University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 8-2010 Density Functional Calculation of X-Ray Absorption Spectra within the Core Hole Approximation: An Implementation in NWChem William Ben Carlen [email protected] Follow this and additional works at: https://trace.tennessee.edu/utk_graddiss Part of the Physical Chemistry Commons Recommended Citation Carlen, William Ben, "Density Functional Calculation of X-Ray Absorption Spectra within the Core Hole Approximation: An Implementation in NWChem. " PhD diss., University of Tennessee, 2010. https://trace.tennessee.edu/utk_graddiss/767 This Dissertation is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a dissertation written by William Ben Carlen entitled "Density Functional Calculation of X-Ray Absorption Spectra within the Core Hole Approximation: An Implementation in NWChem." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Doctor of Philosophy, with a major in Chemistry. Robert J. Harrison, Major Professor We have read this dissertation and recommend its acceptance: Craig E. Barnes, Thomas J. Papenbrock, Charles Feigerle, Robert J. Hinde Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) To the Graduate Council: I am submitting herewith a dissertation written by William Benton Carlen entitled “Den- sity Functional Calculation of X-ray Absorption Spectra within the Core Hole Approxima- tion: An Implementation in NWChem.” I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, with a major in Chemistry. Robert J. Harrison Major Professor We have read this dissertation and recommend its acceptance: Robert J. Hinde Craig E. Barnes Charles Feigerle Thomas F. Papenbrock Accepted for the Council: Carolyn Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official student records.) Density Functional Calculation of X-Ray Absorption Spectra within the Core Hole Approximation: An Implementation in NWChem A Dissertation Presented for the Doctor of Philosophy Degree The University of Tennessee, Knoxville William Benton Carlen August 2010 Acknowledgments I would like to express my deep gratitude to Dr. Robert J. Harrison for the support, guidance, and encouragement given to me througout my academic career. I would also like to thank Drs. Charles Fiegerle, Robert Hinde, Craig Barnes, and Thomas Papenbrock for supporting my doctoral candidacy by serving on my committee. Most importantly, I would like to express my love and appreciation for my family for believing in me and supporting me throughout my career. This work was supported by the Center for Information Technology Research and the Science Alliance Computational Sciences Initiative at the University of Tennessee and by the National Institutes of Health under Grant No. HD52472-01. ii Abstract Density functional theory is used to calculate the core excitation spectra of titanium structures. Specifically, the core-hole approximation is used. In this scenario, the excitation energies of core electrons are calculated using the approximation that the core energy level be frozen throughout the relaxation process of the orbitals. This allows a more acurate determination of the resulting X-ray spectra. The method described has been implemented in an NWChem module. iii Contents 1 Methods in Quantum Chemistry 1 1.1 The Hamiltonian and the Schr¨odingerEquation ................ 1 1.1.1 Variational Theorem ........................... 2 1.2 LCAO ....................................... 3 1.2.1 Basis Sets ................................. 3 1.3 Hartree-Fock Theory ............................... 7 1.3.1 The Slater Determinant ......................... 7 1.3.2 The Hartree-Fock Energy ........................ 7 1.3.3 The Hartree-Fock Equations ...................... 8 1.3.4 Koopmans’ Theorem ........................... 9 1.3.5 Correlation Energy ............................ 9 1.4 Coupled-Cluster Theory ............................. 10 1.4.1 The Exponential Ansatz ......................... 11 1.4.2 Size-Consistency ............................. 11 1.4.3 Size-Extensivity ............................. 12 1.4.4 The Coupled-Cluster Energy Equation ................. 13 1.4.5 Computational Cost ........................... 14 1.4.6 Excited States in CC Theory ...................... 15 1.5 Perturbation Theory ............................... 16 1.6 DFT ........................................ 18 1.6.1 The First Hohenberg-Kohn Theorem .................. 18 1.6.2 N-Representability ............................ 20 iv 1.6.3 The Second Hohenberg-Kohn Theorem ................. 20 1.6.4 The Energy Functional ......................... 21 1.6.5 The Kohn-Sham Equations ....................... 22 1.6.6 Janak’s Theorem ............................. 23 2 X-ray Spectroscopy 25 2.1 X-Ray Spectroscopies .............................. 26 2.2 Discussion of the Theory of XAS ........................ 26 2.3 Multiple Scattering Theory ........................... 29 2.3.1 Derivation of MS Equations ....................... 29 2.3.2 Muffin-Tin Potential ........................... 33 2.3.3 Scattering Path Operators ........................ 34 2.3.4 The Multiple-Scattering Function χL,L0 ................ 35 2.4 State of the Art in XAS Software ........................ 35 2.5 Physical Process of X-ray Absorption ...................... 36 3 Theoretical Treatment of the Core Hole State 38 3.1 Goals and Methods ................................ 38 3.2 The Core Hole State ............................... 40 3.2.1 Limiting Cases of Ionization and Excitation .............. 41 3.3 Core Level Constraint .............................. 41 3.4 The Cross-Section for Absorption ........................ 42 3.4.1 Dipole Approximation .......................... 43 3.4.2 The Excitation Energy .......................... 43 3.4.3 Transition Moments ........................... 45 3.5 Approximations .................................. 47 3.5.1 Systematic Improvement Toward Exact Result ............ 48 4 Methods used in NWChem 49 4.1 The XAS Directive in NWChem ........................ 49 4.2 Enforcing Core-Hole Orthogonality ....................... 51 4.3 Calculating the Cross-Section .......................... 52 v 4.3.1 Subroutine print info ........................... 53 4.3.2 Subroutine Intensity ........................... 55 4.4 Broadening of the Spectra ............................ 58 4.4.1 Lifetime of the Excited State ...................... 59 4.4.2 Thermal Effects .............................. 61 4.5 Identification of Spectral Peaks ......................... 64 4.5.1 Components of MO Vectors ....................... 64 4.5.2 Mulliken Population Analysis ...................... 64 4.6 Parallelization ................................... 66 4.7 Convergence .................................... 67 4.7.1 Damping ................................. 67 4.7.2 Levelshifting ............................... 67 5 Applications and Results 69 5.1 Chemical Motivation ............................... 69 5.2 XAS Studies on Ti Containing Systems .................... 71 5.2.1 Experimental Studies .......................... 71 5.2.2 Theoretical Studies ............................ 72 5.3 Theory and Experiment Working Together ................... 73 5.3.1 Effects of Chlorine on Spectral Features ................ 73 5.3.2 Goals of Research ............................ 73 5.4 Model Systems .................................. 74 5.5 Identification of the Absorption Edge ...................... 74 5.6 Changes Upon Inclusion of Thermal Effects using MD ............ 82 5.7 Convergence with System Size .......................... 83 5.8 Comparison of Excitation and Ionization .................... 84 5.8.1 Energy Gap Comparison for Ionization/Excitation .......... 84 5.8.2 Intensity of Pre-Edge Features ..................... 86 5.8.3 Intensity of In-Edge Feature ....................... 88 5.8.4 Pre-Edge Shift .............................. 89 5.9 Comparison of LDA and B3LYP Functionals ................. 89 vi 5.9.1 Energy Gap Comparison for LDA/B3LYP ............... 93 6 Identification of Spectral Features 94 6.1 Orbital Analysis ................................. 94 6.1.1 Single-Shelled Systems .......................... 94 6.1.2 Two-Shelled Systems ........................... 94 6.2 Mulliken Population Analysis .......................... 99 6.2.1 Single-Shelled Systems .......................... 99 6.2.2 Two-Shelled Systems ........................... 99 6.3 Conclusions .................................... 102 6.3.1 Nature of States in the Absoprtion Edge ................ 102 6.3.2 Chlorine Dependence of Edge Feature ................. 102 6.3.3 Mulliken Population Analysis ...................... 103 6.3.4 Ionization vs. Excitation ......................... 103 6.3.5 Thermal Averaging ............................ 103 6.3.6 LDA vs. B3LYP ............................