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Molecular Symmetry, Super-Rotation, and Semi-Classical Motion – Molecular symmetry, super-rotation, and semi-classical motion – New ideas for old problems Inaugural - Dissertation zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fakultät der Universität zu Köln vorgelegt von Hanno Schmiedt aus Köln Berichterstatter: Prof. Dr. Stephan Schlemmer Prof. Per Jensen, Ph.D. Tag der mündlichen Prüfung: 24. Januar 2017 Awesome molecules need awesome theories DR.SANDRA BRUENKEN, 2016 − Abstract Traditional molecular spectroscopy is used to characterize molecules by their structural and dynamical properties. Furthermore, modern experimental methods are capable of determining state-dependent chemical reaction rates. The understanding of both inter- and intra-molecular dynamics contributes exceptionally, for example, to the search for molecules in interstellar space, where observed spectra and reaction rates help in under- standing the various phases of stellar or planetary evolution. Customary theoretical models for molecular dynamics are based on a few fundamental assumptions like the commonly known ball-and-stick picture of molecular structure. In this work we discuss two examples of the limits of conventional molecular theory: Extremely floppy molecules where no equilibrium geometric structure is definable and molecules exhibiting highly excited rotational states, where the large angular momentum poses considerable challenges to quantum chemical calculations. In both cases, we develop new concepts based on fundamental symmetry considerations to overcome the respective limits of quantum chemistry. For extremely floppy molecules where numerous large-amplitude motions render a defini- tion of a fixed geometric structure impossible, we establish a fundamentally new zero-order description. The ‘super-rotor model’ is based on a five-dimensional rigid rotor treatment depending only on a single adjustable parameter. The respective quantum numbers define a generalized angular momentum which collectively describe internal and overall rotational motion. It consistently predicts the symmetry and energies of (most of) the low-energy + states of the prototype of floppy molecules, protonated methane, CH5 , which were only recently found experimentally. These exciting results suggest that we have opened a new avenue towards a more general understanding of internal large-amplitude motions. For highly excited rotational states, we show a path-integral based semi-classical treatment to be capable of predicting energies and symmetries of quantized states even without sophisticated matrix diagonalization tools. In a proof-of-principle study of sulfur dioxide, we show this analysis to agree with advanced quantum chemical calculations. This encourages a further use of this relatively fast method particularly for molecules where quantum calculations become exceedingly expensive. In addition, we developed a straightforward and mathematically consistent method to determine the nuclear spin permutation and rotational symmetry of molecules consisting of an arbitrary number of identical nuclei with any nuclear spin quantum number. We apply this method to formulate tentative symmetry-based selection rules and statistical state-to-state reaction rates for particular reactive collisions. These reaction rates are calculated independently of internal dynamics, and hence imply the symmetry of reacting, intermediately formed, and final molecular complexes to be an essential ingredient for determining reaction rates. In summary, we show that fundamental symmetry considerations and semi-classical models contribute considerably to the understanding and extension of traditional concepts of molecular theory. Zusammenfassung Traditionelle Molekülspektroskopie wird benutzt, um Moleküle anhand dynamischer und struktureller Eigenschaften zu charakterisieren. Mit modernen experimentellen Methoden ist es außerdem möglich, chemische Reaktionsraten abhängig von einzelnen quanten- mechanischen Zuständen zu bestimmen. Ein Verständnis der internen Dynamik, sowohl einzelner als auch wechselwirkender Moleküle, trägt beispielsweise dazu bei, Moleküle im interstellaren Raum zu identifizieren. Spektroskopische Beobachtungen und Reaktion- sraten helfen insbesondere bei der Untersuchung der verschiedenen Phasen der Stern- und Planetenentstehung. Die theoretische Beschreibung von Molekülen basiert auf einigen wenigen fundamentalen Annahmen wie dem Kugel-Stab-Modell molekularer Struktur. In dieser Arbeit werden zwei Beispiele der Grenzen dieser Beschreibung diskutiert: Extrem elastische Moleküle, bei denen keine feste Gleichgewichtsstruktur definiert werden kann und Moleküle, die so schnell rotieren, dass aufgrund der hohen Drehimpulsquantenzahl quantenmechanische Rechnungen extrem aufwendig sind. In beiden Fällen werden neue Konzepte entwick- elt, die auf fundamentalen Symmetrieüberlegungen basieren und eben jene Grenzen überwinden. Für extrem elastische Moleküle, bei denen die Vielzahl von Großamplitudenbewegungen eine Definition molekularer Struktur unmöglich machen, wird in dieser Arbeit eine funda- mental neue Beschreibung nullter Ordnung entwickelt. Das ‘super-rotor Modell’ basiert auf einem fünfdimensionalen starren Rotors, hängt trotzdem von nur einem einzelnen freien Parameter ab und vereinigt interne und umfassende Rotation in Molekülen. Als erste Anwendung werden Rechnungen für den Prototyp der elastischen Moleküle, protoniertes + Methan, CH5 , vorgestellt. Es zeigt sich, dass dieses Modell sowohl die Energien als auch die Symmetrien der Zustände in sehr guter Übereinstimmung mit jüngsten experimentellen Ergebnissen vorhersagt. Mit dieser Theorie wird der Weg zu einem neuen Verständnis interner molekularer Bewegung geebnet. Es wird außerdem eine semi-klassische Methode zur Beschreibung sehr schnell rotierender Moleküle beschrieben. Sie basiert auf dem Pfadintegralformalismus und sagt sowohl quantisierte Energien als auch Symmetrien von Rotationszuständen voraus, ohne auf anspruchsvolle Techniken zur Matrixdiagonalisierung zurückgreifen zu müssen. Als erste Machbarkeitsstudie werden Rechnungen zu Schwefeldioxid durchgeführt, die gut mit quantenchemischen Rechnungen übereinstimmen. Auch diese Ergebnisse deuten darauf hin, dass diese Methode vielversprechend ist, insbesondere bei Molekülen wo jene quantenchemischen Rechnungen ihre natürlichen (numerischen) Grenzen erreichen. Darüber hinaus wird eine mathematisch konsistente Methode entwickelt, um die Sym- metrie von Kernspinzuständen von Molekülen, unbhängig von Größe und individuellem Kernspin, zu bestimmen. Es werden damit erste zustandsabhängige Auswahlregeln für chemische Reaktionen formuliert, die insbesondere die Symmetrie der beteiligten Moleküle berücksichtigen. Da sie vollständig energieunabhängig berechnet sind, zeigt sich, dass die Molekülsymmetrie eine wesentliche Rolle bei diesen Reaktionen spielt. Zusammenfassend zeigt die vorliegende Arbeit, dass Symmetrieüberlegungen und semi- klassische Ansätze die traditionellen Konzepte der Molekülphysik beträchtlich erweitern. Contents Abstract/Zusammenfassung ........................v List of figures ......................................... ix List of tables .......................................... xi New ideas to old problems – An introduction .................1 Part I Group theory in molecular physics 1 Basic concepts ......................................5 2 Schur-Weyl duality in molecules .................. 21 3 Reactive collisions ................................. 37 Part II Extremely floppy molecules 1 Introducing extreme floppiness ................... 57 2 Symmetry beyond perturbation theory .......... 61 3 The molecular super-rotor ......................... 77 4 Super-rotor states and their symmetry ........... 85 5 Protonated methane ............................... 95 6 Refinements and further applications ........... 105 Part III Semi-classical approach to rotational dynamics 1 Ultrafast rotation ................................... 115 2 Application to sulfur dioxide ..................... 133 3 Discussion .......................................... 139 New ideas to old problems – A conclusion ................. 143 Appendix The semi-classical approach ................................ 147 Bibliography ................................................... 149 Danksagung ................................................... 157 Erklärung und Teilpublikationen ............................. 159 Lebenslauf ..................................................... 161 List of Figures I Group theory in molecular physics 3 1.1 Potential energy for the torsional motion of the methyl group in nitromethane....................................... 14 1.2 Effect of tunnel splitting on the energy term diagram and the according state-to-state transitions................. 15 2.1 Graphical view on mirroring nuclear spin states in terms of Young diagrams....................................... 22 + 3.1 Two distinct reaction pathways for the reaction of H3 and H2..................................................... 38 3.2 Probabilities for state-to-state transitions in the reaction H2 + H+ H+ + H ........................................... 43 ! 2 3.3 State-to-state reaction in the H + + H H + H + reaction. 45 3 2! 2 3 II Extremely floppy molecules 55 2.1 Root vector diagrams of so(3) and so(5)................ 64 2.2 Set of subgroups of the group of rotations in three dimen- sions.................................................. 68 2.3 Pictorial mapping of the permutation-inversion elements of the molecular symmetry group to a subgroup of the group of
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