Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1999 Theoretical study of spin-orbit coupling in molecules Dmitri G. Fedorov Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Analytical Chemistry Commons, and the Physical Chemistry Commons Recommended Citation Fedorov, Dmitri G., "Theoretical study of spin-orbit coupling in molecules " (1999). Retrospective Theses and Dissertations. 12662. https://lib.dr.iastate.edu/rtd/12662 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. 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Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. Bell & Howell Informatton and Learning 300 North Zeeb Road, Ann Artx)r, Ml 48106-1346 USA 800-521-0600 Theoretical study of spin-orbit coupling in molecules by Dmitri G. Fedorov A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Physical Chemistry Major Professor: Mark S. Gordon Iowa State University Ames, Iowa 1999 UMI Niimber: 9940199 UMI Microform 9940199 Copyriglit 1999, by UMI Company. All rights reserved. Tliis microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 ii Graduate College Iowa State University This is to certify that the Doctoral Dissertation of Dmitri G. Fedorov has met the dissertation requirements of Iowa State University Signature was redacted for privacy. Co mitte ^ember Signature was redacted for privacy. ember Signature was redacted for privacy. ommittee Member Signature was redacted for privacy. Committee Member Signature was redacted for privacy. Major Professor Signature was redacted for privacy. Fortne Major Program Signature was redacted for privacy. iii TABLE OF CONTENTS GLOSSARY vi ACKNOWLEDGEMENTS viii ABSTRACT ix CHAPTER I. GENERAL INTRODUCTION 1 L Introduction 1 2. Dissertation organisation 2 References 4 CHAPTER n. A STUDY OF THE RELATIVE IMPORTANCE OF ONE AND TWO ELECTRON CONTRIBUTIONS TO SPIN-ORBIT COUPLING 5 Abstract 5 1. Introduction 5 2. General description of the methods 6 3. Symmetry summary 9 4. Indirect Methods 10 5. Direct Methods 14 6. Comparison of direct and indirect methods 25 7. Discussion and Summary 36 Acknowledgements 37 References 37 CHAPTER m. SPEEDING UP SPIN-ORBIT COUPLING CALCULATIONS 40 Abstract 40 L Introduction 40 2. The Wigner-Eckart theorem 42 3. Hermiticity and time reversal, real valued NR states 44 4. Double group symmetry 46 5. Point group symmetry 50 6. Comparison of the point and double group rules 50 7. Computational speed-up 51 8. Sunmiary and conclusions 54 Acknowledgements 55 iv Appendix 1 55 Appendix II 55 Appendix EH 60 Appendix iV 61 References 62 CHAPTER IV. STUDY OF THE VIBRATIONAL STRUCTURE 65 1. Introduction 65 2. Vibrational wavefunction 65 3. Properties 69 4. Vibronic coupling model 70 5. Results and discussion 71 References 72 CHAPTER V. DL«^ATIC PROPERTIES 74 1. Introduction 74 2. One-electron spin-independent operator (e.g. dipole moment) 75 3. Spin-dependent operator 75 4. Transitions between adiabatic levels 76 References 79 CHAPTER VI. AN EXPERIMENTAL AND THEORETICAL STUDY OF THE SPIN-ORBIT INTERACTION FOR v*=0-41) AND O,* (X-n3;2„^.g, v^-38) 80 Abstract 80 L Introduction 81 2. Experimental and theoretical methods 84 3. Results and discussion 90 4. Conclusions 96 Acknowledgements 97 References 98 Figure Captions 103 CHAPTER vn. A THEORETICAL STUDY OF THE REACTION OF Ti+ WITH ETHANE ill Abstract Ill I. Introduction Ill V n. Computational methods 113 in. Results and discussion 115 rV. Conclusions 123 References 124 Captions to figures 130 CHAPTER vm. GENERAL CONCLUSIONS 146 vi GLOSSARY AO, Atomic Orbitals. One-electron functions whose square is the density of an electron in centred on an atom. CAS, Complete Active Space. A method to simultaneously optimise MOs and CI coeffcients of CSFs (or determinants) built upon these MOs. All occupations within a chosen space are considered (with the possible restriction imposed by synmietry). CI, Configurational Interaction. Wavefunction composed of Slater determinants. CSF, Conflgurational State Function. Spin-adapted (i.e. eigenfunction of S operator) linear combination of Slater determinants. Discoincidence, a pair of Slater determinants that differ by one orbital (in each determinant). This orbital is said to be discoincident. FOCI, First Order CI. Wavefunction composed of all single excitations from CAS into virtual space. Hermiticity, Hermitian character of an operator. Integral convention: physical, same as scalar product: <ab|A|cd>=a^b^Acd. Irrep, irreducible representation. LCAO, Linear Combination of Atomic Orbitals. A generally used approach of expanding MOs in terms of linear combinations of AOs. MO, Molecular Orbitals. One-electron functions whose square is the density of an electron in a molecule. Scalar product convention: physical or Dirac <a|b>=a^. SCF, Self Consistent Field Approach. The backbone of a very large class of modem quantum chemistry methods. vii SOCI, Second Order CI. Wavefunction composed of all single and double excitations from CAS into the virtual space. SOC, Spin-Orbit Coupling. A diabatic relativistic interaction of angular and spin momenta. SOCC, Spin-Orbit Coupling Constant. Determines the magnitude of SOC between a pair of states. viii ACKNOWLEDGMENTS The helpful guidance of my scientific advisor is gratefully acknowledged. The uncountable answered questions and discussions with Michael Schmidt are greatly appreciated. Shiro Koseki, Kenneth Dyall, Peter Pyykkd and David Yarkony are thanked for their collaboration and scientiflc interaction. Thomas Furlani's providing the supplementary notes for his thesis proved to be very beneHcial. ix ABSTRACT Theoretical models to study spin-obit coupling with multi-configurational wavefunctions have been developed both methodologically and implemented into the widely distributed quantum chemistry package GAMESS. Various aspects of making the spin-orbit coupling studies more efficient and thus more available, such as extensive usage of symmetry and parallelisation have been studied. A theoretical development of one, two and partial two electron approaches to spin-orbit coupling is given and tested on a representative set of molecules. A new accurate method to study the vibrational structure of molecules, limited in the current formulation to diatomics, has been proposed. Two chemically interesting systems have been studied, the reaction path of titanium cation and ethane, and the vibrational structure of CO"^ and O,*. 1 CHAPTER 1. GENERAL INTRODUCTION 1. Introduction One of the key ideas in the scientific approach of theoretical understanding of the natural phenomena is the art of making approximations. One can derive beautiful and completely general equations describing nature in its totality, however, it is perfectly clear that any Hnite expression represents a fmite part of the infinite complexity of the natural phenomena. The general equations lack the ability to describe the phenomena exactly. Still, even the limited form of scientiHc wisdom expressed in many beautiful equations does not permit a general and exact solution to these equations to be found. One of the key wheels perpetuating scientific thought is the ability to make and break approximations. An approximation is only a first step intending to probe the bulk of the interaction. The second step necessarily is the understanding of under what conditions this approximation is applicable and what are the forces representing the breakdown of the approximation. Only then can one confidently use the approximation and break it when deemed important. One of the pivotal approximations currently established in quantum chemistry is the adiabatic approximation that proved to be the fundamental step in the quantum theory permitting the transition from study of atomic