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Quantum Dynamics on Potential Energy Surfaces

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Quantum Dynamics on Potential Energy Surfaces Technische Universität München Zentrum Mathematik Wissenschaftliches Rechnen Quantum Dynamics on Potential Energy Surfaces Simpler States and Simpler Dynamics Johannes Friedrich Keller Vollständiger Abdruck der von der Fakultät für Mathematik der Technischen Universität München zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigten Dissertation. Vorsitzender: Univ.-Prof. Dr. Herbert Spohn Prüfer der Dissertation: 1. Univ.-Prof. Dr. Caroline Lasser 2. Prof. George A. Hagedorn, Ph.D., Virginia Polytechnic Institute and State University, USA (nur schriftliche Beurteilung) 3. Univ.-Prof. Dr. Christian Lubich, Eberhard Karls Universität Tübingen Die Dissertation wurde am 14.07.2015 bei der Technischen Universität München eingereicht und durch die Fakultät für Mathematik am 25.09.2015 angenommen. ii Abstract In this dissertation we analyze and simplify wave functions and observables in the context of quantum molecular dynamics. The two main topics we discuss are the structure of Hagedorn wave packets in position and phase space, and semiclassical approximations for the propagation of quantum expectations with nonnegative phase space densities. We provide algorithmic discretizations for these approximations and illustrate their validity and applicability by means of numerical experiments. Zusammenfassung In dieser Dissertation analysieren und vereinfachen wir Wellen- funktionen und Observablen im Kontext der molekularen Quan- tendynamik. Die zwei zentralen, von uns diskutierten Themen sind die Struktur der Hagedornschen Wellenpakete im Orts- und Phasenraum sowie semiklassische Approximationen für die Evolu- tion von Erwartungswerten mit nichtnegativen Phasenraumdichten. Wir präsentieren algorithmische Diskretisierungen für diese Ap- proximationen und illustrieren deren Anwendbarkeit anhand von numerischen Experimenten. iii iv Contents Acknowledgments 1 Introduction 3 I Foundations 9 1 Quantum Systems . .9 2 Hamiltonian Dynamics . 18 3 Potential Energy Surfaces . 28 4 Hellmann-Feynman Theory . 41 5 Weyl Correspondence . 49 Interlude – Simpler States and Simpler Dynamics 59 II Wave Packet Analysis 61 6 Multivariate Polynomials of Hermite Type . 62 7 Hagedorn Wave Packets . 80 8 Wigner-Hagedorn Formula . 94 9 Excursus: The Hagedorn Semigroup . 110 10 Anti-Wick Operators and Spectrograms . 114 III Semiclassical Propagation 133 11 Egorov’s Theorem . 135 12 Propagation with Nonnegative Densities . 141 13 Wave Packet Dynamics . 150 v Contents 14 Ehrenfest Time Scales . 159 15 Localization of Wigner Functions . 165 IV Algorithms and Applications 177 16 Discretization . 178 17 Numerical Experiments . 191 Summary and Outlook 203 Appendix 207 A Notation . 207 B Symbol Classes . 209 C Fundamental Subsets . 210 D L1 Estimates for Hermite cross-Wigner Functions . 215 E Reference Solutions . 218 Bibliography 221 vi Acknowledgments First and foremost, I want to thank my advisor Caroline Lasser for her constant support and strong advice over the last years. With her guidance and encouragement she helped me to become an active member of the mathematical research community. I am grateful to George Hagedorn for sharing his mathematical insights with me on many occasions. It is a great pleasure to thank him and Christian Lubich for the interest they show in my work by writing a report for this thesis. Special thanks go to Helge Dietert, Tomoki Ohsawa, and Stephanie Troppmann for collaborating with me on the analysis of wave pack- ets and the spectrogram approximation, and for many fruitful discussions. I would like to thank Folkmar Bornemann for introducing me to the world of mathematical physics, and Brynjulf Owren for his constant interest in my work as well as for guiding me towards geometric integration. During my work for this dissertation I greatly benefitted from valuable discussions with Dario Bambusi, Semyon Dyatlov, Maurice de Gosson, Lysianne Hari, Roman Schu- bert, Giulio Trigila, and many others. I particularly enjoyed the continuing exchange with my collegues and friends Wolfgang Gaim, Ilja Klebanov, David Sattlegger, Stephanie Troppmann, and Thomas Weiß. I am very grateful to them, and to all members of M3 for creating such a great working atmosphere. In the spring of 2013 I spent three months at the University of California in Berkeley. I would like to express my gratitude towards Maciej Zworski for his warm hospitality and many inspiring con- versations. He made my stay not only scientifically beneficial, but 1 also great fun. During my work on this thesis I have been employed by the SFB Transregio 109 “Discretization in Geometry and Dynamics” of the Deutsche Forschungsgemeinschaft (DFG). I would like to thank the DFG, the TopMath program at the Elite Network of Bavaria, and the TopMath graduate center of the TUM Graduate School for their support. Last but not least, I am very grateful to my parents for their encouragement and unconditional support. 2 Introduction “The underlying physical laws necessary for the mathematical theory of large parts of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complex to be solved. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be de- veloped, which can lead to an explanation of the main features of complex atomic systems without too much computation.” Paul A. M. Dirac in [Dir29] As the above quotation of Dirac from 1929 shows, scientists were already very aware of the immense complexity of quantum mechanical systems well before the advent of computers and the corresponding possibility to run numerical simulations. Since those very first days of modern quantum mechanics, constructing approx- imations that can be used for simulating the dynamics of actual molecules is one of the prevailing aims of chemical physics, com- putational chemistry, and mathematical quantum dynamics. The central equation of quantum molecular dynamics in the Born- Oppenheimer approximation is the time-dependent semiclassical Schrödinger equation # #2 # # # 2 d i#¶ty (t) = − 2 D + V y (t), y (0) = y0 2 L (R ). 3 Introduction It governs the evolution of a wave function y#(t) that represents the state of the molecule’s nuclei on the potential energy surface V : Rd ! R. The small parameter 0 < # 1 arises due to the fact that nuclei are heavy in comparison with electrons, and causes solu- tions y#(t) : Rd ! C to be highly oscillatory in both space and time. This property, together with the high dimensionality of the nuclear configuration space Rd, imposes severe difficulties for computa- tions, and makes it clear that the semiclassical Schrödinger equation is in general “too complex to be solved”. If one even wants to simu- late realistic model systems “without too much computation”, the development of powerful approximations is indispensable. The central aim of this thesis is to pursue and interrelate two lines of simplifications for molecular quantum systems in the spirit of the agenda suggested by Dirac. Firstly, we investigate the structure of Hagedorn wave packets in position and phase space by making a detailed analysis of their polynomial prefactors. The Hagedorn wave packets form a particu- larly simple yet versatile class of highly localized wave functions. They have proven to be well-suited for the semiclassical analysis of quantum systems as well as for the construction of numerical methods for the time-dependent Schrödinger equation. Our main contribution to the analysis of Hagedorn wave packets is to show that their phase space representation via Wigner functions is natural in the sense that it again yields a Hagedorn wave packet in doubled dimension. Moreover, we are able to provide an explanation for the uniform factorization of their Wigner functions in phase space, which has recently been discovered. Secondly, we discuss various semiclassical approximations for the quantum evolution of expectation values and wave functions. Our focus lies on approximate methods for quantum expectations that simultaneously allow for both rigorous error estimates and ap- plicable algorithmic discretizations. We present novel phase space methods for the computation of quantum expectations that utilize 4 Quantum Dynamics on Potential Energy Surfaces smooth probability densities for the representation of the initial state. One of the main approaches we would like to promote with this thesis is the simultaneous semiclassical approximation of states and observables. An example for this beneficial combination of approximations is provided by our proof of a local Egorov theorem for long Ehrenfest times. The two main themes of this dissertation are supported and enframed by a number of side stages. For instance, we present a rigorous proof for the Hellmann-Feynman theorem which is widely applied in chemistry in order to compute gradients of potential energy surfaces. Evaluating these gradients, in turn, is necessary for the computation of the classical trajectories utilized by semiclassical approximations. Outline of the dissertation The main corpus of the dissertation consists of 17 chapters that are arranged into four parts. Many of the chapters come along with their own introductory remarks that include a placement of their contents within the literature. The first part §I is dedicated to a review and exploration of the foundations of molecular quantum dynamics and potential energy surfaces. It motivates the investigations in
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