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Exact Solution of Multi State Problems in Quantum and Statistical Mechanics Exact solution of Multi state problems in Quantum and Statistical Mechanics A THESIS submitted by DIWAKER for the award of the degree of DOCTOR OF PHILOSOPHY SCHOOL OF BASIC SCIENCES INDIAN INSTITUTE OF TECHNOLOGY MANDI. MANDI-175005 (H. P.) INDIA AUGUST, 2015 c 2015 - Diwaker All rights reserved. I dedicate this thesis to my parents, my brothers, my sisters and to my teachers. Indian Institute of Technology BArtFy þODOEgkF s\-TAn Mandi, Mandi (H. P.)-175005, m\ÚF, m\ÚF,(Eh. þ.)-175005, India, Govt. of India BArt, BArt srkAr THESIS CERTIFICATE This is to certify that the thesis titled Exact Solution of Curve Crossing Problems in Quantum Mechanics, submitted by Diwaker, to the Indian Institute of Technology, Mandi, for the award of the degree of Doctor of Philosophy, is a bonafide record of the research work done by him under my supervision. The contents of this thesis, in full or in parts, have not been submitted to any other Institute or University for the award of any degree or diploma. Dr. Aniruddha Chakraborty Research Guide Associate Professor School of Basic Sciences IIT-Mandi, 175005 Phone-01905-237930 Email-achakraborty @iitmandi.ac.in http://www.aniruddhachakraborty.org/ Place: Mandi (H. P.) Date: 1st august 2015 Phone: 01905-237949/237944/267001, Fax: 01905-267009, www.iitmandi.ac.in Indian Institute of Technology BArtFy þODOEgkF s\-TAn Mandi, Mandi (H. P.)-175005, m\ÚF, m\ÚF,(Eh. þ.)-175005, India, Govt. of India BArt, BArt srkAr DECLARATION BY RESEARCH SCHOLAR I hereby declare that the entire work assimilated in this thesis entitled Exact Solution of Curve Crossing Problems in Quantum Mechanics is the result of investigations carried out by me in the School of Basic Sciences, Indian Institute of Technology Mandi, under the supervision of Dr. Aniruddha Chakraborty,and that it has not been submitted elsewhere for any degree or diploma. In keeping with the general practice, due acknowledgments have been made wherever the work described is based on finding of other investigators. Diwaker Enrollment no. D10011 Research Scholar School of Basic Sciences IIT-Mandi, 175005 [email protected] https://sites.google.com/site/diwakeriitmandi/home Place: Mandi (H. P.) Date: 1st august 2015 Phone: 01905-237949/237944/267001, Fax: 01905-267009, www.iitmandi.ac.in ACKNOWLEDGEMENTS First, I would like to express my most sincere thanks and gratitude to my Ph.D. su- pervisor Dr. Aniruddha Chakraborty, Assistant Professor, School of Basic sciences, Indian Institute of Technology Mandi, Mandi (H. P.), for the seemingly endless hours he spent with me going over this work, for his patience when things did not always work out, for his support, encouragement, constant guidance and constructive advice which he has extended to me throughout all the stages of my research work. I also greatly appreciate the confidence he has shown in my abilities and capabilities. Special thanks goes to him for encouraging me to think problems in multiple directions. Next, I am very grateful to Prof. Timothy A Gonsalves, Director, IIT Mandi for setting up a good research environmentin this Himalayan region. I also want to convey thanks to my Doc- toral committee members Dr. Pradumnya Pathak, Dr. Hari Verma, Dr. Prasanth Jose and Dr. Neeraj for their help, support, valuable suggestions and discussions. Iam also grateful to the MHRD India for providing me fellowship to carry out this research work. I am obliged to my parents Shri. Dina Nath and Smt. Brahmi Devi and my younger brother and sister, Akhilesh Kumar and Nivedita and above of all my lovely niece Hrutvika for their love, affection, moral and endless support that has enabled me to achieve this goal. I gratefully concede all my friends specially Manisha thakur and all other persons whose names do not appear here but whose love, cooperation and participation in various forms have motivated me and helped me to complete this work. Diwaker Place: Mandi (H. P.) i ABSTRACT KEYWORDS: Curve crossing, Non adiabatic transitions, two state, three state, Multi state, scattering, Green’s function, Boundary conditions, Dirac Delta Potential, constant potential, linear potential, expo- nential potential, Morse potential, harmonic potential, Transfer matrix, Diabatic orbitals, Diabatization, Non adiabatic coupling matrix elements, MRCI, CASPT2. Non adiabatic transitions due to crossing of the potential energy curves is one of the most probable mechanism responsible for electronic transitions. This is purely an in- terdisciplinary topic which covers a wide range of fields like physics, chemistry and biology. Various spectroscopic, collisions processes and reactions are governed by such kind of transitions. Some of the examples involving such kind of transitions may in- clude radationless transitions in condense matter physics, laser assisted collisions reac- tions, Zener transitions in flux driven metallic rings, super conducting Josephson junc- tions, reactions in nuclear physics and electron proton transfer processes in biological systems. Neutrino conversion in the sun, dissociation of molecules on the metal sur- faces are some of other few examples which explains the importance of non adiabatic transitions. The first report on non adiabatic transitions was published in around 1932, when Landau, Zener, Stuckelberg and Rosen published pioneer work in the area of non adiabatic transitions which opens a pathway for solutions to the problems including such kind of transitions. The approach used by them is purely analytical which can be mapped to problems like electron detachments, ionization in slow atomic and ionic collisions and electronic transitions in crystals where one state of system is interact- ing with a group of states of different nature and many more. From 1932 onwards we have numerous citations in literature based on non adiabatic transitions which involves analytical as well as computational approach. The work presented here in this thesis pays attention to the use of analytical methods for problems involving non adiabatic ii transitions where one state of a system is interacting with a group of states of different nature through Dirac Delta interactions and we provide a simple analytical formula for calculation of transition probability between different interacting states. The present thesis is divided into 7 chapters. Chapter 1 includes the introduction of non-adiabatic transitions/Curve crossing methods in quantum mechanics in which different analytical models are described in brief. Chapter 2 includes the details of the different kind of an- alytical and computational methods used in study of non adiabatic transitions. Chapter 3 is devoted to the exact solution of curve crossing problems using boundary condition and Green’s function method. Chapter 4 includes the exact solution of curve cross- ing problems using Transfer matrix method while in Chapter 5 we have used time dependent approach for the study of non adiabatic transitions/curve crossing problems. Chapter 6 includes the computational approach to study the curve crossing problems in real molecules. Chapter 7 concludes this thesis by providing the summary of all six chapters and future prospects in this area. iii TABLE OF CONTENTS ACKNOWLEDGEMENTS i ABSTRACT ii LIST OF TABLES ix LIST OF FIGURES xiii ABBREVIATIONS xiv NOTATION xv 1 INTRODUCTION 2 1.1 ClampednucleiHamiltonian . 2 1.2 Adiabatic and Born-Oppenheimer approximation . 3 1.3 Adiabatic and Diabatic approaches . 5 1.3.1 Adiabaticapproach . 5 1.3.2 Diabaticapproach . 6 1.4 CrossingofPotentialenergycurves . 8 1.4.1 NaClmolecule-: anexample . 9 1.5 Probability of a non-adiabatic transition . 14 1.6 Importantdefinitions . 17 1.6.1 Avoidedcrossing . 17 1.6.2 Diabaticstaterepresentation. 18 1.6.3 Adiabaticstaterepresentation . 18 1.6.4 AdiabaticityorAdiabaticstates . 19 1.7 Examples of non adiabatic transitions in different fields of science 19 1.7.1 Non adiabatic transitions inPhysics. 19 iv 1.7.2 Non adiabatic transitions in Chemistry . 20 1.7.3 Non adiabatic transitions in Biology and Economics . 21 1.8 Analytical Models of Non Adiabatic Transitions . 21 1.8.1 Landau-Zener-Stuckelberg Theory . 21 1.8.2 Rosen-Zener-DemkovTheory . 26 1.8.3 Nikitin’sExponentialmodel . 27 1.8.4 Non adiabatic Transition due to Coriolis coupling and Dy- namicalstaterepresentation . 29 1.8.5 Curve crossing type of Nonadiabatic transition: . 30 1.8.6 NonadiabticTunnelingtype . 30 1.8.7 Diabatically avoidedcrossing model . 31 1.8.8 ExponentialPotentialModel. 32 1.8.9 LinearPotentialModel . 33 1.8.10 Demkov-kunikemodel . 34 1.8.11 Demkov-Osherovmodel . 35 1.8.12 Zhu-Nakamuratheory . 37 1.8.13 Ouranalyticalmodel:. 37 1.9 Conclusions ............................. 39 2 DIFFERENT APPROACHES USED TO STUDY NON ADI- ABATIC TRANSITIONS/CURVE CROSSING PROBLEMS 41 2.1 Boundaryconditionmethod . 41 2.1.1 Boundary condition method applied to curve crossing prob- lems ............................. 43 2.2 Green’sfunctionMethod. 45 2.2.1 Green’s function method applied to the curve crossing prob- lems ............................. 46 2.3 Transfermatrixmethod . 49 2.3.1 Transfer matrix method applied to the curve crossing prob- lems ............................. 52 2.4 Time dependent Approach to the Non adiabatic transitions/curve crossingproblems . .. .. ... .. .. .. .. ... .. .. .. 54 v 2.4.1 Methodology1 ....................... 54 2.4.2 Methodology2 ....................... 57 2.5 Computational Approach to the Curve Crossing Problems . 58 2.5.1 CASPT2........................... 58 2.6 Conclusions ............................. 60
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