Generalized Quantum Defect Methods in Quantum Chemistry by Serhan Altunata Submitted to the Department of Chemistry in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physical Chemistry at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY May 2006 c Massachusetts Institute of Technology 2006. All rights reserved. Author............................................. ................. Department of Chemistry May 23, 2006 Certified by......................................... ................. Robert W. Field Haslam and Dewey Professor of Chemistry Thesis Supervisor Accepted by......................................... ................ Robert W. Field Chairman, Department Committee on Graduate Students This doctoral thesis has been examined by a Committee of the Department of Chemistry that included Professor Sylvia T. Ceyer (Chairman) Professor Jianshu Cao Professor Robert W. Field (Thesis Supervisor) 2 Generalized Quantum Defect Methods in Quantum Chemistry by Serhan Altunata Submitted to the Department of Chemistry on May 23, 2006, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physical Chemistry Abstract The reaction matrix of multichannel quantum defect theory, K, gives a complete pic- ture of the electronic structure and the electron nuclear dynamics for a molecule. The reaction matrix can be used to examine both↔ bound states and free electron scat- tering properties of molecular systems, which are characterized by a Rydberg/scattering electron incident on an ionic-core. An ab initio computation of the reaction matrix for fixed molecular geometries is a substantive but important theoretical effort. In this thesis, a generalized quantum defect method is presented for determining the reaction matrix in a form which minimizes its energy dependence. This reaction matrix method is applied to the Rydberg electronic structure of Calcium monofluo- ride. The spectroscopic quantum defects for the 2Σ+ ,2Π, 2∆, and 2Φ states of CaF are computed using an effective one-electron calculation. Good agreement with the experimental values is obtained. The Σ-symmetry eigenquantum defects obtained from the CaF reaction matrix are found to have an energy dependence characteristic of a resonance. The analysis shows that the main features of the energy-dependent structure in the eigenphases are a consequence of a broad shape resonance in the 2Σ+ Rydberg series. This short-lived resonance is spread over the entire 2Σ+ Rydberg series and extends well into the ionization continuum. The effect of the shape reso- nance is manifested as a global ”scarring” of the Rydberg spectrum, which is distinct from the more familiar local level-perturbations. This effect has been unnoticed in previous analyses. The quantum chemical foundation of the quantum defect method is established by a many-electron generalization of the reaction matrix calculation. Test results that validate the many-electron theory are presented for the quantum defects of the 1 + 1sσgnpσu Σu Rydberg series of the hydrogen molecule. It is possible that the reaction matrix calculations on CaF and H2 can pave the way for a novel type of quantum chemistry that aims to calculate the electronic structure over the entire bound-state region, as opposed to the current methods that focus on state by state calculations. 3 Thesis Supervisor: Robert W. Field Title: Haslam and Dewey Professor of Chemistry 4 Acknowledgments At the end of this long journey, it is my pleasure to extend my warm gratitude to the many wonderful people I have met over the years. One among these people takes a special precedence. My advisor, Prof. Robert W. Field, has given me the complete freedom to pursue my curiosity since the first day I arrived at MIT. He provided constant guidance, encouraging, smiling, and somehow, always knowing where to go next. He never told me I was wrong, even when he knew I was dead wrong, but always chose to wait and allow me to discover the truth on my own. As I look back upon what I have achieved, I realize that he has been the greatest teacher and mentor in my life. Thank you, Bob, for making this an extraordinary experience. None of this would have been possible without you. I would like to thank my parents for their love, support and raising me the way I am. I devote this thesis to them in our native language Turkish: Sevgili Annem ve Babam, Iste sonunda bu tezide bitirdim! Goztepe’nin kenar mahallesinde futbol oynayan kopuk cocugun buralara gelecegini bir tek siz gordunuz. Gozunuzu kirpmadan, varinizla, yokunuzla beni desteklediniz. Sizin sevginiz olmadan bunlarin hicbiri mumkun olmazdi. Tessekur kelimesi size az kalir. Hazirlanin, daha nice basarilara birlikte yuruyecegiz. Bu arada en buyuk cim bom! I would like to warmly thank Prof. Jianshu Cao for his valuable collaboration and guidance. I have truly enjoyed working with him on my semiclassical Rydberg wavepacket project, which bore my first, and probably the most memorable, paper in Physical Review A. Prof. Sylvia T. Ceyer is also greatly acknowledged for the valuable comments she has provided as my thesis committee chair. I would like to thank the members of the Robert W. Field group for helping me merge my theoretical studies with experimental work. I have learnt much from them and benefited greatly from fruitful discussions on spectroscopy. Specifically, I feel privileged to have worked with Vladimir Petrovic and Bryan Wong on various challenging and interesting theoretical projects. I would also like to thank Ryan Thom, who has patiently answered my endless questions on a wide range of topics 5 in spectroscopy. My special thanks goes to our past member Jason O. Clevenger for welcoming me to the Rydberg project when I first joined the Robert W. Field group. I would like to express my gratitude to the members of the MIT theoretical group. It has been a great pleasure to share a work space with them. They have made many precious contributions to my scientific knowledge through discussions and our com- petitive problem solving sessions. I would like to thank Steve Presse, Jim Witkoskie, Xiang Xia, Shilong Yeng, Jianlan Wu, Chiao-Lun Cheng, and Ophir Flomenbo for making the zoo an enjoyable and welcoming environment. I would like to thank my sister, Selen, for being the wise guide in my life. She has shown me the right choices and helped me to navigate many of the challenges of graduate school. She is a great buddy to have. I would like to thank Cindy for her continuous support and caring. She has been my source of inspiration during both the pleasant times and the hardships that I have experienced during my stay at MIT. I would like to thank her for being part of my life. 6 Contents 1 Introduction 37 1.1 HistoricalAccountofMQDT . 38 1.2 CurrentProgress ............................. 48 1.2.1 Connection between the Effective Hamiltonian and MQDT .. 51 2 Formalism of Multichannel Quantum Defect Theory 67 2.1 Introduction................................ 67 2.2 CoulombScattering............................ 68 2.2.1 AsymptoticLimit......................... 72 2.2.2 Behavior of the Coulomb functions at r =0 .......... 75 2.2.3 ContinuumNormalization . 75 2.2.4 SolutionsatNegativeEnergy . 77 2.2.5 BoundStatesoftheCoulombpotential . 77 2.2.6 Phase-Amplitude Form of the Coulomb Functions . 78 2.2.7 PlotsoftheCoulombFunctions . 79 2.2.8 CoulombScatteringMatrix . 85 2.2.9 Bound States as Poles of the Coulomb Scattering Matrix ... 87 2.3 SingleChannelTheory .......................... 90 2.3.1 BoundStates ........................... 93 2.3.2 CollisionEigenstates . 94 2.3.3 ScatteringWave.......................... 94 2.3.4 Applications of the Single Channel Theory: Alkali Atoms... 95 2.4 MultichannelTheory ........................... 99 7 2.4.1 The Case where All Channels are Closed: Bound States . 101 2.4.2 The Case for All Channels Open: Continuum . 108 2.4.3 The Case of a Mixture of Closed and Open Channels: Autoion- izingResonances ......................... 110 2.4.4 TheFrameTransformation. 111 2.5 ResonanceScatteringTheory . 118 2.5.1 ShapeResonances. 122 2.5.2 FeshbachResonances . 122 2.6 Conclusions ................................ 132 3 Calculation of the Short-Range Reaction Matrix from First Princi- ples 135 3.1 Introduction................................ 135 3.2 Theory................................... 139 3.2.1 SolutionsoftheZero-OrderEquations . 141 3.2.2 CoupledEquations . 143 3.2.3 Iterative Procedure to Calculate the K-Matrix. 144 3.2.4 The Energy Dependence of the Reaction Matrix . 148 3.2.5 The Transformation between K(ǫ) and K¯ ........... 152 3.3 Conclusions ................................ 154 3.4 MathematicalAppendixA . 160 3.5 MathematicalAppendixB . 161 3.6 MathematicalAppendixC . 162 4 Electronic Spectrum of CaF 165 4.1 Introduction................................ 165 4.2 ApplicationandResults . 167 4.2.1 Analytical Continuation of the Integer K Matrix to Negative Energy............................... 183 4.3 PhotoionizationofCaF. 187 4.4 Conclusions ................................ 197 8 4.5 MathematicalAppendix . 200 5 Broad Shape Resonance Effects in CaF Rydberg States 203 5.1 Introduction................................ 203 5.2 ResultsandDiscussion . .. .. 207 5.3 Violation of Mulliken’s Rule Caused by the Shape Resonance..... 216 5.4 Conclusions ................................ 222 6 Many Electron Generalization of the ab initio Reaction Matrix Cal- culation 225 6.1 Introduction................................ 225 6.2 The Variational K-MatrixTheory. 227 6.2.1 The N-electronIon-CoreHamiltonian.