Quantendynamik photochemischer Modellreaktionen entlang konischer Durchschneidungen

This work deals with quantum chemistry and quantum dynamics of cis-trans isomerization via conical intersections for three model systems of reduced dimensionality. The focus is on obtaining clear pictures of the quantum dynamics in the excited electronic state using fulvene, 4-(methyl- cyclohexylidene)-fluoromethane as model systems. The effect of solvent polarity on the conical intersection is also investigated using 4-cyclopentadienylidene-1,4-dihydropyridine as model system. Ab initio potential energy surfaces are built for the ground and first excited electronic states of the molecules 4-(methyl-cyclohexylidene)-fluoromethane and 4-cyclopentadienylidene-1,4-dihydropyridine. For fulvene , model potential energy surfaces are generated based on analysis of the corresponding ab initio potential energy surfaces. To investigate the dynamics of the isomerization process, the nuclear wave packets evolving on one or two dimensional coupled potential energy surfaces are simulated for the fulvene and the 4-(methyl- cyclohexylidene)-fluoromethane. For this purpose, the original adiabatic potential energy surfaces with kinetic couplings are transformed to the diabatic ones with potential couplings. To investigate the effect of laser pulses on the dynamics of the isomerization process, peliminary simulations based on one dimensional ab inito potentials and dipole functions of the 4 -(methyl-cyclohexylidene)-fluoromethane model system are carried out. It is found that for fulvene, radiationless decay due to vibrations along the symmetric allylic stretch is faster than the radiationless decay along the torsional coordinate. For 4-(methyl-cyclohexylidene)-fluoromethane, it is found that torsional/rotational motions can be conserved in the excited as well as in the ground state. Furthermore, the model 4-cyclopentadienylidene-1,4-dihydropyridine, the conical intersection between the ground and first excited electronic shifts in non-polar solvents, whereas in polar solvents the degeneracy is lifted.Die vorliegende Dissertation beschäftigt sich mit der Quantenchemie dreier Modellmoleküle und deren Quantendynamik bei Cis-trans-Isomerisierungen, die über eine konische Durchschneidung verlaufen. Als Modellsysteme für die Untersuchung der Quantendynamik im angeregten elektronischen Zustand dienen die Verbindungen Fulven sowie das 4-(Methyl-cyclohexyliden)-fluoromethan und deren Bewegungen entlang ausgewählter Koordinaten. Anhand des Moleküls 4-Cyclopentadienylidene-1,4-dihydropyridin wird der Einfluss der Polarität des Lösungsmittels auf die konische Durchschneidung aufgezeigt. Ab initio Potenzialenergieflächen werden für den Grundzustand und den ersten angeregten elektronischen Zustand für die Modellverbindungen 4-(Methyl- cyclohexyliden)-fluoromethan and 4-Cyclopentadienylidene-1,4-dihydropyridine berechnet, während für das Fulvenmolekül Modellpotenziale genutzt werden, die auf Ab initio Daten beruhen. Um die Dynamik der Isomerisierung zu erforschen, werden die Bewegungen der Kernwellenpakete des Fulvens sowie des 4-(Methyl- cyclohexyliden)-fluoromethans auf ein- oder zweidimensionalen Potenzialflächen simuliert. Hierfür werden die erhaltenen adiabatischen Potenziale und deren kinetische Kopplungen untereinander in diabatische Potenziale mit Potenzialkopplungen umgewandelt. Der Einfluss der Laseranregung auf die Dynamik der Molekülisomerisierung entlang eines eindimensionalen Reaktionspfades wird am Beispiel des 4-(Methyl-cyclohexyliden)-fluoromethans berechnet. Beim Fulven erfolgt der strahlungslose Zerfall aus dem angeregten elektronischen Zustand durch die symmetrische allylische Streckschwingung des Cyclopentadiengerüsts schneller als über die Torsion entlang der C=C Doppelbindung. Beim 4-(Methyl-cyclohexyliden)-fluoromethan wird gezeigt, wie eine Rotations-Torsions-Bewegung sowohl im Grundzustand als auch im angeregten Zustand erhalten bleiben kann. Die konische Durchschneidung zwischen dem Grundzustand und dem ersten angeregten elektronischen Zustand des 4-Cyclopentadienyliden-1,4-dihydropyridins wird lokalisiert und deren Verschiebung bzw. Aufhebung unter dem Einfluss unpolarer und polarer Lösungsmittel herausgestellt

3nj Morphogenesis and Semiclassical Disentangling

Recoupling coefficients (3nj symbols) are unitary transformations between binary coupled eigenstates of N=(n+1) mutually commuting SU(2) angular momentum operators. They have been used in a variety of applications in spectroscopy, quantum chemistry and nuclear physics and quite recently also in quantum gravity and quantum computing. These coefficients, naturally associated to cubic Yutsis graphs, share a number of intriguing combinatorial, algebraic, and analytical features that make them fashinating objects to be studied on their own. In this paper we develop a bottom--up, systematic procedure for the generation of 3nj from 3(n-1)j diagrams by resorting to diagrammatical and algebraic methods. We provide also a novel approach to the problem of classifying various regimes of semiclassical expansions of 3nj coefficients (asymptotic disentangling of 3nj diagrams) for n > 2 by means of combinatorial, analytical and numerical tools

The Geometric Maximum Traveling Salesman Problem

We consider the traveling salesman problem when the cities are points in R^d for some fixed d and distances are computed according to geometric distances, determined by some norm. We show that for any polyhedral norm, the problem of finding a tour of maximum length can be solved in polynomial time. If arithmetic operations are assumed to take unit time, our algorithms run in time O(n^{f-2} log n), where f is the number of facets of the polyhedron determining the polyhedral norm. Thus for example we have O(n^2 log n) algorithms for the cases of points in the plane under the Rectilinear and Sup norms. This is in contrast to the fact that finding a minimum length tour in each case is NP-hard. Our approach can be extended to the more general case of quasi-norms with not necessarily symmetric unit ball, where we get a complexity of O(n^{2f-2} log n). For the special case of two-dimensional metrics with f=4 (which includes the Rectilinear and Sup norms), we present a simple algorithm with O(n) running time. The algorithm does not use any indirect addressing, so its running time remains valid even in comparison based models in which sorting requires Omega(n \log n) time. The basic mechanism of the algorithm provides some intuition on why polyhedral norms allow fast algorithms. Complementing the results on simplicity for polyhedral norms, we prove that for the case of Euclidean distances in R^d for d>2, the Maximum TSP is NP-hard. This sheds new light on the well-studied difficulties of Euclidean distances.Comment: 24 pages, 6 figures; revised to appear in Journal of the ACM. (clarified some minor points, fixed typos