LBL-34923 Modern Integral Equation Techniques for Quantum Reactive Scattering Theory Scott Michael Auerbach Ph.D. Thesis Department of Chemistry University of California and Chemical Sciences Division Lawrence Berkeley Laboratory University of California Berkeley, California 94720 November 1993 This work was supponod in pan by Ihc Director. Office of Energy Research. Office of Basic Energy Sciences. Chemical Sciences Division of ihe U.S Department of Energy under Contracl No DE-ACO3-76SFD0098. and in pan by ihe National Science Foundation MAWt T ciS~:Lr.' - Modern Integral Equation Techniques for Quantum Reactive Scattering Theory Copyright© 1991 by Scott Michael Auerbach The U.S. Depanment of Energy has lo right to use this thesis for any purpose whatsoever including [he right to reproduce all or any pan thereof 1 Modern Integral Equation Techniques for Quantum Reactive Scattering Theory Scott Michael Auerbach Abstract Rigorous calculations of cross sections and rate constants for elementary gas phase chemical reactions are performed for comparison with experiment, to ensure that our picture of the chemical reaction is complete. We focus on the H/D+H2 —* Ii2/DH + H reaction, and use the time independent integral equation technique in quantum reactive scattering theory. We examine the sensitivity of H+H2 state resolved integral cross sections tTV'j'.vj(E) for the transitions (v = O.j = 0) to (v' = \,j' = 1,3), to the difference between the Liu-Siegbahn-Truhlar-Horowitz (LSTH) and double many body expan­ sion (DMBE) ab initio potential energy surfaces (PES). This sensitivity analysis is performed to determine the origin of a large discrepancy between experimental cross sections with sharply peaked energy dependence and theoretical ones with smooth energy dependence. We find that the LSTH and DMBE PESs give virtually identical cross sections, which lends credence to the theoretical energy dependence. To facilitate quantum calculations on more complex reactive systems, we de%elop a new method to compute the energy Green's function with absorbing bound­ ary conditions (ABC), for use in calculating the cumulative reaction probability. The method is an iterative technique to compute the inverse of a non-Hermitian matrix which is based on Fourier transforming time dependent dynamics, and which requires very little core memory. The Hamiltonian is evaluated in a sine-function based dis­ crete variable representation (DYR). which we argue may often be superior to the fast Fourier transform method for reactive scattering. We apply the resulting power se­ ries Green"s function to the benchmark collinear H—H; system over the energy range 0.37 to 1.27 eV". The convergence of the power series is stable at all energies, and is accelerated by the use of a stronger absorbing potential. The practicality of computing the ABC-DVR Green's function in a polyno­ mial of the Hamiltonian is discussed. We find no feasible expansion which has a fixed and small memory requirement, and is guaranteed to converge. We have found, how­ ever, that exploiting the time dependent picture of the ABC-DVR Green's function leads to a stable and efficient algorithm. The new method, which uses Newton in­ terpolation polynomials to compute the time dependent wavefunction, gives a vastly improved version of the power series Green's function. We show that this approach is capable of obtaining converged reaction probabilities with very straightforward accuracy control. We use the ABC-DVR-Newton method to compute cross sections and rate constants for the initial state selected D+H2(i> = 1, j) —• DH+H reaction. We obtain converged cross sections using no more than 4 Mbytes of core memory, and in as little CPU time as 10 minutes on a small workstation. With these cross sections, we calculate exact thermal rate constants for comparison with experiment. For the first time, quantitative agreement with experiment is obtained for the rotationally 13 3 -1 1 averaged rate constant kv=l(T = 310K) = 1.9 x 1CT cm sec molecule" . The 7—shifting approximation using accurate J = 0 reaction probabilities is tested against the exact results. It reliably predicts £„=i(T) for temperatures up to 700 K, but individual (v = 1, j) —selected rate constants are in error by as much as 41%. Dedication To my beautiful wife Sarah, the light of my life ii iii Acknowledgments I am grateful for the patience and encouragement from my research advi­ sor, Prof. William H. Miller, with whom scientific interaction has always been a most exhilarating experience. Most of all, I would like to thank Bill for keeping me in grad­ uate school. I would also like to acknowledge the friendship and guidance from Prof. Claude Leforestier. In addition, I thank Prof. Andrew Vogt (Georgetown University) for stimulating my interest in Green's functions, Prof. David Chandler for providing a wonderful teaching environment, and Prof. Robert G. Littlejohn for charing his insight on semiclassical mechanics. I have had the pleasure of knowing some very special postdocs m the Miller group. In particular, I would like to thank Dr. John Z. H. Zhang for gently introducing me to the real world of quantum reactive scattering calculations. I am obliged to Dr. Daniel T. Colbert for helping to sustain my interest in this subject. I am grateful to Dr. Gerrit C. Groenenboom for giving me the confidence to hurdle any numerical obstacle. Finally, I thank Dr. Peter Saalfrank for opening my eyes to the greater world of science. I can not express how lucky I feel to have worked with Mrs. Cheryn Gliebe. Her invaluable assistance and friendship throughout my graduate career has made my years at Berkeley truly delightful. I am grateful to the members of the Miller group past and present: Dr. Tamar Seideman. Dr. Gerhard Stock. Dr. Uwe Manthe. Dr. Agathe Untch, Dr. Nancy Makri, Dr. Beverly Grayce. Dr. Yan-Tyng Chang. Dr. Lionel F. X. Gaucher. Dr. Rigoberto Hernandez. Srihari Keshavamurthy. J, Daniel Gezelter. Ward H. Thomp­ son, and Bruce \V Spath for all the help and friendship they have given me. I would like to give special thanks to Dr. Rigoberto Hernandez for skillful assistance with data and text processing, and to J. Daniel Gezelter for superb system administration. It is not possible for me to list by name all the wonderful friends who have sustained me through the years: I am indebted to you all. I woud like to give spe­ cial thanks to Maria C. Longuemare. David A. Heclit. Dr. Avery X. Goldstein, and iv Michael A. Olshavsky for looking after me — I really needed it. I am also grateful to Dr. John N. Gehlen for running a great softball club, and Dr. Joel S. Bader for teaching me my outside topic. I have the best parents in the whole world. Their nurturing love has given me the confidence to make a difference. And their wonderful example has taught me that people always come first. I do not think I can ever know how much I missed Sarah during the first three years in Berkeley. Thankfully, we are together again. I could never have finished this dissertation without her love and encouragement. We did it, honey! This research was supported by a National Science Foundation graduate fellowship, and by the Director, Office of Energy Research, Office of Basic Energy Sciences, Chemical Sciences Division of the U.S. Department of Energy under Con­ tract No. DE-AC03-76SF00098. V Contents Dedication i Acknowledgments iii Table of Contents v List of Tables ix List of Figures xi 1 General Introduction 1 1.1 First Philosophical Principles 1 1.2 Molecular Beams and Collision Theory 3 1.3 Theoretical Practices Old and New 8 1.4 Looking Ahead 13 References 14 2 Sensitivity Analysis for H+H2 17 2.1 Introduction 17 2.2 S—matrix Kohn Formulation 20 2.3 Functional Sensitivity Analysis 24 2.4 The Potentials: LSTH vs. DMBE 26 2.5 The Dynamics: Results and Discussion 27 2.6 Concluding Remarks 35 References 37 3 Power Series Green's Function 41 3.1 Introduction 41 3.2 General Methodology 45 3.3 Power Series Green's Function 47 vi CONTENTS 3.4 The Basis Set 48 3.4.1 SDVR vs. FFT 49 3.4.2 SDVR of the Free Particle Propagator 49 3.5 Multidimensional Generalization 52 3.6 Summary of the Methodology 53 3.7 Results and Discussion 54 3.7.1 The Coordinates 54 3.7.2 The Absorbing Potential 55 3.7.3 Reaction Probabilities 56 3.7.4 Convergence Tests 59 3.8 Concluding Remarks 64 3.8.1 Reactive Scattering 65 3.8.2 Path Integration 66 References 70 4 The Newton Algorithm 77 4.1 Introduction 77 4.2 Quantum Reactive Scattering Formulation 79 4.2.1 Formal Theory 79 4.2.2 Absorbing Boundary Conditions 84 4.3 Polynomial Expansions 86 4.3.1 Direct Expansion 90 4.3.2 Indirect Expansion 112 4.4 Quantum Reactive Scattering Calculations 117 4.4.1 Reaction Probabilites 117 4.4.2 Con-ergence Tests 118 4.5 Concluding Remarks 123 References 126 5 The D+H2(t>= 1) Rate Constant 131 5.1 Introduction 131 5.2 General Methodology 134 5.2.1 General Rate Constant Formulae 135 5.2.2 ABC Formulation of Quantum Reactive Scatteiing 136 5.3 Defining the Linear System 137 5.3.1 The Coordinates )37 5.3.2 The Basis Set ]42 5.3.3 The Reference Scattering State ... 145 5.3.4 The Absorbing Potential 15] 5.3.5 Summary of the Methodology 152 CONTENTS vii 5.4 Results and Discussion 152 5.4.1 Reaction Probabilities and Cross Sections 152 5.4.2 Rate Constants 162 5.5 Concluding remarks 168 References 171 6 General Conclusions 175 6.1 The H/D+R2 System 175 6.2 Integral Equations 176 6.3 And Beyond 179 6.3.1 Quantum Reaction Rate Theory 179 6.3.2 Ga= Phase Reaction Dynamics 180 6.3.3 Physical Chemistry 181 References 182 CONTENTS ix List of Tables 2.1 Fixed angle saddle point comparisons of LSTH/DMBE 27 5.1 Optimized convergence parameters for scattering calculations 153 5.2 Exact and approximate theoretical rate constants for D+H2 167 5.3 Experiment and theory for the D+H2 rate constant 168 LIST OF TABLES xi List of Figures 1.1 CoUinear H+H2 potential energy surface contours 10 2.1 Experiment and theory for the H+H2 cross section 19 2.2 LSTH/DMBE potentials for coUinear geometries 28 2.3 LSTH/DMBE potentials for L-shaped geometries 29 2.4 Partial cross sections for LSTH/DMBE with J = 0, E 31 2.5 Product distributions for LSTH/DMBE with J = 0, E = 1.2 eV .