The development of computationally efficient wavefunction methods that can provide an accurate description of strongly correlated systems and materials is at the heart of electronic structure theory. In general, strong many-electron correlation effects arise from the entanglement of a large number of electrons and are characterized by the unpairing of many electron pairs and their subsequent recoupling to low-spin states, as in the case of Mott metal-insulator transitions where the system traverses from a weakly correlated metallic phase to a strongly correlated insulating one. Although strong correlations have an intrinsically multi-reference nature, multi-reference approaches are not applicable due to the enormous dimensionalities of the underlying model spaces. Therefore, in this dissertation, we focus on single-reference coupled-cluster (CC) approaches, which are widely recognized as the de facto standard for high-accuracy electronic structure calculations and whose size extensivity makes them suitable for the study of extended systems and materials. However, it is well established that the traditional CC methodologies that are based on truncating the cluster operator at a given many-body rank, giving rise to the CCSD, CCSDT, CCSDTQ, etc. hierarchy, fail to provide physically meaningful solutions in the presence of strong correlations. Thus, in this dissertation, we consider unconventional single-reference CC approaches capable of providing an accurate description of the entire spectrum of many-electron correlation effects, ranging from the weakly to the strongly correlated regimes.In the first part of this dissertation, we examine the approximate coupled-pair (ACP) theories. The existing ACP methods and their various modifications retain all doubly excited cluster amplitudes, while using subsets of non-linear diagrams of the CCD/CCSD equations. This eliminates failures of conventional CC approaches, including CCSD and even CCSDT or CCSDTQ, in strongly correlated situations created by the Mott metal-insulator transitions, modeled by linear chains, rings, or cubic lattices of the equally spaced hydrogen atoms, and the π-electron networks described by the Hubbard and Pariser-Parr-Pople Hamiltonians that model one-dimensional metallic systems with periodic boundary conditions. However, typical ACP methods neglect connected triply excited T3 clusters, which are required to produce quantitative results in most chemical applications. Previous attempts to incorporate these clusters using many-body perturbation theory arguments within the ACP framework have only been partly successful. In this dissertation, we address this concern by employing the active-space ideas to incorporate the dominant T3 amplitudes in the ACP methods in a robust, yet computationally affordable, manner. Furthermore, taking into consideration that the various diagram modifications defining ACP approaches were derived using minimum-basis-set models, we introduce a novel ACP scheme utilizing basis-set-dependent scaling factors, denoted as ACCSD(1,3 × no/(no + nu) + 4 × nu/(no + nu)), to extend the ACP methodologies to larger basis sets.In the second part of this dissertation, we discuss a novel approach to extrapolating the exact energetics out of the early stages of full configuration interaction quantum Monte Carlo (FCIQMC) propagations, even in the presence of strong correlations, by merging the ACP approaches with the recently proposed cluster-analysis-driven FCIQMC (CAD-FCIQMC) methodology. In the spirit of externally corrected CC approaches, in the CAD-FCIQMC methodology, one solves CCSD-like equations for the one- and two-body clusters in the presence of their three- and four-body counterparts extracted from the FCIQMC stochastic wavefunction sampling. In this dissertation, we extend CAD-FCIQMC to the strong correlation regime by repartitioning the CC equations so that selected coupled-pair contributions are extracted from FCIQMC as well. For each new methodology described in this thesis, we discuss the relevant mathematical and computer implementation details and provide numerical examples illustrating its performance in challenging strongly correlated situations.
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