Time-Dependent Current Density Functional Theory

Abstract

Materials have been used throughout history for their structural properties, e.g. ductility, elasticity, hardness etc., and later also for their physical properties, i.e., for their characteristic response to external perturbances. These last properties have been investigated in this thesis by using quatumechanics. The mutual interactions of the many-particles which constitute matter complicate its theoretical description enormously. Time-dependent density-functional theory provides a powerful tool to investigate these dynamic properties in atoms, molecules, and clusters. Indeed this theory maps the many-particle problem of interacting electrons in a time-dependent external field onto an auxiliary system of noninteracting particles in an effective field. The many-particle effects enter through so-called exchange-correlation contributions to the effective potentials describing this field. These effective potentials are uniquely determined by the time-dependent density for a given initial state. Essential in this formulation is that the time-dependent density is exactly the same in the real and auxiliary system. The time-dependent density can thus be obtained from the auxiliary system once we know the effective potentials, which in turn are functionals of the time-dependent density. This requires a self-consistent procedure in which one updates the approximate densities and potentials in an iterative way. In this thesis we have only considered systems, initially in the ground state, which are perturbed only weakly by an external electromagnetic field. In this so-called linear response regime the induced density is proportional to the applied field. The exchange-correlation potentials depend on the density in a nontrivial way: the potentials at time t and position r depend in general on the density at all earlier times t' and at all positions r'. In the linear response regime this relation is expressed through a so-called exchange-correlation response kernel. The most common approximation for the exchange-correlation potentials is the adiabatic local density approximation (ALDA), in which these potentials are instantaneous and local functionals of the density. The scenario becomes more complicated for solids, which can be treated as 'open' systems, i.e., systems without boundary. To fully describe these systems, the density is replaced by the current-density leading to time-dependent current-density-functional theory (TDCDFT). In this thesis we have formulated the linear response of metallic crystalline systems to an electric field within this theoretical framework. The method gives already reasonable results for the dielectric and electron energy-loss functions within the ALDA. Although this approximation describes well the interband region of the spectra, it is less satisfying in the intraband region, where relaxation processes, such as electron-electron and electron-phonon scattering, become more important. Indeed the ALDA fails to reproduce the Drude-like tail in the absorption spectra. Approximations to the exchange-correlation potentials which are local-functionals of the current-density and hence nonlocal-functionals of the density can be included in a natural way in our formulation. In these approximations the exchange-correlation kernels can have memory, and thus relaxation processes due to electron-electron scattering can be taken into account. Other types of deviations from experiments can be observed in systems containing heavy elements, if relativistic corrections are neglected. Our method can treat scalar relativistic effects and spin-orbit coupling both in the ground-state and response calculations by using the zeroth-order regular approximation (ZORA). The spin-dependent formulation of the method, which allows to treat spin-orbit coupling, represents a first step to describe the linear response to an external magnetic field.