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A Multireference Density Functional Approach to the Calculation of the Excited States of Uranium Ions

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A Multireference Density Functional Approach to the Calculation of the Excited States of Uranium Ions Air Force Institute of Technology AFIT Scholar Theses and Dissertations Student Graduate Works 3-19-2007 A Multireference Density Functional Approach to the Calculation of the Excited States of Uranium Ions Eric V. Beck Follow this and additional works at: https://scholar.afit.edu/etd Part of the Nuclear Engineering Commons Recommended Citation Beck, Eric V., "A Multireference Density Functional Approach to the Calculation of the Excited States of Uranium Ions" (2007). Theses and Dissertations. 2895. https://scholar.afit.edu/etd/2895 This Dissertation is brought to you for free and open access by the Student Graduate Works at AFIT Scholar. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of AFIT Scholar. For more information, please contact [email protected]. AFIT/DS/ENP/07-01 A MULTIREFERENCE DENSITY FUNCTIONAL APPROACH TO THE CALCULATION OF THE EXCITED STATES OF URANIUM IONS DISSERTATION Eric V. Beck Major, USAF AFIT/DS/ENP/07-01 Approved for public release; distribution unlimited The views expressed in this dissertation are those of the author and do not reflect the official policy or position of the Department of Defense or the United States Government. AFIT/DS/ENP/07-01 A MULTIREFERENCE DENSITY FUNCTIONAL APPROACH TO THE CALCULATION OF THE EXCITED STATES OF URANIUM IONS DISSERTATION Presented to the Faculty of the School of Engineering Physics of the Air Force Institute of Technology Air University In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Eric V. Beck, BS, MS Major, USAF March, 2007 Approved for public release; distribution unlimited Acknowledgements Many people, my wife and family especially, are directly responsible for my successes here at AFIT, in my career, and in my life. While the list is far too long to mention each individual specifically, rest assured, your presence in my life was and is appreciated. Eric V. Beck iii Table of Contents Page Acknowledgements ................................... iii ListofFigures ..................................... vii ListofTables...................................... ix Abstract......................................... xii I. Introduction ................................. 1 StatementofProblem.......................... 3 ResearchObjectives........................... 8 BoundaryConditions .......................... 9 ResearchOverview ........................... 10 II. UraniumIonCalculations.......................... 11 Relativistic Effects in Chemistry . 13 Theory.................................. 16 TheDiracEquation ....................... 17 Relativistic Effective Core Potentials . 20 Basis Sets for Use with Shape-consistent RECPs . 22 Methodology............................... 22 Selection of the Reference Space . 24 Results.................................. 25 Analysis ................................. 30 III. The COLUMBUS-basedDFT/MRCIModel.. .. .. .. .. 34 Theory.................................. 34 Non-relativistic Quantum Theory . 34 iv Page TheKohn-ShamApproachtoDFT . 37 Configuration Interaction and the Graphical Unitary Group Approach............................. 40 HybridDFT/MRCIModel ................... 52 DFT/MRCI Model Implementation with COLUMBUS ......... 60 Density Functional Theory Interface to COLUMBUS ...... 60 Correlation Energy in the Repartitioned Hamiltonian . 67 CIUDG BasedDFT/MRCI.................... 68 Testing the DFT/MRCI Model Implementation within CIUDG 74 Using the COLUMBUS DFT/MRCIModel................ 78 ResultsandAnalysis .......................... 84 CarbonMonoxide ........................ 84 BoronFluoride.......................... 87 BromineAtom .......................... 89 Uranium+5Ion ......................... 91 Uranium+4Ion ......................... 93 2+ Uranyl Ion, UO2 ........................ 94 IV. Conclusions.................................. 98 Uranium Shape-Consistent RECP Accuracy Assessment . 98 DFT/MRCIModel ........................... 100 Research Objective Successes and Failures . 108 FutureWork............................... 111 DFT/MRCI with Other Correlation Density Functionals . 111 Integration of DFT Within SCFPQ using Abelian Point Groups 112 Hybrid Exchange-Correlation Density Functional Implementa- tion ................................ 113 Investigations into the Theoretical Basis of the DFT/MRCI Method.............................. 114 Summary................................. 115 v Page AppendixA. ListofAcronyms.......................... 118 Appendix B. DFT/MRCI Damping Parameter Selection . 120 Hydrogen Molecule, cc-pVDZ Basis . 121 Hydrogen Molecule, cc-pVTZ Basis . 123 Helium Atom, cc-pVDZ Basis . 125 Helium Atom, cc-pVTZ Basis . 127 Lithium Atom, cc-pVDZ Basis . 129 Lithium Atom, cc-pVTZ Basis . 131 Beryllium Atom, cc-pVDZ Basis . 134 Beryllium Atom, cc-pVTZ Basis . 136 Boron Atom, cc-pVDZ Basis . 137 Carbon Atom, cc-pVDZ Basis . 140 Nitrogen Atom, cc-pVDZ Basis . 143 Oxygen atom, cc-pVDZ Basis set . 146 Fluorine Atom, cc-pVDZ Basis . 149 Neon Atom, cc-pVDZ Basis . 151 Beryllium Dimer, cc-pVDZ Basis . 153 Appendix C. A Procedure for Conversion of 3sd Basis Functions into Equiv- alent 1s Functions......................... 156 Vita........................................... 165 Bibliography ...................................... 166 vi List of Figures Figure Page 1. Distinct Row Table Graph for Multi Reference Single and Double Ex- citation Configuration Interaction Expansion (122) . 44 2. OneBodyLoopSegments(124) ..................... 50 3. Non-zero Loop Contributions to the CI Hamiltonian (120:92) . 51 4. Eigenvalue Symmetry Breaking in Two-electron Spin Coupling Example 57 5. Effect of Exponent on Eigenvalue . 58 6. Density Plot of Absoulte Error Between Exact ∆E and Damped ∆E 59 7. DFT/MRCI Averaged, Absolute Error with Respect to Full Configu- rationInteractionResults.. .. .. .. .. .. .. 83 8. H2 cc-pVDZ basis DFT/MRCI Absolute Error with Respect to FCI Results................................... 122 9. H2 cc-pVTZ basis DFT/MRCI Absolute Error with Respect to FCI Results................................... 124 10. He Atom, cc-pVDZ basis DFT/MRCI Absolute Error with Respect to FCIResults ................................ 126 11. He Atom, cc-pVTZ Basis DFT/MRCI Absolute Error with Respect to FCIResults ................................ 128 12. Li Atom, cc-pVDZ Basis DFT/MRCI Absolute Error with Respect to FCIResults ................................ 130 13. Li Atom, cc-pVTZ Basis DFT/MRCI Absolute Error with Respect to FCIResults ................................ 133 14. Be Atom, cc-pVDZ Basis DFT/MRCI Absolute Error with Respect to FCIResults ................................ 135 15. Be Atom, cc-pVTZ Basis DFT/MRCI Absolute Error with Respect to FCIResults ................................ 137 16. B Atom, cc-pVDZ Basis DFT/MRCI Absolute Error with Respect to FCIResults ................................ 139 17. C Atom, cc-pVDZ Basis DFT/MRCI Absolute Error with Respect to FCIResults ................................ 142 vii Figure Page 18. N Atom, cc-pVDZ Basis DFT/MRCI Absolute Error with Respect to FCIResults ................................ 145 19. O Atom, cc-pVDZ Basis DFT/MRCI Absolute Error with Respect to FCIResults ................................ 148 20. F Atom, cc-pVDZ Basis DFT/MRCI Absolute Error with Respect to FCIResults ................................ 150 21. Ne Atom, cc-pVDZ Basis DFT/MRCI Absolute Error with Respect to FCIResults ................................ 152 22. Be Dimer, cc-pVDZ Basis DFT/MRCI Absolute Error with Respect to FCIResults ................................ 155 23. Uranium cc-pVDZ 68 Electron RECP 3sd and Converted 1s Orbitals 159 24. Uranium cc-pVDZ 68 Electron RECP 3sd and Converted 1s Orbital RadialProbabilityDensities . 159 25. Uranium cc-pVDZ 68 Electron RECP 3sd and Converted 1s Radial Probability Density Difference . 160 viii List of Tables Table Page 1. Valence Electrons Included in Uranium PAC-RECPs . 24 2. Basis Sets for Use With Uranium PAC-RECPs . 25 3. U5+ Results, 5f 1 ReferenceSpace .................... 26 4. U5+ Results, (5f6d)1 ReferenceSpace.................. 26 5. U4+ Results, 5f 2 ReferenceSpace .................... 28 6. U4+ Results,(5f6d)2 ReferenceSpace .................. 29 7. U4+ Ionization Potential, 5f n References ................ 30 8. U4+ Ionization Potential, (5f6d)n References.............. 30 9. Optimized DFT/MRCI Parameters for the BHLYP Functional for Sin- glet and Triplet States (63) . 53 10. SCFPQ + Vxc and NWChem DFT Energies, CPBE96 Correlation Func- tional.................................... 66 11. DFT/MRCITestingCases ........................ 76 12. TestingResults .............................. 77 13. Carbon Monoxide cc-pVTZ DFT/MRCI Results . 86 14. BF cc-pVTZ DFT/MRCI Results . 87 15. Bromine Atom cc-pVTZ DFT/MRCI Results . 90 16. U5+ DFT/MRCI Results, (5f6d)1 ReferenceSpace .......... 92 17. U4+ DFT/MRCI and MR-SOCISD Results, 5f2 Reference Space . 93 2+ 18. UO2 DFT/MRCIresults ........................ 95 2+ 19. UO2 Calculated and Measured Fluorescent Series . 96 20. MR-CISD Hartree-Fock Configuration Coefficients For Systems Used for Damping Parameter Determination . 120 21. Hydrogen Molecule, cc-pVDZ Basis Set, Full CI Results . 121 22. DFT/MRCI Error Analysis for Hydrogen Molecule, cc-pVDZ Basis . 121 23. Full CI Results for Hydrogen Molecule, cc-pVTZ Basis . 123 24. DFT/MRCI Error Analysis for Hydrogen Molecule, cc-pVTZ Basis . 123 ix Table Page 25. Full CI Results for Helium Atom, cc-pVDZ Basis . 125 26. DFT/MRCI Error Analysis for Helium Atom, cc-pVDZ
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