Density functional theory (using Becke's exchange and Lee-Yang-Parr's correlation functionals (BLYP)) and ab initio Hartree-Fock calculations were carried out in order to investigate the molecular structure and vibrational spectra of 9,10-anthraquinone and its perdeuterated analog. The calculated structural and spectral features are in good agreement with the available experimental results. Most of the BLYP/6-31G* non-CH(D) stretching frequencies are slightly lower than reliable experimental assignments; the mean absolute deviation is about 14 cm-1. On the basis of agreement between calculated and experimental results, assignments of the fundamental vibrational modes were examined and some reassignments were proposed. The calculated results can serve as a guide for a future experimental search for the missing fundamentals of the target molecules.

The Fast Multipole Method of Greengard and Rokhlin does the seemingly impossible: it approximates the quadratic scaling N-body problem in linear time. The key is to avoid explicitly computing the interactions between all pairs of N points. Instead, by organizing the data in a space-partitioning tree, distant interactions are quickly and efficiently approximated. Similarly, dual-tree algorithms, which approximate or eliminate parts of a computation using distance bounds, are the fastest algorithms for several fundamental problems in statistics and machine learning -- including all nearest neighbors, kernel density estimation, and Euclidean minimum spanning tree construction. We show that this overarching principle -- that by organizing points spatially, we can solve a seemingly quadratic problem in linear time -- can be generalized to problems involving interactions between sets of three or more points and can provide orders-of-magnitude speedups and guarantee runtimes that are asymptotically better than existing algorithms. We describe a family of algorithms, multi-tree algorithms, which can be viewed as generalizations of dual-tree algorithms. We support this thesis by developing and implementing multi-tree algorithms for two fundamental scientific applications: n-point correlation function estimation and Hartree-Fock theory. First, we demonstrate multi-tree algorithms for n-point correlation function estimation. The n-point correlation functions are a family of fundamental spatial statistics and are widely used for understanding large-scale astronomical surveys, characterizing the properties of new materials at the microscopic level, and for segmenting and processing images. We present three new algorithms which will reduce the dependence of the computation on the size of the data, increase the resolution in the result without additional time, and allow probabilistic estimates independent of the problem size through sampling. We provide both empirical evidence to support our claim of massive speedups and a theoretical analysis showing linear scaling in the fundamental computational task. We demonstrate the impact of a carefully optimized base case on this computation and describe our distributed, scalable, open-source implementation of our algorithms. Second, we explore multi-tree algorithms as a framework for understanding the bottleneck computation in Hartree-Fock theory, a fundamental model in computational chemistry. We analyze existing fast algorithms for this problem, and show how they fit in our multi-tree framework. We also show new multi-tree methods, demonstrate that they are competitive with existing methods, and provide the first rigorous guarantees for the runtimes of all of these methods. Our algorithms will appear as part of the PSI4 computational chemistry library.

Nuclear spinodal instabilities are investigated in non-relativistic and relativistic stochastic mean-field approaches for charge asymmetric and charge symmetric nuclear matter. Quantum statistical effect on the growth of instabilities are calculated in non-relativistic approach. Due to quantal effects, in both symmetric and asymmetric matter, dominant unstable modes shift towards longer wavelengths and modes with wave numbers larger than the Fermi momentum are strongly suppressed. As a result of quantum statistical effects, in particular at lower temperatures, amplitude of density fluctuations grows larger than those calculated in semi-classical approximation. Relativistic calculations in the semi-classical limit are compared with the results of non-relativistic calculations based on Skyrme-type effective interactions under similar conditions. A qualitative difference appears in the unstable response of the system: the system exhibits most unstable behavior at higher baryon densities around $rho_{B}=0.4 rho_{0}$ in the relativistic approach while most unstable behavior occurs at lower baryon densities around $rho_{B}=0.2 rho_{0}$ in the non-relativistic calculations.

Les travaux de recherche présentés dans cette thèse portent sur la méthode de décomposition de l'interaction d'échange ($J$) entre les centres magnétiques d'une molécule portant des électrons célibataires par l'approche à brisure de symétrie. Le but de cette méthode est d'extraire à l'aide des calculs non-restreints les différentes contributions physiques au couplage magnétique : l'échange direct ($J_0$) entre les orbitales magnétiques, l'échange cinétique ($\Delta J_{KE}$) permettant la délocalisation des orbitales magnétiques et la polarisation des orbitales de cœur ($\Delta J_{CP}$). La première partie de ce travail est consacrée à la théorie de la méthode de décomposition dans le cas le plus simple de deux électrons dans deux centres magnétiques d'un système centro-symétrique. La physique des contributions est expliquée en lien avec la présentation des outils méthodologiques et théoriques utilisés dans cette approche. La deuxième partie de cette thèse concerne l'implémentation de cette méthode dans le logiciel Orca et son application à des systèmes non-centrosymétriques. La dernière partie de ces travaux porte sur les développements méthodologiques que j'ai pu mener au cours de ces trois années de thèse. Une nouvelle approche permettant d'extraire la contribution de polarisation en spin est présentée. De plus, une proposition de généralisation de la décomposition de l'interaction d'échange magnétique basée sur la théorie des hamiltoniens effectifs est discutée. Un effort particulier a été porté sur l'explication et la signification physique de l'approche à brisure de symétrie dans le formalisme Hartree-Fock et la théorie de la fonctionnelle de la densité / This work is focused on the decomposition of the magnetic exchange coupling ($J$) between magnetic centers in the broken-symmetry approach. The purpose of this method is to extract from unrestricted calculations the different contributions to the magnetic coupling: the direct exchange ($J_0$) between the magnetic orbitals, the kinetic exchange interaction ($\Delta J_{KE}$) allowing the delocalization of the magnetic orbitals and the core polarization ($\Delta J_{CP}$) of non-magnetic electrons. The first part of this thesis is centered on the theory of the decomposition method in the simplest case of a centro-symmetric system with two electrons in two magnetic centers. The physical meaning is explained in relation with methodological and theoretical tools used in this approach. The second part presents the implementation of the method in the Orca package and its application to non centro-symmetric systems. In particular, this application highlights the interest of a such automatic method in standard quantum package. The last part of this work is focused on the methodological developments carried out during these three years. An innovative method avoiding the spin contamination problem is presented to extract the spin polarization effects. Furthermore, a generalization of the decomposition of $J$ to more complicated systems with more than two electrons in two magnetic centers is discussed. Special attention was given to the explanation and physical meaning of broken symmetry approach in Hartree-Fock and Density Functional Theory

This thesis concerns quadratic response properties and their application to properties in Jablonski diagrams such as resonant two-photon absorption and excited state absorption. Our main interest lies in optical power limiting applications, and in this context, molecules containing heavy metal atoms prove superior. Therefore, we are interested in how relativity affects these properties, and in order to assess this, a four-component relativistic framework is adopted. To properly address the molecular properties of interest, both relativistic effects and electron correlation need to be accounted for. These two properties are not additive, and, therefore, correlation needs to be incorporated into the four-component framework. We present the implementation of quadratic response properties at the four-component density functional level of theory. For second-harmonic generation, we have, with numerical examples, demonstrated that correlation and relativity are indeed not additive and that the inclusion of noncollinear magnetization is of little importance. We report that both electron correlation as well as relativity strongly affect results for second-harmonic generation. For example, relativity alone reduces the µβ-response signal by 62% and 75% for meta- and ortho-bromobenzene, respectively, and enhances the same response by 17% and 21% for meta- and ortho-iodobenzene, respectively. In the four-component framework, we present the implementations of single and double residues of the quadratic response function, which allows for the evaluation of resonant two-photon absorption cross sections and excited state properties. Using these tools, we discuss different levels of approximation to the relativistic Hamiltonian and we demonstrate that for two-photon absorption, a proper treatment of relativistic effects qualitatively alters the spectrum. For example, already for an element as light as neon, significant differences are seen between the relativistic and nonrelativistic spectra as triplet transitions acquire substantial absorption cross sections in the former case. Finally, quantum mechanics in conjunction with electrodynamics is applied to determine clamping levels in macroscopic samples. The microscopic properties of the optically active chromophores are determined by response theory, and then, electrodynamics is used to describe the interactions between the chromophores and incident laser pulses. Using this approach a series of molecules have been investigated and their performances have been compared and ranked in order to find novel materials for optical power limiting applications.