<p>The computational success of density functional theory relies on the construction of suitable approximations to the exchange-correlation energy functional. Use of functional approximations depending explicitly upon the density alone appear unable to address all aspects of many-body interactions, such as the fundamental constraint that the ground state energy is a piecewise linear function of the total number of electrons, and the ability to model nonlocal effects. Functionals depending explicitly upon occupied and unoccupied Kohn–Sham orbitals are considered necessary to address these and other issues. This dissertation considers certain issues relevant to the successful implementation of explicitly orbital-dependent functionals through the optimized effective potential (OEP) approach, as well as extending the potential functional formalism that provides the formal basis for the OEP approach to systems in the presence of noncollinear magnetic fields.</p><p>The self-consistent implementation of orbital-dependent energy functionals is correctly done through the optimized effective potential approach—minimization of the ground state energy with respect to the Kohn–Sham potential that generates the set of orbitals employed in the energy evaluation. The focus on the potential can be problematic in finite basis set approaches as determining the exchange-correlation potential in this manner is an inverse problem which, depending upon the combination of orbital and potential basis sets employed, is often ill-posed. The ill-posed nature manifests itself as nonphysical exchange-correlation potentials and total energies. We address the problem of determining meaningful exchange-correlation potentials for arbitrary combinations of orbital and potential basis sets through an L-curve regularization approach based on biasing towards smooth potentials in the energy minimization. This approach generates physically reasonable potentials for any combination of basis sets as shown by comparisons with grid-based OEP calculations on atoms, and through direct comparison with DFT calculations employing functionals not depending on orbitals for which OEP can also be performed. This work ensures that the OEP methodology can be considered a viable many-body computational methodology.</p><p>A separate issue of our OEP implementation is that it can suffer from a lack of size-extensivity—the total energy of a system of infinitely separated monomers may not scale linearly with the total number of monomers depending upon how we construct the Kohn–Sham potential. Typically, a fixed reference potential is employed to aid in the convergence of a finite basis set expansion of the Kohn–Sham potential. This reference potential can be utilized to ensure other desirable properties of the resulting potential. In particular, it can enforce the correct asymptotic behavior. The Fermi–Amaldi potential is often used for this purpose but suffers from size-nonextensivity owing to the explicit dependence of the potential on the total number of electrons. This error is examined and shown to be rather small and rapidly approaches a limiting linear behavior. A size-extensive reference potential with the correct asymptotic behavior is suggested and examined.</p><p>We also consider a formal aspect of the potential-based approach that provides the underlying justification of the OEP methodology. The potential functional formalism of Yang, Ayers, and Wu is extended to include systems in the presence of noncollinear magnetic fields. In doing so, a solution to the nonuniqueness issue associated with mapping between potentials and wave functions in such systems is provided, and a computational implementation of the OEP in noncollinear systems is suggested.</p><p>Finally, as an example of an issue for which orbital-dependent functionals seem necessary to obtain a correct description, we consider the ground state structures of C_{4<italic>N</italic> + 2} rings which are believed to exhibit a geometric transition from angle-alternation (<italic>N</italic> ≤ 2) to bond-alternation (<italic>N</italic> > 2). So far, no published DFT approach has been able to reproduce this behavior owing to the tendency of common density functional approximations to bias towards delocalized electron densities. Calculations are presented with the rCAM-B3LYP exchange-correlation functional that correctly predict the structural evolution of this system. This is rationalized in terms of the recently proposed delocalization error for which rCAM-B3LYP explicitly attempts to address.</p> / Dissertation

Chemische Reaktionen in der Natur sowie Prozesse in synthetischen Materialien werden oft erst durch die Wechselwirkung von Licht mit Materie ausgelöst. Üblicherweise werden diese komplexen Prozesse mit Hilfe von Näherungen beschrieben. Im ersten Teil der Arbeit wird die Gültigkeit der Born-Oppenheimer Näherung in einem vibronischen Modellsystem (Trans-Polyacetylene) unter Photoelektronenspektroskopie im Gleichgewicht sowie zeitaufgelöster Photoelektronenspektroskopie im Nichtgleichgewicht überprüft. Die vibronische Spektralfunktion zeigt aufgrund des faktorisierten Anfangs- und Endzustandes in der Born-Oppenheimer Näherung zusätzliche Peaks, die in der exakten Spektralfunktion nicht auftreten. Im Nichtgleichgewicht zeigen wir für eine Franck-Condon Anregung und eine Anregung mit Pump-Probe Puls, wie die Bewegung des vibronischen Wellenpaktes im zeitabhängigen Photoelektronenspektrum verfolgt werden kann. Im zweiten Teil der Arbeit werden sowohl die Materie als auch das Licht quantisiert behandelt. Für eine volle quantenmechanische Beschreibung des Elektron-Licht Systems, verwenden wir die kürzlich entwickelte quantenelektrodynamische Dichtefunktionaltheorie (QEDFT) für gekoppelte Elektron-Photon Systeme. Wir zeigen erste numerische QEDFT-Berechnungen voll quantisierter Atome und Moleküle in optischen Kavitäten, die an das quantisierte elektromagnetische Feld gekoppelt sind. Mit Hilfe von Fixpunktiterationen berechnen wir das exakte Kohn-Sham Potential im diskreten Ortsraum, wobei unser Hauptaugenmerk auf dem Austausch-Korrelations-Potential liegt. Wir zeigen die erste Näherung des Austausch-Korrelations-Potentials mit Hilfe eines optimierten effektiven Potential Ansatzes angewandt auf einen Jaynes-Cummings-Dimer. Die dieser Arbeit zugrunde liegenden Erkenntnisse und Näherungen ermöglichen es neuartige Phänomene an der Schnittstelle zwischen den Materialwissenschaften und der Quantenoptik zu beschreiben. / Many natural and synthetic processes are triggered by the interaction of light and matter. All these complex processes are routinely explained by employing various approximations. In the first part of this work, we assess the validity of the Born-Oppenheimer approximation in the case of equilibrium and time-resolved nonequilibrium photoelectron spectra for a vibronic model system of Trans-Polyacetylene. We show that spurious peaks appear for the vibronic spectral function in the Born-Oppenheimer approximation, which are not present in the exact spectral function of the system. This effect can be traced back to the factorized nature of the Born-Oppenheimer initial and final photoemission states. In the nonequilibrium case, we illustrate for an initial Franck-Condon excitation and an explicit pump-pulse excitation how the vibronic wave packet motion can be traced in the time-resolved photoelectron spectra as function of the pump-probe delay. In the second part of this work, we aim at treating both, matter and light, on an equal quantized footing. We apply the recently developed quantum electrodynamical density-functional theory, (QEDFT), which allows to describe electron-photon systems fully quantum mechanically. We present the first numerical calculations in the framework of QEDFT. We focus on the electron-photon exchange-correlation contribution by calculating exact Kohn-Sham potentials in real space using fixed-point inversions and present the performance of the first approximate exchange-correlation potential based on an optimized effective potential approach for a Jaynes-Cummings-Hubbard dimer. This work opens new research lines at the interface between materials science and quantum optics.