Dynamic inter-electronic correlations are exquisitely local effects responsible for many interesting phenomena in molecular physics, including much or all of mutual attractions between molecules. The main thrust of this work is the development of high-accuracy, low-scaling models of electron correlation exploiting locality in the context of non-covalent interactions.

In this thesis, we present our work in pursuit of black-box, \textit{ab initio} methods for computing positions and widths of molecular resonances. The method of complex basis functions is efficiently implemented and applied in the context of various electronic structure approximations. Within the static exchange approximation, basis set effects are investigated and the method is applied to a series of N-containing hetercycles. The extension to Hartree-Fock theory allows for more accurate calculations. These methods have been applied to several small molecules, and the computation of properties within this framework is discussed. The application of complex basis functions to shape and Feshbach resonances at correlated levels of theory including M\o ller-Plesset perturbation theory at second order and equation of motion coupled cluster singles and doubles is also investigated from a practical perspective, and the prospect of using these methods for computing accurate potential energy surfaces is explored. Finally, we describe some theoretical and practical aspects of computing positions and widths of low-energy shape resonances by analytic continuation in the coupling constant. We find that the properties of attenuated Coulomb potentials make them ideal for such calculations.

A radical molecule contains one or more electrons that are unpaired. A radicaloid may be defined as a molecule in which there are that are partially unpaired. As a result, the electronic structure of the radicaloid can be quite complicated for a variety of reasons. For a singlet biradicaloid, the singlet and triplet wavefunction can be quite close energetically which can lead to problems when trying to describe the system with a single determinant. The simplest solution to this problem is to allow the wavefunction to break spin-symmetry in order to get a lower energy. Unfortunately this action can lead to wavefunctions that are no longer eigenfunctions of the operator.

In the second chapter we investigate a distannyne which has a biradicaloid resonance structure. By examining the orbital Hessian, it is discovered that the spin-symmetric solution is a saddle-point in wavefunction space and is structurally different than the spin-polarized solution. We then increase the complexity of the model system and see that the spin-symmetric solution is only a minimum for the exact experimental system and not for a simplified model system in which bulky organic substituents are replaced by simpler phenyl groups. Therefore, the breaking of spin-symmetry is absolutely critical in the small model systems and the full substituents play a non-trivial role.

However, the breaking of the spin-symmetry can have consequences for physical quantities when correlated methods are used. At the point of spin polarization or unrestriction the orbital Hessian will have one eigenvalue which is zero. Since the relaxed density matrix in correlated methods like Second-Order M\o ller-Plesset theory (MP2) depend on the inverse of the Hessian, at the unrestriction point this quantity will be undefined. Some unphysical artifacts are identified as a direct consequence of this fact. First, discontinuities in first order molecular properties such as the dipole moment are seen at the geometries associated with unrestriction. Second, the relaxed density matrix itself fails to be N-representable, with natural orbital occupation numbers less than zero and greater than one. Therefore, it is desirable to use a method that is not dependent on the inverse of the Hessian like orbital optimized MP2 (O2).

Another system which requires the use of orbital optimization is a neutral soliton on a polyacetylene chain. In this system, the Hartree-Fock reference suffers from severe spin-polarization making the wavefunction physically unreasonable unless a very sophisticated treatment of electron correlation is used to correct this problem. Originally, it was found that computationally expensive methods like CCSD(T) and CASSCF could adequately describe small model chain but not the full system. The O2 method is found to be an dramatic improvement over traditional MP2 which can be feasibly applied to polyenyl chains long enough to characterize the soliton. It is also discovered that density functionals are generally inadequate in describing the half-width of the soliton.

Finally, the last chapter takes a slightly different perspective and focuses on the addition of correlation energy to a successful energy decomposition analysis based on absolutely localized molecular orbitals. It is discovered that the resulting new method can adequately describe systems with dispersive intermolecular interactions and large amounts of charge transfer. This scheme is then applied to the water dimer systems and it is found that all of the intermolecular interactions similar in size with the electrostatic interaction being the largest and the dispersive interaction being the smallest. This method is also contrasted with other EDA schemes.

In this dissertation, we will investigate two different areas of electronic structure theory.

In the first portion, we shall focus on the study of molecular transition metal systems.

Specifically, we will be modeling oxidation and reduction catalysts, which are potentially useful in creating new clean energy sources.

Secondly, we will develop a wave-function based method for computing electronic excited states, with a focus on multielectron excitations.

These excitations play an important role in many areas of physics and chemistry, including photosynthesis, vision, and organic solar cells.

Quantum mechanics tells one how to compute all the properties of a system which correspond to operators, but what if a property of a given system has no corresponding operator and what if this property provides synthetic chemists a collectively obtained historical intuition about reactivity?

This set of questions exactly applies to the case of oxidation state, a property which represents the hypothetical charge on a given atom.

This makes calculating oxidation states from electronic structure a matter of creating a recipe which respects both the physics of the system and matches "known'' values.

We develop one such recipe which we call the localized orbital bonding analysis (LOBA).

This method is based on two standard electronic structure methods: localization of molecular orbitals and population analysis.

We show that LOBA matches "known'' oxidation states of a large set of transition metal systems, we compare it with other recipes for calculating oxidation states, and use it to show a ligand reduction in a proposed transition-metal water-oxidation catalyst.

Now that we have a method which allows us to present computational results in a chemically intuitive framework, we applied LOBA and Kohn-Sham Density Functional Theory (KS-DFT) to study the electro-catalytic generation of hydrogen from protons in an aqueous solution by a molecular molybdenum-oxo complex.

We mapped out the thermodynamics of the catalytic cycle including both the reduction steps, which by comparison with experimental evidence were found to be coupled with a proton transfer.

We also evaluated the kinetics of the release of hydrogen gas, using transition state theory.

It was found experimentally that the reduction overpotential would be the limiting factor in using this catalyst for real world applications

By analyzing the LOBA results across this cycle, we found that one of the reductions partially occurred on the ligand, because of this we supposed that substitutions on the ligand could provide changes in the reduction potentials.

With this evidence we calculated the same path for the complex with electron withdrawing (fluorides) and electron donating (methyl-) groups located on the ligand and found that the fluorides reduced the potential making the catalyst theoretically more efficient.

Additionally, we present an alternative mechanism for H₂ release which has a lower energy barrier by the inclusion of a bridging water, propose a mechanism for the stoichiometric reaction which produces the catalyst, and discuss alternative catalytic cycles.

Our second major advance was the development of an excited state method, Non-orthogonal Configuration Interaction (NOCI).

This method has many advantages including the ability to treat double excitations at the same level as single excitations and low computational scaling when compared with other methods.

NOCI is a configuration interaction method where the basis in which one diagonalizes the Hamiltonian can be any Slater determinant, each of which may be non-orthogonal with the rest of the basis.

To provide proof of concept, we choose systems for which the character of the wavefunctions of excited states are known; the linear polyenes and β-carotene.

This also showed that NOCI is able to treat double excitations which are important for these systems.

We demonstrate that NOCI, when applied to systems with extended conjugation, provides a feasible way to obtain a qualitatively correct wavefunction.

We also present a new extension to this method allowing for purification of higher-order spin states by utilizing Generalized Hartree--Fock (GHF) Slater determinants and the details for computing < S² > for the ground and excited states.

In order to apply NOCI more straightforwardly to other systems, we develop a procedure for automatically generating the Slater determinants used in the expansion of the wavefunction.

This method utilizes information from the Hartree--Fock Hessian to generate the basis for diagonalizing the Hamiltonian.

We validate this method by considering a subset of the systems studied in the previous section, showing that the method performs almost the same as when prior knowledge of the system is used to choose the basis.

We also corroborate this success by applying the method to a wide range of well-characterized chromophores, resulting in errors very similar to CIS for single excitations and much improved for double excitations.

This thesis is concerned with the development of minimally-parameterized and highly-transferable density functionals. A methodology for searching a given functional space is developed and used to parameterize three novel functionals: ωB97X-V -- a 10-parameter, range-separated hybrid, generalized gradient approximation density functional with VV10 nonlocal correlation, B97M-V -- a 12-parameter, local meta-generalized gradient approximation density funcitonal with VV10 nonlocal correlation, and ωB97M-V -- a 12-parameter, range-separated hybrid, meta-generalized gradient approximation density functional with VV10 nonlocal correlation. These three functionals are validated by comparisons to the best existing density functionals in their class, and their proper usage (with respect to basis sets and integration grids) is documented to facilitate use in chemical applications.

This thesis is concerned with the application, development, and assessment of computational approaches to treat chemical reactions from an electronic structure theory perspective. In Chapter 2, the recently developed density functional, wB97X-V, and coupled cluster with single, double and perturbative triple excitations [CCSD(T)] are applied to study the reaction networks of excited state silicon atoms with allene and methylacetylene in the circumstellar envelope of carbon stars. In Chapter 3, another recently developed density functional, wB97M-V, and CCSD(T) are used to benchmark the performance of the ReaxFF reactive force field as applied to hydrogen combustion systems. In Chapter 4, an extension of third-order Moller-Plesset perturbation theory (MP3) and regularized perturbation theory (k-OOMP2) is developed and used to calculate thermochemistry, barrier heights, and noncovalent interaction energies. In Chapter 5, the use of k-OOMP2 optimized orbitals is extended to CCSD(T) and used to investigate the vibrational frequencies of diatomic molecules.

This dissertation contributes to three different areas in electronic structure theory. The first part of this thesis advances the fundamentals of orbital active spaces. Orbital active spaces are not only essential in multi-reference approaches, but have also become of interest in single-reference methods as they allow otherwise intractably large systems to be studied. However, despite their great importance, the optimal choice and, more importantly, their physical significance are still not fully understood. In order to address this problem, we studied the higher-order singular value decomposition (HOSVD) in the context of electronic structure methods. We were able to gain a physical understanding of the resulting orbitals and proved a connection to unrelaxed natural orbitals in the case of Moller-Plesset perturbation theory to second order (MP2). In the quest to find the optimal choice of the active space, we proposed a HOSVD for energy-weighted integrals, which yielded the fastest convergence in MP2 correlation energy for small- to medium-sized active spaces to date, and is also potentially transferable to coupled-cluster theory.

In the second part, we studied monomeric and dimeric glycerol radical cations and their photo-induced dissociation in collaboration with Prof. Leone and his group. Understanding the mechanistic details involved in these processes are essential for further studies on the combustion of glycerol and carbohydrates. To our surprise, we found that in most cases, the experimentally observed appearance energies arise from the separation of product fragments from one another rather than rearrangement to products.

The final chapters of this work focus on the development, assessment, and application of the spin-flip method, which is a single-reference approach, but capable of describing multi-reference problems. Systems exhibiting multi-reference character, which arises from the (near-) degeneracy of orbital energies, are amongst the most interesting in chemistry, biology and materials science, yet amongst the most challenging to study with electronic structure methods. In particular, we explored a substituted dimeric BPBP molecule with potential tetraradical character, which gained attention as one of the most promising candidates for an organic conductor. Furthermore, we extended the spin-flip approach to include variable orbital active spaces and multiple spin-flips. This allowed us to perform wave-function-based studies of ground- and excited-states of polynuclear metal complexes, polyradicals, and bond-dissociation processes involving three or more bonds.

This dissertation is concerned with the development and applications of approaches to the electron correlation problem. We start with an introduction that summarizes modern approaches to the electron correlation problem. In our view, there are two remaining challenges that modern density functional theory cannot satisfactorily solve. The first challenge is due to self-interaction error and the second is due to strong correlation. We discuss two methods developed by the author that attempt to make progress to address the second challenge.

The first approach is useful in distinguishing strong and weak correlation in a computationally economical way. It is based on orbital optimization in the presence of regularized second-order Moller-Plesset perturbation theory (k-OOMP2), which is an approximate method to obtain Brueckner orbitals. k-OOMP2 includes weak correlation while attenuating strong correlation. As such, it distinguishes artificial and essential symmetry breaking which occur at the Hartree-Fock (HF) level. Artificial symmetry breaking appears due to the lack of weak correlation, not due to the lack of strong correlation. Therefore, the common wisdom in quantum chemistry, which equates symmetry breaking at the HF level and strong correlation, can result in a wrong understanding of the system. Essential symmetry breaking, on the other hand, signals strong correlation that is beyond the scope of simple perturbation theory. k-OOMP2 has been shown to reliably distinguish these two: symmetry breaking in the k-OOMP2 orbitals is essential. This has been applied to a recent controversy about whether C60 is strongly correlated. Starting from a broken-symmetry HF solution, k-OOMP2 restores every symmetry. As such, C60 is not strongly correlated. Moreover, k-OOMP2 successfully predicts strong correlation for a known biradicaloid, C36, by showing essential symmetry breaking in its orbitals. We also exploited essential symmetry breaking in singlet biradicaloids using k-OOMP2 and showed quantitative accuracy in obtaining singlet-triplet gaps of various molecules. This new approach should be helpful for redefining the common wisdom in quantum chemistry.

The second method is an exact, spin-pure, polynomial-scaling way to describe strong spin-correlation (SSC). SSC is present when there are many spatially separate open-shell electrons with small spin-flip energy cost. Describing SSC exactly requires the inclusion of all essential spin-couplings. The number of such spin-couplings scales exponentially with the number of electrons. Because of this, SSC was thought to require an exponential number of wavefunction parameters in general. However, new development suggests that there is an efficient way to obtain all these spin-couplings with only a quadratic number of wavefunction parameters, which is called the coupled-cluster valence-bond (CCVB) method. We discuss different challenges in CCVB: (1) its non-black-box nature and (2) its inability to describe SSC in spin-frustrated systems. We present two improved CCVB approaches that address these two challenges. These approaches were applied to describe emergent strong correlation in oligoacenes and SSC in spin-frustrated systems such as single molecular magnets and metalloenzymes. The remaining challenges in CCVB are the inclusion of ionic excitations which are not relevant for SSC, but crucial for obtaining quantitative accuracy.

This thesis is primarily concerned with the development of tools aiming to improve the efficiency of electronic structure calculations using modern density functional theory (DFT), and to utilize the results of DFT-based energy decomposition analysis (EDA) for probing and modeling intermolecular interactions. It covers three distinct while also related topics: (i) development of a novel scheme to perform self-consistent field (SCF) calculations using a minimal adaptive basis (MAB) that is optimized on-the-fly; (ii) development and assessment of energy decomposition analysis (EDA) methods, including a new EDA scheme that is capable of probing the observable effects of intermolecular interactions, and a thorough discussion on the definition of the charge-transfer term in EDA methods; (iii) application of EDA to the development of potential energy surfaces for molecular simulations, including an assessment of the AMOEBA polarizable force field, and the formulation, implementation, and analysis of a mutually polarizable QM/MM scheme employing AMOEBA.

Correlation methods within electronic structure theory focus on recovering the exact electron-electron interaction from the mean-field reference. For most chemical systems, including dynamic correlation, the correlation of the movement of electrons proves to be sufficient, yet exact methods for capturing dynamic correlation inherently scale polynomially with system size despite the locality of the electron cusp. This work explores a new family of methods for enhancing the locality of dynamic correlation methodologies with an aim toward improving accuracy and scalability. The introduction of range-separation into ab initio wavefunction methods produces short-range correlation methodologies, which can be supplemented with much faster approximate methods for long-range interactions.

First, I examine attenuation of second-order Møller-Plesset perturbation theory (MP2) in the aug-cc-pVDZ basis. MP2 treats electron correlation at low computational cost, but suffers from basis set superposition error (BSSE) and fundamental inaccuracies in long-range contributions. The cost differential between complete basis set (CBS) and small basis MP2 restricts system sizes where BSSE can be removed. Range-separation of MP2 could yield more tractable and/or accurate forms for short- and long-range correlation. Retaining only short-range contributions proves to be effective for MP2 in the small aug-cc-pVDZ (aDZ) basis. Using one range-separation parameter within either the complementary error function (erfc) or a sum of two error functions (terfc), superior behavior is obtained versus both MP2/aDZ and MP2/CBS for inter- and intra-molecular test sets. Attenuation of the long-range helps to cancel both BSSE and intrinsic MP2 errors. Direct scaling of the MP2 correlation energy (SMP2) proves useful as well. The resulting SMP2/aDZ, MP2(erfc, aDZ), and MP2(terfc, aDZ) methods perform far better than MP2/aDZ across systems with hydrogen-bonding, dispersion, and mixed interactions at a fraction of MP2/CBS computational cost.

Second, attenuated MP2 is developed within the larger aug-cc-pVTZ (aTZ) basis set for inter- and intramolecular non-bonded interactions. A single attenuation parameter is optimized on the S66 database of 66 intermolecular interactions, leading to a very large RMS error reduction by a factor of greater than 5 relative to standard MP2/aTZ. Attenuation introduces an error of opposite sign to basis set superposition error (BSSE) and overestimation of dispersion interactions in finite basis MP2. A variety of tests including the S22 set, conformer energies of peptides, alkanes, sugars, sulfate-water clusters, and the coronene dimer establish the transferability of the MP2(terfc, aTZ) model to other inter and intra-molecular interactions. Direct comparisons against attenuation in the smaller aug-cc-pVDZ basis shows that MP2(terfc, aTZ) often significantly outperforms MP2(terfc, aDZ), although at higher computational cost. MP2(terfc, aDZ) and MP2(terfc, aTZ) often outperform MP2 at the complete basis set limit. Comparison of the two attenuated MP2 models against each other and against attenuation using non-augmented basis sets gives insight into the error cancellation responsible for their remarkable success.

Third, I present an improved algorithm for single-node multi-threaded computation of the correlation energy using resolution of the identity second-order Møller-Plesset perturbation theory (RI-MP2). This algorithm is based on shared memory parallelization of the rate-limiting steps and an overall reduction in the number of disk reads. The requisite fifth-order computation in RI-MP2 calculations is efficiently parallelized within this algorithm, with improvements in overall parallel efficiency as the system size increases. Fourth-order steps are also parallelized. As an application, I present energies and timings for several large, noncovalently interacting systems with this algorithm, and demonstrate that the RI-MP2 cost is still typically less than 40% of the underlying self consistent field (SCF) calculation. The attenuated RI-MP2 energy is also implemented with this algorithm, and some new large-scale tests of this method are reported. The attenuated RI-MP2(terfc, aug-cc-pVDZ) method yields excellent agreement with benchmark values for the L7 database (R. Sedlak et al., J. Chem. Theory Comput. 2013, 9, 3364) and 10 tetrapeptide conformers (L. Goerigk et al., Phys. Chem. Chem. Phys. 2013, 15, 7028), with at least a 90% reduction in the root-mean-squared (RMS) error versus RI-MP2/aug-cc-pVDZ.

Fourth, semi-empirical spin-component scaled (SCS) attenuated MP2 is developed for treating both bonded and nonbonded interactions. SCS-MP2 improves the treatment of thermochemistry and noncovalent interactions relative to MP2, although the optimal scaling coefficients are quite different for thermochemistry versus noncovalent interactions. This work reconciles these two different scaling regimes for SCS-MP2 by using two different length scales for electronic attenuation of the two spin components. The attenuation parameters and scaling coefficients are optimized in the aug-cc-pVTZ (aTZ) basis using the S66 database of intermolecular interactions and the W4- 11 database of thermochemistry. Transferability tests are performed for atomization energies and barrier heights, as well as on further test sets for inter- and intramolecular interactions. SCS dual- attenuated MP2 in the aTZ basis, SCS-MP2(2terfc, aTZ), performs similarly to SCS-MP2/aTZ for thermochemistry while frequently outperforming MP2 at the complete basis set limit (CBS) for nonbonded interactions.

Finally, I examine the performance of attenuated MP2 for noncovalent interactions using basis sets that range as high as augmented triple (T) and quadruple (Q) zeta with TQ extrapolation towards the complete basis set (CBS) limit. By comparing training and testing performance as a function of basis set size, the effectiveness of attenuation as a function of basis set can be assessed. While attenuated MP2 with TQ extrapolation improves systematically over MP2, there are at most small improvements over attenuated MP2 in the aug-cc-pVTZ basis. Augmented functions are crucial for the success of attenuated MP2.