Syracuse University SURFACE Dissertations - ALL SURFACE December 2016 Development of multicomponent coupled-cluster theory and its application to nanoclusters and molecular systems Benjamin Harrison Ellis Syracuse University Follow this and additional works at: https://surface.syr.edu/etd Part of the Physical Sciences and Mathematics Commons Recommended Citation Ellis, Benjamin Harrison, "Development of multicomponent coupled-cluster theory and its application to nanoclusters and molecular systems" (2016). Dissertations - ALL. 572. https://surface.syr.edu/etd/572 This Dissertation is brought to you for free and open access by the SURFACE at SURFACE. It has been accepted for inclusion in Dissertations - ALL by an authorized administrator of SURFACE. For more information, please contact [email protected]. Abstract Many theoretical and computational methods are based upon the Born-Oppenheimer approximation. This approximation greatly simplifies the search for a wave function that describes all electrons and nuclei in a chemical system. This is accomplished by assuming that the motion of nuclei and electrons are vastly different; the motion of the two particle types is decoupled. While the BO approximation is ubiquitous in computational and theoretical studies, it is not always justifiable. There are two main cases where this approximation is not valid. The first is when nuclear and electronic motion cannot be decoupled. Decoupling the motion leads to incorrect observations and conclusions drawn. The second case is when a chemical system has more than one type of particle to be treated without the Born-Oppenheimer approximation. For these types of systems, a different and more general interpreta- tion of the Born-Oppenheimer approximation must be made where multiple particle types can be investigated whose motion is not decoupled from one another. In order to investigate systems that are classified in this more inclusive interpretation, new computational theories and methods are needed. To accomplish this task, the mul- ticomponent coupled-cluster method has been developed. In its present form, this new computational method is capable of treating two types of particles without de- coupling their motion. The fundamental theories and methods for multicomponent coupled-cluster theory are discussed before the derivation and resulting multicompo- nent coupled-cluster equations are discussed. This method was then used to study excited electronic states in molecular systems and semiconductor quantum dots via the electron-hole representation. It was also used to calculated ground state energy of the positronium hydride system. These projects sparked further interest in the con- sequences of the Born-Oppenheimer approximation's application to chemical systems and how it compares to a non Born-Oppenheimer treatment. Development of multicomponent coupled-cluster theory and its application to nanoclusters and molecular systems by Benjamin H Ellis B.S., University of Wisconsin-Madison, 2011 M.Phil., Syracuse University, 2013 Dissertation Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Chemistry Syracuse University December 2016 c Copyright by Benjamin H Ellis, 2016. All rights reserved. iv Acknowledgements First and foremost, I would like to thank my advisor Professor Arindam Chakraborty. Throughout my time in graduate school, Ari has guided my academic growth and inspired me to follow my intellectual passions. I am grateful for his tutelage and support. I may not have fully convinced you on Python just yet, but your time will come! I also want to thank my parents, Tom and Maureen. They always urged me to follow my passions and to do what I found most interesting in life. They have supported and encouraged me in my journey to graduate school, and continue to do so as I move on to the next phase of my life. Thank you both, so much, for all that you do for me. I am also thankful for the support of my brother, Lann, and sister-in-law, Jackie. They have been great friends, great visitors, and great hosts to me during my years in the program. I also want to thank my girlfriend, Julie, and her family for their support. Thank you to Daisy for being a great companion, hiking buddy, movie enthusiast, and couch potato with me. I'd like to also thank my lab mates and friends at Syracuse. Specifically I'd like to thank Chris Blanton, Jen Elward, Mike Bayne, and Jeremey Scher. You were all great lab mates. You were all great at providing help on projects when I needed it. More importantly, you all made being in lab fun. I will fondly remember gathering around the coffee pot, excitedly smelling freshly opened beans, and doing our best to play coffee critic. v Let us think the unthinkable, let us do the undoable, let us prepare to grapple with the ineffable itself, and see if we may not eff it after all. Douglas Adams vi Contents Abstract.....................................i Acknowledgements............................... iv List of Tables..................................x List of Figures.................................. xiii 1 Multicomponent systems in chemistry1 1.1 Introduction................................1 1.2 Single component description......................2 1.3 Multicomponent description.......................4 1.3.1 Subatomic particles bound to molecular systems.......5 1.3.2 Quasiparticle representation...................5 1.3.3 Non Born-Oppenheimer affects.................7 1.4 This work.................................7 2 Single component quantum chemistry9 2.1 The Hartree-Fock method........................ 10 2.2 Second quantization........................... 16 2.2.1 Creation and annihilation operators.............. 17 2.2.2 Anticommutation relationships................. 18 2.2.3 Fermi vacuum and normal ordering............... 20 2.2.4 Wick's theorem.......................... 21 vii 2.2.5 Normal ordered Hamiltonian................... 24 2.2.6 Slater-Condon rules....................... 25 2.3 Configuration interaction theory..................... 26 2.3.1 Full configuration interaction.................. 27 2.3.2 Configuration interaction singles and doubles.......... 29 2.3.3 Consequences of truncating the CI wave function....... 30 2.3.4 Recent developments in configuration interaction....... 31 2.4 Coupled-cluster theory.......................... 32 2.4.1 Choosing an excitation operator................ 33 2.4.2 Similarity transformation and the Baker-Campbell-Hausdorff expansion............................. 35 2.4.3 The CCSD equations...................... 36 2.4.4 Size consistency and the CC wave function.......... 38 2.4.5 Relating CC to CI........................ 38 3 Multicomponent methods in quantum chemistry 41 3.1 Multicomponent Hartree-Fock...................... 43 3.1.1 Choice of the multicomponent Fock operator......... 46 3.2 Multicomponent second quantization.................. 49 3.3 Multicomponent configuration interaction............... 51 3.3.1 Multicomponent full configuration interaction wave function. 52 3.3.2 Truncating the multicomponent configuration interaction wave function.............................. 53 4 Multicomponent coupled-cluster theory 55 4.1 Introduction............................... 55 4.2 Theory.................................. 60 4.2.1 Construction of the vacuum states................ 60 viii 4.2.2 Effective normal-ordered Hamiltonian.............. 62 4.2.3 The mcCC Equations....................... 65 4.3 Implementation details......................... 78 4.4 Computational details.......................... 82 4.5 Results.................................. 82 4.5.1 Model-A single component Hooke's atom............ 82 4.5.2 Model-B multicomponent Hooke's atom............ 84 4.5.3 Positronium hydride system................... 85 4.5.4 Excitonic systems......................... 86 4.5.5 Biexcitonic system........................ 88 4.6 Conclusion................................ 92 4.A Calculation parameters.......................... 93 5 Investigating biexcitons in seminconductor quantum dots 95 5.1 Introduction............................... 95 5.2 Theory.................................. 97 5.3 Results.................................. 105 5.4 Conclusion................................ 111 5.A Material and basis parameters...................... 112 6 Applying multicomponent coupled-cluster method to molecular sys- tems 115 6.1 Traditional methods for studying excited electronic states....... 116 6.1.1 Configuration interaction singles................. 117 6.1.2 Equation-of-motion coupled-cluster............... 119 6.1.3 Time dependent Hartree-Fock.................. 120 6.2 Electron-hole representation....................... 124 6.3 Linear response theory.......................... 125 ix 6.3.1 Time dependent many body systems.............. 125 6.3.2 General linear response theory.................. 127 6.3.3 Density linear response...................... 129 6.3.4 Perturbation theory approach.................. 131 6.4 Preliminary results............................ 133 7 Investigating mass and confinement effects in multicomponent sys- tems 136 7.1 Introduction................................ 136 7.2 Theory................................... 138 7.2.1 The Born-Oppenheimer Approximation for single component systems.............................. 140 7.2.2 The Born-Oppenheimer Approximation for multicomonent sys- tems................................ 143 7.3 Method.................................. 147 7.4 Computational details.......................... 150 7.5 Results..................................