published_or_final_version / abstract / Chemistry / Doctoral / Doctor of Philosophy

published_or_final_version / Chemistry / Doctoral / Doctor of Philosophy

published_or_final_version / abstract / Chemistry / Doctoral / Doctor of Philosophy

In 1909, F. Riesz succeeded in giving an integral represntation for continuous linear functionals on C[0,1]. Although other authors, notably Hadamard and Frechet, had given representations for continuous linear functionals on C[0,1], their results lacked the clarity, elegance, and some of the substance (uniqueness) of Riesz's theorem. Subsequently, the integral representation of continuous linear functionals has been known as the Riesz Representation Theorem. In this paper, three different proofs of the Riesz Representation Theorem are presented. The first approach uses the denseness of the Bernstein polynomials in C[0,1] along with results of Helly to write the continuous linear functionals as Stieltjes integrals. The second approach makes use of the Hahn-Banach Theorem in order to write the functional as an integral. The paper concludes with a detailed presentation of a Daniell integral development of the Riesz Representation Theorem.

In this thesis, we use first-principles methods to study a class of systems known as strongly correlated materials in which exceptionally strong electron-electron repulsion in the d or f electron shell can lead to intriguing physical properties. The focus is on transition metal oxide and phosphate intercalation materials such as LiₓCoO₂ and LiₓFePO₄, which are employed as the positive electrode in rechargeable Li ion batteries. We also study the transition metal dichalcogenide system VS₂ as a candidate for strong correlation physics with analogous features to the cuprate high-temperature superconductors. Density functional theory (DFT), the standard theory of materials science which can be viewed as an effective single-electron theory, often breaks down for strongly correlated materials. In this thesis, we augment DFT with a more sophisticated many-electron approach known as dynamical mean-field theory (DMFT). We use the resultant DFT+DMFT approach with the numerically exact continuous-time quantum Monte Carlo solver to explore the physics of the materials studied here and probe compositional phase stability and related observables within DFT+DMFT for the first time. The elementary but efficient Hartree-Fock solver for the DMFT equations (i.e., DFT+U) is also utilized in order to cleanly separate the role of dynamical correlations and to better understand the respective methods. With these ab initio methods, we predict the compositional phase stability, average intercalation voltage, Li order-disorder transition temperature, structural phase stability, phonons, magnetic properties, and other important characteristics of strongly correlated materials. At the DFT+U level of theory, electronic correlations destabilize the intermediate-x compounds of cathode materials via enhanced ordering of the endmember d orbitals. DFT+U is qualitatively consistent with experiments for phase stable LixCoO₂, phase separating LiₓFePO₄, and phase stable LiₓCoPO₄. In Li₁/₂CoO₂, which is not charge ordered in experiments, the charge ordering predicted by DFT+U primarily stems from the approximate interaction, is necessary to qualitatively capture the phase stability, and erroneously predicts an insulating state and an overestimated Li order-disorder transition temperature. DFT+DMFT calculations describe LiCoO₂ as a band insulator with appreciable correlations within the Eg states and CoO₂ as a moderately correlated Fermi liquid; for both these systems we find evidence for appreciable charge and spin fluctuations. Dynamical correlations substantially dampen changes in the number of d electrons per site and the total energy as compared to DFT+U, which alters the predicted battery voltage between the two methods. We find that our DFT+DMFT results underestimate the average intercalation voltage for LiₓCoO₂ and discuss possible reasons for the discrepancy. In monolayer VS₂, a combination of crystal field splitting and direct V-V hopping leads to an isolated low-energy band for the trigonal prismatic phase within non-spin-polarized DFT. Ferromagnetism spin splits this band within spin DFT and leads to a S=1/2 ferromagnetic Stoner insulator. DFT+U opens this gap and leads to Mott insulating behavior, though for sufficiently high U an octahedral phase becomes favored. Using the known charge density wave of this octahedral phase, we assess the validity of DFT and DFT+U in this class of materials. If realized, trigonal prismatic VS₂ could be experimentally probed in an unprecedented fashion due to its monolayer nature.

Strongly correlated materials are an important class of materials for research in condensed matter physics. Other than ordinary solid-state physical systems, which can be well described and analyzed by the energy band theory, the electron-electron correlation effects in strongly correlated materials are far more significant. So it is necessary to develop theories and methods that are beyond the energy band theory to describe their rich and varied behaviors. Not only are there electron-electron correlations, typically the multiple degrees of freedom in strongly correlated materials, such as charge distribution, orbital occupancies, spin orientations, and lattice structure exhibit cooperative or competitive behaviors, giving rise to rich phase diagrams and sensitive or non-perturbative responses to changes in external parameters such as temperature, strain, electromagnetic fields, etc. This thesis is divided into two parts. In the first part, we use the density functional theory (DFT) plus U correction, i.e., the DFT+U method, to calculate the equilibrium and nonequilibrium phase transitions of LuNiO3 and VO2. The effect of adding U is manifested in both materials as the change of band structure in response to the change of orbital occupancies of electrons, i.e., the soft band effect. This effect bring about competitions of electrons between different orbitals by lowering the occupied orbitals and raising the empty orbitals in energy, giving rise to multiple metastable states. In the second part, we study the dynamic mean field theory (DMFT) as a beyond band-theory method. This is a Green's function based theory for open quantum systems. By selecting one lattice site of an interacting lattice model as an open system, the other lattice sites as the environment are equivalently replaced by a set of non-interaction orbitals according to the hybridization function, so the whole system is transformed into an Anderson impurity model. We studied how to use the density matrix renormalization group (DMRG) method to perform real-time evolutions of the Anderson impurity model to study the non-equilibrium dynamics of a strongly correlated lattice system. We begin in Chapter 1 with an introduction to strongly correlated materials, density functional theory (DFT) and dynamical mean-field theory (DMFT). The Kohn-Sham density functional theory and its plus U correction are discussed in detail. We also demonstrate how the DMFT reduces the lattice sites other than the impurity site as a set of non-interacting bath orbitals. Then in Chapters 2 and 3, we show material-related studies of LuNiO3 as an example of rare-earth nickelates under substrate strain, and VO2 as an example of a narrow-gap Mott insulator in a pump-probe experiment. These are two types of strongly correlated materials with localized 3d orbitals (for Ni and V). We use the DFT+U method to calculate their band structures and study the structural phase transitions in LuNiO3 and metal-insulator transitions in both materials. The competition between the charge-ordered and Jahn-Teller distorted phases of LuNiO3 is studied at various substrate lattice constants within DFT+U. A Landau energy function is constructed based on group theory to understand the competition of various distortion modes of the NiO6 octahedra. VO2 is known for its metal-insulator transition at 68 degree C, above which temperature it's a metal and below which it's an insulator with a doubled unit cell. For VO2 in a pump-probe experiment, a metastable metal phase was found to exist in the crystal structure of the equilibrium insulating phase. Our work is to understand this novel metastable phase from a soft-band picture. We also use quantum Boltzmann equation to justify the prethermalization of electrons over the lifetime of the metastable metal, so that the photoinduced transition can be understood in a hot electron picture. Finally, in Chapters 4 and 5, we show a focused study of building a real-time solver for the Anderson impurity model out of equilibrium using the density matrix renormalization group (DMRG) method, towards the goal of building an impurity solver for nonequilibrium dynamical mean-field theory (DMFT). We study both the quenched and driven single-impurity Anderson models (SIAM) in real time, evolving the wave function written in a form with 4 matrix product states (MPS) in DMRG. For the quenched model, we find that the computational cost is polynomial time if the bath orbitals in the MPSs are ordered in energy. The same energy-ordering scheme works for the driven model in the short driving period regime in which the Floquet-Magnus expansion converges. In the long-period regime, we find that the computational time grows exponentially with the physical time, or the number of periods reached. The computational cost reduces in the long run when the bath orbitals are quasi-energy ordered, which is discussed in further detail in the thesis.

Chan Sze-ki. / Thesis submitted in: December 2004. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (leaves 73-76). / Abstracts in English and Chinese. / ABSTRACT (English Version) --- p.iii / ABSTRACT (Chinese Version) --- p.iv / ACKNOWLEDGEMENTS --- p.v / TABLE OF CONTENTS --- p.vi / LIST OF TABLES --- p.viii / LIST OF FIGURES --- p.xi / Chapter CHAPTER 1 --- INTRODUCTION / Chapter 1.1. --- Structure of Deoxyribonucleic acid (DNA) / Chapter 1.1.1. --- Configuration and Conformation of Deoxyribonucleic acid (DNA) --- p.1 / Chapter 1.1.2. --- Torsion Angle --- p.2 / Chapter 1.1.3. --- Base Pairing --- p.5 / Chapter 1.2. --- DNA Damage --- p.6 / Chapter 1.3. --- The Objective of this Project --- p.11 / Chapter CHAPTER 2 --- theory and Computational Details / Chapter 2.1. --- Computational Theory / Chapter 2.1.1. --- Density Functional Theory (DFT) --- p.12 / Chapter 2.1.2. --- Closed-shell and Open-shell Determinantal Wavefunctions --- p.13 / Chapter 2.1.3. --- Calculation Method --- p.13 / Chapter 2.1.4. --- Basis Set Details --- p.14 / Chapter 2.2. --- Ionization Potential and Electron Affinity --- p.15 / Chapter 2.3. --- Charge Distribution --- p.16 / Chapter 2.4. --- Molecular Orbital --- p.16 / Chapter 2.5. --- Computation Details in this Project / Chapter 2.5.1. --- Calculation Method --- p.17 / Chapter 2.5.2. --- Studied Model --- p.17 / Chapter CHPATER 3 --- Results and Discussion / Chapter 3.1. --- Neutral State / Chapter 3.1.1. --- Bond Length --- p.19 / Chapter 3.1.2. --- Torsion Angle of DNA backbone --- p.19 / Chapter 3.1.3. --- Sugar Ring Puckering Mode --- p.25 / Chapter 3.1.4. --- Natural Population Analysis (NAP) --- p.28 / Chapter 3.1.5. --- Molecular Orbitals --- p.31 / Chapter 3.2. --- Cationic State / Chapter 3.2.1. --- Ionization Potential --- p.33 / Chapter 3.2.2. --- Bond Length --- p.34 / Chapter 3.2.3. --- Backbone Torsion Angles --- p.38 / Chapter 3.2.4. --- Puckering Mode of Sugar Ring --- p.40 / Chapter 3.2.5. --- Charge Distribution --- p.43 / Chapter 3.2.6. --- Molecular Orbitals --- p.43 / Chapter 3.2.7. --- Summary --- p.47 / Chapter 3.3. --- Anionic State / Chapter 3.3.1. --- Ionization Potential --- p.51 / Chapter 3.3.2. --- Bond Lengths --- p.52 / Chapter 3.3.3. --- Torsion Angles of Backbone --- p.54 / Chapter 3.3.4. --- Sugar Ring Puckering Mode --- p.54 / Chapter 3.3.5. --- Charge Distribution --- p.58 / Chapter 3.3.6. --- Molecular Orbital --- p.63 / Chapter 3.3.7. --- Summary --- p.66 / Chapter CHAPTER 4 --- CONCLUSION AND FUTURE WORK / Chapter 4.1. --- Conclusion --- p.68 / Chapter 4.2. --- Future Work --- p.71 / REFERENCE --- p.73

Yim Wai-leung. / "September 2002." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (p. 117-123). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest Information and Learning Company, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

Au Yuen-yee. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves 140-152). / Abstracts in English and Chinese. / Abstract --- p.iii / Acknowledgements --- p.v / Chapter Chapter One: --- General Background and Introduction --- p.1 / Chapter 1 -1 --- Introduction --- p.1 / Chapter 1-2 --- Three-Bond Coupling Constants (3J) --- p.1 / Chapter 1-2-1 --- Development of the Karplus Equation --- p.1 / Chapter 1-2-2 --- Application of3J in the Conformational Analysis of Nucleic Acid --- p.4 / Chapter 1-2-3 --- Problem of Accuracy for3 J Measurement --- p.7 / Chapter 1-3 --- Two-Bond Coupling Constants (2J) --- p.7 / Chapter 1-3-1 --- The Use of the Projection Method --- p.7 / Chapter 1-3-2 --- 2J Coupling Constant Involving Hydrogen Bonds --- p.8 / Chapter 1-4 --- One-Bond Coupling Constants (1J) --- p.10 / Chapter 1-5 --- Conclusion --- p.11 / Chapter Chapter Two: --- Experimental Section / Chapter 2-1 --- Introduction --- p.12 / Chapter 2-2 --- Heteronuclear Multiple-Quantum Coherence (HMQC) Experiment --- p.12 / Chapter 2-3 --- Experimental Section --- p.15 / Chapter 2-3-1 --- Sample Preparation --- p.15 / Chapter 2-3-2 --- NMR Spectroscopy --- p.16 / Chapter Chapter Three: --- Theory of Nuclear Spin-Spin Coupling Constants --- p.18 / Chapter 3-1 --- Introduction --- p.18 / Chapter 3-2 --- Application of Finite Perturbation Theory on Nuclear Spin-Spin Coupling --- p.18 / Chapter 3-3 --- Methodology --- p.22 / Chapter Chapter Four: --- DFT and NMR Study of1JCH Coupling Constants --- p.28 / Chapter 4-1 --- Introduction --- p.28 / Chapter 4-2 --- Nomenclature and Definition of Structural Parametersin DNA and RNA --- p.28 / Chapter 4-2-1 --- "Nomenclature, Symbols and Atomic Numbering Schemes" --- p.28 / Chapter 4-2-2 --- Definition of Torsion Angles and their Rangesin Nucleotides --- p.31 / Chapter 4-2-3 --- Description of the Furanose Ring --- p.31 / Chapter 4-3 --- Results and Discussion --- p.37 / Chapter 4-3-1 --- Basis Set Effect --- p.37 / Chapter 4-3-2 --- Relative Conformational Energy Profiles --- p.37 / Chapter 4-3-3 --- Comparison of the Dependence of 1JCH Coupling Constants on Conformational Changes With and Without the DNA Backbone --- p.40 / Chapter 4-3-4 --- Effect of Backbone 3'- and 5'-Phosphate --- p.42 / Chapter 4-3-5 --- Effect of Glycosidic Torsion Anglex --- p.49 / Chapter 4-3-6 --- Effect of Ring Conformation with Fixed Glycosidic Torsion Anglex --- p.52 / Chapter 4-3-7 --- Effect of Torsion Angle α --- p.52 / Chapter 4-3-8 --- Effect of Torsion Angle β --- p.53 / Chapter 4-3-9 --- Effect of Torsion Angle γ --- p.56 / Chapter 4-3-10 --- Effect of Torsion Angle ε --- p.59 / Chapter 4-3-11 --- Effect of Torsion Angle ζ --- p.61 / Chapter 4-3-12 --- Effect of Base Pairing --- p.65 / Chapter 4-3-13 --- Effect of Base Stacking from the (n-1) and (n+1) Base --- p.65 / Chapter 4-3-14 --- Comparison of Experimental and Theoretical Data --- p.68 / Chapter 4-4 --- Conclusion --- p.74 / Chapter Chapter Five: --- DFT Study of 2JCH and 3JCH Coupling Constants --- p.79 / Chapter 5-1 --- Introduction --- p.79 / Chapter 5-2 --- Results and Discussion on 2JCH Coupling Constants --- p.79 / Chapter 5-2-1 --- Effect of Backbone 3'- and 5'-Phosphate --- p.79 / Chapter 5-2-2 --- Effect of Ring Conformation with Fixed Glycosidic Torsion Anglex --- p.82 / Chapter 5-2-3 --- Effect of Glycosidic Torsion Anglex --- p.87 / Chapter 5-2-4 --- Effect of Torsion Angleγ --- p.87 / Chapter 5-2-5 --- Effect of Torsion Angle ε --- p.90 / Chapter 5-2-6 --- Effect of Base Pairing --- p.90 / Chapter 5-2-7 --- Effect of Base Stacking from the (n-1) and (n+1) Base --- p.90 / Chapter 5-3 --- Results and Discussion on 3JCH Coupling Constants --- p.95 / Chapter 5-3-1 --- Effect of Backbone 3'- and 5'-Phosphate --- p.95 / Chapter 5-3-2 --- Effect of Ring Conformation with Fixed Glycosidic Torsion Anglex --- p.95 / Chapter 5-3-3 --- "Effect of Different Torsion Angles (X,α,β,γ,ε,and ζ)" --- p.100 / Chapter 5-3-4 --- Effect of Base Pairing --- p.100 / Chapter 5-3-5 --- Effect of Base Stacking from the (n-1) and (n+1) Base --- p.105 / Chapter 5-4 --- Conclusion --- p.105 / Chapter Chapter Six: --- Conclusion --- p.111 / Appendix A Product Operator Formalism on HMQC Pulse Scheme --- p.113 / Appendix B Finite Perturbation Theory --- p.115 / Appendix C Supplementary Figures of Chapter Four --- p.118 / Appendix D Some of the NMR Spectra --- p.134 / References --- p.140

In modelling the energetics of molecules and solids, the need for practical electron density functionals that seamlessly include the van der Waals interaction is growing. Such functionals are still in their infancy, and there is yet much experimentation to be performed in the formulation and numerical testing of the requisite approximations. A ground-state density functional approach that uses the exact relations of the adiabatic connection formula and the fluctuation-dissipation theorem to obtain the xc energy from the density-density response function seems promising, though a direct local density approximation for the interacting susceptibility will fail to yield the vdW interaction. Significant nonlocality can be built into the interacting susceptibility by screening a 'bare' susceptibility, for which a carefully chosen constraint-obeying local approximation is sufficient to yield a non-trivial van der Waals energy [6]. The constraints of charge conservation, and no response to a constant potential, are guaranteed by expressing the bare susceptibility in terms of the double gradients of a nonlocal bare polarisability. for which it should be easier to make an approximation based on physical principles than it would be for the susceptibility. The 'no-flow' condition is also deemed important. In this work, a simple delta-function approximation for the nonlocal polarisability is fully constrained by a new version of a recently-discovered force theorem (sum rule), requiring the additional input of the independent-electron Kohn-Sham potential. This constrained polarisability cannot be used as input for the seamless vdW scheme, which requires a non-delta-function bare polarisability, and is instead applied to systems containing spherical fragments in a perturbative/asymptotic fashion for calculation of the widely-separated van der Waals interaction. The main thrust of this work is an investigation of the efficacy of the force theorem to constrain simple approximations for response quantities. Many recent perturbative vdW density functionals are based on response functions that are electron-hydrodynamical approximations to the response of the uniform electron gas. These schemes require their response functions to be 'cut off' at low density and high density-gradient, where the approximation overestimates the true response. The imposition of the cut-off is crucial to the success of such schemes. Here, we replace the cut-off with an exact theorem (the force theorem) which naturally 'ties down' the response, based on the potential- and density-functions of the system. This is the first time that the force theorem has been directly applied as a constraint upon a model response function (its original use, by Vignale and Kohn (7), was as an exact identity in time-dependent DFT). Also new in this work is the orbital-by-orbital Kohn-Sham version of the force theorem, and its proof (differing significantly from Vignale's original derivation (8) of the interacting theorem) by directly appealing to the Kohn-Sham orbitals makes its first appearance here. For quantum dots, our constrained response-approximation exactly recovers the net linear dipole response, due mainly to the force theorem's ideal applicability to harmonically confined systems. For angularly-averaged atoms, reasonable static dipole polarisabilities are obtained for the independent-electron Kohn-Sham (bare) case. The results are poor for the fully-interacting case, attributable to the local nature of the approximation. This lends weight to the assertion that it is better to approximate a bare quantity, then screen it, than it is to directly approximate a fully-interacting quantity. Dynamic net polarisabilities constrained by the force theorem are guaranteed to have the correct high-frequency asymptotic convergence to the free electron response. It is seen that the calculated dynamic polarisabilities for atoms are too small at intermediate frequencies, since the calculated vdW C6 coefficients (Hamaker constants) of atomic dimers are up to an order of magnitude too small, even without the use of a low-density cutoff. It is seen that our constrained local model response is non-analytic along the imaginary-frequency axis, and this is very detrimental to the C6 calculations, even though the integrated net polarisability is analytic. Improvement of the polarisability ansatz is indicated, perhaps to a non-deltafunction uniform-gas-based approximation. The use of pseudopotentials may improve the force theorem results, by softening the extreme nature of the bare Coulomb potential.