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<p>Understanding the chemical and physical properties of molecules and materials at a fundamental level often requires quantum-mechanical models for these substance's electronic structure. This type of many body quantum mechanics calculation is computationally demanding, hindering its application to substances with more than a few hundreds atoms. The supreme goal of many researches in quantum chemistry-and the topic of this dissertation-is to develop more efficient computational algorithms for electronic structure calculations. In particular, this dissertation develops two new numerical integration techniques for computing molecular and atomic properties within conventional Kohn-Sham-Density Functional Theory (KS-DFT) of molecular electronic structure. </p> <p>The first of these grid-based techniques is based on the transformed sparse grid construction. In this construction, a sparse grid is generated in the unit cube and then mapped to real space according to the pro-molecular density using the conditional distribution transformation. The transformed sparse grid was implemented in program deMon2k, where it is used as the numerical integrator for the exchange-correlation energy and potential in the KS-DFT procedure. We tested our grid by computing ground state energies, equilibrium geometries, and atomization energies. The accuracy on these test calculations shows that our grid is more efficient than some previous integration methods: our grids use fewer points to obtain the same accuracy. The transformed sparse grids were also tested for integrating, interpolating and differentiating in different dimensions (n = 1, 2, 3, 6).</p> <p> The second technique is a grid-based method for computing atomic properties within QTAIM. It was also implemented in deMon2k. The performance of the method was tested by computing QTAIM atomic energies, charges, dipole moments, and quadrupole moments. For medium accuracy, our method is the fastest one we know of.</p> / Thesis / Doctor of Philosophy (PhD)

<p> The material presented in this thesis is an attempt to obtain an increased understanding of the electronic structure and chemical binding in molecular systems. The one-electron charge distribution in methane, which is derived by considering only one-electron dependent properties of the system, is used to analyze the chemical binding in this molecule. A theoretical method, which allows one to determine the effect of the Pauli exclusion principle on the one-electron density distribution, is used to test the concepts underlying the electron pair repulsion theory as applied to H2O and NH3. Kinetic energy distributions are defined in order to examine the relationship between the topographical features of the molecular one-electron charge distribution and the kinetic energy of the system.</p> / Thesis / Doctor of Philosophy (PhD)

Optical and spectroscopic molecular properties are key to characterizing the behavior of molecules interacting with an applied electromagnetic field of light. Response theory has been used for a long time to calculate such properties in the frequency domain. Real-time (RT) methods solve for the frequency-dependent properties in the time domain by explicitly propagating the time-dependent wave function. Various quantum chemical methods can be incorporated with the RT formalism, including Hartree-Fock, density functional theory, configurational interaction, coupled cluster, etc. Among these, coupled cluster (CC) methods provide high accuracy for systems with strong electron correlation, making RT-CC implementations intriguing. All applications of CC methods face a substantial challenge due to their high-order polynomial scaling. For RT-CC methods, two aspects may be explored to improve the efficiency, the numerical techniques regarding the RT propagation and the reduced-scaling methods regarding CC itself. In this work, we start with the exploration of the hardware used for the calculations and the numerical integration methods for propagating the wave function parameters. Firstly, a GPU-enabled Python implementation has been developed by conducting the tensor contractions on GPUs utilizing PyTorch, a machine learning package, that has similar syntax as NumPy for tensor operations. A speedup of a factor of 14 is obtained for the RT-CCSD/cc-pVDZ absorption spectrum calculation of the water tetramer. Furthermore, to optimize the performance on GPUs, single-precision arithmetic is added to the implementation to achieve an additional speedup of a factor of two. Lastly, a group of integrators for solving differential equations are introduced to the RT framework, including regular explicit integrators, adaptive integrators, and a mixed-step-size approach customized for strong-field simulations. The optimal choice of the integrator depends on the requiring accuracy, stability and efficiency. In addition to being highly accurate, CC methods are also systematically improvable and provide a hierarchy of accuracy. Based upon the RT-CCSD implementation, the coupled cluster singles, doubles and approximate triples (CC3) method, favorable for calculating frequency-dependent properties, is tailored to the RT framework for high excitation and approximate orbital relaxation. The calculation is tested on both CPUs and GPUs, with a significant speedup gained from GPUs for the water cluster test cases. To further expand the range of applications of our RT-CC implementation, dynamic polarizabilities, first hyperpolarizabilities, and the G' tensor are calculated from induced electric and magnetic dipole moments using finite-difference methods. A discussion has also been conducted to compare RT-CC3 with RT-CCSD, and time-dependent nonorthogonal orbital-optimized coupled cluster doubles (TDNOCCD) method. Additionally, electron dynamics, including the Rabi oscillation and exited state to excited state transitions, have also been explored utilizing the well-developed RT-CC framework. / Doctor of Philosophy / Theoretical studies aim to match experiments, but more importantly, provide insights to interpret and predict experimental data. Calculating optical properties related to light-matter interactions is one of the most crucial tasks for characterizing molecular properties. In experiments, electromagnetic radiation in the form of light is applied to the system. The absorption or emission of light can be measured to identify, for example, the electronic structure of the molecule. In theoretical simulations, this applied radiation is represented by a perturbation operator that is added to the Hamiltonian in the Schrödinger equation. Quantum chemists are dedicated to developing methods that provide a better description of the spectroscopy. In the current work, the frequency, shape and the intensity of the radiation can all be finely-tuned, similar to experimental setups. The framework for extracting optical properties from time-dependent trajectories of induced dipole moments is established for accurate and efficient simulations. To improve efficiency and make the method feasible for real-world applications, a strong understanding of light-matter interactions on a quantum level and proper utilization of computational resources are both necessary. Improvements achieved and presented in this dissertation demonstrate a powerful tool for a better understanding of the nature of the interaction between the system and the electromagnetic radiation.

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The central objective of this dissertation is developing new methods for calculating higher-order nonlinear optical responses of atoms, molecules, and ions, and discussing the relevant physical mechanisms that give rise to harmonic generation, Kerr effect, and higher-order Kerr effect. The applications of nonlinear optical properties in development of predictive models for femtosecond laser filamentation dynamics, photoemission spectroscopy, imaging, and design of new molecular systems have motivated the theoretical investigations in advancing methods for calculating nonlinear optical properties and finding the optimum conditions for controlling the nonlinearities. The time-dependent nonlinear refractive index coefficient 4 n is investigated for argon and generalized for all noble gas atoms helium, neon, krypton, and xenon in the wavelengths ranging from 250 nm to 2000 nm, using ab initio methods. The secondorder polynomial fitting of DC-Kerr, electric-field-induced second-harmonic generation (ESHG), and static second-order hyperpolarizability have been performed, using an auxiliary electric field approach to obtain the corresponding fourth-order optical properties. An expression on the basis of static, DC-Kerr, DFWM fourth-order hyperpolarizability is derived, which allows the calculations of the DSWM coefficients with considerably reduced error. The results of the calculations suggest that filament stabilization is most likely to be induced by the generation of free electrons. Applications of these calculations resolve the HOKE controversy and are important for the development of predictive models for femtosecond laser filamentation dynamics. In a series of proof-of-concept studies, the approach was employed for calculating dynamic linear and nonlinear hyperpolarizability of the radical cations. In this regard, the polarizability and second-order hyperpolarizability of nitrogen radical cation were investigated, using density functional theory (DFT) and multi-configurational self-consistent field (MCSCF) methods. The open-shell electronic system of nitrogen radical cation provides negative second-order optical nonlinearity, suggesting that the hyperpolarizability coefficient for nitrogen radical cation, in the non-resonant regime is mainly composed of combinations of virtual one-photon transitions rather than two-photon transitions. The calculations of second-order optical properties for nitrogen radical cation as a function of bond length have been investigated to study the effect of internuclear bond distance on optical process. The variation of nonlinear responses versus bond length shows the potential application in finding optimum conditions for higher values of nonlinear coefficients. Furthermore, the computation of dynamic second-order hyperpolarizabilities for multiply ionized noble gases have been studied in the wavelength ranging from 100 nm to the red of the first multi-photon resonance all the way toward the static regime, using the MCSCF method. The results indicate that the second-order hyperpolarizability coefficients decrease when the electrons are removed from the systems. As the atoms reach higher ionization states, the second-order hyperpolarizability responses as a function of wavelength, become less dispersive. The second-order hyperpolarizability coefficients for each ionized species have also been investigated in terms of quantum state symmetries; the results suggest that the sign of the optical responses for each ionized atom depends on the spin of the quantum states defined for the ionized species. The calculations are of value for predictive models of high-harmonic generation in multiply ionized plasma at X-ray photon energies. This research also focuses on investigating possible mechanisms for photodissociation of polyatomic molecules (acetophenone and the substituted derivatives) ionized through strong field infrared laser pulses. In this regard, quantum mechanical methods are combined with pump-probe spectroscopy to understand and control the dissociation dynamics in strong field regime. The applications of quantum mechanical models in interpreting time-resolved wavepacket dynamics and achieving coherent control has stimulated the interest to explore the PESs and investigate the role of conical intersections in wavepacket dynamics in strong field regime. The electronic ground and excited states for acetophenone radical cation and the substituted derivatives have been investigated to probe the resonance features observed in measurements at 1370 nm with laser intensity of 1013 W cm-2. The ten lowest lying ionic potential energy surfaces (PESs) of the acetophenone radical cation were explored, and the three-state conical intersection was mapped onto the PES, using MCSCF model to propose a photo-dissociation mechanism for acetophenone undergoing tunnel ionization and elucidate the potential dissociation pathways for formation of benzoyl fragment ion, as well as phenyl, acylium, and butadienyl small fragment ions. Similar calculations are presented for propiophenone radical cation which support the existence of a one-photon transition from the ground ionic to a bright dissociative D2 state, where motion of the acetyl group from a planar to nonplanar structure within the pulse duration enables the otherwise forbidden transition. The wavepacket dynamics in acetophenone molecular ion is modeled using the classical wavepacket trajectory calculations, to propose the mechanism wherein the 790 nm probe pulse excites a wavepacket on the ground surface D0 to the excited D2 surface at a delay of 325 fs. The innovations of this research are used to design control strategies for selective bond-breaking in acetophenone radical cation, as well as design control schemes for other molecules. / Chemistry

Thermoelectric semiconductors can convert temperature differences into electricity or electricity into temperature differences. This offers great potential for the use of wasted heat or cooling. These materials can be used in a variety of fields, from healthcare to space exploration. The effectiveness of the materials is evaluated by their thermoelectric properties such as the Seebeck coefficient, electrical conductivity, and thermal conductivity. The aim of this PhD thesis is to investigate the electronic structure using first-principle methods for potential thermoelectric applications. Materials of interest include Copper and Tin based ternary /quaternary compounds, and monolayers of SnS2, SnSe2 and Janus SnSSe. Density functional theory, ab initio molecular dynamics and Boltzmann transport theory are used to study the electronic and phonon transport properties. In the first part of the thesis, electronic structure calculations were performed on both monoclinic and disordered cubic forms of Cu2SnS3(CTS). The impact of structural disorder on thermoelectric properties was examined through these simulations. The results, obtained through first-principle calculations, revealed the existence of band tails in the electronic density of states for the disordered structure, and low-lying optical modes in the disordered cubic structure. This was found to be caused by a significant variation in Sn bonding, leading to strong anharmonicity as measured by the Grüneisen parameter. The findings from the first principle calculations were supported by Nuclear inelastic scattering experiments. Furthermore, the effect of grain size on Cu2SnS3 was studied using first-principles calculations on various ordered and disordered surfaces. The density of states (DOS) revealed that the surface of CTS is conductive due to the presence of dangling bonds. Furthermore, calculations of the formation energy showed that the stoichiometric CTS, Cu-vacant and Cu-rich systems are energetically more favourable, while the formation of Sn-vacant and Sn-rich systems is less likely. In the subsequent study, the impact of Ag substitution at the Sn site at various concentrations was investigated. The Fermi level for Ag-substituted systems was found to lie deep within the valence band, with the shift of the Fermi level inside the valence band increasing with substitution increasing the carrier concentration. The incorporation of Ag into the system decreases the root mean squared displacement of the other cations and anions, which reduces the scattering of phonons and thereby increases the lattice thermal conductivity. A comparative study of various polymorphs of CTS, Cu2ZnSnS4 and Cu2ZnSnSe4 was done. Ab-initio molecular dynamics was performed on CTS, CZTS and CZTSe. The root mean squared displacement value for the disordered polymorph was higher than for the ordered phase, indicating increased static disorder. This corresponds to the static (temperature-independent) distortion of the crystalline lattice due to the disorder of the cations and is associated with higher anharmonicity and bond inhomogeneity in the disordered phase, which is then directly responsible for the ultra-low thermal conductivity. In the final part of the thesis, thermoelectric properties of dichalcogenide monolayer of SnS2, SnSe2 and Janus SnSSe was performed. Density functional theoretical calculations points out the hexagonal Janus SnSSe monolayer as a potential high-performing thermoelectric material. Results for the Janus SnSSe monolayer show an ultra-low thermal conductivity originating from the low group velocity of the low-lying optical modes, leading to superior zT values of 0.5 and 3 at 300 K and 700 K for the p-type doping, respectively. The successful calculation of properties for materials shows that the computational work done in this thesis can be used for further research into thermoelectricity.

Over the past few decades, single-molecule magnets (SMMs) have been an area of significant interest due to their plethora of potential uses, including possible applications to quantum computing and compact data storage devices. Although theoretical chemistry calculations could aid our understanding of the magnetic couplings present in these types of systems, they are often multiconfigurational in nature, making them difficult to model with tradi- tional single-reference approaches. Methods to handle these types of strongly correlated systems have been developed but often have significant drawbacks, and so these molecules remain difficult to model computationally. In this work, we discuss the application of Fock-space CI approaches to large transition metal complexes. First, we introduce a novel formalism which combines the spin-flip (SF), ioniza- tion potential (IP), and electron affinity (EA) approaches. This redox spin-flip approach, the restricted active space spin-flip and ionization potential/electron affinity (RAS-SF-IP/EA) method, is applied to several molecules exhibiting double exchange behavior. Model Hamil- tonian parameters are extracted from energy gaps and found to be in qualitative agreement with experiment. Having shown the efficacy of this approach, we move on to optimization, using a diagrammatic approach to derive equations for several RAS-1SF-IP/EA schemes. These equations allow direct construction of the most expensive intermediates in the David- son algorithm and should provide significant speedup, allowing application of Fock-space CI approaches to larger systems than ever before. The derived equations are implemented in the LibRASSF package in Q-Chem, as well as in an open-source PyFockCI code, avail- able on GitHub. A Bloch effective Hamiltonian formalism is also utilized to extract model Hamiltonian parameters from RAS-1SF calculations, allowing more nuanced studies of the Heisenberg J couplings present in many molecules with magnetically coupled sites. Over- all, our work with Fock-space CI provides a way to study magnetic couplings in very large strongly correlated systems at relatively low computational cost. This work was supported by a grant from the U.S. Department of Energy: DE-SC0018326. / Doctor of Philosophy / Humans have been familiar with magnets for thousands of years, and we have found a variety of useful applications for them. Magnets are used in everything from navigational devices to credit cards to data storage. Most people are familiar with large, solid magnets, but in the 1990s, it was discovered that individual molecules, called single-molecule magnets (SMMs), could also exhibit magnetic behavior. This means that in the presence of some external magnetic field, like the field caused by the presence of another magnet, the electrons in a SMM will align themselves with the field, and the electrons will maintain that alignment for some period of time after the field is removed. These SMMs have been a significant area of interest to scientists because they have a variety of interesting applications, including applications to quantum computing. In cases such as these, theoretical chemistry can offer useful insight. Broadly, the purpose of theoretical chemistry is to describe chemical problems using mathematical equations. We can use computational models to obtain information about the behavior of electrons in a particular system (the so-called electronic structure) and consequently, we can model the magnetic couplings in a given molecule. However, SMMs are difficult to model using tra- ditional theoretical methods because they often contain multiple orbitals which have nearly the same energy. In these cases, it often becomes ambiguous which orbitals ought to be oc- cupied by electrons; the effect this has on the energy is called "strong correlation". Ideally, one ought to consider all possible fillings of the orbitals, but most methods do not account for this and assume only one configuration is important when solving for the shapes of the orbitals. In this work, we combine two previously-introduced approaches, the spin-flip (SF) and ioniza- tion potential/electron affinity (IP/EA) approaches, to handle strongly correlated systems. In the SF-IP/EA approach, one adds or removes electrons and flips their spins in order to remove all of the ambiguity in orbital occupations. Once we determine the shapes of the or- bitals for this unambiguous state, the electrons are added, removed, or spin-flipped in order to obtain the desired strongly correlated state. We then solve for the energy of the system while considering all possible configurations within the set of ambiguously-occupied orbitals, allowing us to treat them on equal footing. We also study the effect of adding additional configurations to account for contributions from other orbitals, which provides more accu- rate results, albeit at a higher computational cost. Our method is less expensive than many other wavefunction-based methods used for these systems, and it yields qualitatively correct results, allowing theoreticians to study magnetic couplings in SMMs in a straightforward and inexpensive way. We also discuss optimization of our code, as well as an extension of our code that allows us to obtain coupling information for systems containing multiple magnetic sites. It is our hope that these developments will provide useful insights into the electronic structure of these SMM systems.

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.].