As a microscopic theory to express the electromagnetic response of the system to the external fields, we present the time-dependent current-density-functional theory (td-CDFT), taking into account the effect of the energy dissipation from the system. In order to express such an effect, the non-Hermitian term which corresponds to the lifetime of the higher-energy component of the many-electron state is added to the Hamiltonian of the system. Specifically, the occupation number of the Bloch states, the energies of which are higher than the Fermi energy, is chosen as a basic variable in addition to the electron density and paramagnetic current density. In order to confirm its validity, the present td-CDFT is applied to the two-level system. We successfully revisit the electric susceptibility of the Drude-Lorentz model via the present theory.
As a microscopic theory to express the electromagnetic response of the system to the external fields, we present the time-dependent current-density-functional theory (td-CDFT), taking into account the effect of the energy dissipation from the system. In order to express such an effect, the non-Hermitian term which corresponds to the lifetime of the higher-energy component of the many-electron state is added to the Hamiltonian of the system. Specifically, the occupation number of the Bloch states, the energies of which are higher than the Fermi energy, is chosen as a basic variable in addition to the electron density and paramagnetic current density. In order to confirm its validity, the present td-CDFT is applied to the two-level system. We successfully revisit the electric susceptibility of the Drude-Lorentz model via the present theory.
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.
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:
The dynamics of a many-body system coupled to an external environment represents a fundamentally important problem. To this class of open quantum systems pertains the study of energy transport and dissipation, dephasing, quantum measurement and quantum information theory, phase transitions driven by dissipative effects, etc. Here, we discuss in detail an extension of time-dependent current-density-functional theory (TDCDFT), we named stochastic TDCDFT [Phys. Rev. Lett. 98, 226403 (2007)], which allows the description of such problems from a microscopic point of view. We discuss the assumptions of the theory, its relation to a density-matrix formalism, and the limitations of the latter in the present context. In addition, we describe a numerically convenient way to solve the corresponding equations of motion and apply this theory to the dynamics of a one-dimensional gas of excited bosons confined in a harmonic potential and in contact with an external bath.
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.
We adapt time-dependent current density functional theory to allow for a fragment-based solution of the many-electron problem of molecules in the presence of time-dependent electric and magnetic fields. Regarding a molecule as a set of non-interacting subsystems that individually evolve under the influence of an auxiliary external electromagnetic vector-scalar potential pair, the partition 4-potential, we show that there are one-to-one mappings between this auxiliary potential, a sharply-defined set of fragment current densities, and the total current density of the system. The partition electromagnetic (EM) 4-potential is expressed in terms of the real EM 4-potential of the system and a gluing EM 4-potential that accounts for exchange-correlation effects and mutual interaction forces between fragments that are required to yield the correct electron dynamics. We prove the zero-force theorem for the fragmented system, establish a variational formulation in terms of action functionals, and provide a simple illustration for a charged particle in a ring.
Given the time-evolution of an electron charge density, the local potential in Kohn-Sham time-dependent density functional theory (KS-TDDFT) can be modeled as a sum of instantaneous and dynamic contributions by assuming a certain form of the time-dependent Kohn-Sham (TD-KS) orbitals. The instantaneous part is obtained numerically using methods from ground-state density functional theory (DFT) and the dynamic part is expressed in terms of a velocity potential that depends on the electron current density. The suggested form of the TD-KS orbitals satisfies several known constraints (orthonormality, N-representability, J-representability), and the domain of validity is shown to depend on the evolution of the instantaneous quantities. Through this decomposition, we can relate time-dependent and ground -state V-representability. The resulting potentials are shown numerically to approximate the exact time-dependent Kohn-Sham potentials for a set of 3 non-singlet two-particle systems (a Kohn-mode, a Coulomb explosion, and a double quantum well) where the exact solutions and reference densities are known or obtained through configuration interaction (CI) approaches.