Constraint-Guided Construction of Exchange-Correlation Approximations in Density-Functional Theory

The further advancement of alternative energy sources such as photovoltaics based on organic semiconductor materials requires, in addition to experimental efforts, a deeper understanding of the underlying physical processes in the organic electronic devices at the theoretical level. In order to access the electronic structure of the relevant systems, i.e., molecules with up to several hundreds of electrons, a numerically feasible yet reliable theoretical framework is in great demand. Density-functional theory provides such an efficient and, in principle, exact formalism to calculate the electronic structure of matter from first principles. However, a practical application of density-functional theory requires an approximate expression for the exchange-correlation energy as a functional of the electron density, which leads to an approximate description of physical observables. In fact, it is observed that the quality of results crucially depends on the approximation of the exchange-correlation energy that is used. In particular, functionals that describe ground-state properties such as molecular structures and binding energies reasonably well, often fail to predict quantities related to ionization and photoemission processes with a comparable quality. In the course of this thesis I investigate this issue with a special focus on the class of hybrid functionals, which use nonlocal exact exchange in combination with semilocal functional parts. I present a novel hybrid functional that, in contrast to the traditional hybrid approach, uses a space- and density-dependent mixing of the nonlocal and semilocal components. Guided by the principle of combining exact exchange with compatible correlation, the presented functional is constructed to fulfill exact constraints on the exchange-correlation energy. Furthermore, it is designed to effectively counteract electronic self-interaction, a fundamental problem with serious implications for the reliability of density-functional methods. I discuss to what extent this generalized hybrid ansatz leads to results that are similar to or different from the standard hybrids. In particular, I address the asymptotic behavior of the exchange-correlation potential and its connection to the problem of simultaneously describing thermochemistry in contrast to ionization properties with comparable accuracy. Further, I evaluate the performance of the novel hybrid functional for applications that are drastically influenced by self-interaction such as, for instance, the interpretation of the eigenvalue spectrum as a physical density of states in the context of photoemission experiments. My investigations reveal unexpected similarities between this novel and the standard hybrid concept, and provide insights into the construction of functional approximations for the characterization of organic semiconductor molecules. Additionally, I present an analysis of density-functional methods that are generalized to ensemble states with fractional numbers of electrons. This formalism was found to…