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As natural light-harvesting mechanisms are more efficient and robust than artificial solar technology, a deeper understanding of the energy absorption and conversion processes in plants and bacteria is at the center of a lot of current research. The theoretical prediction and interpretation of these phenomena requires methods that facilitate a quantum mechanical description of systems made from several thousands of electrons. Density functional theory in its Kohn-Sham formulation is by far the most popular method for the study of molecules, clusters and solids due to its beneficial ratio of accuracy to computational cost, and its time-dependent extension (TDDFT) is commonly used for the description of dynamical properties of molecular and nanostructures. The reliability and efficiency of Kohn-Sham density functional theory is determined by the approximation to the exchange-correlation energy (or potential) employed in practical applications. For the study of electronic excitations one usually resorts to TDDFT in the linear regime. Linear response calculations in the Casida formulation are routinely done with most quantum chemistry codes. While the Casida approach is technically highly developed and quite efficient for medium-sized systems, it involves virtual orbitals, which leads to an unfavorable scaling with the system size. Additionally, as an eigenvalue problem with a dense matrix, it is not suited for a high degree of parallelization. Therefore, the Casida scheme is not optimal for the study of the larger systems occurring in natural light-harvesting complexes, which prompts the development of alternative linear response methods. The investigation of the relevant processes in these systems through TDDFT is further complicated by the fact that they involve certain types of electronic excitations that most commonly used exchange-correlation approximations cannot predict reliably. This includes, among others, charge-transfer excitations, which play an important role in, e. g., photosynthetic reaction centers. More involved approximations that give a qualitatively correct description of charge-transfer exist, but are too expensive for applications in larger systems. This has motivated the development of various cheaper approximations that aim at mimicking the decisive features of, e. g., expensive exact-exchange based range-separated hybrid functionals. Meta-generalized gradient approximations seem to particularly well-suited for this task, but like most other approximations that might improve the description of charge transfer and other difficult excitations, they are orbital dependent approximations, which makes their application in time-dependent Kohn-Sham calculations highly nontrivial. In order to address these problems, this thesis is focused on advancing an alternative, lesser-known linear response scheme, the Sternheimer method. While only relatively few applications of the scheme have been reported so far, it is particularly promising for the study of large systems since it only involves occupied orbitals, scales favorably with the system size, and can be parallelized massively, most notably because different frequencies can be treated completely independently. In the first part of the thesis the scheme is developed further, regarding both formal and technical aspects. Among other things, a new derivation is presented, it is extended to the treatment of triplet excitations, and novel strategies for the efficient evaluation of excitation energies are put forward. Then the scheme is employed to study an orbital independent exchange-correlation approximation designed to mimic properties of exact exchange, the Armiento-Kümmel generalized gradient approximation. To be able to study more flexible approximations, a new and efficient way of treating orbital dependent exchange-correlation potentials in the Sternheimer approach is developed, which suggests that the Sternheimer method might be better suited for the application for orbital functionals then linear response schemes based on Casida’s equations or on real-time propagation. Finally, this method is applied to the recently developed TASK meta-generalized gradient approximation, and TASK’s performance in the description of charge transfer in a donor-acceptor-donor system of experimentally relevant size is studied.

The description of systems of interacting electrons in the presence of magnetic fields within density-functional theory requires the current and magnetization densities to be used as basic variables besides the particle density. However, electron-gas-based approximations for the exchange and correlation energies exhibit derivative discontinuities as a function of the magnetic field whenever a new Landau level is occupied, which makes them impractical to use as local approximation. The same approximations also assume a collinear configuration respectively between the magnetization, vorticity density and corresponding conjugate fields. Orbital dependent functionals -- which depend on Kohn-Sham spinors -- allow to overcome both limitations. To make use of such expressions, we have derived the integral equations for the exchange-correlation components of the scalar and vector potential, and the magnetic field in the fashion of the optimized effective potential method. We suggest a practical scheme for their solution as well as a simplifying approximation. The formalism is then applied to several systems and problems: we consider the degeneracy problem for open-shell atoms; orbital magnetic moments and band-splittings induced by spin-orbit coupling for extended systems are calculated; and quantum dots in an external magnetic field are studied. The exact-exchange functional is employed both in collinear and non- collinear fashion within density-functional theory, spin-density-functional theory and current-spin-density-functional theory. The effect of the exchange vector potential is verified to be minor. An expression for the Colle and Salvetti approximation for the correlation energy has been derived which explicitly includes orbital currents and their relevance has been verified. New orbital and density functionals for the exchange energy of two-dimensional systems are proposed, which are seen to perform significantly better than the corresponding local-spin-density approximation in two-dimensional systems with few electrons., Die Beschreibung von Systemem wechselwirkender Elektronen in magnetischen Feldern im Rahmen der Dichtefunktionaltheorie erfordert die Verwendung der Strom- und Magnetisierungsdichten als zusaetzliche fundamentale Variablen neben der Teilchendichte. Fuer ein homogenes Elektronengas im aeusseren Magnetfeld weist die Austausch-Korrelations-energie immer dann Ableitungsunstetigkeiten als Funktion des Magnetfelds auf, wenn ein neues Landau-Niveau mit Elektronen besetzt wird. Im Rahmen einer auf diesem Modell beruhenden Lokaldichtenaeherung fuehrt dies zu von lokalen Werten der Dichte bestimmten Unstetigkeiten der entsprechenden Austausch-Korrelationspotentiale, die die Lokaldichtenaeherung in numerischen Anwendungen unpraktikabel macht. Des weiteren nimmt man in der Lokaldichtenaeherung eine kollineare Konfiguration zwischen der lokalen Magnetsierungs- bzw. Vortizitaetsdichte und den entsprechenden konjugierten Potentialen an. Mit Hilfe von Funktionalen, die explizit von Kohn-Sham Spinororbitalen abhaengen, kann man beide Einschraenkungen ueberwinden. Zur Anwendung solcher Funktionale wird hier eine Verallgemeinerung der sogenannten Methode des Optimierten Effektiven Potentials vorgestellt und die entsprechenden gekoppelten Integralgleichungen fuer die Austausch-Korrelationspotentiale (skalares Potential, Magnetfeld und Vektorpotential) hergeleitet. Ein praktisches Verfahren zu ihrer Loesung sowie eine vereinfachende Naeherung werden vorgeschlagen. Anwendungen des Formalismus fuer verschiedene Systeme und Probleme werden untersucht: wir betrachten das Problem entarteter Grundzustaende bei Atomen mit offenen Schalen; magnetische Orbitalmomente und spin-bahn-induzierte Bandaufspaltungen fuer ausgedehnte Festkoerper werden berechnet; Quantenpunkte im externen Magnetfeld werden untersucht. Bei diesen Anwendungen wird das exakte Austauschfunktional sowohl in kollinearer als auch in nicht-kollinearer Situation im Rahmen der Dichte-, Spindichte- und Stromdichtefunktionaltheorie verwendet. In allen Beispielen zeigt sich, dass der Effekt des Austausch- Vektorpotentials gering ist. Fuer die Korrelationsenergie wird eine Verallgemeinerung des Colle-Salvetti-Funktionals hergeleitet, die Orbitalstroeme explizit beruecksichtigt, und der Effekt dieser Stroeme wird untersucht. Weiterhin werden neue Orbital- und Dichtefunktionale fuer die Austauschenergie zweidimensionaler Systeme vorgeschlagen und es wird gezeigt, dass diese fuer Systeme mit wenigen Elektronen deutlich verbesserte Ergebnisse im Vergleich zur 2D-Lokaldichtenaeherung liefern.

We introduce orbital functionals $\int \boldsymbol{\beta} $ simultaneously for each commensurability class of orbital surfaces. They are realized on infinitely dimensional \emph{orbital} divisor spaces spanned by (arithmetic-geodesic real $2$-dimensional) orbital curves on any orbital surface. We discover infinitely many of them on each commensurability class of orbital Picard surfaces, which are real $4$-spaces with cusps and negative constant Kähler-Einstein metric degenerated along an orbital cycle. For a suitable (Heegner) sequence $\int \mathbf{h}_N$, $N \in \mathbb{N}$, of them we investigate the corresponding formal orbital $q$-series $\mathop{\sum}\limits_{N=0}^\infty (\int \mathbf{\mathbf{h}}_N)q^N$. We show that after substitution $q = \re^{2\pi\ri\tau}$ and application to arithmetic orbital curves $\mathbf{\hat{C}}$ on a fixed Picard surface class the series $\mathop{\sum}\limits_{N=0}^{\infty} (\int_{\mathbf{\hat{C}}} \mathbf{\mathbf{h}}_N)\re^{2\pi\ri\tau}$ define modular forms of well-determined fixed weight, level and Nebentypus. The proof needs a new orbital understan-ding of orbital hights introduced in \cite{Ho1} and Mumford-Fulton's rational intersection theory on singular surfaces in Riemann-Roch-Hirzebruch style. It has to be connected with Zeta and Theta functions of hermitian lines, indefinit quaternionic fields and of a matrix algebra along a research marathon over $75$ years represented by Cogdell, Kudla, Hirzebruch, Zagier, Shimura, Schoeneberg and Hecke. Our aim is to open a door to an effective enumerative geometry for complex geodesics on orbital varieties with nice metrics.