Accurately predicting the effects of correlations caused by strong interactions between electrons represents one of the key unsolved challenges in quantum chemistry. This thesis details theoretical developments addressing two aspects of this challenge. First, advances in Canonical Transformation (CT) theory address the task of describing weak interactions between the chemically active valence electrons and the chemically inert core electrons when the valence electrons are strongly correlated. Second, new methodologies for Jastrow factor wavefunctions are explored in an attempt to describe strong interactions amongst the valence electrons. In its original form, CT theory suffered from poorly understood numerical instabilities in strongly interacting systems and poor accuracy relative to coupled cluster theory in weakly interacting systems. Here the numeric issues are identified as consequences of CT theory's central approximation and shown to be similar to the intruder states of many-body perturbation theory. It is demonstrated that these instabilities can be avoided by placing particular restrictions on the theory's degrees of freedom. In weakly interacting systems, we show that using a quadratic, rather than linear, commutator approximation improves accuracy, and that the accuracy in any system can be improved by making use of the system's 3-body reduced density matrix. These improvements are demonstrated in a variety of systems, including N2 , H2 O, NiO, and free base porphin. Jastrow factor wavefunctions have traditionally been handled using quan- tum Monte Carlo techniques such as variational Monte Carlo (VMC). Here we explore how VMC can be made to accommodate the unconventionally large numbers of variational parameters present in many-body Jastrow factors and show that the stochastic reconfiguration algorithm can be thus improved by employing the method of conjugate gradients to solve its central linear equation. In addition to extending the reach of VMC, we have also developed nonstochastic methods for working with Jastrow factor wavefunctions that do not rely on random sampling. These methods, similar in design to those used with the coupled cluster wavefunction, produce energies very close to those of VMC in applications to the Heisenberg, spinless Hubbard, and full Hubbard model Hamiltonians.
Accurately predicting the effects of correlations caused by strong interactions between electrons represents one of the key unsolved challenges in quantum chemistry. This thesis details theoretical developments addressing two aspects of this challenge. First, advances in Canonical Transformation (CT) theory address the task of describing weak interactions between the chemically active valence electrons and the chemically inert core electrons when the valence electrons are strongly correlated. Second, new methodologies for Jastrow factor wavefunctions are explored in an attempt to describe strong interactions amongst the valence electrons. In its original form, CT theory suffered from poorly understood numerical instabilities in strongly interacting systems and poor accuracy relative to coupled cluster theory in weakly interacting systems. Here the numeric issues are identified as consequences of CT theory's central approximation and shown to be similar to the intruder states of many-body perturbation theory. It is demonstrated that these instabilities can be avoided by placing particular restrictions on the theory's degrees of freedom. In weakly interacting systems, we show that using a quadratic, rather than linear, commutator approximation improves accuracy, and that the accuracy in any system can be improved by making use of the system's 3-body reduced density matrix. These improvements are demonstrated in a variety of systems, including N2 , H2 O, NiO, and free base porphin. Jastrow factor wavefunctions have traditionally been handled using quan- tum Monte Carlo techniques such as variational Monte Carlo (VMC). Here we explore how VMC can be made to accommodate the unconventionally large numbers of variational parameters present in many-body Jastrow factors and show that the stochastic reconfiguration algorithm can be thus improved by employing the method of conjugate gradients to solve its central linear equation. In addition to extending the reach of VMC, we have also developed nonstochastic methods for working with Jastrow factor wavefunctions that do not rely on random sampling. These methods, similar in design to those used with the coupled cluster wavefunction, produce energies very close to those of VMC in applications to the Heisenberg, spinless Hubbard, and full Hubbard model Hamiltonians.
Achieving materials-specific predictions for large, realistic molecules and solids with strong electron correlations remains a long-standing challenge in quantum chemistry. This dissertation develops scalable ab initio quantum many-body methods for crystalline materials, demonstrating that rigorous optimization of theoretical approximations and numerical algorithms enables predictive computation even in strongly correlated systems. The work tackles three interconnected challenges, namely describing material-specific properties of strongly correlated superconductors, achieving linear scaling with k-points for large-scale periodic calculations, and establishing reliable methods for electron-phonon interactions. The following chapters address each challenge in turn. Chapter 2 demonstrates that ab initio density matrix embedding theory, combined with symmetry-breaking strategies for superconducting order parameters, reproduces experimental trends in cuprate superconductivity, including pressure and layer effects on transition temperatures. By directly solving the electronic Schrödinger equation from material structures, these calculations capture multi-orbital covalency effects that simplified model Hamiltonians neglect, revealing their essential role in the spin fluctuations driving pairing. This predictive capability establishes a foundation for efficient computational screening of superconducting materials. Extending these calculations to larger k-point meshes and broader phase diagram explorations, however, demands more efficient computational algorithms. To overcome this computational bottleneck, Chapter 3 introduces interpolative separable density fitting framework. This development enables thermodynamic limit calculations with up to 1000 k-points, as validated on diamond, carbon dioxide, nickel monoxide, and cuprate systems. Integration with density matrix embedding theory and local natural orbital methods further yields converged ground-state energies from accurate correlated wavefunction methods such as CCSD(T) for crystalline materials in the thermodynamic limit. With computational efficiency established, the remaining challenge lies in extending the theoretical framework beyond purely electronic degrees of freedom. Chapter 4 addresses this need by establishing exponential ansatz wavefunctions as competitive approaches for electron-phonon systems. Unlike previous treatments based on perturbation theory, systematic benchmarking of coupled cluster theory and variational Lang-Firsov methods on the Holstein model demonstrates, for the first time, that these approaches accurately describe polaron formation across coupling regimes when combined with appropriate reference state optimization. These results provide a foundation for incorporating phonon effects into ab initio studies of superconducting materials. In addition to these methodological advances, GPU acceleration and MPI parallelization enhance computational throughput, enabling efficient large-scale calculations. Together, these developments constitute a computational framework for predictive ab initio quantum many-body calculations in crystalline materials, opening new avenues for understanding and designing strongly correlated quantum materials.
Our overarching goal is the development of wavefunction-based quantum chemistry methods that give good results for both weakly- and strongly-correlated systems, that preserve as many fundamental properties of the exact wavefunction as possible, and that are computationally affordable and black-box for the end user. Many of our contributions in this area are based on the archetypal quantum chemistry method, coupled-cluster theory, and they have resulted from our efforts to understand and remedy the failure of symmetry-adapted coupled-cluster theory in the presence of strong correlation. In this thesis, we recount these investigations and discuss the key insights and results that they have produced. We describe in detail the novel wavefunction approaches we have devised, as well as the approximations and considerations one must make in order to arrive at useful equations. We believe that this work could be important for the continued development of affordable methods capable of accurately describing a wide variety of quantum chemical systems.
Computer simulation is central to understanding the behaviour of matter. Yet, more than a century after the development of quantum theory, first-principles simulation of chemistry remains a daunting challenge, owing to the exponential growth in memory required to represent interacting quantum particles such as electrons in molecules. Quantum computing offers, in principle, a fundamentally more efficient paradigm for simulating quantum systems, with quantum chemistry often cited as a flagship application. However, many quantum algorithms face intrinsic limitations, and any genuine quantum advantage must surpass the strong baseline set by decades of progress in classical algorithms and hardware. While classical methods often suffice for simulating weakly correlated systems, systems with many strongly correlated electrons remain a challenge for both classical and quantum approaches. In this thesis, we develop quantum algorithms for solving the time-independent Schrödinger equation, with a focus on molecular systems with strongly correlated electrons. We introduce a framework for identifying a compact set of states that parametrise the many-body wavefunction by capturing the correlation due to spin coupling between localised electrons. This representation requires very few variational parameters but incurs a very high memory cost on classical computers. To overcome this, we design quantum circuits that efficiently prepare such states by exploiting their symmetry, encoding the dominant entanglement structure of electronic wavefunctions into the initial qubit states. Through theoretical analysis and classical simulations of quantum computers, we show that this substantially reduces the quantum resources required to accurately compute ground- and excitedstate properties. Finally, we develop an approach for approximating the optimal ansatz within the variational quantum eigensolver, an algorithm designed for near-term quantum devices. Our method constructs wavefunctions from products of symmetrypreserving unitary operators and couples discrete and continuous optimisation to yield highly compact representations of strongly correlated electrons using shallow quantum circuits. Together, these results establish complementary directions for embedding problemspecific structure into quantum simulation algorithms, advancing the practical simulation of strongly correlated electrons.
We present a class of correlated wavefunctions generated by exponentials of two-body on-site Hermitian operators that can be evaluated with polynomial computational cost via a Hamiltonian similarity transformation. Wavefunctions of this form have been studied with variational Monte Carlo methods, but we present a formalism to perform non-stochastic calculations. The Hausdorff series generated by these Jastrow factors can be summed exactly without truncation resulting in a set of equations with polynomial computational cost. The correlators include the density-density, collinear spin-spin, spin-density cross terms, and on-site double occupancy operators. The resulting non-Hermitian many-body Hamiltonian can be solved in a biorthogonal mean-field approach with only a small set of correlation terms required for accurate calculations in systems with local interactions. Although the energy of the model is unbound, projective equations in the spirit of coupled cluster theory lead to well-defined solutions. The theory is tested on the one and two-dimensional repulsive Hubbard model where it yields accurate results for large systems with low computational cost. Symmetry projection methods are included to further improve the reference wavefunction and results under strong correlation without sacrificing good quantum numbers resulting in very accurate energies for small systems and producing a better ground state for the calculation of other properties.
Standard approaches to approximating solutions to the Schroedinger equation rely upon the assumption that there is a single dominant electronic configuration with a small number of low-lying excited states. As such, they are referred to as single reference approaches and are accurate only when the band gap is small in comparison with the inter-electronic repulsion of the valence electrons. Systems for which this is the case are weakly-correlated. If the inter-electronic repulsion larger than the band gap, then a single electronic reference is not an appropriate ansatz for the physical wavefunction. As the strength of the interaction is increased many Slater determinants become near-degenerate, each providing non-negligible contributions. Such systems are strongly-correlated. The goal of the thesis was to employ wavefunctions which are: non-trivial, tractable, size-consistent and extensive, and not given in an active space. We have outlined approaches based on wavefunction forms more suited to describe strong correlation. Built on the Lie algebra su(2), our first trial wavefunction was the Bethe ansatz solution to the reduced BCS Hamiltonian derived by Richardson. This wavefunction is expressed as a product of quasi-particles each parametrized in terms of quasi-momenta. Consistency among the quasi-momenta is ensured by a set of non-linear equations to be solved numerically. This approach is feasible in both variational and projected flavours. We next considered a wavefunction of the same form without requiring it to be an eigenvector of the reduced BCS Hamiltonian. A variational approach in this case is only feasible provided there are two equivalent descriptions in terms of particles and holes. In any case, a bivariational principle and a projected approach are both feasible. Also based on su(2), there is AP1roG (coupled cluster pair doubles). Open-shell singlets and triplets were constructed with the Lie algebras sp(N) and so(4N). The resulting formulae appear to be infeasible unless it is possible to efficiently compute generalizations of determinants to three-dimensional objects. Treating each spatial orbital individually, all four possible occupations close the Lie superalgebra gl(2|2).
This thesis develops stochastic many-body methodologies for the electronic structure description of chemically relevant systems with strong electron correlation and nonequilibrium dynamics. Many phenomena of interest in chemistry and materials science, including excited-state processes, charge transfer at catalytic interfaces, and plasmon-mediated photocatalysis, require an accurate treatment of interacting electrons beyond mean-field theory. However, direct solutions of the many-body Schrodinger equation remain computationally intractable for realistic systems due to the exponential growth of the configurational Hilbert space with system size. This challenge motivates the development of reduced representations and embedding strategies capable of retaining essential correlation effects at lower computational cost. The work presented here focuses on stochastic matrix compression and downfolding techniques for many-body electronic structure theory. In particular, new stochastic compression approaches are introduced for the efficient representation of many-body Hamiltonians and correlated wavefunctions within reduced active subspaces. These methods extend stochastic electronic structure techniques beyond their traditional applications in many-body perturbation theory and provide a framework for treating correlated subsystems embedded within large chemical environments. Applications of these approaches are demonstrated across several classes of problems. First, stochastic downfolding methodologies are developed for the computation of charged excitations and Dyson orbitals relevant to catalytic interfaces and molecular spectroscopy. These techniques enable the treatment of large orbital spaces while maintaining accurate descriptions of electron correlation in the active region. Second, stochastic compression methods are applied to the computation of total energies in large solvated and reactive systems, where conventional correlated approaches become prohibitively expensive. Finally, the thesis explores nonequilibrium electronic dynamics at driven catalytic interfaces, including plasmon-assisted charge transfer processes. In this context, time-dependent many-body simulations reveal that transient driving fields can dynamically modify molecular spectral structure, opening additional resonant pathways for hot-carrier injection that are absent in equilibrium descriptions. Taken together, the methods developed in this thesis provide new strategies for reducing the computational complexity of correlated electronic structure calculations while preserving the essential physics of many-body interactions. These results broaden the applicability of stochastic and embedding-based approaches to large-scale equilibrium and nonequilibrium problems in chemistry, condensed matter physics, and materials science.