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Model Wavefunction Forms to Describe Strong Correlation in Quantum Chemistry MODEL WAVEFUNCTION FORMS to DESCRIBE STRONG

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Model Wavefunction Forms to Describe Strong Correlation in Quantum Chemistry MODEL WAVEFUNCTION FORMS to DESCRIBE STRONG Model Wavefunction Forms to Describe Strong Correlation in Quantum Chemistry MODEL WAVEFUNCTION FORMS TO DESCRIBE STRONG CORRELATION IN QUANTUM CHEMISTRY BY PAUL ANDREW JOHNSON, B.Sc. a thesis submitted to the department of chemistry & chemical biology and the school of graduate studies of mcmaster university in partial fulfilment of the requirements for the degree of Doctor of Philosophy c Copyright by Paul Andrew Johnson, June 2014 All Rights Reserved Doctor of Philosophy (2014) McMaster University (Chemistry & Chemical Biology) Hamilton, Ontario, Canada TITLE: Model Wavefunction Forms to Describe Strong Correla- tion in Quantum Chemistry AUTHOR: Paul Andrew Johnson B.Sc., (Chemistry) Carleton University, Ottawa, Ontario, Canada SUPERVISORS: Dr. Paul W. Ayers, Prof. Dr. Dimitri Van Neck, Prof. Dr. Patrick Bultinck NUMBER OF PAGES: xvi, 240 ii Never create anything, it will be misinterpreted, it will chain you and follow you for the rest of your life. { Hunter S. Thompson Abstract Standard approaches to approximating solutions to the Schr¨odingerequation rely upon the assumption that there is a single dominant electronic configuration with a small number of low-lying excited states. As such, they are referred to as single reference approaches and are accurate only when the band gap is small in comparison with the inter-electronic repulsion of the valence electrons. Systems for which this is the case are weakly-correlated. If the inter-electronic repulsion larger than the band gap, then a single electronic reference is not an appropriate ansatz for the physical wavefunction. As the strength of the interaction is increased many Slater determinants become near-degenerate, each providing non-negligible contributions. Such systems are strongly-correlated. The goal of the thesis was to employ wavefunctions which are: • non-trivial • tractable • size-consistent and extensive • not given in an active space We have outlined approaches based on wavefunction forms more suited to describe iv strong correlation. Built on the Lie algebra su(2), our first trial wavefunction was the Bethe ansatz solution to the reduced BCS Hamiltonian derived by Richardson. This wavefunction is expressed as a product of quasi-particles each parametrized in terms of quasi-momenta. Consistency among the quasi-momenta is ensured by a set of non-linear equations to be solved numerically. This approach is feasible in both variational and projected flavours. We next considered a wavefunction of the same form without requiring it to be an eigenvector of the reduced BCS Hamiltonian. A variational approach in this case is only feasible provided there are two equiva- lent descriptions in terms of particles and holes. In any case, a bivariational principle and a projected approach are both feasible. Also based on su(2), there is AP1roG (coupled cluster pair doubles). Open-shell singlets and triplets were constructed with the Lie algebras sp(N) and so(4N). The resulting formulae appear to be infea- sible unless it is possible to efficiently compute generalizations of determinants to three-dimensional objects. Treating each spatial orbital individually, all four possible occupations close the Lie superalgebra gl(2j2). v Nederlandstalige samenvatting De klassieke manier om een benaderende oplossing te vinden voor de Schr¨odingerverge- lijking is, om aan te nemen dat ´e´enenkele elektronische configuratie dominant is, samen met een klein aantal laag liggende aangeslagen toestanden. Deze benadering staat bekend als 'single reference'. Het is een goede benadering indien de bandkloof klein is in vergelijking met de inter-elektron afstoting van de valentie elektronen. Een systeem waarvoor dit geldt staat gekend als een zwak gecorreleerd systeem. Als de inter-elektron afstoting groter is dan de bandkloof, kan de golffunctie van het sys- teem niet meer correct beschreven worden door ´e´enenkele Slater determinant. Als de sterkte van de interactie stijgt, dan heb je een groot aantal bijna-ontaarde Slater determinaten nodig om een correct beschrijving van de golffunctie te krijgen. Al deze Slater determinaten zullen een niet verwaarloosbare bijdrage leveren tot de golffunc- tie. Dit soort systemen staat gekend als sterk gecorreleerd. Het doel van dit proefschrift was om golffuncties te gebruiken die: • niet triviaal zijn • computationeel handelbaar • size-consistent en extensive zijn vi • niet gebruik maken van een actieve ruimte We hebben verschillende benaderingen onderzocht die meer geschikt zijn om een sterk gecorreleerd systeem te beschrijven. Onze eerste test golffunctie was de Bethe ansatz oplossing van de gereduceerde BCS hamiltoniaan afgeleid door Richardson. Deze golffunctie is een product van quasideeltjes die geparametriseerd worden voor een quasimomentum. Consistentie tussen de quasimomentum wordt bereikt via de numerieke oplossing van een stel niet-lineaire vergelijkingen. Je kunt deze benader- ing zowel variationeel als via projectoren berekenen. De volgende golffunctie die we onderzochten had dezelfde vorm maar was geen oplossing meer van de gereduceerde BCS hamiltoniaan. Daardoor is een variationele oplossing enkel mogelijk als er 2 equivalente beschri- jvingen zijn in functie van deeltjes en gaten. In het algemene geval kun je een pro- jectoren methode of een bivariationele oplossingen gebruiken. Er zijn nog andere op su(2) gebaseerde benaderingen mogelijk, namelijk AP1roG (coupled cluster pair dou- bles). Open-schil singlet en triplets werden opgebouwd uit de Lie algebra sp(N) en so(4N). De resulterende golffunctie is helaas niet bruikbaar tenzij er een eficiente manier wordt gevonden om de veralgemeende determinant van een drie-dimensionaal object uit te rekenen. Als je alle ruimtelijke orbitalen individueel behandeld, dan zullen de 4 mogelijk bezettingen per orbitaal de Lie superalgebra gl(2j2) sluiten. vii Acknowledgments Time flies when you do most of your real work after midnight. Five months can go by and it feels like one long sleepless night. { Hunter S. Thompson First and foremost: NSERC and the Vanier Canada Graduate Scholarships The friends from the old days at Carleton that I see every time I go home: Dr. Se´an T. Barry, Dr. Peter Gordon, Jason Coyle, Peter Pallister, Julie Pallister, Agnieszka Kurek, and David Mandia. Those who were so kind as to have me as a visitor: Prof. Dr. Evert-Jan Baerends, Dr. Patrick Cassam-Chena¨ı, Prof. Dr. Jean-S´ebastienCaux, Prof. Dr. Werner Kutzelnigg, Dr. Paola Gori-Giorgi, Dr. Oleg Gritsenko, Prof. Dr. Christof H¨attig, Dr. Fr´ed´ericPatras, Prof. Dr. Markus Reiher, Dr. Andreas Savin, and Dr. Julien Toulouse. On my extended visits those that made me feel so welcome in Ghent: Ward Poel- mans, Dr. Sebastian Wouters, Dr. Brecht Verstichel, Dr. Matthias Degroote, Mario Van Raemdonck, Pieter Claeys, Dr. Veerle Hellemans, Prof. Dr. Toon Verstraelen, Prof. Dr. Karen Hemelsoet, Dr. An Ghysels, Dr. Marc Van Houteghem, Dr. Sofie Van Damme, Dr. Helen van Aggelen, Lennart Joos, Thierry De Meyer. Of course viii the CMM as a whole and Prof. Dr. Michel Waroquier and Prof. Dr. Veronique Van Speybroeck. Our group here has certainly changed over the years. I thought I was the new guy. The postdocs: Dr. Peter Limacher, Dr. Pawe l Tecmer, Dr. Katharina Boguslawski, Dr. Steven Burger, Dr. Carlos C´ardenas,Dr. Alfredo Guevara. The former graduate students: Dr. James Anderson, Dr. Rogelio Cuevas, Dr. Annie Liu, Dr. Debajit Chakraborty, Dr. Sandra Rabi. The current graduate students: Farnaz Heidar Zadeh, Ahmed Mohammed, Matt Chan, Cristina Gonz´alez,David Kim, Derrick Yang, Yilin Zhao. Also Eleonora Echegaray, Pavel Kulikov, Ivan Vinogradov, and Chunying Rong. For all of their help, and more importantly, their support: Dr. Brian McCarry, Dr. Willie Leigh, Dr. Alex Adronov, Connie Carrabs, Christine Cosgrove, Jane Garneau, Linda Spruce, Amin Ilmane, Pierre Van Bocxstaele, and Wim Dewitte. The friends that I have made at McMaster, in particular Greg Bahun, Patricia Edem, and Dr. Todd Whitehorne. My committee members Dr. Randy Dumont and Dr. Sung-Sik Lee. The various friends and well-wishers: Dr. Marcel Nooijen, Dr. Artur Izmaylov, Dr. Viktor Staroverov, Dr. Erin Johnson, and Dr. Gino DiLabio. Without doubt those that have had the largest impact on my career and education in many respects: Prof. Dr. Patrick Bultinck, Dr. Stijn De Baerdemacker, and Prof. Dr. Dimitri Van Neck. Mom, Dad, Sarah, I love you guys. Don't ever forget it. And of course, who could forget Dr. Paul W. Ayers. Paul your support and friendship have meant more to me than you'll ever know. ix Contents Abstract iv Nederlandstalige samenvatting vi Acknowledgments viii 1 Introduction 1 1.1 Bras and kets . .2 1.2 The Schr¨odingerequation in Chemistry . .4 1.3 Many-Electron States: First Quantization . .7 1.4 Many-Electron States: Second Quantization . .9 1.4.1 Electron Creators and Annihilators . 10 1.4.2 Upper and Lower Indices . 11 1.4.3 Operators . 12 1.4.4 Scalar products and Wick's Theorem . 13 1.5 Many-Electron States: Hopf Algebra . 17 1.5.1 Fock Space: Algebra Structure . 17 1.5.2 Fock Space: Coalgebra Structure . 19 1.5.3 Fock Space: Hopf Algebra Structure . 20 x 1.6 The Standard Approaches . 21 1.6.1 Hartree-Fock . 24 1.6.2 Møller-Plesset Perturbation Theory . 27 1.6.3 Coupled-Cluster . 28 1.6.4 Configuration Interaction . 30 1.7 Density Functional Theory . 31 1.7.1 Hohenberg-Kohn Theorems . 31 1.7.2 Kohn-Sham Density Functional Theory . 35 1.8 Reduced Density Matrix Approaches . 37 1.9 Weak and Strong Electron Correlation . 41 1.10 Size-consistency and Extensivity . 42 1.11 Geminals . 43 1.11.1 Seniority . 44 1.11.2 Geminal Flavours . 44 2 Bethe Ansatz Wavefunction Form: su(2) 50 2.1 Quasi-spin representation of su(2) . 50 2.2 Integrability . 52 2.2.1 Definitions . 52 2.2.2 Gaudin Algebra . 53 2.3 Reduced BCS Hamiltonian: Bethe Ansatz . 57 2.4 Richardson's Equations . 61 2.5 Scalar Products . 69 2.5.1 Slavnov's Theorem . 69 2.5.2 Norm .
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