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The development of quantum mechanics has historically allowed researchers to theoretically explore the fundamental physical properties of atoms and molecules. Although quantum mechanics has been around for almost a century, its use was largely limited by the computational complexity it demanded. In the past decade, computer technology has evolved to the point where it is possible to perform calculations on biologically relevant systems. This has allowed us to corroborate results obtained from experiment as well as predict and explain phenomena that experiment cannot. Unfortunately, the field as a whole has not progressed to the point where high level methods, such as Multi-Reference Configuration Interaction (MRCI), are applicable to large molecular systems. Thus, to effectively study these systems, compromises must be made. In this work, two different approaches are taken to study the photophysical properties of systems such as DNA. In the first approach, a model system is formulated and studied in lieu of the larger target system. The excited state dynamics of 8-oxoguanine (8-oG) and its anion are studied in order to assess the possibility of taking part in an electron transfer mechanism to repair a nearby cyclobutane pyrimidine dimer (CPD). It is found that barriers on the anion S1 excited state surface prohibits easy access to conical intersections with the ground state, causing the anion to have a much longer excited state lifetime than the neutral form. Although much insight can be gained by this method, it is not uncommon for crucial interactions to be lost through simplification. In this case, when 8-oG is placed in an adenine dinucleotide, the π stacking interaction allows it to form a long lived radical base pair, which may be fundamental to its role in CPD repair. Unfortunately, it is impossible to carry out the same excited state calculations for the 8-oG/adenine dinucleotide due to computational cost. For reasons such as these, we also implement and benchmark a new approach to carrying out high level configuration interaction calculations in which the MRCI is expanded in the basis of high multiplicity natural orbitals (HMNOs). Specifically, the HMNO approach is implemented by expanding the MRCI wavefunction in the basis of natural orbitals generated from a ground state high multiplicity Configuration Interaction Singles and Doubles (CISD) calculation. Excited state calculations both at and away from the Franck-Condon region were performed to benchmark the ability of the HMNO approach using CISD and MRCI to reproduce standard MRCI energies. The ability of the HMNOs to be truncated was also explored, yielding efficient truncation criteria and guidelines for choosing the best basis set. It is found that the MRCI/HMNO approach yields energies that are in excellent agreement with standard MRCI while only requiring a fraction of the computational effort, possibly allowing it to be applied to larger systems such as nucleotide dimers. / Chemistry

Electronic structure methods aim to accurately describe the behaviour of the electrons in molecules and materials. To be applicable to arbitrary systems, these methods cannot depend on observations of specific chemical phenomena and must be derived solely from the fundamental physical constants and laws that govern all electrons. Such methods are called ab initio methods. Ab initio methods directly solve the electronic Schrödinger equation to obtain the electronic energy and wavefunction. For more than one electron, solving the electronic Schrödinger equation is impossible, so it is imperative to develop approximate methods that cater to the needs of their users, which can vary depending on the chemical systems under study, the available computational resources and time, and the desired level of accuracy. The most accessible ab initio approaches, including Hartree-Fock methods and Kohn-Sham density functional theory methods, assume that only one electronic configuration is needed to describe the system. While these single-reference methods are successful when describing systems where a single electron configuration dominates, like most closed-shell ground-state organic molecules in their equilibrium geometries, single-reference methods are unreliable for molecules in nonequilibrium geometries (e.g., transition states) and molecules containing unpaired electrons (e.g., transition metal complexes and radicals). For these types of multireference systems, accurate results can only be obtained if multiple electronic configurations are accounted for. Wavefunctions that incorporate many electronic configurations are called multideterminant wavefunctions. This thesis presents a systematic approach to developing multideterminant wavefunctions. First, we establish a framework that outlines the structural components of a multideterminant wavefunction and propose several novel wavefunction ansätze. Then, we present a software package that is designed to aid the development of new wavefunctions and algorithms. Using this approach, we develop an algorithm for evaluating the geminal wavefunctions, a class of multideterminant wavefunctions that are expressed with respect to electron pairs. Finally, we explore using machine learning to solve the Schrödinger equation by presenting a neural network wavefunction ansatz and optimizing its parameters using stochastic gradient descent. / Thesis / Doctor of Science (PhD)

Despite the fact that higher-order tensors (HOTs) plague electronic structure methods and severely limits the modeling of interesting chemistry problems, introduction and application of higher-order tensor (HOT) decompositions, specifically the canonical polyadic (CP) decomposition, is fairly limited. The CP decomposition is an incredibly useful sparse tensor factorization that has the ability to disentangle all correlated modes of a tensor. However the complexities associated with CP decomposition have made its application in electronic structure methods difficult. Some of the major issues related to CP decomposition are a product of the mathematics of computing the decomposition: determining the exact CP rank is a non-polynomially hard problem, finding stationary points for rank-R approximations require non-linear optimization techniques, and inexact CP approximations can introduce a large degree of error into tensor networks. While other issues are a result of the construction of computer architectures. For example, computer processing units (CPUs) are organized in a way to maximize the efficiency of dense linear algebra and, thus, the performance of routine tensor algebra kernels, like the Khatri-Rao product, is limited. In this work, we seek to reduce the complexities associated with the CP decomposition and create a route for others to develop reduced-scaling electronic structure theory methods using the CP decomposition. In Chapter 2, we introduce the robust tensor network approximation. This approximation is a way to, in general, eliminate the leading-order error associated with approximated tensors in a network. We utilize the robust network approximation to significantly increase the accuracy of approximating density fitting (DF) integral tensors using rank-deficient CP decompositions in the particle-particle ladder (PPL) diagram of the coupled cluster method with single and double substitutions (CCSD). We show that one can produce results with negligible error in chemically relevant energy differences using a CP rank roughly the same size as the DF fitting basis; which is a significantly smaller rank requirement than found using either a nonrobust approximation or similar grid initialized CP approximations (the pseudospectral (PS) and tensor hypercontraction (THC) approximations). Introduction of the CP approximation, formally, reduces the complexity of the PPL diagram from 𝓞(N⁶) to 𝓞(N⁵) and, using the robust approximation, we are able to observe a cost reduction in CCSD calculations for systems as small as a single water molecule. In Chapter 3, we further demonstrate the utility of the robust network approximation and, in addition, we construct a scheme to optimize a grid-free CP decomposition of the order-four Coulomb integral tensor in 𝓞(N⁴) time. Using these ideas, we reduce the complexity of ten bottleneck contractions from 𝓞(N⁶) to 𝓞(N⁵) in the Laplace transform (LT) formulation of the perturbative triple, (T), correction to CCSD. We show that introducing CP into the LT (T) method with a CP rank roughly the size of the DF fitting basis reduces the cost of computing medium size molecules by a factor of about 2.5 and introduces negligible error into chemically relevant energy differences. Furthermore, we implement these low-cost algorithms using newly developed, optimized tensor algebra kernels in the massively-parallel, block-sparse TiledArray [Calvin, et. al Chemical Reviews 2021 121 (3), 1203-1231] tensor framework. / Doctor of Philosophy / Electronic structure methods and accurate modeling of quantum chemistry have developed alongside the advancements in computer infrastructures. Increasingly large and efficient computers have allowed researchers to model remarkably large chemical systems. Sadly, for as fast as computer infrastructures grow (Moores law predicts that the number of transistors in a computer will double every 18 months) the cost of electronic structure methods grows more quickly. One of the least expensive electronic structure methods, Hartree Fock (HF), grows quartically with molecular size; this means that doubling the size of a molecule increase the number of computer operations by a factor of 16. However, it is known that when chemical systems become sufficiently large, the amount of physical information added to the system grows linearly with system size.[Goedecker, et. al. Comput. Sci. Eng., 2003, 5, (4), 14-21] Unfortunately, standard implementations of electronic structure methods will never achieve linear scaling; the disparity between actual cost and physical scaling of molecules is a result of storing and manipulating data using dense tensors and is known as the curse of dimensionality.[Bellman, Adaptive Control Processes, 1961, 2045, 276] Electronic structure theorists, in their desire to apply accurate methods to increasingly large systems, have known for some time that the cost of conventional algorithms is unreasonably high. These theorists have found that one can reveal sparsity and develop reduced-complexity algorithms using matrix decomposition techniques. However, higher-order tensors (HOTs), tensors with more than two modes, are routinely necessary in algorithm formulations. Matrix decompositions applied to HOTs are not necessarily straight-forward and can have no effect on the limiting behavior of an algorithm. For example, because of the positive definiteness of the Coulomb integral tensor, it is possible to perform a Cholesky decomposition (CD) to reduce the complexity of tensor from an order-4 tensor to a product of order-3 tensors.[Beebe, et. al. Int. J. Quantum Chem., 1977, 12, 683-705] However, using the CD approximated Coulomb integral tensors it is not possible to reduce the complexity of popular methods such as Hartree-Fock or coupled cluster theory. We believe that the next step to reducing the complexity of electronic structure methods is through the accurate application of HOT decompositions. In this work, we only consider a single HOT decomposition: the canonical polyadic (CP) decomposition which represents a tensor as a polyadic sum of products. The CP decomposition disentangles all modes of a tensor by representing an order-N tensor as N order-2 tensors. In this work, we construct the CP decomposition of tensors using algebraic optimization. Our goal, here, is to tackle one of the biggest issues associated with the CP decomposition: accurately approximating tensors and tensor networks. In Chapter 2, we develop a robust formulation to approximate tensor networks, a formulation which removes the leading-order error associated with tensor approximations in a network.[Pierce, et. al. J. Chem. Theory Comput., 2021 17 (4), 2217- 2230] We apply a robust CP approximation to the coupled cluster method with single and double substitutions (CCSD) to reduce the overall cost of the approach. Using this robust CP approximation we can compute CCSD, on average, 2.5-3 times faster and introduce negligibly small error in chemically relevant energy values. Furthermore in Chapter 3, we again use the robust CP network approximation in conjunction with a novel, low cost approach to compute order-four CP decompositions, to reduce the cost of 10 high cost computations in the the perturbative triple, (T), correction to CCSD. By removing these computations, we are able to reduce the cost of (T) by a factor of about 2.5 while introducing significantly small error.

Accurate correlated electronic structure methods involve a significant amount of computations and can be only employed to small molecular systems. For example, the coupled-cluster singles, doubles, and perturbative triples model (CCSD(T)), which is known as the ``gold standard" of quantum chemistry for its accuracy, usually can treat molecules with 20-30 atoms. To extend the reach of accurate correlated electronic structure methods to larger molecular systems, we work towards two directions: parallel computing and reduced-cost/scaling approaches. Parallel computing can utilize more computational resources to handle systems that demand more substantial computational efforts. Reduced-cost/scaling approaches, which introduce approximations to the existing electronic structure methods, can significantly reduce the amount of computation and storage requirements. In this work, we introduce a new distributed-memory massively parallel implementation of standard and explicitly correlated (F12) coupled-cluster singles and doubles (CCSD) with canonical bigO{N^6} computational complexity ( C. Peng, J. A. Calvin, F. Pavov{s}evi'c, J. Zhang, and E. F. Valeev, textit{J. Phys. Chem. A} 2016, textbf{120}, 10231.), based on the TiledArray tensor framework. Excellent strong scaling is demonstrated on a multi-core shared-memory computer, a commodity distributed-memory computer, and a national-scale supercomputer. We also present a distributed-memory implementation of the density-fitting (DF) based CCSD(T) method. (C. Peng, J. A. Calvin, and E. F. Valeev, textit{in preparation for submission}) An improved parallel DF-CCSD is presented utilizing lazy evaluation for tensors with more than two unoccupied indices, which makes the DF-CCSD storage requirements always smaller than those of the non-iterative triples correction (T). Excellent strong scaling is observed on both shared-memory and distributed-memory computers equipped with conventional Intel Xeon processors and the Intel Xeon Phi (Knights Landing) processors. With the new implementation, the CCSD(T) energies can be evaluated for systems containing 200 electrons and 1000 basis functions in a few days using a small size commodity cluster, with even more massive computations possible on leadership-class computing resources. The inclusion of F12 correction to the CCSD(T) method makes it converge to basis set limit much more rapidly. The large-scale parallel explicitly correlated coupled-cluster program makes the accurate estimation of the coupled-cluster basis set limit for molecules with 20 or more atoms a routine. Thus, it can be used rigorously to test the emerging reduced-scaling coupled-cluster approaches. Moreover, we extend the pair natural orbital (PNO) approach to excited states through the equation-of-motion coupled cluster singles and doubles (EOM-CCSD) method. (C. Peng, M. C. Clement, and E. F. Valeev, textit{submitted}) We simulate the PNO-EOM-CCSD method using an existing massively parallel canonical EOM-CCSD program. We propose the use of state-averaged PNOs, which are generated from the average of the pair density of excited states, to span the PNO space of all the excited states. The doubles amplitudes in the CIS(D) method are used to compute the state-averaged pair density of excited states. The issue of incorrect states in the state-averaged pair density, caused by an energy reordering of excited states between the CIS(D) and EOM-CCSD, is resolved by simply computing more states than desired. We find that with a truncation threshold of $10^{-7}$, the truncation error for the excitation energy is already below 0.02 eV for the systems tested, while the average number of PNOs is reduced to 50-70 per pair. The accuracy of the PNO-EOM-CCSD method on local, Rydberg and charge transfer states is also investigated. / Ph. D.

Numerous pigment protein complexes exist in natural systems to harvest light energy such as photosystem II and Nanosalina xenorhodopsin. However, the mechanisms of these lipid embedded photo-active complexes have yet to be fully understood. Photosystem II is of interest due to being a compact complex which can perform the three initial key steps of photosynthesis: absorb light, transfer the excitation from the antennae to reaction center, and perform efficient charge separation. Despite considerable theoretical and experimental effort the exact mechanism of this process remains uncertain. Nanosalina xenorhodopsin is a more recently discovered inwards proton pump with minimal studies into the inwards proton pumping mechanism. Nanosalina xenorhodopsin is of interest as it contrasts with other known and well studied rhodopsins which serve as outwards proton pumps, moving H+ ions out of a cell. In this work, we use the Hamiltonian ensemble method to construct the first fully ab initio computational models of these systems which will be used to determine the mechanisms of these systems. To construct these models we first investigated the effect of the modeled surrounding membrane and simulated temperature. The effect of the extended modeled environment on calculated results is often overlooked but important for the construction of an accurate ab initio model. Our models showed that both membrane composition and temperature result in significant changes in the behavior of the extended membrane system, relative excitation energies of chromophores, and energy dynamics of a pigment protein complex. The absolute excitation energies of chromophores, absorption spectra, and linear dichroism spectra were comparatively insensitive to changes in the modeled environment. With the effect of the environment established, we present a preliminary method to extend our photosystem II model to include charge transfer states, and a preliminary model of Nanosalina xenorhodopsin which can determine the photocycle states through validation of calculated spectra against experimental results.