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In this work, multi-particle quantum optimal control problems are studied in the framework of time-dependent density functional theory (TDDFT).
Quantum control problems are of great importance in both fundamental research and application of atomic and molecular systems. Typical applications are laser induced chemical reactions, nuclear magnetic resonance experiments, and quantum computing.
Theoretically, the problem of how to describe a non-relativistic system of multiple particles is solved by the Schrödinger equation (SE). However, due to the exponential increase in numerical complexity with the number of particles, it is impossible to directly solve the Schrödinger equation for large systems of interest. An efficient and successful approach to overcome this difficulty is the framework of TDDFT and the use of the time-dependent Kohn-Sham (TDKS) equations therein.
This is done by replacing the multi-particle SE with a set of nonlinear single-particle Schrödinger equations that are coupled through an additional potential.
Despite the fact that TDDFT is widely used for physical and quantum chemical calculation and software packages for its use are readily available, its mathematical foundation is still under active development and even fundamental issues remain unproven today.
The main purpose of this thesis is to provide a consistent and rigorous setting for the TDKS equations and of the related optimal control problems.
In the first part of the thesis, the framework of density functional theory (DFT) and TDDFT are introduced. This includes a detailed presentation of the different functional sets forming DFT. Furthermore, the known equivalence of the TDKS system to the original SE problem is further discussed.
To implement the TDDFT framework for multi-particle computations, the TDKS equations provide one of the most successful approaches nowadays. However, only few mathematical results concerning these equations are available and these results do not cover all issues that arise in the formulation of optimal control problems governed by the TDKS model.
It is the purpose of the second part of this thesis to address these issues such as higher regularity of TDKS solutions and the case of weaker requirements on external (control) potentials that are instrumental for the formulation of well-posed TDKS control problems. For this purpose, in this work, existence and uniqueness of TDKS solutions are investigated in the Galerkin framework and using energy estimates for the nonlinear TDKS equations.
In the third part of this thesis, optimal control problems governed by the TDKS model are formulated and investigated. For this purpose, relevant cost functionals that model the purpose of the control are discussed.
Henceforth, TDKS control problems result from the requirement of optimising the given cost functionals subject to the differential constraint given by the TDKS equations. The analysis of these problems is novel and represents one of the main contributions of the present thesis.
In particular, existence of minimizers is proved and their characterization by TDKS optimality systems is discussed in detail.
To this end, Fréchet differentiability of the TDKS model and of the cost functionals is addressed considering \(H^1\) cost of the control.
This part is concluded by deriving the reduced gradient in the \(L^2\) and \(H^1\) inner product.
While the \(L^2\) optimization is widespread in the literature, the choice of the \(H^1\) gradient is motivated in this work by theoretical consideration and by resulting numerical advantages.
The last part of the thesis is devoted to the numerical approximation of the TDKS optimality systems and to their solution by gradient-based optimization techniques.
For the former purpose, Strang time-splitting pseudo-spectral schemes are discussed including a review of some recent theoretical estimates for these schemes and a numerical validation of these estimates.
For the latter purpose, nonlinear (projected) conjugate gradient methods are implemented and are used to validate the theoretical analysis of this thesis with results of numerical experiments with different cost functional settings.
Kaum ein Bereich der menschlichen Tätigkeit hat sich jemals so stürmisch entwickelt, wie die Mikro- und Nanoelektronik in den letzten Jahrzehnten. Der rasche Fortschritt dieser Gebiete war möglich, weil die Vorteile in der Anwendung der Mikroelektronik den gewaltigen Entwicklungs- und Forschungsaufwand rechtfertigten. Eine besondere Rolle spielt dabei die Herstellung von Halbleiterbauelementen durch Kristallzüchtungsmethoden. In dieser Arbeit wurden Prozesse untersucht, die sich auf der Kristalloberfläche abspielen und somit das Wachstum von hochgeordneten Kristallstrukturen bestimmen. In den vergangenen Jahren wurden mehrere Methoden zur Untersuchung dieser Prozesse entwickelt, deren Präzision sich von Jahr zu Jahr unablässig steigerte. In der Reihe der theoretischen Ansätze stehen quantenchemische Methoden im Vordergrund. Eine von diesen Methoden, die Dichtefunktionaltheorie, ist aufgrund ihrer Anschaulichkeit und des relativ niedrigen Rechenaufwands das Hauptwerkzeug der vorliegenden Arbeit. Im ersten Teil dieser Arbeit wurden die Wanderungsmöglichkeiten eines Adsorbatatoms (Cd oder Te) auf der (001) Oberfläche von CdTe (Substrat) auf DFT-Niveau im Rahmen der GGA-Näherung untersucht. Dies erforderte es, die Gesamtenergie des Systems Adsorbat-Kristall an verschiedenen Adsorptionsstellen zu berechnen. Dabei wurde nur ein Teil des Kristalls - das Adsorbat selbst und die nächste Umgebung der Adsorptionsstelle (Quantencluster) - auf DFT-Niveau berechnet. Der Einfluss des übrigen Kristalls auf den Cluster wurde mit einem Gitter aus Punktladungen angenähert, wobei die Te- und Cd-Atome die Ladungen −2 bzw. +2 trugen. Bei dem Einsatz dieses Modells ergab sich allerdings das Problem, dass es eigentlich nur auf Ionenkristalle anwendbar ist, die in guter Näherung volle Ionizität besitzen. CdTe stellt aber laut vielen experimentellen und theoretischen Untersuchungen eine Abstufung zwischen ionischen und kovalenten Kristallen dar, was eine gründliche Analyse der Abhängigkeit unserer Ergebnisse von der Clustergröße und der Entfernung der Adsorptionsstelle von den Clusterrändern erforderte. Als Ergebnis wurde ein Modell entworfen, das dazu in der Lage ist, die Struktur der (2X1) Te-terminierten CdTe Oberfläche mit ausreichender Genauigkeit wiederzugeben. Durch geeignete Wahl des Quantenclusters (ausreichende Größe in den Richtungen parallel zur Oberfläche und Platzierung der weniger polarisierbaren Cd-Kationen an den Außenflächen) gelang es, den Einfluss der Clusteroberfläche auf die untersuchten Eigenschaften auf ein akzeptables Maß zu verringern. Die durchgeführten Berechnungen der Cd-Potentialenergiefläche zeigen zwei Potentialtöpfe, mit den Tiefen 2.1 eV und 1.7 eV. Die Existenz dieser beiden Minima ist eng mit der Dimerisierung von Te-Atomen an der adsorbatfreien Te-Oberfläche verbunden. Das erste, der Struktur =Te-Cdad-Te= entsprechende Minimum entsteht durch den Bruch einer Te-Te Dimerbindung beim Cd-Angriff an diese Stelle. Der zweite Potentialtopf kommt dadurch zustande, dass das Cd-Adsorbatatom mit zwei entlang der [110]-Richtung angeordneten Te2-Dimeren reagiert. Die Potentialenergiefläche des Te-Adsorbats unterscheidet sich zwar wesentlich von der des Cd-Atoms, es gibt aber auch Ähnlichkeiten. Das gilt vor allem für das der Struktur =Te-Tead-Te= entsprechende Minimum, das ungefähr 2.8 eV tief ist. Wie im Fall der Cd-Adsorption entsteht diese Struktur infolge der Wechselwirkung eines adsorbierten Te-Atoms mit einem Te2-Dimer auf der Oberfläche. Die Ergebnisse unserer Berechnungen bestätigen experimentelle Hinweise, gemäß denen Te- und Cd-Atome aus dem Teilchenfluss, dem die (2X1)Te Oberfläche während der MBE ausgesetzt ist, leicht adsorbiert werden. Außerdem wurden die relativ genauen Werte der Potentialbarrieren bekommen, die für ein besseres Verständnis des Wachstumsprozesses zum Beispiel mit Hilfe von Monte-Carlo-Simulationen notwendig sind. Im Vordergrund des zweiten Teils der vorliegenden Arbeit stand die Strukturbestimmung von ZnO-Nanoclustern, die durch spezielle Kristallisationsprozesse erzeugt werden und wegen ihrer eigenartigen optischen und elektronischen Eigenschaften von großem Interesse sind. Zwei grundsätzlich unterschiedliche Atomanordnungen wurden betrachtet, wobei festgestellt werden sollte, welche dieser Strukturen in Abhängigkeit von der Clustergröße und der Umgebung stabiler ist. Angenommen wurde dabei, dass diese Tendenz bei der weiteren Vergrößerung der Atomanzahl von Hundert bis mehreren Tausenden erhalten bleibt. Die Clustermodelle erster Art besaßen die für ZnO-Verbindungen typische Wurtzitstruktur, die anderen, sogenannten Käfigcluster, bestanden aus Zn3O3- und (oder) Zn2O2-Ringen, die so verknüpft sind, dass sie kugel- oder zylinderförmige Strukturen bilden. Charakteristisch für letztere Cluster ist eine Homogenität der Atomumgebung, da alle Zn- und O-Atome dreifach koordiniert sind, während sie in Wurtzitstrukturen im Wesentlichen vierfach koordiniert sind. Durch Knüpfung zusätzlicher Zn-O Bindungen konnte die Anzahl der in Frage kommenden Strukturen nennenswert vergrößert werden. Dabei entstehen vierfach koordinierten Atome und, laut den Berechnungen, deutlich stabilere Cluster. Die Rechnungen wurden sowohl im Vakuum als auch im Rahmen des COSMO Verfahrens (im „Wasser“) durchgeführt. Sie ergaben, dass die Wurtzitstrukturen bei der Zunahme der Atomanzahl stabiler werden als ihre Käfig-Analoge. Dieses Ergebnis ist allerdings eher von theoretischem Interesse, da die experimentell in einer Lösung gezüchteten ZnO-Nanocluster an ihrer Oberfläche mit Molekülen aus der Lösung bedeckt sind. Ein weiterer Schritt war daher, den Einfluss der Umgebung auf die Bildungsenergie durch die Absättigung der Oberfläche mit H+- und OH−-Ionen zu simulieren. Als Bezugspunkt für die Berechnung der Bildungsenergie der verschiedenen Cluster wurde der Molekülkomplex Zn(OH)2(H2O)2 verwendet. Mit anderen Worten wurde angenommen, dass ein freies Zn2+-Ion in der Lösung von zwei OH−-Gruppen und zwei H2O-Molekülen umgeben ist. Die Ergebnisse zeigen, dass die Absättigung einen starken Einfluss auf die Randbereiche der wurtzitartigen Cluster ausübt. Bei fast allen Clustermodellen sind diese stark verformt, während bei den Käfigstrukturen nur deutlich geringere Verzerrungen beobachtet werden. Ebenso stark ist der Einfluss auf die Bildungsenergie: Verglichen mit ihren unabgesättigten Analogen werden alle abgesättigte Strukturen erheblich stabiler, was auf die Tatsache zurückzuführen ist, dass durch die OH− -Gruppen und H+-Kationen die freien Valenzen an der Clusteroberfläche abgesättigt werden. Ansonsten lassen sich bei den abgesättigten Strukturen dieselben Tendenzen erkennen, wie bei nicht abgesättigten. So werden Wurtzitstrukturen mit zunehmender Clustergröße energetisch günstiger als Käfigstrukturen mit der gleichen Anzahl an Atomen. Da es die im Rahmen dieser Arbeit festgestellten Regelmäßigkeiten ermöglichen, die stabilsten ZnO-Atomanordnungen auf die hier Betrachteten einzuschränken, ergibt sich, dass die stabilste Struktur für Nanocluster wurtzitartig ist. Dies stimmt auch mit allen verfügbaren experimentellen Daten überein.
This work presents excited state investigations on several systems with respect to experimental
spectroscopic work. The majority of projects covers the temporal evolution of
excitations in thin films of organic semiconductor materials. In the first chapters, thinfilm
and interface systems are build from diindeno[1,2,3-cd:1’,2’,3’-lm]perylene (DIP)
and N,N’-bis-(2-ethylhexyl)-dicyanoperylene-3,4:9,10-bis(dicarboximide) (PDIR-CN2)
layers, in the third chapter bulk systems consist of 4,4’,4”-tris[(3-methylphenyl)phenylamino]
triphenylamine (m-MTDATA), 4,7-diphenyl-1,10-phenanthroline (BPhen) and
tris-(2,4,6-trimethyl-3-(pyridin-3-yl)phenyl)borane (3TPYMB). These were investigated
by aggregate-based calculations. Careful selection of methods and incorporation
of geometrical relaxation and environmental effects allows for a precise energetical assignment
of excitations. The biggest issue was a proper description of charge-transfer
excitations, which was resolved by the application of ionization potential tuning on
aggregates. Subsequent characterization of excitations and their interplay condenses
the picture. Therefore, we could assign important features of the experimental spectroscopic
data and explain differences between systems.
The last chapter in this work covers the analysis of single molecule spectroscopy on
methylbismut. This poses different challenges for computations, such as multi-reference
character of low-lying excitations and an intrinsic need for a relativistic description.
We resolved this by combining complete active space self-consistent field based methods
with scalarrelativistic density-functional theory. Thus we were able to confidently
assign the spectroscopic features and explain underlying processes.
In this thesis a systematic analysis of the correlation effects between lattice dynamics and magnetism in the Multiferroic Manganites RMnO3 with Pnma structure was conducted. For this task, Raman and FT-IR Spectroscopy were employed for an investigation of all optically accessible lattice vibrations, i.e. phonons. To study the correlation effects as well as their specific connections to symmetry and compositional properties of the Multiferroic Manganites, the polarisation and temperature dependence of the phonons were considered explicitly. In combination with lattice dynamical calculations based on Density Functional Theory, two coupling effects - Spin-Phonon Coupling and Electromagnon-Phonon Coupling - were systematically analysed.
In this work we utilized Density Functional Theory to calculate EPR parameters and spin-density distributions of several transition metal complexes. To demonstrate the performance of our theoretical approach several validation studies were performed (Chapters 3-5). In contrast, the last three chapters of the thesis deal with specific chemical problems regarding several classes of biologically relevant transition metal complexes.
Die vorliegende Dissertation behandelt zum einen die mechanistischen Details von Bindungsaktivierungs-Reaktionen an Disauerstoff und weißem Phosphor mit den Komplexfragmenten 1[Ni(iPr2Im)2] und 3[(η5-C5H5)Co(iPr2Im)] und zum anderen die Regioselektivität von oxidativen Insertionsreaktionen des 1[Ni(iPr2Im)2]-Komplexfragments in C–X-Bindungen substituierter Fluoraromaten (X = F, OCH3, CN, H).
This work focuses on theoretical approaches for predicting the valence and core excited states of aggregate systems. For the valence excitations, TD-HF and TD-DFT with different functionals have been tested at the Perylene bisimide (PBI) system. A simple character analysis method based on the calculated transition dipole moments is proposed. However, this method does not work for excited states without any transition dipole moment. Thus, we proposed a more general and more valid method based on a calculated CIS type wavefunction for the character analysis. Furthermore, a model Hamiltonian method is derived from a localized picture. The energies of the diabatic states and the corresponding coupling parameters were also determined on the basis of ab initio calculations. For the core excitation, three different methods were validated for C 1s-excited and ionized states if several small molecules. Also we tested the basis sets dependence of these core excited states. Based on those results, we chose the frozen core approximation method to evaluate the core excited states of NTCDA molecules. In order to explain the findings in the experiments, we developed an algorithm to evaluate the exciton coupling parameter where non-orthogonal MOs are used.
This thesis describes the inclusion of dynamical effects in the theoretical calculation of Electron Paramagnetic Resonance (EPR) spectroscopic parameters. The studies were performed using Density Functional Theory (DFT) methodology and a perturbation-theoretical approach to g-tensor calculations. Hydrogen atoms trapped in octasilasesquioxane cages display unexpectly high, positive g-values. Computational simulation of these systems successfully reproduced the positive g-values and found them to arise from spin-orbit coupling around the oxygen nuclei. Dynamical effects were estimated by calculating the potential well in which the hydrogen atom moves. Semiquinone radical anions are important bioradicals that play a role in photosynthesis and respiration. The simplest and most prototypical, benzosemiquinone anion, was simulated both in the gas phase and in aqueous solution by Car-Parrinello Molecular Dynamics (CPMD). The neutral benzoquinone was also simulated for comparison. The solvation environments of both the anionic and neutral molecules were analysed and compared. EPR parameters were calculated for the semiquinone, providing the first example of full inclusion of dynamic effects in g-tensor calculation. The effects of different solvation interactions on the g-tensor and hyperfine interactions were extensively examined. Additionally, static calculations (i.e., calculations not incorporating any dynamical effects) were performed. Comparison between these (and prior computational studies) and the dynamical system allowed an assessment of the effects of dynamics on solvation and EPR parameters. Ubisemiquinone radical anion, one of the most widely-occurring semiquinone radicals, was simulated in the aqueous phase using CPMD. The solvation environment was analysed and EPR parameters were calculated. The motion of the side-chain, and its effects on solvation and EPR parameters, were examined.
Quantum chemical modeling of electron paramagnetic resonance (EPR) parameters, in combination with data from the modern high-field/high-frequency EPR (HF-EPR) techniques, constitutes an invaluable analytical tool for gaining insight into radical-protein interactions, which determine the specificity and directionality of the radical-mediated biochemical processes. This thesis reports a series of density functional (DFT) studies on EPR parameters of several biologically relevant radicals and a series of molecular devices inspired by radical-protein interaction in photosystem I (PS-I). We demonstrate our methodology’s accuracy and capacity to provide insight into the in vivo environment and reactivity of bioradicals. Our DFT approach for the calculation of electronic g-tensors has been applied to semiquinone radical anions in the different protein environments of photosynthetic reaction centers. Supermolecular models have been constructed, based on combined crystallographic and quantum chemical structure data, for the QA and QB active sites of bacterial reaction centers, for the A1 site of PS-I, as well as for ubisemiquinone in frozen 2-propanol. After scaling of the computed gx components by 0.92, both gx and gy components computed at gradient-corrected DFT level with accurate spin-orbit operators agree with HF-EPR reference data essentially to within experimental accuracy in all four systems studied. The influence of the various semiquinone-protein non-covalent interactions has been studied by successive removal of individual residues from the models. The effects of hydrogen bonding to the two carbonyl oxygen atoms of the semiquinones was found to be nonadditive, due to compensating spin-polarization effects. The effects of tryptophan-semiquinone -stacking are different for QA and A1 sites. This may be traced back to a different alignment of the interacting fragments and to differential spin polarization. In the next part of this work our DFT methodology has been applied to the semiquinone in the environment of the “high-affinity” binding site of quinol oxidase (QH site). Recent multi-frequency EPR studies of the QH binding site of quinol oxidase have suggested a very asymmetric hydrogen-bonding environment for the semiquinone radical anion state. Single-sided hydrogen bonding to the O1 carbonyl position was one of the proposals, which contrasts with some previous experimental indications. The density functional calculations of the EPR parameters (g-tensors, 13C, 1H, and 17O hyperfine tensors) for a wide variety of supermolecular model complexes have been used to provide insight into the detailed relations between structure, environment and EPR parameters of ubisemiquinone radical anions. A single-sided binding model is not able to account for the experimentally observed low gx component of the g-tensor nor for the observed magnitude of the asymmetry of the 13C carbonyl hyperfine coupling (HFC) tensors. Based on the detailed comparison between computation and experiment, a model with two hydrogen bonds to O1 and one hydrogen bond to O4 was suggested for the QH site, but a model with one more hydrogen bond on each side could not be excluded. Additionally, several general conclusions on the interrelations between EPR parameters and hydrogen bond patterns of ubisemiquinones in proteins were provided. The computational studies related to the mechanism of electron transfer in PS-I gave an impetus to the theoretical design, based on quantum-chemical calculations, of relatively small rotational molecular motors made up from intramolecularly connected dyads consisting of a quinone unit and a pyrrole or indole moiety. It was shown computationally for several systems, depending on the length and attachment points of the interconnecting chains, that a reduction of the quinone to the semiquinone radical anion or quinolate dianion states leads to a reversible intramolecular reorientation from a -stacked to a T-stacked arrangement. In the rearranged structures, a hydrogen bond from the pyrrole or indole N-H function to the semiquinone or quinolate -system is created upon reduction. In some systems, hydrogen bonds to the semiquinone or quinolate oxygen atoms are partly feasible and will be preferred over T-stacking. It was shown that the intramolecular interactions modify the quinone redox potentials. The electronic g-tensors computed for the semiquinone states reflected characteristically the presence and nature of hydrogen bonds to the semiquinone and were suggested as suitable EPR spectroscopic probes for the preferred structures. Intramolecular proton transfer was observed to be possible in the dianionic state. In contrast to semiquinones, which represent paramagnetic states of enzyme cofactors, glycyl radicals are genuine protein radicals. As a step towards an in-depth understanding of the EPR parameters of glycyl radicals in proteins, the hyperfine- tensors and, particularly, the g-tensor of N-acetylglcyl in the environment of a single crystal of N-acetylglycine have been studied by systematic state-of-the-art quantum chemical calculations on various suitable model systems. The quantitative computation of the g-tensors for such glycyl-derived radicals is a veritable challenge, mainly due to the very small g-anisotropy combined with a non-symmetrical, delocalized spin-density distribution and several atoms with comparable spin-orbit contributions to the g-tensors. The choice of gauge origin of the magnetic vector potential, and of approximate spin-orbit operators, both turn out to be more critical than found in previous studies of g-tensors for organic radicals. Environmental effects, included by supermolecular hydrogen-bonded models, were found to be moderate, due to a partial compensation between the influences from intramolecular and intermolecular hydrogen bonds. The largest effects on the g-tensor are caused by the conformation of the radical. The DFT methods employed systematically overestimate both the gx and gy components of the g-tensor. This is important for investigations on the protein-glycyl radicals (see next paragraph). The 1H and 13C hyperfine couplings depend only slightly on the supermolecular model chosen and appear less sensitive probes of detailed structure and environment. The number of enzymes that require a glycyl-based radical for their function is growing. Here we provide systematic quantum-chemical studies of spin-density distributions, electronic g-tensors, and hyperfine couplings of various models of protein-bound glycyl radicals. Similarly to what was found for N-acetylglycyl (see previous paragraph), the small g-anisotropy for this delocalized, unsymmetrical system presents appreciable challenges to state-of-the-art computational methodology. This pertains to the quality of structure optimization, as well as to the choice of spin-orbit Hamiltonian and gauge origin of the magnetic vector potential. Environmental effects due to hydrogen bonding are complicated and depend in a subtle fashion on the different intramolecular hydrogen bonding for different conformations of the radical. Indeed, the conformation has the largest overall effect on the computed g-tensors (less so on the hyperfine-tensors). We discuss this in the context of different g-tensors obtained by recent HF-EPR measurements for three different enzymes. Based on results of calibration study for N-acetylglycyl, we support that the glycyl radical observed for E.coli anaerobic ribonucleotide reductase (ARNR) has a fully extended conformation, which differs from those of the corresponding radicals in pyruvate formate-lyase (PFL) or benzylsuccinate synthase (BSS).
Die absoluten Konfigurationen von mehr als 20 neuartigen Naturstoffen und Syntheseprodukten mit unterschiedlichen Chiralitätselementen (stereogene Zentren, chirale Achsen und chirale Ebenen) wurden durch Vergleich ihrer experimentellen CD-Spektren mit den quantenchemisch berechneten der jeweils möglichen Stereoisomere aufgeklärt. Zur Simulation des molekularen CD kamen dabei semiempirische Verfahren (CNDO/S und OM2) und die zeitabhängige Dichtefunktionaltheorie (TDDFT) zum Einsatz.