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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.].

The efficient computation of dynamic (time-dependent) molecular properties is a broad field with numerous applications in aiding molecular synthesis and design, with a particular preva- lence in spectroscopic predictions. Typical methods for computing the response of a molecu- lar system to an electromagnetic field (EMF) considers a quantum mechanical description of the molecule and a classical approximation for the EMF. Methods for describing light-matter interactions with high-accuracy electronic structure methods, such as coupled cluster (CC), are discussed, with a focus on improving the efficiency of such methods. The CC method suffers from high-degree polynomial scaling. In addition to the ground-state calculation, computing dynamic properties requires the description of sensitive excited-state effects. The cost of such methods often prohibits the accurate calculation of response prop- erties for systems of significant importance, such as large-molecule drug candidates or chiral species present in biological systems. While the literature is ripe with reduced-scaling meth- ods for CC ground-state calculations, considerably fewer approaches have been applied to excited-state properties, with even fewer still providing adequate results for realistic systems. This work presents three studies on the reduction of the cost of molecular property evalu- ations, in the hopes of closing this gap in the literature and widening the scope of current theoretical methods. There are two main ways of simulating time-dependent light-matter interactions: one may consider these effects in the frequency domain, where the response of the system to an EMF is computed directly; or, the response may be considered explicitly in the time domain, where wave function (or density) parameters can be propagated in time and examined in detail. Each methodology has unique advantages and computational bottlenecks. The first two studies focus on frequency-domain calculations, and employ fragmentation and machine- learning techniques to reduce the cost of single-molecule calculations or sets of calculations across a series of geometric conformations. The third study presents a novel application of the local correlation technique to real-time CC calculations, and highlights deficiencies and possible solutions to the approach. / Doctor of Philosophy / Theoretical chemistry plays a key role in connecting experimental results with physical inter- pretation. Paramount to the success of theoretical methods is the ability to predict molecular properties without the need for costly high-throughput synthesis, aiding in the determina- tion of molecular structure and the design of new materials. Light-matter interactions, which govern spectroscopic techniques, are particularly complicated, and sensitive to the theoreti- cal tools employed in their prediction. Compounding the issue of accuracy is one of efficiency — accurate theoretical methods typically incur steep scaling of computational cost (memory and processor time) with respect to the size of the system. An important aspect in improving the efficiency of these methods is understanding the nature of light-matter interactions at a quantum level. Many unanswered questions still remain, such as, "Can light-matter interactions be thought of as a sum of interactions be- tween smaller fragments of the system?" and "Can conventional methods of accelerating ground-state calculations be expected to perform well for spectroscopic properties?" The present work seeks to answer these questions through three studies, focusing on improving the efficiency of these techniques, while simultaneously addressing their fundamental flaws and providing reasonable alternatives.

One of the most promising methods for surmounting the high-degree polynomial scaling wall associated with electron correlating wave function methods is the local correlation technique of Pulay and Saeb&oslash;. They have proposed using a set of localized occupied and virtual orbitals free of the canonical constraint commonly employed in quantum chemistry, resulting in a method that scales linearly (in the asymptotic limit) with molecular size. Pulay and Saeb$oslash; first applied their methods to configuration interaction and later to M$oslash;ller-Plesset perturbation theory. Werner et al. have have extended the local correlation scheme of Pulay and Saeb$oslash; to coupled-cluster theory. One of the pitfalls of the local correlation methods developed by Pulay and Saeb$oslash; is the dependence of domain selection on the molecular geometry. In other words, as the geometry changes the domain structure of the local correlation calculation can change also, leading to discontinuities in the potential energy surface. We have examined the size of these discontinuities for the homolytic bond cleavage of fluoromethane and the heterolytic bond dissociation of singlet ketene and propadienone. Properties such as polarizabilities and optical rotation are realized through linear response theory, where the Hamiltonian is subject to an external perturbation and the wave function is allowed to respond to the applied perturbation. Within the context of local correlation it is necessary to understand how the domain structure alters in response to an applied perturbation. We have proposed using solutions to the CPHF equations (coupled-perturbed Hartree-Fock) in order to predict the correlation response to an applied perturbation. We have applied this technique to the calculation of polarizabilities, with very favorable results, and also to optical rotation, with mixed results. / Ph. D.

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.

In this work we present the first implementation of the incremental scheme for coupled cluster linear-response frequency-dependent dipole polarizabilities. The implementation is fully automated and makes use of the domain-specific basis set approach. The accuracy of the approach is determined on the basis of a test suite of 47 molecules and small clusters. The local approximation in the coupled cluster singles and doubles polarizability exhibits a mean error of 0.02% and a standard deviation of 0.32% when using a third-order incremental expansion. With the proposed approach, it is possible to compute polarizabilities with larger basis sets compared to the canonical implementation and thus it is possible to obtain higher total accuracy. The incremental scheme yields the smallest errors for weakly-bound and quasi-linear systems, while two- and three-dimensional (cage-like) structures exhibit somewhat larger errors as compared to the full test set. / Dieser Beitrag ist aufgrund einer (DFG-geförderten) Allianz- bzw. Nationallizenz frei zugänglich.

Die effiziente Kompression von Daten wird eine wichtige Rolle für mehrere bevorste- hende und geplante Weltraummissionen spielen, die Helioseismologie betreiben werden, wie beispielsweise Solar Orbiter. Solar Orbiter ist die nächste Mission, die Helioseismologie beinhaltet, und soll im Oktober 2018 gestartet werden. Das Hauptmerkmal von Solar Orbiter ist der Orbit. Die Umlaufbahn des Satelliten wird zur Ekliptik geneigt sein, sodass der Satellit einen solaren Breitengrad von bis zu 33 Grad erreichen wird. Dies wird erstmals ermöglichen, die Pole der Sonne mit Hilfe von lokaler Helioseismologie zu studieren. Zusätzlich dazu können kombinierte Beobachtungen von Solar Orbiter und einem anderen Instrument dazu benutzt werden, die tiefen Schichten der Sonne mittels stereoskopischer Helioseismologie zu erforschen. Die Aufnahmen der Dopplergeschwindigkeit und der Kontinuumsintensität, die für Helioseismologie benötigt werden, werden vom Polarimetric and Helioseismic Imager (PHI) geliefert werden. Große Hindernisse für Helioseismologie mit Solar Orbiter sind die niedrige Datenüber- tragungsrate und die (wahrscheinlich) kurzen Beobachtungszeiten. Außerdem erfordert die Untersuchung der Pole der Sonne Beobachtungen in der Nähe des Sonnenrandes, sogar von dem geneigten Orbit von Solar Orbiter aus. Dies kann zu systematischen Fehlern führen. In dieser Doktorarbeit gebe ich eine erste Einschätzung ab, wie stark Helioseismologie von verlustbehafteter Datenkompression beeinflusst wird. Mein Schwerpunkt liegt dabei auf der Solar Orbiter Mission, die von mir erzielten Ergebnisse sind aber auch auf andere geplante Missionen übertragbar. Zunächst habe ich mit Hilfe synthetischer Daten die Eignung des PHI Instruments für Helioseismologie getestet. Diese basieren auf Simulationen der Konvektion nahe der Sonnenoberfläche und einem Modell von PHI. Ich habe eine sechs Stunden lange Zeitreihe synthetischer Daten erstellt, die die gleichen Eigenschaften wie die von PHI erwarteten Daten haben. Hierbei habe ich mich auf den Einfluss der Punktspreizfunktion, der Vibrationen des Satelliten und des Photonenrauschen konzentriert. Die von diesen Daten abgeleitete spektrale Leistungsdichte der solaren Oszillationen legt nahe, dass PHI für Helioseismologie geeignet sein wird. Aufgrund der niedrigen Datenübertragungsrate von Solar Orbiter müssen die von PHI für die Helioseismologie gewonnenen Daten stark komprimiert werden. Ich habe den Einfluss von Kompression mit Hilfe von Daten getestet, die vom Helioseismic and Magnetic Imager (HMI) stammen. HMI ist ein Instrument an Bord des Solar Dynam- ics Observatory Satelliten (SDO), der 2010 gestartet worden ist. HMI erstellt mit hoher zeitlicher Abfolge Karten der Kontinuumsintensität, der Dopplergeschwindigkeit und des kompletten Magnetfeldvektors für die komplette von der Erde aus sichtbare Hemispäre der Sonne. Mit Hilfe mit von HMI aufgenommenen Karten der Dopplergeschwindigkeit konnte ich zeigen, dass das Signal-zu-Rausch Verhältnis von Supergranulation in der Zeit-Entfernungs Helioseismologie nicht stark von Datenkompression beeinflusst wird. Außerdem habe ich nachgewiesen, dass die Genauigkeit und Präzision von Messungen der Sonnenrotation mittels Local Correlation Tracking von Granulation durch verlust- behaftete Datenkompression nicht wesentlich verschlechtert werden. Diese Ergebnisse deuten an, dass die niedrige Datenübertragungsrate von Solar Orbiter nicht unbedingt ein großes Hinderniss für Helioseismologie sein muss.

No description available.

In this work we present the first implementation of the incremental scheme for coupled cluster linear-response frequency-dependent dipole polarizabilities. The implementation is fully automated and makes use of the domain-specific basis set approach. The accuracy of the approach is determined on the basis of a test suite of 47 molecules and small clusters. The local approximation in the coupled cluster singles and doubles polarizability exhibits a mean error of 0.02% and a standard deviation of 0.32% when using a third-order incremental expansion. With the proposed approach, it is possible to compute polarizabilities with larger basis sets compared to the canonical implementation and thus it is possible to obtain higher total accuracy. The incremental scheme yields the smallest errors for weakly-bound and quasi-linear systems, while two- and three-dimensional (cage-like) structures exhibit somewhat larger errors as compared to the full test set. / Dieser Beitrag ist aufgrund einer (DFG-geförderten) Allianz- bzw. Nationallizenz frei zugänglich.

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.