| Summary: | With the rapid increase in computing power Quantum Chemists are looking towards larger and larger molecules. This thesis presents new ways to reduce the expensive scaling of computational cost with system size, thus allowing the advances in computer science to be utilized. The first chapter is an introduction to Hartree-Fock theory and the traditional methods of calculating electron correlation. This is followed by an introduction to Density Functional Theory, concentrating on Kohn-Sham density functional theory. Chapter 3 presents a new way of assessing the accuracy of a density functional by partitioning the density and examining the energy of the component pieces. Chapter 4 describes a new density functional (EDF1), designed especially for small basis sets, thus making it ideal for large systems. The functional is formed from several other common functionals, grouped together in a way to minimize the error of the chemical energetics of a selection of molecules. Chapter 5 gives an introduction to modern two-electron integral theory and then describes a new method for efficiently calculating integrals arising from charges that are well separated. The new algorithm does not scale with the concentration of the basis set. The efficient algorithm of Chapter 5 is <I>O(N</I><SUP>2</SUP>) overall, and therefore still too slow. To truly examine large molecules <I>O(N) </I>methods are required. Chapter 6 provides an introduction to these linear methods and also presents a new method, the CASE approximation, which neglects long-range interactions. How to implement this new method (in<I> O(N)</I> work) is described in Chapter 7. The method is extended to density functional theory in Chapter 8 by attenuating the Dirac functional. Chapter 9 presents a second way to reduce the magnitude (and speed) of the approximation, and also a correction for the main failure of the original approximation. Chapter 10 examines the accuracy of the approximation on a variety of chemical properties. The final chapter describes a way to improve the accuracy of CASE by correcting for the neglected terms in only <I>O(N)</I> work. This correction however is not without its own problems and work continues in this area.
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