Optical Response and Control of Molecular Systems by Porscha Louise McRobbie A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Physics) in The University of Michigan 2010 Doctoral Committee: Professor Eitan Geva, Co-Chair Professor Roseanne J. Sension, Co-Chair Associate Professor Luming Duan Assistant Professor Kevin J. Kubarych Assistant Professor Jennifer P. Ogilvie Porscha Louise McRobbie c 2010 ! All Rights Reserved This work is dedicated to my parents, who taught me to dream big and refuse to give up. ii ACKNOWLEDGEMENTS I would like to acknowledge the help of many individuals whom I have been fortu- nate enough to know during both my undergraduate and graduate studies. Without them, this thesis would not exist. I would like to thank my undergraduate mentors at Seattle University: Dr. John Meany for his sense of humor and for being the first to nudge me toward gradu- ate studies; Dr. David Boness, for his attempts to explain the precise difference between physical chemistry and chemical physics, and for his generous advice re- garding academia and job searches long after I had graduated; and Dr. Michael Morgan who served as my primary undergraduate research advisor, for showing me the beauty of classical mechanics and mathematics, and whose infectious passion for knowledge is unforgettable. I would like to thank Dr. Jose Alvarez for serving as my surrogate adviser and spending countless hours with me during my first summer of graduate research, and Sung-Kwan Mo for his kindness and cheery disposition, which made our shared of- fice environment quiet and pleasant. I would also like to thank my classmates Kevin Landmark and Damian Khan for keeping it all light while suffering through electro- dynamics and condensed matter courses. I would especially like to thank Stephen Reed, who assured me that the world wouldn’t crumble if I stopped calculating and ate cookies once in a while, and who became one of my best friends. A special thanks to Anson Cheung, for all of the late night laughs and whose encouragement iii and support over the years were much appreciated. I would like to thank members of the Geva group, past and present, who made the office environment both a colorful and pleasant place to be: Frank Vazquez, Gabriel Hanna, Irina Navrotskaya, and Henry Boateng. I also thank Anne Vazquez for her inspiring teaching insights, and overall positive attitude. Finally, I consider myself unbelievably fortunate to have crossed paths with my thesis adviser, Prof. Eitan Geva. His willingness to offer guidance and advice in matters of research, teaching, and career advancement are truly unparalleled. I will miss the morning espresso tradition, and am grateful for having been taught to argue about phase factors and tensor algebra “Israeli style”. I leave his group after five years transformed and inspired both scientifically and personally. In terms of computing, I am indebted to Todd Raeker and David Braun for their abilities to make computing as painless as possible. An additional thanks to my fellow Ann Arbor Ashtangis, who have inspired me beyond words both on and offthe mat, and who contributed greatly to my overall sanity and peace of mind. Namaste! And lastly, a special thanks goes to Atsushi Yagi for his endless support at all stages of my graduate tenure. I have learned much from his calm and level-headed approach to both studies and life, and for his patience and kindness toward me I am truly grateful. iv TABLE OF CONTENTS DEDICATION .......................................... ii ACKNOWLEDGEMENTS .................................. iii LIST OF FIGURES ...................................... vii LIST OF APPENDICES ................................... x ABSTRACT ........................................... xi CHAPTER I. Prologue .......................................... 1 1.1 Introduction . 1 1.1.1 Coherent Control . 2 1.1.2 Multidimensional Spectroscopy . 5 1.2 Model Systems . 6 1.2.1 Two-State Model for Electronic Spectroscopy . 7 1.2.2 Multistate Model for Vibrational Spectroscopy . 10 1.3 Overview . 12 II. Coherent Control of Population Transfer in the Presence of Dephasing .. 15 2.1 Introduction . 15 2.2 Chirped Laser Pulses and Control Parameters . 17 2.3 Two-State Model in a Rotating Frame and the RWA . 19 2.4 Quantum Mechanical Calculations for Population Transfer . 22 2.4.1 Wavepacket Propagation via Fourier Grid Methods . 24 2.4.2 Energy Representation . 29 2.5 Dephasing Effects . 32 2.5.1 Stochastic model for fluctuating energy levels . 33 2.5.2 Solving the Stochastic Schr¨odingerEquation . 35 2.6 Results and Discussion . 36 2.6.1 Dephasing-Free Population Transfer . 36 2.6.2 The Limit of Slow Fluctuations . 43 2.6.3 The Limit of Fast Fluctuations . 47 2.7 Summary and Future Outlook . 50 III. Methods for Calculating One- and Two-Dimensional Optical Spectra ... 52 3.1 Introduction . 52 3.2 Theory of Optical Response for a Two-State System . 54 3.2.1 Linear Response . 54 v 3.2.2 Nonlinear Response . 55 3.3 Benchmark Model and Exact Solutions . 57 3.4 Forward-Backward Initial-Value Representation Method (FB-IVR) . 63 3.5 Linearized Semiclassical Method (LSC) . 69 3.6 Standard Second Order Cumulant Method (2OC) . 75 3.7 Alternative Second Order Cumulant Method (2OCa) . 78 3.8 Results and Discussion . 82 3.8.1 Linear Spectra . 83 3.8.2 Nonlinear Spectra . 87 3.9 Summary and Future Outlook . 92 IV. Two-dimensional Infrared Spectra for Multistate Systems .......... 94 4.1 Introduction . 94 4.2 Polarization-Selective Response: Optical Response Tensors . 97 4.2.1 Orientational Averaging . 98 4.2.2 Optical Response Tensors for a Multistate Vibrational System . 100 4.3 ab initio 2DIR Spectra . 104 4.3.1 Field-free Anharmonic Hamiltonian in Normal Coordinates . 104 4.3.2 Field-Matter Interaction: Polarized Transition Dipoles . 109 4.3.3 Electronic Structure Inputs . 112 4.4 Application to Metal Carbonyl Compound Mn(CO)5 .............113 4.4.1 Preliminary Results and Discussion . 114 V. Summary ......................................... 117 APPENDICES .......................................... 121 BIBLIOGRAPHY ........................................ 148 vi LIST OF FIGURES Figure 1.1 A 2D experiment with pulse sequence E1,E2 and E3, which stimulates the emission (3) of the third-order signal field Esig. The signal is a function of three time delays, (t1,t2,t3), where t1 is the time delay between the first and second pulses (evolution), t2 is the delay between the second and third pulses (evolution), and t3 is the time delay between the third pulse and detection. 6 1.2 Schematic of the two-state model. ω0 and ω are the 0 0 and vertical transition eg ver − frequencies, respectively. Ve/g is the Born-Oppenheimer potential associated with the e/g electronic state, and Qd is displacement between the minima. Each surface has associated vibrational states labeled by n, g/e for n =0, 1, 2,.......... 9 {| #} 1.3 Schematic energy level diagram for a multi-state system with eigenstates and en- ergy levels mj ,εmj . The index m =0, 1, 2,... labels the band, where m =0 corresponds{| to the# ground} state, and j =0, 1, 2,... labels the state within a given band. 11 2.1 Energy fluctuation statistics for σ = 1. Sample noise trajectories for correlation 3 times (a) τc = 10 τ0 and (b) τc = 10 τ0, corresponding histogram (c) with FWHM is 2√2 log 2σ×, and (d) the noise correlation× function C(t)= ξ(t)ξ(0) from (2.38). .& . .# . 35 2.2 Final excited state population contour as a function of the frequency domain field amplitude E˜0 and the linear spectral chirp α!, for an isolated molecule at 0K. 38 2.3 Franck-Condon overlap factors (absolute value) for horizontal displacement (a) Qd = 1 used for the population maps presented in this chapter, and (b) Qd =2 for comparison. Integers jg and je label the ten lowest vibrational eigenstates of the ground and excited electronic states, respectively, and the height of the bar corresponds to the absolute value of the overlap. 39 2.4 Intrapulse “pump-dump” mechanism between two electronic states Vg and Ve. The top part of the figure represents a negatively chirped pulse, where the high frequency components arrive earlier in time than the low frequency ones. Population is first transferred upward by the blue edge of the pulse, followed by time evolution of the wavepacket for a time on the order of the pulse duration tf tqi, and finally population is tranferred downward by the red edge of the pulse.∼ − . 41 vii 2.5 Ground and excited state populations and wave function probability densities for zero chirp α! = 0. Left panels: The pulse envelope E(t) (black dashed), ground | | state population pg(t) (blue) and excited state population pe(t) (red). Right panels: Probability densities of the ground state before the pulse (black), the ground state after the pulse at tf (blue) and the excited state after the pulse at tf (red). Lower intensity E˜0 =3.7 is shown in (a) and (b), while higher intensity E˜0 =4.8 is shown in| (c)| and (d). .| . .| . 43 2.6 Population maps showing the approach to the inhomogeneously broadened limit. 4 The correlation time is τc = 10 τ0, where τ0 =0.79 is the transform limited pulse duration, and increasing values× of the fluctuation amplitude are shown in subfigures (a)-(f). All maps were averaged over 2000 realizations of the stochastic process to ensure convergence. 45 2.7 Population maps showing the approach to the homogeneously broadened (motional narrowing) limit. The fluctuation amplitude is σ = 2, and decreasing values of the solvent correlation time are shown in subfigures (a)-(f). All maps were averaged over 2000 realizations of the stochastic process to ensure convergence. 49 3.1 Propagation pathway for classical trajectories of R2 in the FB-IVR method.