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A Combined Experimental and Theoretical Study of The A COMBINED EXPERIMENTAL AND THEORETICAL STUDY OF THE ELECTRONIC STRUCTURE OF MOLECULES BY ELECTRON MOMENTUM SPECTROSCOPY AND DENSITY FUNCTIONAL THEORY By Patrick Duffy B.Sc. (Hons.), U.B.C. , 1987 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF CHEMISTRY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 1995 © Patrick Duffy, 1995 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Chemistry The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1Z1 Date: WMCL if IK Abstract New theoretical models for the instrumental angular resolution function and for the experimental cross-section for electron momentum spectroscopy (EMS) are proposed, tested, and evaluated. For the experimental resolution function, the existing Gaussian Ap method is discussed and found inadequate as it is not representative of the spectrometer itself. A new model for the experimental resolution function, first proposed by Bawagan and Brion (Chem. Phys. 144, 167 (1990)), is considered in detail. An analytic form for the resulting resolution function, which involves the use of Gaussians in 6 and (f>, is developed, tested, and evaluated, and found to be both more realistic physically and to cause high-level quantum-mechanical calculations of the EMS angular cross-section to come into better agreement with experiment than when the Ap method (commonly employed previously) is used. The new experimental resolution function is further tested, in conjunction with exist­ ing theoretical methods, in an EMS study of acetylene. Three common ah initio methods for the calculation of approximate EMS angular cross-sections (Roothaan-Hartree-Fock, configuration interaction, and Green function) are considered. It is found that, for this molecule, the comparatively simple Hartree-Fock method provides an adequate descrip­ tion of the outer-valence electron angular cross-sections, and that the Green Function and multi-reference singles and doubles configuration interaction calculations also dis­ cussed are therefore not required when calculating angular cross-sections for acetylene. However, because of the pronounced breakdown of the single-particle model of ionization in the inner-valence region, these latter methods are found to be necessary for a quantita­ tive prediction of the angular cross-sections in that region. Likewise, simple Koopmans' ii ionization potentials predict the binding energy spectrum of acetylene relatively accu­ rately in the outer-valence region, while correlated treatments are found to be necessary to successfully predict the observed breakdown of the single-particle picture of ionization in the inner-valence region. Because of the general complexity of the calculations necessary for the calculation of EMS binding energies and cross-sections by conventional ab initio methods, and because electron momentum spectroscopy studies are evolving towards more complex systems, including biomolecules, a new method, less computationally intensive than conventional ab initio methods, is proposed for use: density functional theory (DFT). Binding energies and EMS cross-sections are calculated using DFT and compared with the appropriate ab initio (SCF and CI) results. DFT is found to provide very good predictions of binding energies for a wide range of molecules, and to provide EMS cross-sections of comparable quality to Hartree-Fock. The overall charge density provided by DFT within the local density approximation is further tested, in order to better determine the reliability of DFT for both EMS calculations and the prediction of other properties based on the electron density. It is found that while DFT (within the local density approximation) generally predicts outer-spatial properties based on the density with reasonable accuracy, it is unreliable for calculation of properties which depend on an accurate description of the electron density near the nucleus. Use of (spatially) non-local functionals is found to not improve calculated results significantly. iii Table of Contents Abstract ii List of Tables ix List of Figures xi Acknowledgements xiv Preface xvi Quotation xvii 1 Models Used in Electronic Structure 1 1.1 Theoretical Models for Atomic and Molecular Structure 1 1.1.1 Early Concepts 2 1.2 Quantum Mechanics 3 1.2.1 The Born-Oppenheimer Approximation 4 1.3 The Hartree-Fock Approximation 5 1.3.1 The LCAO approximation 6 1.3.2 Post Hartree-Fock Methods 6 1.3.3 Limitations of Hartree-Fock and Post Hartree-Fock Methods ... 7 1.4 Density Functional Theory 8 1.4.1 Two Early Models 8 1.4.2 Hohenberg and Kohn 9 iv 1.4.3 Kohn and Sham 10 1.4.4 Better Approximations to the Exchange-Correlation Potential. 11 1.5 Electron Momentum Spectroscopy 11 1.5.1 A Brief History of EMS 12 1.6 Scope of This Thesis 14 2 Modeling the Experimental Cross-Section 15 2.1 A Brief Overview of the EMS Experiment 15 2.2 The Distorted Wave Impulse Approximation 17 2.2.1 Assumptions in the Model 17 2.2.2 Limitations of the Model 18 2.3 The Plane Wave Impulse Approximation 19 2.3.1 Assumptions in the Model 19 2.3.2 Limitations of the PWIA 21 2.4 Information Obtained Within the PWIA 21 2.4.1 Binding Energy Spectra 22 2.4.2 Experimental Momentum Profiles 22 2.5 Calculation of the PWIA Cross-Section: The Generalized Overlap and the Target Hartree-Fock Approximation 23 2.5.1 Configuration Interaction 24 2.5.2 Green Functions 25 2.5.3 The Spectroscopic Sum Rule 25 2.5.4 The Target Hartree-Fock Approximation 26 2.6 Considerations for Calculations 27 2.6.1 Basis Sets 27 2.6.2 Computational Requirements of the Methods 28 v 3 Experimental Details 30 3.1 The Vacuum System 30 3.2 Electron Optics 32 3.3 Detection Circuitry and Coincidence Electronics 34 3.4 Computer Control 37 4 Modeling Finite Resolution Effects 39 4.1 Introduction 39 4.2 Conceptual Framework 45 4.3 Computational Details 53 4.3.1 Monte Carlo Method 54 4.3.2 Analytic Method 56 4.4 Results 58 4.4.1 Evaluation of the Ap, UW-PG, and GW-PG Models 59 4.4.2 Comparison of the GW-PG and MAGW Methods 62 4.4.3 Non-Gaussian Angular Resolution Functions 64 4.5 Conclusions 65 5 Evaluation of Existing EMS Models: A Study of Acetylene 67 5.1 Introduction 67 5.2 Experimental Method 70 5.3 Computational Details 71 5.3.1 Preliminary Details 71 5.3.2 Post-Target Hartree-Fock Calculations and the Effects of Correla­ tion and Relaxation 73 5.4 Results and Discussion 76 5.4.1 Binding Energy Spectra 77 vi 5.4.2 Experimental and Theoretical Momentum Profiles for the Valence Orbitals of C2H2 79 5.4.3 Detailed Studies of the Binding Energy Spectra and Momentum Distributions in the Inner Valence Region 88 5.5 Conclusions 95 6 Density Functional Theory 96 6.1 Hartree-Fock Theory 96 6.1.1 Computational Implementation 98 6.2 Density Functional Theory 99 6.3 Computational Implementation 101 6.3.1 Approximations to the Exchange-Correlation Potential 101 6.3.2 The Xa Potential 102 6.3.3 Local Potentials Including Correlation 103 6.3.4 Non-local Exchange and Correlation Potentials 103 6.3.5 Evaluation of the Exchange and Correlation Potential 105 7 EMS from DFT: The Target Kohn-Sham Approximation 107 7.1 Introduction 107 7.2 Theoretical Background 108 7.2.1 Dyson's equation and target approximations 109 7.2.2 Kohn-Sham density-functional theory 114 7.2.3 Approximate Dyson orbitals from DFT 116 7.3 Computational details 123 7.4 Results and discussion 130 7.4.1 Target Kohn-Sham Approximation in the LDA 132 7.4.2 Ad hoc DFT approximations for Dyson orbitals 144 vii 7.4.3 Effect of the Functional 148 7.5 Conclusions 152 8 Calculation of Ionization Potentials from Density Functional Theory 157 8.1 Introduction 157 8.2 Density Functional Computations 159 8.3 Results and Discussion 160 9 Testing the Accuracy of the DFT Electron Density 166 9.1 Introduction 166 9.2 Computational Details 167 9.2.1 Preliminary Considerations 171 9.3 Results and Discussion 174 9.3.1 Non-local Functionals 185 9.4 Conclusions 186 10 Conclusions 188 List of Abbreviations 190 Bibliography 193 A Quadrature for the Analytic Method 207 B Contributions to the Width of the Momentum Resolution Function 208 C Orbital Energy Shifts 211 viii List of Tables 5.1 Total energies of C2H2 calculated using several basis sets and methods. 72 5.2 Vertical ionization potentials for the principal ionic states of C2H2". ... 75 7.1 Basis set dependence of the total energy, dipole moment, and principle components of the dipole polarizability of H20 127 7.2 Absolute peak heights and positions of the maxima of MDs for the valence orbitals of CH4, NH3, H20, HF, Ne, and C2H2 137 7.3 Symmetry breaking in DFT calculations on excited states of C2H2 as in­ dicated by the nonzero dipole moment 146 7.4 Comparison of the negative of Kohn-Sham orbital energies at full (KS orbital energy) and half (transition state method) occupancy with exper­ imental ionization potentials, for C2H2 and H20, using the LDAxc func­ tional and the ANO+ basis set 147 7.5 EiFect of exchange-correlation functional on the dipole moments and po- larizabilities of H20, using the ANO+ basis set 151 8.1 Ionization potentials of C2H2 as calculated using different methods in den­ sity functional theory.
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