Quantum chemical methods to predict experimental chiroptical properties by solving the time-dependent Schrödinger equation are useful in the assignment of absolute configurations. Chiroptical properties, being very sensitive to the electronic structure of the system, require highly-accurate methods on the one hand and on the other, need to be able to be computed with limited computational resources. The calculation of the optical rotation in the solution phase is complicated by solvent effects. In order to capture those solvent effects, we present a study that uses conformational averaging and time-dependent density functional theory calculations that incorporate solvent molecules explicitly in the quantum mechanical region. While considering several controllable parameters along which the system's optical rotation varies, we find that the sampling of the dynamical trajectory and the density functional chosen have the largest impact on the value of the rotation. In order to eliminate the arbitrariness of the choice of density functional, we would prefer to use coupled cluster theory, a robust and systematically improvable method. However, the high-order polynomial scaling of coupled cluster theory makes it intractable for numerous large calculations, including the conformational averaging required for optical rotation calculations in solution. We therefore attempt to reduce the scaling of a linear response coupled cluster singles and doubles (LR-CCSD) calculation via a perturbed pair natural orbital (PNO++) local correlation approach which uses an orbital space created using a perturbed density matrix. We find that by creating a "combined PNO++" space, incorporating a set of orbitals from the unperturbed pair natural orbital (PNO) space into the PNO++ space, we can obtain well-behaved convergence behavior for both CCSD correlation energies and linear response properties, including dynamic polarizabilities and optical rotations, for the small systems considered. The PNO++ and combined PNO++ methods require aggressive truncation to keep the computational cost low, due to an expensive two-electron integral transformation at the beginning of the calculation. We apply the methods to larger systems than previously studied and refine them for more aggressive truncation by exploring an alternative form of the perturbed density and a perturbation-including weak pair approximation. / Doctor of Philosophy / Theoretical chemistry attempts to provide connections between the structure of molecules and their observable properties. One such family of observables are chiroptical properties, or the effect of the medium on the light which passes through it. These properties include the scattering, absorption and change in polarization of light. Light being classically an electromagnetic field, chiroptical properties can be derived by treating molecules quantum mechanically and the light classically. The prediction of chiroptical properties on computers using the principles of quantum mechanics is still a growing field, being very sensitive to the method used, and requiring considerations of factors such as conformations and anharmonic corrections. Matching experimental properties is an important step in the creation of a reliable method of predicting properties of systems in order to provide more information than can be obtained through experimental observation. This work begins by addressing the problem of matching experimentally obtained quantities. Our results show that current time-intensive methods still fall short in the matching of experimental data. Thus, we then move on to approximating a more robust but computationally expensive method in order to be able to use a more accurate method on a larger scale than is currently possible. On obtaining positive results for small test systems, we test the new method on larger systems, and explore possible improvements to its accuracy and efficiency.

One of the central problems limiting the application of accurate {em ab initio} methods to large molecular systems is their high computational costs, i.e., their computing and storage requirements exhibit polynomial scaling with the size of the system. For example, the coupled cluster singles and doubles method with the perturbative inclusion of triples: the CCSD(T) model, which is considered to be the ``gold standard'' of quantum chemistry scales as 𝑂(N⁷) in its canonical formulation, where $N$ is a measure of the system size. However, the steep scaling associated with these methods is unphysical since the property of dynamic electron correlation or dispersion (for insulators) is local in nature and decays as R⁻⁶ power of distance. Different reduced-scaling techniques which attempt to exploit this inherent sparsity in the wavefunction have been used in conjunction with the coupled cluster theory to calculate ground-state properties of molecular systems with hundreds of heavy atoms in reasonable computational time. However, efforts towards extension of these methods for describing response properties like polarizabilities, optical rotations, etc., which are related to the derivative of the wavefunction with respect to external electric or/and magnetic fields, have been fairly limited and conventional reduced-scaling algorithms have been shown to yield large and often erratic deviations from the full canonical results. Accurate simulation of response properties like optical rotation is highly desirable as it can help the experimental chemists in understanding the structure-activity relationship of different chiral drug candidates. In this work, we identify the reasons behind the unsatisfactory performance of the pair natural orbital (PNO) based reduced-scaling approach for calculating linear response properties at the coupled cluster level of theory and propose novel modifications, which we refer to as PNO++, (A. Kumar and T. D. Crawford. Perturbed Pair Natural Orbitals for Coupled-Cluster Linear-Response Theory. 2018, {em manuscript in preparation}) that can provide the necessary accuracy at significantly lower computational costs. The motivation behind the PNO++ approach came from our works on the (frozen) virtual natural orbitals (FVNO), which can be seen as a precursor to the concept of PNOs (A. Kumar and T. D. Crawford. Frozen Virtual Natural Orbitals for Coupled-Cluster Linear-Response Theory. {em J. Phys. Chem. A}, 2017, 121(3), pp 708 716) and the improved FVNO++ method (A. Kumar and T. D. Crawford. Perturbed Natural Orbitals for Coupled-Cluster Linear-Response Theory. 2018, {em manuscript in preparation}). The essence of these modified schemes (FVNO++ and PNO++) lie in finding suitable field perturbed one-electron densities to construct ``perturbation aware" virtual spaces which, by construction, are much more compact for describing response properties, making them ideal for applications on large molecular systems. / Ph. D.

Through the exploitation of data-sparsity ---a catch all term for savings gained from a variety of approximations--- it is possible to reduce the computational cost of accurate electronic structure calculations to linear. Meaning, that the total time to solution for the calculation grows at the same rate as the number of particles that are correlated. Multiple techniques for exploiting data-sparsity are discussed, with a focus on those that can be systematically improved by tightening numerical parameters such that as the parameter approaches zero the approximation becomes exact. These techniques are first applied to Hartree-Fock theory and then we attempt to design a linear scaling massively parallel electron correlation strategy based on second order perturbation theory. / Ph. D.

Over the course of the last century, electronic structure theory (or, alternatively, computational quantum chemistry) has grown from being a fledgling field to being a "full partner with experiment" [Goddard Science 1985, 227 (4689), 917--923]. Numerous instances of theory matching experiment to very high accuracy abound, with one excellent example being the high-accuracy ab initio thermochemical data laid out in the 2004 work of Tajti and co-workers [Tajti et al. J. Chem. Phys. 2004, 121, 11599] and another being the heats of formation and molecular structures computed by Feller and co-workers in 2008 [Feller et al. J. Chem. Phys. 2008, 129, 204105]. But as the authors of both studies point out, this very high accuracy comes at a very high cost. In fact, at this point in time, electronic structure theory does not suffer from an accuracy problem (as it did in its early days) but a cost problem; or, perhaps more precisely, it suffers from an accuracy-to-cost ratio problem. We can compute electronic energies to nearly any precision we like, as long as we are willing to pay the associated cost. And just what are these high computational costs? For the purposes of this work, we are primarily concerned with the way in which the computational cost of a given method scales with the system size; for notational purposes, we will often introduce a parameter, N, that is proportional to the system size. In the case of Hartree-Fock, a one-body wavefunction-based method, the scaling is formally N⁴, and post-Hartree-Fock methods fare even worse. The coupled cluster singles, doubles, and perturbative triples method [CCSD(T)], which is frequently referred to as the "gold standard" of quantum chemistry, has an N⁷ scaling, making it inapplicable to many systems of real-world import. If highly accurate correlated wavefunction methods are to be applied to larger systems of interest, it is crucial that we reduce their computational scaling. One very successful means of doing this relies on the fact that electron correlation is fundamentally a local phenomenon, and the recognition of this fact has led to the development of numerous local implementations of conventional many-body methods. One such method, the DLPNO-CCSD(T) method, was successfully used to calculate the energy of the protein crambin [Riplinger, et al. J. Chem. Phys 2013, 139, 134101]. In the following work, we discuss how the local nature of electron correlation can be exploited, both in terms of the occupied orbitals and the unoccupied (or virtual) orbitals. In the case of the former, we highlight some of the historical developments in orbital localization before applying orbital localization robustly to infinite periodic crystalline systems [Clement, et al. 2021, Submitted to J. Chem. Theory Comput.]. In the case of the latter, we discuss a number of different ways in which the virtual space can be compressed before presenting our pioneering work in the area of iteratively-optimized pair natural orbitals ("iPNOs") [Clement, et al. J. Chem. Theory Comput. 2018, 14 (9), 4581--4589]. Concerning the iPNOs, we were able to recover significant accuracy with respect to traditional PNOs (which are unchanged throughout the course of a correlated calculation) at a comparable truncation level, indicating that our improved PNOs are, in fact, an improved representation of the coupled cluster doubles amplitudes. For example, when studying the percent errors in the absolute correlation energies of a representative sample of weakly bound dimers chosen from the S66 test suite [Řezác, et al. J. Chem. Theory Comput. 2011, 7 (8), 2427--2438], we found that our iPNO-CCSD scheme outperformed the standard PNO-CCSD scheme at every truncation threshold (τ_PNO) studied. Both PNO-based methods were compared to the canonical CCSD method, with the iPNO-CCSD method being, on average, 1.9 times better than the PNO-CCSD method at τ_PNO = 10⁻⁷ and more than an order of magnitude better for τ_PNO < 10⁻¹⁰ [Clement, et al. J. Chem. Theory Comput 2018, 14 (9), 4581--4589]. When our improved PNOs are combined with the PNO-incompleteness correction proposed by Neese and co-workers [Neese, et al. J. Chem. Phys. 2009, 130, 114108; Neese, et al. J. Chem. Phys. 2009, 131, 064103], the results are truly astounding. For a truncation threshold of τ_PNO = 10⁻⁶, the mean average absolute error in binding energy for all 66 dimers from the S66 test set was 3 times smaller when the incompleteness-corrected iPNO-CCSD method was used relative to the incompleteness-corrected PNO-CCSD method [Clement, et al. J. Chem. Theory Comput. 2018, 14 (9), 4581--4589]. In the latter half of this work, we present our implementation of a limited-memory Broyden-Fletcher-Goldfarb-Shanno (BFGS) based Pipek-Mezey Wannier function (PMWF) solver [Clement, et al. 2021 }, Submitted to J. Chem. Theory Comput.]. Although orbital localization in the context of the linear combination of atomic orbitals (LCAO) representation of periodic crystalline solids is not new [Marzari, et al. Rev. Mod. Phys. 2012, 84 (4), 1419--1475; Jònsson, et al. J. Chem. Theory Comput. 2017, 13} (2), 460--474], to our knowledge, this is the first implementation to be based on a BFGS solver. In addition, we are pleased to report that our novel BFGS-based solver is extremely robust in terms of the initial guess and the size of the history employed, with the final results and the time to solution, as measured in number of iterations required, being essentially independent of these initial choices. Furthermore, our BFGS-based solver converges much more quickly and consistently than either a steepest ascent (SA) or a non-linear conjugate gradient (CG) based solver, with this fact demonstrated for a number of 1-, 2-, and 3-dimensional systems. Armed with our real, localized Wannier functions, we are now in a position to pursue the application of local implementations of correlated many-body methods to the arena of periodic crystalline solids; a first step toward this goal will, most likely, be the study of PNOs, both conventional and iteratively-optimized, in this context. / Doctor of Philosophy / Increasingly, the study of chemistry is moving from the traditional wet lab to the realm of computers. The physical laws that govern the behavior of chemical systems, along with the corresponding mathematical expressions, have long been known. Rapid growth in computational technology has made solving these equations, at least in an approximate manner, relatively easy for a large number of molecular and solid systems. That the equations must be solved approximately is an unfortunate fact of life, stemming from the mathematical structure of the equations themselves, and much effort has been poured into developing better and better approximations, each trying to balance an acceptable level of accuracy loss with a realistic level of computational cost and complexity. But though there has been much progress in developing approximate computational chemistry methods, there is still great work to be done. Many chemical systems of real-world import (particularly biomolecules and potential pharmaceuticals) are simply too large to be treated with any methods that consistently deliver acceptable accuracy. As an example of the difficulties that come with trying to apply accurate computational methods to systems of interest, consider the seminal 2013 work of Riplinger and co-workers [Riplinger, et al. J. Chem. Phys. 2013, 139, 134101]. In this paper, they present the results of a calculation performed on the protein crambin. The method used was DLPNO-CCSD(T), an approximation to the "gold standard" computational method CCSD(T). The acronym DLPNO-CCSD(T) stands for "`domain-based local pair natural orbital coupled cluster with singles, doubles, and perturbative triples." In essence, this method exploits the fact that electron-electron interactions ("electron correlation") are a short-range phenomenon in order to represent the system in a mathematically more compact way. This focus on the locality of electron correlation is a crucial piece in the effort to bring down computational cost. When talking about computational cost, we will often talk about how the cost scales with the approximate system size N. In the case of CCSD(T), the cost scales as N⁷. To see what this means, consider two chemical systems A and B. If system B is twice as large as system A, then the same calculation run on both systems will take 2⁷ = 128 times longer on system B than on system A. The DLPNO-CCSD(T) method, on the other hand, scales linearly with the system size, provided the system is sufficiently large (we say that it is "asymptotically linearly scaling"), and so, for our example systems A and B, the calculation run on system B should only take twice as long as the calculation run on system A. But despite the favorable scaling afforded by the DLPNO-CCSD(T) method, the time to solution is still prohibitive. In the case of crambin, a relatively small protein with 644 atoms, the calculation took a little over 30 days. Clearly, such timescales are unworkable for the field of biochemical research, where the focus is often on the interactions between multiple proteins or other large biomolecules and where many more data points are required. In the work that follows, we discuss in more detail the genesis of the high costs that are associated with highly accurate computational methods, as well as some of the approximation techniques that have already been employed, with an emphasis on local correlation techniques. We then build off this foundation to discuss our own work and how we have extended such approximation techniques in an attempt to further increase the possible accuracy to cost ratio. In particular, we discuss how iteratively-optimized pair natural orbitals (the PNOs of the DLPNO-CCSD(T) method) can provide a more accurate but also more compact mathematical representation of the system relative to static PNOs [Clement, et al. J. Chem. Theory Comput. 2018, 14 (9), 4581--4589]. Additionally, we turn our attention to the problem of periodic infinite crystalline systems, a class of materials less commonly studied in the field of computational chemistry, and discuss how the local correlation techniques that have already been applied with great success to molecular systems can potentially be applied in this domain as well [Clement, et al. 2021, Submitted to J. Chem. Theory Comput.].

Quantum mechanical methods are becoming increasingly useful and applicable tools to complement and support experiment. Nonetheless, some barriers to further applications of theoretical models still remain. A coupled cluster singles and doubles (CCSD) calculation, a reliable ab initio method, scales approximately on the order of 𝑂(𝑁⁶), where 𝑁 is a measure of the system size. This unfortunately limits the use of such high-accuracy methods to relatively small systems. Coupled cluster property calculations must be used in conjunction with reduced-scaling methods in order to broaden the range of applications to larger systems. In this work, we introduce some of the underlying theory behind such calculations and test the performance of several local correlation techniques for polarizabilities, optical rotations, and excited state properties. In general, when the computational cost is significantly reduced, the necessary accuracy is lost. Polarizabilities are less sensitive to the truncation schemes than optical rotations, and the excitation data is often only in agreement with the canonical result for the first few excited states. Additionally, we present a novel application of equation-of-motion coupled cluster singles and doubles to simulated circularly polarized luminescence spectra of eight chiral ketones. Both the absorption in the ground state and emission from the excited states were examined. Extensive geometry analyses were performed, revealing that optimized structures at the density functional theory were adequate for the calculation accurate coupled cluster excitation data. / Ph. D.

Wave function based electronic structure methods have became a robust and reliable tool for the prediction and interpretation of the results of chemical experiments. However, they suffer from very steep scaling behavior with respect to an increase in the size of the system as well as very slow convergence of the correlation energy with respect to the basis set size. Thus these methods are limited to small systems of up to a dozen atoms. The first of these issues can be efficiently resolved by exploiting the local nature of electron correlation effects while the second problem is alleviated by the use of explicitly correlated R12/F12 methods. Since R12/F12 methods are central to this work, we start by reviewing their modern formulation. Next, we present the explicitly correlated second-order Mo ller-Plesset (MP2-F12) method in which all nontrivial post-mean-field steps are formulated with linear computational complexity in system size [Pavov{s}evi'c et al., {em J. Chem. Phys.} {bf 144}, 144109 (2016)]. The two key ideas are the use of pair-natural orbitals for compact representation of wave function amplitudes and the use of domain approximation to impose the block sparsity. This development utilizes the concepts for sparse representation of tensors described in the context of the DLPNO-MP2 method by Neese, Valeev and co-workers [Pinski et al., {em J. Chem. Phys.} {bf 143}, 034108 (2015)]. Novel developments reported here include the use of domains not only for the projected atomic orbitals, but also for the complementary auxiliary basis set (CABS) used to approximate the three- and four-electron integrals of the F12 theory, and a simplification of the standard B intermediate of the F12 theory that avoids computation of four-index two-electron integrals that involve two CABS indices. For quasi-1-dimensional systems (n-alkanes) the bigO{N} DLPNO-MP2-F12 method becomes less expensive than the conventional bigO{N^{5}} MP2-F12 for $n$ between 10 and 15, for double- and triple-zeta basis sets; for the largest alkane, C$_{200}$H$_{402}$, in def2-TZVP basis the observed computational complexity is $N^{sim1.6}$, largely due to the cubic cost of computing the mean-field operators. The method reproduces the canonical MP2-F12 energy with high precision: 99.9% of the canonical correlation energy is recovered with the default truncation parameters. Although its cost is significantly higher than that of DLPNO-MP2 method, the cost increase is compensated by the great reduction of the basis set error due to explicit correlation. We extend this formalism to develop a linear-scaling coupled-cluster singles and doubles with perturbative inclusion of triples and explicitly correlated geminals [Pavov{s}evi'c et al., {em J. Chem. Phys.} {bf 146}, 174108 (2017)]. Even for conservative truncation levels, the method rapidly reaches near-linear complexity in realistic basis sets; e.g., an effective scaling exponent of 1.49 was obtained for n-alkanes with up to 200 carbon atoms in a def2-TZVP basis set. The robustness of the method is benchmarked against the massively parallel implementation of the conventional explicitly correlated coupled-cluster for a 20-water cluster; the total dissociation energy of the cluster ($sim$186 kcal/mol) is affected by the reduced-scaling approximations by only $sim$0.4 kcal/mol. The reduced-scaling explicitly correlated CCSD(T) method is used to examine the binding energies of several systems in the L7 benchmark data set of noncovalent interactions. Additionally, we discuss a massively parallel implementation of the Laplace transform perturbative triple correction (T) to the DF-CCSD energy within density fitting framework. This work is closely related to the work by Scuseria and co-workers [Constans et al., {em J. Chem. Phys.} {bf 113}, 10451 (2000)]. The accuracy of quadrature with respect to the number of quadrature points has been investigated on systems of the 18-water cluster, uracil dimer and pentacene dimer. In the case of the 18-water cluster, the $mu text{E}_{text{h}}$ accuracy is achieved with only 3 quadrature points. For the uracil dimer and pentacene dimer, 6 or more quadrature points are required to achieve $mu text{E}_{text{h}}$ accuracy; however, binding energy of $<$1 kcal/mol is obtained with 4 quadrature points. We observe an excellent strong scaling behavior on distributed-memory commodity cluster for the 18-water cluster. Furthermore, the Laplace transform formulation of (T) performs faster than the canonical (T) in the case of studied systems. The efficiency of the method has been furthermore tested on a DNA base-pair, a system with more than one thousand basis functions. Lastly, we discuss an explicitly correlated formalism for the second-order single-particle Green's function method (GF2-F12) that does not assume the popular diagonal approximation, and describes the energy dependence of the explicitly correlated terms [Pavov{s}evi'c et al., {em J. Chem. Phys.} {bf 147}, 121101 (2017)]. For small and medium organic molecules the basis set errors of ionization potentials of GF2-F12 are radically improved relative to GF2: the performance of GF2-F12/aug-cc-pVDZ is better than that of GF2/aug-cc-pVQZ, at a significantly lower cost. / Ph. D.

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.