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

This dissertation focuses on computational study of the geometry and energetics small molecules and nanoclusters involving transition metals (TM). These clusters may be used for various industrial applications including catalysis and photonics. Specifically, in this work we have studied hydrides and carbides of 3d-transition metal systems (Sc through Cu), small nickel and gold clusters. Qualitatively correct description of the bond dissociation is ensured by allowing the spatial and spin symmetry to break. We have tested applicability of new exchange-correlation functional and alternative theoretical descriptions (spin-contamination correction in broken symmetry DFT and ensemble Kohn-Sham (EKS)) as well. We studies TM hydrides and carbides systems to understand the importance of underlying phenomenon of bond breaking in catalytic processes. We have tested several exchange-correlation functionals including explicit dependence on kinetic energy density for the description of hydrides (both neutral and cationic) and carbides formed by 3d-transition metals. We find M05-2x and BMK dissociation energies are in better agreement with experiment (where available) than those obtained with high level wavefunction theory methods, published previously. This agreement with experiment deteriorates quickly for other functionals when the fraction of the Hartree-Fock exchange in DFT functional is decreased. Higher fraction of HF exchange is also essential in EKS formalism, but it does not help when spin-adapted unrestricted approach is employed. We analyze the electron spin densities using Natural Bond Orbital population analysis and find that simple description of 3d electrons as non-bonding in character is rarely correct. Unrestricted formalism results in appreciable spin-contamination for some of the systems at equilibrium, which motivated us to investigate it further in details. In order to correct the spin contamination effect on the energies, we propose a new scheme to correct for spin contamination arising in broken-symmetry DFT approach. Unlike conventional schemes, our spin correction is introduced for each spin-polarized electron pair individually and therefore is expected to yield more accurate energy values. We derive an expression to extract the energy of the pure singlet state from the energy of the broken-symmetry DFT description of the low spin state and the energies of the high spin states (pentuplet and two spin-contaminated triplets in the case of two spin-polarized electron pairs). We validate our spin-contamination correction approach by a simple example of H2 and applied to more complex MnH system. Ensemble KS formalism is also applied to investigate the dissociation of C2 molecule. We find that high fraction of HF exchange is essential to reproduce the results of EKS treatment with exact exchange-correlation functional. We analyze the geometry and energetics of small nickel clusters (Ni2-Ni5) for several lowest energy isomers. We also study all possible spin states of small nickel cluster isomers and report observed trends in energetics. Finally we determine the geometry and energetics of ten lowest energy isomers of four small gold clusters (Au2, Au4, Au6, and Au8). We have also investigated the influence of cluster geometry, ligation, solvation and relativistic effects on electronic structure of these gold clusters. The effect of one-by-one ligand attachment in vacuum and solvent environment is also studied. Performance of five DFT functionals are tested as well; Local Spin Density Approximation (SVWN5), Generalized Gradient Approximation (PBE), kinetic energy density-dependent functional (TPSS), hybrid DFT (B3LYP), and CAM-B3LYP which accounts for long-range exchange effects believed to be important in the analysis of metal bonding in gold complexes and clusters. Our results exhibit the ligand induced stability enhancement of otherwise less stable isomers of Au4, Au6 and Au8. Ligands are found to play a crucial role in determining the 2D to 3D transition realized in small gold clusters. In order to select an appropriate theory level to use in this study, we investigate the effect of attachment of four different ligands (NH3, NMe3, PH3, PMe3) on cluster geometry and energetics of Au2 and Au4 in vacuum and in solution. Our results benchmark the applicability of DFT functional model and polarization functions in the basis set for calculations of ligated gold cluster systems. We employ five different basis sets with increasing amount of polarization and diffuse functions; LANL2DZ, LANL2DZ-P, def2-SVP, def2-TZVP, and def2-QZVP. We obtain NMe3 = NH3 > PH3 > PMe3 order of ligand binding energies and observe shallow potential energy surfaces in all molecules. Our results suggest appropriate quantum-chemical methodologies to model small noble metal clusters in realistic ligand environment to provide reliable theoretical analysis in order to complement experiments.