Motivated by our frustration with the lack of quantum chemistry methods for strongly-correlated open-shell systems, we develop quantitative methods for computing the electronic structure of such systems and qualitative tools for analyzing their chemical properties and reactivity. Specifically, we present a modern framework for performing sparse configuration interaction (CI) computations with arbitrary (Slater determinant) N-electron basis sets, using restricted or generalized spin-orbitals, and including computation of spin-polarized 1- and 2- electron reduced density matrices (RDMs). This framework is then used to implement the flexible ansätze for N-electron CI (FanCI) method more efficiently, via increased vectorization in the FanCI equations and use of sparse CI algorithms. We also extended the FanCI approach, including least-squares and stochastic optimization techniques, the computation of spin-polarized 1- and 2- electron RDMs, and transition energies (ionization potentials, electron affinities, and excitation energies). We use these tools to compare various open-shell CI methods and FanCI methods based on various antisymmetrized product of nonorthogonal geminals ansätze. To translate the vast amount of quantitative data present in the energies and (spin-polarized) density matrices of multiple open-shell states, we present a new, internally consistent and unambiguous framework for spin-polarized conceptual density-functional theory (SP-DFT) that reduces to a sensible formulation of spin-free CDFT in an appropriate limit. Using this framework, we were able to generalize the (non-spin-polarized) Parr function. We can also, using this framework, construct promolecules with proatoms having non-integer charges and multiplicities. Finally, we describe an equations-of-motion-based method for computing spin-polarized reactivity descriptors of a chemical system from only the ground state energy and the 1- and 2- electron RDMs from a single-point electronic structure computation, and show some benchmark computations for this method based on various CI and FanCI electronic structure methods. / Thesis / Doctor of Philosophy (PhD)

"We cannot afford the luxury any longer of ignoring the nature of the bonding in these interesting compounds .... " P.E.D. Morgan, (1974). An understanding of bonding is paramount to furthering our understanding of materials (Morgan, 1974). The properties of materials are governed by the interactions between atoms. These interactions are governed by the nature of the bonds. In this study, two methods are explored which provide insight into chemical interactions. First, promolecule radii, calculated for nitride, oxide, and sulfide coordinated polyhedra with bond lengths fixed at the sums of effective ionic and crystal radii, are analyzed. Radii calculated for transition and non-transition cations for the first four rows of the periodic table are highly correlated with crystal radii derived for oxide and sulfide crystals and with ionic radii derived for nitride crystals. Promolecule radii calculated for the coordination polyhedra match experimentally determined bonded radii to within ~0.02Å, on average. Calculated radii anions tend to match ionic radii when bonded to highly electropositive cations and atomic radii when bonded to highly electronegative cations. In the second study, molecular orbital calculations were completed on a series of small molecules containing the nitride anion. Bond type can be characterized by studying the systematics of parameters derived from the bond critical point properties of the electron density distributions. A set of criteria is established to suggest how covalent or ionic a bond is. This criteria is based on bond critical point properties such as the Laplacian of the electron density distribution evaluated at the bond critical point, the electron density distribution at the critical point, the local energy density at the critical point, the relative electronegativity of the cation, the curvatures of the electron density distribution, and the distance from the nucleus of the nitride anion to the bond critical point, (the bonded radius of the nitrogen atom). Parameters computed for promolecule data indicate that these easily obtained results offer a method of calculating bond critical properties which are close in value to the more extensive results derived from molecular orbital calculations. / Ph. D.