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Dynamical Symmetry Breaking in Molecules and Molecular Dynamical Symmetry Breaking in Molecules and Molecular Aggregates by Giinter Maximilian Schmid Diplomchemiker, Technische Universitat Miinchen, 1991 Submitted to the Department of Chemistry in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 1994 © Massachusetts Institute of Technology 1994. All rights reserved. Author .................................. August 12, 1994 Certified by ........... ............... .......... ~obert J. Silbey Professor of Chemistry Thesis Supervisor Accepted by...... ........C ' ............ Dietinar Seyferth Chairman, Departmental Committee on Graduate Students MASSACHUSETTS INSTITUTE OF TFCHPJnl Ony OICT 051994 LIBRARIES ArsniTW This doctoral thesis has been examined by a Committee of the De- partment of Chemistry as follows: Professor Irwin Oppenheim ............... p. ................... (Chairman) Professor Robert J. Silbey ... ,........ Thesis Supervis/ Professor Robert W . Field . ·......... ........................... 2 Dynamical Symmetry Breaking in Molecules and Molecular Aggregates by Giinter Maximilian Schmid Submitted to the Department of Chemistry on August 12, 1994, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract In this thesis we explain dynamical symmetry breaking as an important and practically rel- evant mechanism of localizing excitation in molecules and molecular aggregates. Explicitly treated are the normal to local mode transition in triatomic molecules AB 2 of symmetry C2v, modelled by two harmonically coupled anharmonic oscillators, and the self-trapping transition of an excess charge in a molecular crystal dimer with site exchange symmetry, modelled by Holstein's molecular crystal. At these transitions, localization of excitation occurs in violation of symmetry requirements. The potentials that confine the excitations appear upon excitation and thus are called dynamic potentials. Therefore localization oc- curs under dynamical symmetry breaking. We show that all qualitative and quantitative aspects of dynamical symmetry breaking in these systems can be successfully discussed in the language of quasiparticles. For the triatomic molecules, the quasiparticle is referred to as "local mode" and consists of a bare vibrational excitation and an induced distortion of the molecular equilibrium configuration; for the molecular crystal dimer, the quasiparticle is known as "small polaron" and consists of an excess charge and an induced distortion of the crystal equilibrium configuration. On the basis of timescale arguments, we obtain the equations of motion for these quasiparticles, called Duffing's equation, and solve them analytically. In addition, we employ geometrical methods from classical mechanics to discuss various dynamical aspects. In the discussion of the triatomic molecules we put emphasis on the connection of classical, semiclassical and quantum mechanical modes of description. In particular, we for- mulate a stability criterion for eigenstates of the quantum system in terms of susceptibilities to symmetry breaking perturbations and show that parametric instability in the classical correspondent can be deduced in the classical limit. We discuss the practical relevance of the susceptibilities for the control of intramolecular dynamics by lasers. In the discussion of the molecular crystal we focus on the derivation of the equations of motion in two different "adiabatic" limits, which we distinguish as the standard adia- batic limit and the Duffing limit. We establish the connection to the commonly employed adiabatic approximation. Thesis Supervisor: Robert J. Silbey Title: Professor of Chemistry 3 Acknowledgments This thesis was presented in its final form with the help and advice of my mentors, friends and colleagues. First and foremost I thank my supervisor Bob Silbey for creating, with his unique ability to wittingly intertwine the professional and personal aspects of life, a very pleasant and stimulating working environment. He taught me to be open to ideas from different branches of science and boldness in borrowing from these. I owe to my teacher Irwin Oppenheim the valuable insight that apparently difficult problems are often "very easy". His early morning pat on the shoulder gave me the momentum to penetrate the sometimes very difficult issues of the day,some of which he presented in his immaculate lectures. Highly infectious in his incomparable enthusiasm for science is my teacher Bob Field, whose feel for scientific and social issues guided me through graduate student terrain. With his enormous skill and patience in explaining experimental and theoretical issues, as well as U.S.- American social culture, Steve Coy kept me going at decisive steps of our research. My friend Sophia Yaliraki read through the entire manuscript and made many valuable suggestions. For this and her good sense of humor in dealing with me over the last two years I thank her very much. My friends and house-mates Mauricio "Maybe" Barahona and Robert "Breyers" Grotzfeld always gave me cultivated com- pany and never shied away from a scientific argument. My friend and office-mate Luis Cruz always had at least one open ear for my worries of the day and played often the "advocatus diaboli" when I couldn't. My "older" friends David Chasman "the Caleian" , David Dab, Bih-Yah Jin, Alex Orsky and Alberto Suarez were always pa- tient with me and my "younger" friends Cliff Liu, Dave Reichmann, Welles Morgado and Joan Shea. I thank all of them. 4 Contents 1 Introduction 13 1.1 Motivation . .. 13 1.2 Basic notions. 14 1.2.1 Localized and delocalized excitations ........ 14 1.2.2 Transitions from delocalized to localized excitations . 16 1.3 Realizations of dynamical symmetry breaking ....... 21 1.3.1 Formation of a polaron ................ 21 1.3.2 Formation of a local mode .............. 23 1.4 Theory of dynamical symmetry breaking .......... .. 24 1.4.1 Model systems. 25 1.4.2 The discrete nonlinear Schrbdinger equation .... .. 26 1.4.3 Classical and quantum mechanical aspects ..... 26 1.5 Posed problems. .. 28 2 The local to normal mode transition in polyatomic molecules 32 2.1 Introduction . .. 32 2.2 The classical Darling-Dennison system .............. ... 37 2.2.1 Derivation of the classical Darling-Dennison Hamiltonian .. 37 2.2.2 Equivalency to Duffing's oscillator ............. 41 2.2.3 Geometrical construction of the polyad phase-sphere . 45 2.2.4 Adaptation to experimental data ............. 58 2.2.5 Analytical solution of the equations of motion ...... .. 67 2.2.6 Panoptic view of different dynamic representations . 78 5 2.2.7 Extension to damped systems .................. 78 2.2.8 Preliminary review of the classical results and conclusions . 83 2.3 Semiclassical interpretation ....................... 87 2.4 Quantum mechanical aspects of the normal to local mode transition . 89 2.4.1 The eigenvalue problem . .................... 90 2.4.2 The energetic splitting of eigenstates of the Darling-Dennison Hamiltonian ............................ 94 2.4.3 Recovery of the classical bifurcation parameter ......... 97 2.4.4 Perturbative analysis and stability ............. 100 2.4.5 Preliminary review of the quantum mechanical results and con- clusions .............................. 120 2.4.6 A stability criterion for eigenstates ............... 121 2.5 Relevance for control of intramolecular dynamics . ........... 137 2.6 Review ................................... 142 3 Duffing's oscillator and the nonlinear dimer 148 3.1 Introduction . ... ... 148 3.2 Derivation of the reduced dynamic equations .............. 151 3.2.1 Variational procedure . 152 3.2.2 Phase space analysis . .. .. 155 3.2.3 Elimination of fast variables in the Duffing limit ....... 159 3.2.4 Elimination of fast variables in the standard adiabatic limit 160 3.3 Solution and interpretation of the reduced dynamic equations .... 162 3.3.1 Charge density dynamics in the Duffing limit .......... 162 3.3.2 Remarks on the dynamics of molecular vibrations in the stan- dard adiabatic limit ........................ 177 3.4 Conclusion ............ ..................... 180 A Discrete fourier analysis of dynamical systems 184 B The spectroscopic condition number 188 6 C The discrete self-trapping equation 196 7 List of Figures 1-1 Delocalized vibrational excitation in normal modes v and /3 of AB 2 systems ................................... 16 1-2 Localized vibrational excitation in local modes 11 and 12 of AB 2 systems. 16 1-3 Coherent degenerate local modes 11 and 12 form normal modes vl and V3 of AB 2 systems. ............................ 16 1-4 Probability density of a delocalized excess charge in a molecular crystal of diatomic monomers. Translational symmetry is conserved ...... 17 1-5 Probability density of a localized excess charge in a molecular crystal of diatomic monomers. Translational symmetry is broken due to the induced distortion. ............. ......... 17 1-6 Localization in a dynamic potential: Particle moving in gravitational field on a deformable surface under friction. .............. 20 1-7 Localization of vibrational excitation leads to dynamical symmetry lowering, here from D3h to C2 symmetry. ............... 21 2-1 Bifurcation diagram for the undamped Duffing oscillator without driv- ing. Dashed lines (- -) indicate neutrally stable branches, dotted lines (...) unstable branches. At I = 1 there occurs a pitchfork bifurcation and at n = 0 there occurs a transcritical bifurcation. ........
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