We evaluate the density matrix of an arbitrary quantum mechanical system in terms of the quantities pertinent to the solution of the time-dependent density functional theory (TDDFT) problem. Our theory utilizes the adiabatic connection perturbation method of G\"orling and Levy, from which the expansion of the many-body density matrix in powers of the coupling constant $\ensuremath{\lambda}$ naturally arises. We then find the reduced density matrix ${\ensuremath{\rho}}_{\ensuremath{\lambda}}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}},t)$, which, by construction, has the $\ensuremath{\lambda}$ independent diagonal elements ${\ensuremath{\rho}}_{\ensuremath{\lambda}}(\mathbf{r},\mathbf{r},t)=n(\mathbf{r},t)$,
n(r,t) being the particle density. The off-diagonal elements of ${\ensuremath{\rho}}_{\ensuremath{\lambda}}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}},t)$ contribute importantly to the processes unaccessible via the density, directly or by the use of the known TDDFT functionals. Of those, we consider the momentum-resolved photoemission, doing this to the first order in $\ensuremath{\lambda}$, i.e., on the level of the exact exchange theory. In illustrative calculations of photoemission from the quasi-2D electron gas and isolated atoms, we find quantitatively strong and conceptually far-reaching differences with the independent-particle Fermi's golden rule formula.
Two interacting electrons that are confined to the surface of a sphere have a uniform ground-state (surface) density. The Schr\"odinger equation of this helium-type two-electron system is solved here accurately for different values $\ensuremath{\alpha}∊\mathbb{R}$ of a constant that is multiplied to the electron-electron repulsion ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}}_{\mathit{ee}}$. The correlation structure in the resulting wave functions is analyzed for different values of $\ensuremath{\alpha}$. The asymptotic limits $\ensuremath{\alpha}\ensuremath{\rightarrow}0$ and $\ensuremath{\alpha}\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}$ are treated analytically. Using these results, the ISI (interaction-strength interpolation) model for the density-functional ${E}_{\mathit{xc}}[\ensuremath{\rho}]$ of the exchange-correlation energy in the real system with $\ensuremath{\alpha}=1$ is tested against the exact adiabatic connection in density-functional theory.
Density functional theory provides the basis for uncounted studies of ground-state properties of many-body systems. However, the connection between the energy density functional and the underlying microscopic interactions of a given theory is not fully understood. In this thesis we use renormalization-group techniques in combination with density functional theory to study many-body systems from microscopic interactions. We apply our formalism to different one-dimensional systems. We start with systems of identical fermions interacting via a short-range repulsive and long-range attractive interaction which serves as a simple one-dimensional toy model for nuclei. After that we study systems of spin-1/2 fermions, where we assume interactions only between fermions with different spins and consider both a non-local interaction as in the previous case and a contact interaction. In particular, the contact interaction plays a prominent role for ultracold Fermi gases. We calculate ground-state properties such as the energy, density, intrinsic density and density correlation functions and compare our results to values obtained from other approaches. Moreover, we show how energies of excited states and the absolute square of the ground-state wavefunction can be extracted from the density-density correlation function. The relation between our formalism and conventional density functional theory as well as many-body perturbation theory is discussed which may help to guide the development of ab initio functionals for quantitative studies of nuclei in the future.
Within density-functional theory a time-dependent perturbation theory is derived that yields exact orbital-dependent expressions for the time-dependent exchange-correlation potential in the form of a perturbation series. This leads to an exact time-dependent Kohn-Sham scheme based on perturbation theory. In first order an ``exchange-only'' Kohn-Sham procedure is obtained, that is a generalization of a recently presented time-dependent optimized potential method. The exact, in higher order, but quite complicated expressions for the exchange-correlation potential may serve as a starting point for the construction of approximate exchange-correlation potentials. An adiabatic connection or coupling constant path between a time-dependent physical system and its corresponding Kohn-Sham system is introduced.
A density-functional theory is developed for fermions in one dimension, interacting via a $\ensuremath{\delta}$ function. Such systems provide a natural testing ground for questions of principle, as the local-density approximation should be highly accurate since for this interaction type the exchange contribution to the local-density approximation is intrinsically self-interaction-free. The exact-exchange contribution to the total energy is a local functional of the density. A local-density approximation for correlation is obtained using perturbation theory and Bethe ansatz results for the one-dimensional contact-interacting uniform Fermi gas. The ground-state energies are calculated for two finite systems, the analogs of helium and of Hooke's atom. The local-density approximation is shown to be excellent as expected.
We introduce a method of evaluating the density matrix of an arbitrary quantum mechanical system in terms of the quantities pertinent to the solution of the time-dependent density functional theory (TDDFT) problem. Our theory utilizes the adiabatic connection perturbation method [G\{o}rling and Levy, Phys. Rev. A {\bf 50}, 196 (1994), G\{o}rling, ibid. {\bf 55}, 2630 (1997)], from which the expansion of the density matrix in powers of the interaction constant
λ naturally arises. By this, we obtain the one-density-matrix $\rho_\lambda(\rv,\rv',t)$, which, by construction, has the
λ-independent diagonal elements $\rho_\lambda(\rv,\rv,t)=n(\rv,t)$, where $n(\rv,t)$ is the particle density. The off-diagonal elements of $\rho_\lambda(\rv,\rv',t)$ importantly contribute to the processes beyond the reach of TDDFT, of which we consider the momentum-resolved photoemission, doing this to the first order in
λ (exact exchange). In an illustrative calculation of photoemission from the quasi-2D electron gas with {\it one filled subband} we find strong deviations from the independent-particle Fermi golden rule formula.