The focus of this dissertation research has been on technical improvements to computational quantum chemistry methods. Integration grids for evaluating the numerical exchange-correlation (XC) integrals in density functional theory (DFT) were developed, which improved upon the numerical stability of the currently-used Lebedev grids. A recursive 2-center integration technique was developed for semiempirical quantum chemistry methods, particularly the density functional tight binding (DFTB) method. This allows for all needed 2-center integrals and integral-derivatives to be evaluated during runtime of the code, allowing better flexibility and transferability compared to pre-tabulated integrals (i.e. the Slater-Koster tables for DFTB) without a significant increase in computational cost. In the process of searching for 4-index 2-center integral formulas, a closed-form method of approaching molecular integral evaluation was developed based on Fourier Transform Cartesian Separation (FTCS). Basic semiempirical quantum chemistry methods were implemented as a proof for the 2-center integral techniques and were tested for their ability to optimize molecular geometries. Due to the unimpressive performance of the extended-Hückel type methods chosen, we worked towards a systemic investigation of the errors of semiempirical approximations. Towards this goal, an atomic Kohn-Sham program was developed to determine atomic densities, orbitals, and potentials. The program utilized numerical orbitals and a novel high order difference approximation to almost exactly solve the Schr?dinger equation numerically. The general 2-center integral techniques and the technique for determining numerical orbitals may prove to be useful elements for the development of future semiempirical quantum chemistry methods.
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