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Numerical Treatment of Schrödinger's Equation for One-Particle and Two-Particle Systems Using Matrix Method

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Numerical Treatment of Schrödinger's Equation for One-Particle and Two-Particle Systems Using Matrix Method Dissertations and Theses 12-2019 Numerical Treatment of Schrödinger’s Equation for One-Particle and Two-Particle Systems Using Matrix Method Spatika Dasharati Iyengar Follow this and additional works at: https://commons.erau.edu/edt Part of the Aerospace Engineering Commons This Dissertation - Open Access is brought to you for free and open access by Scholarly Commons. It has been accepted for inclusion in Dissertations and Theses by an authorized administrator of Scholarly Commons. For more information, please contact [email protected]. NUMERICAL TREATMENT OF SCHRÖDINGER’S EQUATION FOR ONE-PARTICLE AND TWO-PARTICLE SYSTEMS USING MATRIX METHOD A Dissertation Submitted to the Faculty of Embry-Riddle Aeronautical University by Spatika Dasharati Iyengar In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Aerospace Engineering December 2019 Embry-Riddle Aeronautical University Daytona Beach, Florida iii ACKNOWLEDGMENTS I would like to express my deepest gratitude to my Dissertation Advisor, Dr. Eric Royce Perrell for his continuous support, patience, motivation and immense knowledge. Besides my advisor, I would like to thank the rest of my committee: Dr. John Ekaterinaris, Dr. Bereket Berhane, Dr William Engblom and Dr. Yechiel Crispin for their encouragement and insightful comments. Last but not the least, I would like to thank my family for everything they have done to help me fulfill my dream of attaining a PhD degree. iv TABLE OF CONTENTS Page LIST OF TABLES :::::::::::::::::::::::::::::::: vi LIST OF FIGURES ::::::::::::::::::::::::::::::: vii SYMBOLS :::::::::::::::::::::::::::::::::::: xi ABBREVIATIONS :::::::::::::::::::::::::::::::: xv ABSTRACT ::::::::::::::::::::::::::::::::::: xix 1. Introduction :::::::::::::::::::::::::::::::::: 1 1.1. Background ::::::::::::::::::::::::::::::: 1 1.2. Ab initio Calculations ::::::::::::::::::::::::: 4 1.3. Potential Energy Surfaces ::::::::::::::::::::::: 7 2. Literature Review ::::::::::::::::::::::::::::::: 12 2.1. Analytical Solutions :::::::::::::::::::::::::: 12 2.2. Numerical Solutions ::::::::::::::::::::::::::: 21 2.3. Approximate solutions ::::::::::::::::::::::::: 41 2.3.1. Hybrid solutions :::::::::::::::::::::::: 79 2.4. Quantum Chemistry Softwares ::::::::::::::::::::: 83 3. Numerical Approach ::::::::::::::::::::::::::::: 85 3.1. One-Particle System :::::::::::::::::::::::::: 86 3.1.1. Non-dimensional form of Schr¨odinger'sEquation ::::::: 86 3.1.2. Setting up the Matrix Eigenvalue Problem :::::::::: 87 3.1.3. Singularity Treatment ::::::::::::::::::::: 101 3.2. Two-particle system :::::::::::::::::::::::::: 116 3.2.1. Modeling the averaged Repulsion potential term ::::::: 118 4. Code Structure and Solver :::::::::::::::::::::::::: 120 4.1. Code description :::::::::::::::::::::::::::: 121 4.1.1. Header Files ::::::::::::::::::::::::::: 121 4.1.2. C++ functions ::::::::::::::::::::::::: 122 4.1.3. Post-processing ::::::::::::::::::::::::: 127 4.2. How to run the code :::::::::::::::::::::::::: 127 5. Results and Discussion :::::::::::::::::::::::::::: 130 5.1. One-particle systems :::::::::::::::::::::::::: 130 5.1.1. Uniform Grid Spacing with Second-order differences ::::: 130 5.1.2. Uniform Grid Spacing with Fourth-order differences ::::: 149 5.1.3. Non-Uniform Grid Spacing ::::::::::::::::::: 158 v Page 5.1.4. Other Singularity Treatments ::::::::::::::::: 172 5.2. Two-particle system :::::::::::::::::::::::::: 175 5.3. Parallel Scaling ::::::::::::::::::::::::::::: 176 6. Conclusion ::::::::::::::::::::::::::::::::::: 179 7. Recommendations ::::::::::::::::::::::::::::::: 181 REFERENCES :::::::::::::::::::::::::::::::::: 182 vi LIST OF TABLES Table Page 2.1 Real Hydrogen-like wavefunctions ::::::::::::::::::::: 17 5.1 Comparison of numerical and analytical energies for Hydrogen atom with a uniform mesh and second-order differences. :::::::::::: 132 5.2 Comparison of numerical and analytical energies for Helium ion with a uniform mesh and second-order differences. :::::::::::::::: 139 5.3 Comparison of numerical and analytical energies for Lithium ion with a uniform mesh and second-order differences. :::::::::::::::: 144 5.4 Comparison of numerical and analytical energies for Hydrogen atom with a uniform mesh and fourth-order differences. :::::::::::: 149 5.5 Comparison of numerical and analytical energies for Helium ion with a uniform mesh and fourth-order differences. :::::::::::::::: 153 5.6 Comparison of numerical and analytical energies for Lithium ion with a uniform mesh and fourth-order differences. :::::::::::::::: 156 5.7 Comparison of numerical and analytical energies for Hydrogen Atom with a non-uniform mesh having 61 grid points in each direction. ::: 159 5.8 Comparison of numerical and analytical energies for Hydrogen Atom with a non-uniform mesh having 77 grid points in each direction. ::: 163 5.9 Comparison of numerical and analytical energies for Hydrogen Atom with a non-uniform mesh having 91 grid points in each direction. ::: 168 5.10 Comparison of numerical and analytical energies for the case with no singularity treatment :::::::::::::::::::::::::::: 173 5.11 Comparison of numerical and analytical energies for the case with LS fit. 174 5.12 Comparison of numerical and experimental energies for a Helium Atom. 176 5.13 Parallel Scaling capability of the code. :::::::::::::::::: 178 vii LIST OF FIGURES Figure Page 1.1 A sample PES showing the minimum energy path or IRC (top) and the formation of transition state and product along the IRC (bottom). ::: 9 2.1 1s, 2s, 2px, 2py,and 2pz hydrogen-like wavefunctions. Adapted from Chapter 7 The Hydrogen atom; Atomic Orbitals University of Michigan, Chemistry 461/570, S.M. Blinder, 2002, p. 18, Retrieved October 15, 2019, from, http://www.umich.edu/ chem461/QMChap7.pdf . :::::::::::: 18 2.2 Pople Diagram for ab initio computations. Adapted from Electron Correlation - Methods beyond Hartree-Fock how to approach chemical accuracy, Max-Planck-Institute for Chemical Energy Conversion, Alexander Auer, 2014, p. 5, Retrieved October 15, 2019, from, https://cec.mpg.de/ fileadmin/media/Presse/Medien /Auer Electron Correlation.pdf. Reprinted with permission. :::::: 79 3.1 3D Cartesian Grid with the nucleus at the origin :::::::::::: 88 3.2 Coefficient matrix A for a uniform mesh with second-order differences : 92 3.3 Coefficient matrix A for a uniform mesh with fourth-order differences. 95 3.4 Non-uniform Cartesian grid in one-dimension. ::::::::::::: 96 3.5 Coefficient matrix A for a non-uniform mesh. :::::::::::::: 100 3.6 Coefficient matrix A with the origin row containing values obtained from Taylor series extrapolation method and second-order differences : 106 4.1 Makefile for test cases ::::::::::::::::::::::::::: 128 4.2 Sample PBS script for test cases :::::::::::::::::::::: 129 5.1 1s orbital of a Hydrogen Atom for a uniformp mesh with second-order differences (Scaling factor for numerical is 5). ::::::::::: 133 5.2 vs r for 1s orbital of a Hydrogen Atom for a uniform meshp with second-order differences (Scaling factor for numerical is 5). :::: 133 5.3 2s orbital of a Hydrogen Atom for a uniformp mesh with second-order differences (Scaling factor for numerical is 1:5). ::::::::::: 134 5.4 vs r for 2s orbital of a Hydrogen Atom for a uniform meshp with second-order differences (Scaling factor for numerical is 1:5). :::: 134 viii Figure Page 5.5 2px orbital of a Hydrogen Atom for a uniformp mesh with second-order differences (Scaling factor for numerical is 5). :::::::::::: 135 5.6 2py orbital of a Hydrogen Atom for a uniform mesh with second-order differences (No scaling required for numerical ). :::::::::::: 135 5.7 2pz orbital of a Hydrogen Atom for a uniform mesh with second-order differences (No scaling required for numerical ). :::::::::::: 136 5.8 1s orbital of a Helium ion for a uniform meshp with second-order differences (Scaling factor for numerical is 12:5). :::::::::: 140 5.9 vs r for 1s orbital of a Helium ion for a uniform mesh withp second-order differences (Scaling factor for numerical is 12:5). :: 140 5.10 2s orbital of a Helium ion for a uniform meshp with second-order differences (Scaling factor for numerical is 8:5). ::::::::::: 141 5.11 vs r for 2s orbital of a Helium ion for a uniform mesh withp second-order differences (Scaling factor for numerical is 8:5). :::: 141 5.12 2px orbital of a Helium ion for a uniform mesh with second-order differences (No scaling required for numerical ). :::::::::::: 142 5.13 2py orbital of a Helium ion for a uniform mesh with second-order differences (No scaling required for numerical ). :::::::::::: 142 5.14 2pz orbital of a Helium ion for a uniform mesh with second-order differences (No scaling required for numerical ). :::::::::::: 143 5.15 1s orbital of a Lithium ion for a uniform meshp with second-order differences (Scaling factor for numerical is 17:5). :::::::::: 145 5.16 vs r for 1s orbital of a Lithium ion for a uniform mesh withp second-order differences (Scaling factor for numerical is 17:5). :: 146 5.17 2s orbital of a Lithium ion for a uniform meshp with second-order differences (Scaling factor for numerical is 10). ::::::::::: 146 5.18 vs r for 2s orbital of a Lithium ion for a uniform mesh withp second-order differences (Scaling factor for numerical is 10). :::: 147 5.19 2px orbital of a Lithium ion for a uniform meshp with second-order differences (Scaling factor for numerical is 2). :::::::::::: 147 5.20 2py orbital of a Lithium ion for a uniform mesh with second-order differences (No scaling required for numerical ). :::::::::::: 148 5.21 2pz
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