certain nonlinear lattice Schr\"odinger equation, to which the standard existence and uniqueness results for nonlinear differential equations are applicable. For two versions of the lattice TDCDFT, we prove that any continuous-in-time current density is locally V\mathcal{V}V-representable (both interacting and noninteracting), provided that in the initial state the local kinetic energy is nonzero everywhere. In most cases of physical interest, the V\mathcal{V}V-representability should also hold globally in time. These results put the application of TDCDFT to any lattice model on a firm foundation, and pave the way for studying exact properties of exchange-correlation potentials.
The logical structure and basic theorems of time-dependent current density functional theory (TDCDFT) are analyzed and reconsidered from the point of view of recently proposed time-dependent deformation functional theory (TDDefFT). It is shown that the formalism of TDDefFT allows to avoid a traditional external potential-to-density/current mapping. Instead the theory is formulated in a form similar to the constrained search procedure in the ground state DFT. Within this formulation of TDCDFT all basic functionals appear from the solution of a constrained universal many-body problem in a comoving reference frame, which is equivalent to finding a conditional extremum of a certain universal action functional. As a result the physical origin of the universal functionals entering the theory, as well as their proper causal structure become obvious. In particular, this leaves no room for any doubt concerning the predictive power of the theory.