Over the years electronic structure theory has proven to be a powerful method with which one can probe the behaviour of materials, making it possible to describe and predict material properties. The numerical tools needed for these methods are always in need of development, since the desire to calculate more complex materials pushes this field forward. This thesis contains work on both this implementational and developmental aspects. It begins by reviewing density functional theory and dynamical mean field theory, with the aim of merging these two methods. We point out theoretical and technical issues that may occur while doing this. One issue is the Padé approximant, which is used for analytical continuation. We assess the approximant and point out difficulties that can occur, and propose and evaluate methods for their solution. The virial theorem is assessed within the framework of density functional theory merged with many-body methods. We find that the virial theorem is extended from its usual form, and confirm this by performing practical calculations. The unified theory of crystal structure for transition metals has been established a long time ago using early electronic structure calculations. Here we implement the first- principles exact muffin-tin orbitals method to investigate the structural properties of the 6d transition metals. The goal of our study is to verify the existing theory for the mostly unknown 6d series and the performance of the current state-of-the art in the case of heavy d metals. It is found that these elements behave similarly to their lighter counterparts, except for a few deviations. In these cases we argue that it is relativistic effects that cause this anomalous behaviour. Palladium is then studied, taking many-body effects into account. We find that we can reproduce experimental photoemission spectra by these methods, as well as the Fermi surface. The thesis ends with an investigation of the stacking fault energies of the strongly correlated metal cerium. In addition to providing the first ab-initio stacking fault data for the two cubic phases of Ce, we discuss how these results could have an impact on the interpretation of the phase diagram of cerium / <p>QC 20150522</p>

This thesis is concerned with the development of a Projector Quantum Monte Carlo method for non-linear wavefunction ansatzes and its application to strongly correlated materials. This new approach is partially inspired by a prior application of the Full Configuration Interaction Quantum Monte Carlo (FCIQMC) method to the three-band (p-d) Hubbard model. Through repeated stochastic application of a projector FCIQMC projects out a stochastic description of the Full Configuration Interaction (FCI) ground state wavefunction, a linear combination of Slater determinants spanning the full Hilbert space. The study of the p-d Hubbard model demonstrates that the nature of this FCI expansion is profoundly affected by the choice of single-particle basis. In a counterintuitive manner, the effectiveness of a one-particle basis to produce a sparse, compact and rapidly converging FCI expansion is not necessarily paralleled by its ability to describe the physics of the system within a single determinant. The results suggest that with an appropriate basis, single-reference quantum chemical approaches may be able to describe many-body wavefunctions of strongly correlated materials. Furthermore, this thesis presents a reformulation of the projected imaginary time evolution of FCIQMC as a Lagrangian minimisation. This naturally allows for the optimisation of polynomial complex wavefunction ansatzes with a polynomial rather than exponential scaling with system size. The proposed approach blurs the line between traditional Variational and Projector Quantum Monte Carlo approaches whilst involving developments from the field of deep-learning neural networks which can be expressed as a modification of the projector. The ability of the developed approach to sample and optimise arbitrary non-linear wavefunctions is demonstrated with several classes of Tensor Network States all of which involve controlled approximations but still retain systematic improvability towards exactness. Thus, by applying the method to strongly-correlated Hubbard models, as well as ab-initio systems, including a fully periodic ab-initio graphene sheet, many-body wavefunctions and their one- and two-body static properties are obtained. The proposed approach can handle and simultaneously optimise large numbers of variational parameters, greatly exceeding those of alternative Variational Monte Carlo approaches.

Quantum chemistry is plagued by the presence of high-rank quantities, stemming from the N-body nature of the electronic Schrödinger equation. These high-rank quantities present a significant mathematical and computational barrier to the computation of chemical observables, and also drastically complicate the pedagogical understanding of important interactions between particles in a molecular system. The application of physically-motivated rank reduction approaches can help address these to problems. This thesis details recent efforts to apply rank reduction techniques in both of these arenas. With regards to computational tractability, the representation of the 1/r Coulomb repulsion between electrons is a critical stage in the solution of the electronic Schrödinger equation. Typically, this interaction is encapsulated via the order-4 electron repulsion integral (ERI) tensor, which is a major bottleneck in terms of generation, manipulation, and storage. Many rank reduction techniques for the ERI tensor have been proposed to ameliorate this bottleneck, most notably including the order-3 density fitting (DF) and pseudospectral (PS) representations. Here we detail a new and uniquely powerful factorization - tensor hypercontraction (THC). THC decomposes the ERI tensor as a product of five order-2 matrices (the first wholly order-2 compression proposed for the ERI) and offers great flexibility for low-scaling algorithms for the manipulations of the ERI tensor underlying electronic structure theory. THC is shown to be physically-motivated, markedly accurate, and uniquely efficient for some of the most difficult operations encountered in modern quantum chemistry. On the front of chemical understanding of electronic structure theory, we present our recent work in developing robust two-body partitions for ab initio computations of intermolecular interactions. Noncovalent interactions are the critical and delicate forces which govern such important processes as drug-protein docking, enzyme function, crystal packing, and zeolite adsorption. These forces arise as weak residual interactions leftover after the binding of electrons and nuclei into molecule, and, as such, are extremely difficult to accurately quantify or systematically understand. Symmetry-adapted perturbation theory (SAPT) provides an excellent approach to rigorously compute the interaction energy in terms of the physically-motivated components of electrostatics, exchange, induction, and dispersion. For small intermolecular dimers, this breakdown provides great insight into the nature of noncovalent interactions. However, SAPT abstracts away considerable details about the N-body interactions between particles on the two monomers which give rise to the interaction energy components. In the work presented herein, we step back slightly and extract an effective 2-body interaction for each of the N-body SAPT terms, rather than immediately tracing all the way down to the order-0 interaction energy. This effective order-2 representation of the order-N SAPT interaction allows for the robust assignment of interaction energy contributions to pairs of atoms or functional groups (the A-SAPT or F-SAPT partitions), allowing one to discuss the interaction in terms of atom- or functional-group-pairwise interactions. These A-SAPT and F-SAPT partitions can provide deep insight into the origins of complicated noncovalent interactions, e.g., by clearly shedding light on the long-contested question of the nature of the substituent effect in substituted sandwich benzene dimers.

Kyoto University (京都大学) / 0048 / 新制・課程博士 / 博士(工学) / 甲第14639号 / 工博第3107号 / 新制||工||1462(附属図書館) / 26991 / UT51-2009-D351 / 京都大学大学院工学研究科分子工学専攻 / (主査)教授 榊 茂好, 教授 田中 庸裕, 教授 村上 正浩 / 学位規則第4条第1項該当

Kyoto University (京都大学) / 0048 / 新制・課程博士 / 博士(工学) / 甲第15396号 / 工博第3275号 / 新制||工||1493(附属図書館) / 27874 / 京都大学大学院工学研究科分子工学専攻 / (主査)教授 榊 茂好, 教授 白川 昌宏, 教授 北川 進 / 学位規則第4条第1項該当

Kyoto University (京都大学) / 0048 / 新制・課程博士 / 博士(工学) / 甲第17162号 / 工博第3652号 / 新制||工||1555(附属図書館) / 29901 / 京都大学大学院工学研究科分子工学専攻 / (主査）教授 佐藤 啓文, 教授 横尾 俊信, 教授 梶 弘典 / 学位規則第4条第1項該当

Optical and spectroscopic molecular properties are key to characterizing the behavior of molecules interacting with an applied electromagnetic field of light. Response theory has been used for a long time to calculate such properties in the frequency domain. Real-time (RT) methods solve for the frequency-dependent properties in the time domain by explicitly propagating the time-dependent wave function. Various quantum chemical methods can be incorporated with the RT formalism, including Hartree-Fock, density functional theory, configurational interaction, coupled cluster, etc. Among these, coupled cluster (CC) methods provide high accuracy for systems with strong electron correlation, making RT-CC implementations intriguing. All applications of CC methods face a substantial challenge due to their high-order polynomial scaling. For RT-CC methods, two aspects may be explored to improve the efficiency, the numerical techniques regarding the RT propagation and the reduced-scaling methods regarding CC itself. In this work, we start with the exploration of the hardware used for the calculations and the numerical integration methods for propagating the wave function parameters. Firstly, a GPU-enabled Python implementation has been developed by conducting the tensor contractions on GPUs utilizing PyTorch, a machine learning package, that has similar syntax as NumPy for tensor operations. A speedup of a factor of 14 is obtained for the RT-CCSD/cc-pVDZ absorption spectrum calculation of the water tetramer. Furthermore, to optimize the performance on GPUs, single-precision arithmetic is added to the implementation to achieve an additional speedup of a factor of two. Lastly, a group of integrators for solving differential equations are introduced to the RT framework, including regular explicit integrators, adaptive integrators, and a mixed-step-size approach customized for strong-field simulations. The optimal choice of the integrator depends on the requiring accuracy, stability and efficiency. In addition to being highly accurate, CC methods are also systematically improvable and provide a hierarchy of accuracy. Based upon the RT-CCSD implementation, the coupled cluster singles, doubles and approximate triples (CC3) method, favorable for calculating frequency-dependent properties, is tailored to the RT framework for high excitation and approximate orbital relaxation. The calculation is tested on both CPUs and GPUs, with a significant speedup gained from GPUs for the water cluster test cases. To further expand the range of applications of our RT-CC implementation, dynamic polarizabilities, first hyperpolarizabilities, and the G' tensor are calculated from induced electric and magnetic dipole moments using finite-difference methods. A discussion has also been conducted to compare RT-CC3 with RT-CCSD, and time-dependent nonorthogonal orbital-optimized coupled cluster doubles (TDNOCCD) method. Additionally, electron dynamics, including the Rabi oscillation and exited state to excited state transitions, have also been explored utilizing the well-developed RT-CC framework. / Doctor of Philosophy / Theoretical studies aim to match experiments, but more importantly, provide insights to interpret and predict experimental data. Calculating optical properties related to light-matter interactions is one of the most crucial tasks for characterizing molecular properties. In experiments, electromagnetic radiation in the form of light is applied to the system. The absorption or emission of light can be measured to identify, for example, the electronic structure of the molecule. In theoretical simulations, this applied radiation is represented by a perturbation operator that is added to the Hamiltonian in the Schrödinger equation. Quantum chemists are dedicated to developing methods that provide a better description of the spectroscopy. In the current work, the frequency, shape and the intensity of the radiation can all be finely-tuned, similar to experimental setups. The framework for extracting optical properties from time-dependent trajectories of induced dipole moments is established for accurate and efficient simulations. To improve efficiency and make the method feasible for real-world applications, a strong understanding of light-matter interactions on a quantum level and proper utilization of computational resources are both necessary. Improvements achieved and presented in this dissertation demonstrate a powerful tool for a better understanding of the nature of the interaction between the system and the electromagnetic radiation.

The efficient computation of dynamic (time-dependent) molecular properties is a broad field with numerous applications in aiding molecular synthesis and design, with a particular preva- lence in spectroscopic predictions. Typical methods for computing the response of a molecu- lar system to an electromagnetic field (EMF) considers a quantum mechanical description of the molecule and a classical approximation for the EMF. Methods for describing light-matter interactions with high-accuracy electronic structure methods, such as coupled cluster (CC), are discussed, with a focus on improving the efficiency of such methods. The CC method suffers from high-degree polynomial scaling. In addition to the ground-state calculation, computing dynamic properties requires the description of sensitive excited-state effects. The cost of such methods often prohibits the accurate calculation of response prop- erties for systems of significant importance, such as large-molecule drug candidates or chiral species present in biological systems. While the literature is ripe with reduced-scaling meth- ods for CC ground-state calculations, considerably fewer approaches have been applied to excited-state properties, with even fewer still providing adequate results for realistic systems. This work presents three studies on the reduction of the cost of molecular property evalu- ations, in the hopes of closing this gap in the literature and widening the scope of current theoretical methods. There are two main ways of simulating time-dependent light-matter interactions: one may consider these effects in the frequency domain, where the response of the system to an EMF is computed directly; or, the response may be considered explicitly in the time domain, where wave function (or density) parameters can be propagated in time and examined in detail. Each methodology has unique advantages and computational bottlenecks. The first two studies focus on frequency-domain calculations, and employ fragmentation and machine- learning techniques to reduce the cost of single-molecule calculations or sets of calculations across a series of geometric conformations. The third study presents a novel application of the local correlation technique to real-time CC calculations, and highlights deficiencies and possible solutions to the approach. / Doctor of Philosophy / Theoretical chemistry plays a key role in connecting experimental results with physical inter- pretation. Paramount to the success of theoretical methods is the ability to predict molecular properties without the need for costly high-throughput synthesis, aiding in the determina- tion of molecular structure and the design of new materials. Light-matter interactions, which govern spectroscopic techniques, are particularly complicated, and sensitive to the theoreti- cal tools employed in their prediction. Compounding the issue of accuracy is one of efficiency — accurate theoretical methods typically incur steep scaling of computational cost (memory and processor time) with respect to the size of the system. An important aspect in improving the efficiency of these methods is understanding the nature of light-matter interactions at a quantum level. Many unanswered questions still remain, such as, "Can light-matter interactions be thought of as a sum of interactions be- tween smaller fragments of the system?" and "Can conventional methods of accelerating ground-state calculations be expected to perform well for spectroscopic properties?" The present work seeks to answer these questions through three studies, focusing on improving the efficiency of these techniques, while simultaneously addressing their fundamental flaws and providing reasonable alternatives.

Strong electron correlations in electronic structure theory are purely quantum effects arising as a result of degeneracies in molecules and materials, and exhibit significantly different yet interesting characters than do weak correlations. Although weak correlations have recently been able to be described very efficiently and accurately within single particle pictures, less known are good prescriptions for treating strong correlations efficiently. Brute-force calculations of strong correlations in wave function theories tend to be very computationally-intensive, and are usually limited to small molecules for applications. Breaking symmetry in a mean-field approximation is an efficient alternative to acquire strong correlations with, in many cases, qualitatively accurate results. The symmetry broken in quantum chemistry has been traditionally of spin, in so-called unrestricted methods, which typically break spatial symmetry as a consequence, and vice versa, in most situations. In this work, we present a novel approach to accurately describing strong correlations with a mean-field cost by means of Hartree- Fock-Bogoliubov (HFB) theory. We are inspired by the number-symmetry-breaking in HFB, which, with an attractive particle interaction, accounts for strong correlations, while maintaining spin and spatial symmetry. We show that this attractive interaction must be restricted to the chemically-relevant orbitals in an active space to obtain physically meaningful results. With such constraints, our constrained pairing mean-field theory (CPMFT) can accurately describe potential energy curves of various strongly-correlated molecular systems, by cleanly separating strong and weak correlations. To achieve the correct dissociation limits in hetero-atomic molecules, we have modified our CPMFT functional by adding asymptotic constraints. We also include weak correlations by combining CPMFT with density functional theory for chemically accurate results, and reveal the connection between CPMFT and traditional unrestricted methods. The similarity between CPMFT and unrestricted methods leads us to the idea of constrained active space unrestricted mean-field approaches. Motivated by CPMFT, we partially retrieve spin-symmetry that has been fully broken in unrestricted methods. We allow symmetry breaking only in an active space. This constrained unrestricted Hartree-Fock (CUHF) is an interpolation between two extrema: the fully broken-symmetry solution and the symmetry preserved solution. This thesis defines the theory behind and reports the results of CUHF. We first show that, if an active space is chosen to include only open-shell electrons, CUHF reduces to restricted open-shell Hartree-Fock (ROHF), and such CUHF proves in many ways significantly

Density-functional theory has become an indispensible tool for studying matter on the atomic level, being routinely applied across diverse disciplines from solid-state physics to chemistry and molecular biology. Its failure to account for dispersion interactions has spurred intensive research over the past decade. In this thesis, a semilocal density-functional approximation including dispersion is developed by combining standard functionals for exchange and correlation with the nonempirical “exchange-hole dipole moment“ (XDM) dispersion model of Becke and Johnson. With a minimum of empiricism, the method accurately describes all types of noncovalent interactions, from the extremely weak dispersion forces in rare-gas systems to the hydrogen bonding and stacking interactions responsible for the structure and function of biological macromolecules such as DNA and proteins. The method is compatible with a wide variety of standard Gaussian basis sets, and is easily applied to any system that can be modeled with density-functional theory.