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<p>This thesis is part of an effort to enable large-scale Hartree-Fock/Kohn-Sham (HF/KS) calculations. The objective is to model molecules and materials containing thousands of atoms at the quantum mechanical level. HF/KS calculations are usually performed with the Self-Consistent Field (SCF) method. This method involves two computationally intensive steps. These steps are the construction of the Fock/Kohn-Sham potential matrix from a given electron density and the subsequent update of the electron density usually represented by the so-called density matrix. In this thesis the focus lies on the representation of potentials and electron density and on the density matrix construction step in the SCF method. Traditionally a diagonalization has been used for the construction of the density matrix. This diagonalization method is, however, not appropriate for large systems since the time complexity for this operation is σ(n³). Three types of alternative methods are described in this thesis; energy minimization, Chebyshev expansion, and density matrix purification. The efficiency of these methods relies on fast matrix-matrix multiplication. Since the occurring matrices become sparse when the separation between atoms exceeds some value, the matrix-matrix multiplication can be performed with complexity σ(n).</p><p>A hierarchic sparse matrix data structure is proposed for the storage and manipulation of matrices. This data structure allows for easy development and implementation of algebraic matrix operations, particularly needed for the density matrix construction, but also for other parts of the SCF calculation. The thesis addresses also truncation of small elements to enforce sparsity, permutation and blocking of matrices, and furthermore calculation of the HOMO-LUMO gap and a few surrounding eigenpairs when density matrix purification is used instead of the traditional diagonalization method.</p>

<p>Time-dependent density functional theory calculations have been carried out to determine the complex first-order hyperpolarizability in the two-photon resonance region of the molecule IDS-Cab. Calculations show that three strongly absorbing states, in the ultraviolet region, are separated to the extent that no significant interference of the imaginary parts of the tensor elements of the first-order hyper-polarizability occurs. Consequently, and in contrast to experimental findings [27], no reduced imaginary parts of the first-order hyperpolarizability in the two-photon resonant region can be seen.</p>

Density functional theory (DFT) is not only an accurate but also a widely used theory for describing the quantum-mechanical electronic structure of matter. In this approach, the intractable problem of interacting electrons is simplified to a tractable problem of non-interacting electrons moving in an effective potential. Even with this simplification, DFT remains extremely computationally expensive. In particular, DFT scales cubically with respect to the number of atoms, which restricts the size of systems that can be studied. Orbital free density functional theory (OF-DFT) represents a simplification of DFT applicable to metallic systems that behave like a free-electron gas. Current implementations of OF-DFT employ the plane-wave basis, the global nature of the basis prevents the efficient use of modern high-performance computer archi- tectures. We present a real-space formulation and higher-order finite-difference implementation of periodic Orbital-free Density Functional Theory (OF-DFT). Specifically, utilizing a local reformulation of the electrostatic and kernel terms, we develop a gener- alized framework suitable for performing OF-DFT simulations with different variants of the electronic kinetic energy. In particular, we develop a self-consistent field (SCF) type fixed-point method for calculations involving linear-response kinetic energy functionals. In doing so, we make the calculation of the electronic ground-state and forces on the nuclei amenable to computations that altogether scale linearly with the number of atoms. We develop a parallel implementation of our method using Portable, Extensible Toolkit for scientific computations (PETSc) suite of data structures and routines. The communication between processors is handled via the Message Passing Interface(MPI). We implement this formulation using the finite-difference discretization, us- ing which we demonstrate that higher-order finite-differences can achieve relatively large convergence rates with respect to mesh-size in both the energies and forces. Additionally, we establish that the fixed-point iteration converges rapidly, and that it can be further accelerated using extrapolation techniques like Anderson mixing. We verify the accuracy of our results by comparing the energies and forces with plane-wave methods for selected examples, one of which is the vacancy formation energy in Aluminum. Overall, we demonstrate that the proposed formulation and implementation is an attractive choice for performing OF-DFT calculations.

A molecule can crystallise in more than one crystal structure, a common phenomenon in organic compounds known as polymorphism. Different polymorphic forms may have significantly different physical properties, and a reliable prediction would be beneficial to the pharmaceutical industry. However, crystal structure prediction (CSP) based on the knowledge of the chemical structure had long been considered impossible. Previous failures of some CSP attempts led to speculation that the thermodynamic calculations in CSP methodologies failed to predict the kinetically favoured structures. Similarly, regarding the stabilities of co-crystals relative to their pure components, the results from lattice energy calculations and full CSP studies were inconclusive. In this thesis, these problems are addressed using the state-of-the-art CSP methodology implemented in the GRACE software. Firstly, it is shown that the low-energy predicted structures of four organic molecules, which have previously been considered difficult for CSP, correspond to their experimental structures. The possible outcomes of crystallisation can be reliably predicted by sufficiently accurate thermodynamic calculations. Then, the polymorphism of 5- chloroaspirin is investigated theoretically. The order of polymorph stability is predicted correctly and the isostructural relationships between a number of predicted structures and the experimental structures of other aspirin derivatives are established. Regarding the stabilities of co-crystals, 99 out of 102 co-crystals and salts of nicotinamide, isonicotinamide and picolinamide reported in the Cambridge Structural Database (CSD) are found to be more stable than their corresponding co-formers. Finally, full CSP studies of two co-crystal systems are conducted to explain why the co-crystals are not easily obtained experimentally.

The interfacial electronic structure of the organic material- tris(8-hydroxyquinolinato)aluminum (Alq3) forming an interface with cobalt metal has been investigated in this research. The primary characterization method used in this research was near-edge X-ray absorption fine structure (NEXAFS) spectroscopy which probes the unoccupied molecular orbitals of a material. Density functional theory (DFT) calculations have also been employed to calculate the partial density of states (PDOS) of all constituent elements present in Alq3 molecule. The DFT calculations helped to determine the molecular orbital structure of Alq3 and to understand how the orbital structure is influenced by forming an interface with ferromagnetic Co layer. The experimental NEXAFS spectra measured in total fluorescence yield (TFY) showed that the lowest unoccupied molecular orbital (LUMO) and LUMO+1 states of Alq3 were not affected by the presence of Co when Co is deposited onto Alq3. On the other hand, a charge transfer between Co and Alq3 led the loss or reduction of LUMO+2 state for a Co(top)/Alq3 bilayer sample when compared to pristine Alq3 reference sample (without Co deposition). This selective effect of Co on the orbital configuration of Alq3 suggests that Co atoms diffuse into Alq3 and interact with preferred sites in Alq3. By comparing the spectral change in the experimental NEXAFS spectra to the calculated PDOS of Alq3, the preferred interaction sites between Co and Alq3 could be successfully determined. This work suggests that the spectroscopic approach using synchrotron-radiation X-ray spectroscopy can serve as a powerful means for studying the interfacial electronic structure between magnetic metals and organic semiconductors and can contribute to the research and development of high performance organic spintronics.

London or dispersion interactions are weak van der Waals (vdW) interactions. They are important in determining the structure and properties of many chemical and biochemical systems. In this thesis, an optimizer using the nonempirical generalized gradient approximation (GGA) functional PW86+PBE+XDM, to capture van der Waals interactions, is presented. The work in this thesis covers the assessment of a variety of basis sets for their ability to reproduce accurate GGA repulsive and binding energies. Selected basis sets were then used to compute binding energies of 65 vdW complexes at equilibrium. This functional was also tested for binding energies of two sets of vdW complexes at distorted geometries. The last part deals with forces to investigate their accuracy using PW86+PBE+XDM in order to build an optimizer for vdW complexes using a nonempirical DFT method. Eventually, after confirming a high reproducibility of the optimizer on the geometries and binding energies, it was used in two biologically relevant applications. This optimizer is a unique tool to compute deformation energies with a nonempirical DFT method. The second part of this thesis covers a biologically relevant application where a conventional DFT is used. This application is related to the carrier of the genetic codes in living cells, DNA. DNA undergoes harmful mutations under external perturbations such as applied external electric fields. In this study, DNA base pairs were first mimicked by a simpler model, namely, the formic acid dimer. The effect of applied external electric fields on the geometries of the formic acid dimer is studied. The effect of these applied fields on the potential energy surface, the barrier height and the frequency of the double proton transfer in the formic acid dimer are also investigated. The study was then repeated on DNA base pairs to study the effect of an external applied electric field on the tunneling corrected rate constants of the double proton transfer reactions in AT and GC.

This work presents a study of the electronic structure of four transition metal oxides (TMOs) using spectroscopic data and a variety of theoretical models. TMOs are a class of materials made from d-block metals in the periodic table, and one or more oxygen atoms. The nature of d-electrons is examined and theoretical models used to treat d-electron systems are tested against experimental data. Background theory of condensed matter physics is outlined. An overview of density functional theory (DFT) as a theoretical model for calculating the electronic structure of materials is presented. A variety of exchange-correlation (XC) functionals used within the DFT framework are outlined and tested for their applicability to the TMO systems in question. X-ray spectroscopy is briefly outlined and used to test the validity of the different XC functionals. All four compounds, AgO, Ag2O, CuO, and Cu2O require a Hubbard U term in the XC functional to most accurately reproduce experimental results. The effects of varying the value of U is examined in depth. The oxygen K-edge X-ray emission spectra (XES) exhibits a“two peak” structure for all compounds; the effect of varying the U value is to change the intensity ratio of the two peaks. The ratio of the two peaks as a function of U shows a linear trend in all compounds. A simple line is fit to the peak ratio vs. U curve. A common line between all compounds would provide an important metric with which to predict the appropriate U value needed in similar materials based on simple experimental data. However, the parameters of the fitted line were not common between the four compounds and any metric derived from this method would be system-dependent and not widely applicable to other systems. There are, however, interesting trends in the data when the U value is varied that provide subjects for future research. A number of fundamental quantities are determined both from experiment and theoretical calculations. Calculated bandgap values are shown to be lower than the experimental values for most functionals tested. This is not unexpected as DFT methods are known to predict much smaller bandgaps than expected. The Heyd-Scuseria-Ernzerhof (HSE) functional used for Ag2O and Cu2O does predict the bandgaps very accurately. The core-hole effect is estimated and proven to be negligible in these systems. Charge transfer and on-site Coulomb repulsion energies, important quantities in the electronic behaviour of TMOs, are determined and compared to previously reported values.

A summary of research conducted since September 2007 at the University of Toronto in the laboratory of Professor Robert A. Batey is presented in this thesis, which is divided into four chapters. The first chapter contains a two-part introduction, where aryl- and aliphatic-Claisen rearrangements are discussed in part 1, and the nucleophilic addition of organoboron reagents to unsaturated C–N functionalities is described in part 2. Chapter 2 contains research involving synthetic and theoretical studies of aryl-Claisen rearrangements and other sigmatropic reactions. The work towards developing the lanthanide-catalyzed domino aryl-Claisen rearrangement for the synthesis of contiguous aryl–C(sp³) moieties is presented first. This is followed by computational studies involving E/Z-selectivity differences for the aryl-Claisen rearrangement, which was an issue noted for the domino aryl-Claisen reaction of a linear substrate. The mechanistic origins of E/Z-selectivity differences for the mono aryl-Claisen rearrangement, which was experimentally ambiguous for over 40 years, is resolved through computational methods. A theoretical analysis of selectivity differences for the allylic azide rearrangement is also described. The third section contains a discussion of Eu(fod)3-catalyzed aryl-Claisen rearrangements on vinyl bromide systems and preliminary studies involving application of the substrates in cross-coupling reactions, and other attempted mono- and domino sigmatropic rearrangements are presented in the fourth section. In chapter 3, the search for computational methods that can accurately predict experimental free energy of activation barriers for the aliphatic-Claisen rearrangement through benchmarking studies with a priori kinetic barrier and kinetic isotope effect data is described. Methods were found to predict new valid transition states and predict ΔG‡ values with a mean unsigned error of 0.3 kcal/mol relative to experimental values. In chapter 4, the development of new allylboration reaction is outlined, involving the double allylboration of nitriles and anhydrides, and initial studies towards the first aminoallylboration reactions of N-aluminoaldimines to form 1,2-diamines.

Metal-cyclic hydrocarbon complexes were prepared in a laser-vaporization molecular beam source and studied by single-photon zero electron kinetic energy (ZEKE) and IR-UV resonant two-photon ionization (R2PI) spectroscopy. The ionization energies and vibrational frequencies of the metal complexes were measured from the ZEKE spectra. Metal-ligand bonding and low-lying electronic states of the neutral and ionized complexes were analyzed by combining the ZEKE measurements with density functional theory (DFT) calculations. In addition, C-H stretching frequencies were measured from the R2PI spectra. In this dissertation, metal complexes of 1, 3, 5, 7-cyclo-octatetraene (COT), toluene, p-xylene, mesitylene, hexamethylbenzene, biphenyl, naphthalene, pyrene, perylene, and coronene were studied. For each metal-ligand complex, different effects from the metal coordination have been identified. Although free COT is a nonaromatic molecule with a tub-shaped structure, the group III transition metal atoms (Sc, Y, and La) donate two electrons to a partially filled π orbital of COT, making the ligand a dianion. As a result, metal coordination converts COT into a planar, aromatic structure and the resulting complex exhibits a half-sandwich structure. For the Sc(methylbenzene) complexes, the benzene rings of the ligands are bent and the π electrons are localized in a 1, 4-diene fashion due to differential Sc binding with the carbon atoms of the rings. Due to differential metal binding, the degenerate d orbitals split and the Sc-methylbenzene complexes prefer the low-spin ground electronic states. In addition, as the number of methyl group substituents in the ligand increases, the ionization energies (IEs) of the Sc-methylbenzene complexes decrease. However, Ti, V, or Co coordination does not disrupt the delocalized π electron network within the carbon skeleton in the high-spin ground states of the metal complexes. For group VI metal (Cr, Mo, and W)-bis(toluene) complexes, methyl substitution on the benzene ring yields complexes with four rotational conformers of 0°, 60°, 120°, and 180° conformation angles between two methyl groups. In addition, variable-temperature ZEKE spectroscopy using He, Ar, or their mixtures has determined the totally eclipsed 0° rotamer to be the most stable. When there are two equivalent benzene rings, the metal (Ti, Zr or Hf) binds to both the benzene rings of biphenyl, or the metal (Li) binds to one of the benzene rings of naphthalene. On the other hand, the metal (Li) favors the ring-over binding site of the benzene ring with a higher π electron content and aromaticity in pyrene, perylene, and coronene.