Approaching exact quantum chemistry by stochastic wave function sampling and deterministic coupled-cluster computations

One of the main goals of quantum chemistry is the accurate description of ground- and excited-state energetics of increasingly complex polyatomic systems, especially when non-equilibrium structures and systems with stronger electron correlations are examined. Size extensive methods based on coupled-cluster (CC) theory and their extensions to excited electronic states via the equation-of-motion (EOM) framework have become the de facto standards for addressing this goal. In the vast majority of chemistry problems, the traditional single-reference CC hierarchy, including CCSD, CCSDT, CCSDTQ, etc., and its EOM counterpart provide the fastest convergence toward the exact, full configuration interaction (FCI), limit, allowing one to capture the leading many-electron correlation effects, in a systematic manner, by employing particle-hole excitations from a single reference determinant. Unfortunately, computational costs associated with the incorporation of higher-than-two-body components of the cluster and excitation operators of CC and EOMCC, which are required to achieve a fully quantitative description of molecular systems, are usually prohibitive. This has exacerbated the need for new ideas in quantum chemistry leading to computationally affordable methodologies that do not suffer from failures of lower-order, less expensive approximations. In this dissertation, I propose two novel and computationally cost-effective strategies for acquiring accurate electronic energetics, equivalent to those obtained with high-level CC/EOMCC methods, such as CCSDT, EOMCCSDT, and CCSDTQ, and the exact, FCI, theory, even when wave function quasi-degeneracies and other higher-order correlation effects become significant.The first strategy consists in merging the deterministic CC framework, abbreviated as, CC(P;Q), with the stochastic CI Quantum Monte Carlo (QMC) approaches, such as CISDT-MC, CISDTQ-MC, FCIQMC, and their CCSDT-MC counterpart with the intention of using wave functions resulting from QMC propagations to identify the leading higher-than-doubly excited components entering deterministic CC and EOMCC calculations.The second proposed methodology provides a direct way of recovering FCI energetics following the ideas of the externally corrected CC theories. In the resulting approach, that we denominate CAD-FCIQMC, we utilize the cluster analysis of FCIQMC wave functions to obtain the connected three- and four-body cluster components, which are the only clusters that directly couple to the one- and two-body components of the cluster operator. In general, the externally corrected CC methods are guaranteed to attain the exact, FCI, ground-state energy, if the triply and quadruply excited cluster components extracted from a non-CC source entering the suitably corrected CCSD-like equations approach their exact values. Since the long-time FCIQMC propagations converge the exact FCI wave functions, our new CAD-FCIQMC method is assured to produce numerically exact energies.All of the above approaches, including the semi-stochastic CC(P;Q) and CAD-FCIQMC methodologies, are discussed in this dissertation. This includes their mathematical foundations, computer implementation, and numerical tests using bond breaking in the F2, H2O, and CH+ molecules and the automerization of cyclobutadiene.