Neste trabalho mostramos de que maneira o princípio variacional dependente do tempo pode ser usado para se estudar correções quânticas ao limite clássico, particularmente, no contexto do modelo de Lipkin SU(2). Mostramos que tais correções podem ser colocadas na forma Hamiltoniana, acoplando-se a dinâmica clássica um conjunto de variáveis associadas às flutuações quânticas, nos levando à uma dinâmica efetiva com o número de graus de liberdade dobrado em relação ao sistema clássico. Como conseqüência o comportamento caótico emerge. Mostramos que este caos semiquântico é o mecanismo através do qual o tunelamento se manifesta no espaço de fase. Mostramos que tais correções melhoram sistematicamente o resultado c1ássico, propondo um critério para quantificar esta melhora. / We show how the time dependent variational principle can be used to study quantum corrections to the classical limit, in particular of the SU(2) Lipkin Model. We show how much corrections can be cast in Hamiltonian form, coupling to the classical dynamics a set of variables associated to the quantum fluctuations. This leads to an effective dynamics which has the number of degrees of freedom doubled with respect to the classical system. As a consequence chaotic behavior emerges. We show that this semiquantal chaos is the mechanism through which tunneling is effected, and also, that these corrections systematically improve the classical results and propose some quantitative measure of this improvement.

We study the classical nonlinear dynamics and the quantum vibrational energy eigenstates of the molecule HOCl. The classical vibrational dynamics, at energies below the HO+Cl dissociation energy, contains several saddle-center and period doubling bifurcations. The saddle-center bifurcations are shown to be due to a 2:1, and at higher energies a 3:1, nonlinear resonance between bend and stretch motions in various periodic orbits. The sequence of bifurcations takes the system from nearly integrable at low energies to almost completely chaotic at energies near the HO+Cl dissociation energy. At energies above dissociation we study the chaotic scattering of the Cl atom off the HO dimer. This scattering is governed by a homoclinic tangle formed by the stable and unstable manifolds of a parabolic periodic orbit at infinity. We construct the first three segments of the homoclinic tangle in phase space and use scattering functions to investigate its higher-order structure. For the quantum system we use a discrete variable representation to efficiently calculate the Hamiltonian matrix. We find 365 even and 357 odd parity eigenstates with energies below the dissociation energy. By plotting the eigenstates in configuration space we show that almost every quantum eigenstate can be associated with one or more of the classical periodic orbits. The classical bifurcations that give rise to new periodic orbits are manifest quantum mechanically through the sudden appearance of new classes of eigenstates. Despite the high degree of chaos in the classical dynamics at energies near the dissociation energy most quantum eigenstates remain highly ordered with recognizable nodal patterns. We use R-matrix theory together with a discrete variable representation to calculate quasibound states with energies above the dissociation energy. We find quasibound states with lifetimes ranging over 5 orders of magnitude. Using configuration space plots and Husimi distributions we show that the long-lived quasibound states are supported by unstable periodic orbits in the classical dynamics and medium-lived quasibound states are spread throughout the chaotic region of the classical phase space. Short-lived quasibound states show some similarity to unstable periodic orbits that stretch along the dissociation channel. / text

Neste trabalho mostramos de que maneira o princípio variacional dependente do tempo pode ser usado para se estudar correções quânticas ao limite clássico, particularmente, no contexto do modelo de Lipkin SU(2). Mostramos que tais correções podem ser colocadas na forma Hamiltoniana, acoplando-se a dinâmica clássica um conjunto de variáveis associadas às flutuações quânticas, nos levando à uma dinâmica efetiva com o número de graus de liberdade dobrado em relação ao sistema clássico. Como conseqüência o comportamento caótico emerge. Mostramos que este caos semiquântico é o mecanismo através do qual o tunelamento se manifesta no espaço de fase. Mostramos que tais correções melhoram sistematicamente o resultado c1ássico, propondo um critério para quantificar esta melhora. / We show how the time dependent variational principle can be used to study quantum corrections to the classical limit, in particular of the SU(2) Lipkin Model. We show how much corrections can be cast in Hamiltonian form, coupling to the classical dynamics a set of variables associated to the quantum fluctuations. This leads to an effective dynamics which has the number of degrees of freedom doubled with respect to the classical system. As a consequence chaotic behavior emerges. We show that this semiquantal chaos is the mechanism through which tunneling is effected, and also, that these corrections systematically improve the classical results and propose some quantitative measure of this improvement.

Nowadays, reactions on surfaces are attaining great scientific interest because of their diverse applications. Some well known examples are production of ammonia on metal surfaces for fertilizers and reduction of poisonous gases from automobiles using catalytic converters. More recently, also photoinduced reactions at surfaces, useful, textit{e.g.}, for photocatalysis, were studied in detail. Often, very short laser pulses are used for this purpose. Some of these reactions are occurring on femtosecond (1 fs=$10^{-15}$ s) time scales since the motion of atoms (which leads to bond breaking and new bond formation) belongs to this time range. This thesis investigates the femtosecond laser induced associative photodesorption of hydrogen, H$_2$, and deuterium, D$_2$, from a ruthenium metal surface. Many interesting features of this reaction were explored by experimentalists: (i) a huge isotope effect in the desorption probability of H$_2$ and D$_2$, (ii) the desorption yield increases non-linearly with the applied visible (vis) laser fluence, and (iii) unequal energy partitioning to different degrees of freedom. These peculiarities are due to the fact that an ultrashort vis pulse creates hot electrons in the metal. These hot electrons then transfer energy to adsorbate vibrations which leads to desorption. In fact, adsorbate vibrations are strongly coupled to metal electrons, textit{i.e.}, through non-adiabatic couplings. This means that, surfaces introduce additional channels for energy exchange which makes the control of surface reactions more difficult than the control of reactions in the gas phase. In fact, the quantum yield of surface photochemical reactions is often notoriously small. One of the goals of the present thesis is to suggest, on the basis of theoretical simulations, strategies to control/enhance the photodesorption yield of H$_2$ and D$_2$ from Ru(0001). For this purpose, we suggest a textit{hybrid scheme} to control the reaction, where the adsorbate vibrations are initially excited by an infrared (IR) pulse, prior to the vis pulse. Both textit{adiabatic} and textit{non-adiabatic} representations for photoinduced desorption problems are employed here. The textit{adiabatic} representation is realized within the classical picture using Molecular Dynamics (MD) with electronic frictions. In a quantum mechanical description, textit{non-adiabatic} representations are employed within open-system density matrix theory. The time evolution of the desorption process is studied using a two-mode reduced dimensionality model with one vibrational coordinate and one translational coordinate of the adsorbate. The ground and excited electronic state potentials, and dipole function for the IR excitation are taken from first principles. The IR driven vibrational excitation of adsorbate modes with moderate efficiency is achieved by (modified) $pi$-pulses or/and optimal control theory. The fluence dependence of the desorption reaction is computed by including the electronic temperature of the metal calculated from the two-temperature model. Here, our theoretical results show a good agreement with experimental and previous theoretical findings. We then employed the IR+vis strategy in both models. Here, we found that vibrational excitation indeed promotes the desorption of hydrogen and deuterium. To summarize, we conclude that photocontrol of this surface reaction can be achieved by our IR+vis scheme. / Heutzutage werden Reaktionen auf Oberflächen wegen ihrer vielfältigen Anwendungen intensiv untersucht. Einige der bekannten Beispiele sind die Herstellung von Ammoniak auf Metalloberflächen für die Kunstdüngerproduktion und die Reduktion giftiger Abgase in Autokatalysatoren. In letzter Zeit wurden auch photoinduzierte Reaktionen an Oberflächen eingehender untersucht, die z.B. für die Photokatalyse verwandt werden können. Häufig werden in diesen Untersuchungen sehr kurze Laserpulse benutzt. Einige der Reaktionen finden auf einer Femtosekunden-Zeitskala mbox{(1 fs =10$^{-15}$ s)} statt, da die Bewegungen einzelner Atome in derart kurzen Zeitspannen ablaufen (durch die der Bindungsbruch und das Knüpfen neuer Bindungen verursacht wird). Diese Arbeit untersucht die femtosekunden-laserinduzierte assoziative Photodesorption von Wasserstoff, H$_2$, und Deuterium, D$_2$, von einer Rutheniumoberfläche. Viele interessante Eigenschaften dieser Reaktion wurden in Experimenten entdeckt: (i) ein großer Isotopeneffekt in der Desorptionswahrscheinlichkeit von H$_2$ und D$_2$, (ii) die Desorptionsausbeute steigt nicht-linear mit der (vis) Laserfluenz an und (iii) eine Nicht-Gleichverteilung der Energie auf die einzelnen Freiheitsgrade. Diese Auffälligkeiten sind durch den Umstand verursacht, dass der ultrakurze vis-Laserpuls heiße Elektronen im Metall erzeugt. Die heißen Elektronen transferieren dann Energie in die Schwingungen des Adsorbats, was zur Desorption führt. Tatsächlich sind die Adsorbatschwingungen stark an die Elektronen gekoppelt, nämlich durch nicht-adiabatische Kopplungen. Dies bedeutet, dass durch Oberflächen neue Kanäle für den Energietransfer geöffnet werden, was die Kontrolle von Oberflächenreaktionen im Vergleich zu solchen in der Gasphase erschwert. In der Tat sind die Quantenausbeuten von photochemischen Oberflächenreaktionen bekannterweise klein. Eines der Ziele in der vorliegenden Arbeit ist es auf der Basis von theoretischen Simulationen Strategien vorzuschlagen, um die Photodesorptionsausbeute von H$_2$ und D$_2$ von Ru(0001) zu kontrollieren bzw. zu verbessern. Zu diesem Zweck schlagen wir ein gemischtes Kontrollschema für die Reaktion vor, bei dem zunächst die Adsorbatschwingungen vor dem vis-Puls durch einen infraroten (IR) Puls angeregt werden. Sowohl adiabatische als auch nicht-adiabatische Repräsentationen für photoinduzierte Desorptionsprozesse werden dabei benutzt. Die adiabatische Repräsentation ist in klassischen Molekulardynamik-Simulationen mit elektronischer Reibung verwirklicht. In einer quantenmechanischen Beschreibung werden nicht-adiabatische Repräsentationen innerhalb der Dichtematrixtheorie für offene Quantensysteme verwandt. Die zeitliche Entwicklung des Desorptionsprozesses wird in einem Zwei-Modenmodell reduzierter Dimensionalität mit einer Schwingungs- und einer Translationskoordinate des Adsorbats beschrieben. Die Potentiale für den elektronische Grundzustand und den angeregten Zustand sind abgeleitet aus quantenchemischen Rechnungen (textsl{first principles}). Die IR-getriebene Schwingungsanregung der Adsorbatmoden mit moderatem Wirkungsgrad wird mit (modifizierten) $pi$-Pulsen und/oder der Theorie der optimalen Kontrolle erreicht. Die Abhängigkeit der Desorption von der Fluenz wird mit Hilfe der elektronischen Temperatur des Metalls berechnet, welche im Rahmen des Zwei-Temperatur-Modells bestimmt wird. Dabei weisen unsere Ergebnisse eine gute Übereinstimmung mit experimentellen und früheren theoretischen Arbeiten auf. Daraufhin wandten wir die IR+vis Strategie in beiden Modellen an. Dadurch konnten wir zeigen, dass Schwingungsanregung in der Tat die Desorption von Wasserstoff und Deuterium begünstigt. Zusammenfassend stellen wir fest, dass die Photokontrolle dieser Oberflächenreaktion durch unser IR+vis Schema erreichbar ist.

Les travaux présentés dans cette thèse peuvent être divisés en deux parties. Dans une première partie, nous avons étudié le processus de photoisomérisation dans plusieurs systèmes. Une analyse de structure électronique accompagnée d’un calcul préliminaire de dynamique semi-classique ont été appliqué à un modèle minimal du rétinal afin d’extraire les degrés de libertés les plus importants lors de l’isomérisation. Cela dans le but de construire des surfaces d’énergie potentielle diabatiques pour effectuer une étude de dynamique quantique. Une approche de type dynamique semi-classique a été appliquée à un modèle de moteur moléculaire dans le but d’étudier l’origine de l’uni-directionalité de sa rotation. Finalement, une étude de structure électronique d’un complexe de Rhénium contenant un ligand de type rétinal a été effectué pour étudier l’influence du métal sur la spectroscopie du ligand rétinal. Dans une deuxième partie nous nous sommes intéressés à l’étude des croisements intersystème dans un complexe de Rhénium. Afin de pouvoir apporter une explication à un comportement contrintuitif de ce complexe, nous avons développé un Hamiltonien modèle capable de tenir compte des couplages vibroniques interétats et spin-orbit. Cet Hamiltonien a été testé sur ce-dit système, et nous a permis, grâce à une étude de structure électronique de proposer un mécanisme de relaxation différent de celui proposé expérimentalement. / This thesis can be divided in two parts.In the first one, we have studied the photoisomerization process in several systems. An electronic structure analysis mixed with a preliminary semi-classical dynamics investigation has been applied to a minimal model of the retinal chromophore in order to select the most important degrees of freedom involved in the process. The goal of this is to build diabatic potential energy surfaces in order to conduct quantum dynamics simulations. A semi-classical approach has also been applied to a molecular motor model to study the origin of the unidirectionality of its rotary motion. Finally, an electronic structure of a rhenium complex with a retinal-like ligand has been performed to study the effect of the coordination to a metallic atom on the spectroscopy of the retinal ligand. In the second part, we have investigated the intersystem crossings in a rhenium complex. In order to bring an explanation to an experimentally observed conterintuitive behavior of this complex, we have developed a model Hamiltonian that includes both interstate vibronic coupling and spin-orbit coupling. This Hamiltonian has been tested on the said complex and, in complement to an electronic structure study, allowed us to formulate a decay mechanism different from the one proposed based on experiments.

Ce mémoire étudie le modèle quantique d’Ising-Kawasaki en une dimension. Cette chaîne quantique de spin-1/2 décrit la dynamique de Kawasaki hors d’équilibre d’une chaîne d’Ising classique couplée à un bain thermique. L’Hamiltonien est obtenu pour le cas général désor- donné avec des couplages d’Ising et champs magnétiques non-uniformes. Quand les champs magnétiques sont nuls, la chaîne de spin quantique est stochastique, et dépend des couplages d’Ising normalisés par la température du bain de chaleur. Dans le cas de couplages uniformes, nous donnons les états fondamentaux exacts de la chaîne de spin, ainsi que ses excitations à 1-magnon. Les solutions pour les spectres à deux magnons sont dérivées via une variante de l’Ansatz de Bethe. Dans le régime antiferromagnétique, les états de branche à deux magnons présentent un comportement complexe, notamment en ce qui concerne l’hybridation avec le continuum. L’analyse faite dans ce mémoire, combinée aux études précédentes, suggère que le système manifeste des dynamiques multiples à basse énergie, comme le montre la présence de plusieurs exposants critiques dynamiques. La distribution de l’espacement de l’ensemble des niveaux d’énergie est évaluée en fonction du couplage d’Ising. On conclut que le sys- tème est non-intégrable pour des paramètres génériques, ou de manière équivalente, que la dynamique classique hors équilibre correspondante est ergodique. / We study a quantum spin-1/2 chain that is dual to the non-equilibrium Kawasaki dynamics of a classical Ising chain coupled to a thermal bath. The Hamiltonian is obtained for the general disordered case with non-uniform Ising couplings. The quantum spin chain is stoquastic, and depends on the Ising couplings normalized by the bath’s temperature. Proceeding with uniform couplings, we give the exact groundstates of the gapless spin chain, as well as its single-magnon excitations. Solutions for the two-magnon spectra are derived via a Bethe Ansatz scheme. In the antiferromagnetic regime, the two-magnon branch states show intricate behavior, especially regarding hybridization with the continuum. Our analysis, when combined with previous studies, suggests that the system hosts multiple dynamics at low energy as seen via the presence of multiple dynamical critical exponents. Finally, we analyze the full energy level spacing distribution as a function of the Ising coupling. We conclude that the system is non-integrable for generic parameters, or equivalently, that the corresponding non-equilibrium classical dynamics are ergodic.