Mathematical Analysis of Quantum Chemical Models for Small Atoms

Quantum chemical methods are of great use in predicting chemical and physical properties of atoms and molecules. Whilst many of the methods are built upon a rigorous mathematical foundation, there are very few chemically-specific mathematical results, and many aspects, such as the choice of optimal basis sets, are poorly understood. This thesis centres on the choice of a minimal basis for the second row atoms, motivated by perturbation theory and physical screening arguments, for which a full Configuration Interaction (CI) calculation is performed. This calculation requires no numerical integration and can essentially be performed ‘by hand’. As well as demonstrating that this method leads to an accurate prediction of the atomic spectra, a number of computationally useful results concerning the Hamiltonian matrix and the spectra themselves are rigorously proven. A rigorous rate of convergence for the CI method applied to the ground state of the Helium atom, with a basis ordered by angular momentum, is also derived. This includes a derivation of the leading order constant, enabling extrapolation of computational results. Another interesting area of computational chemistry is the prediction of molecular geometries. This thesis investigates the AH2 trimers, where A is a second row atom. The relationship between the maximum of the pair density of the central atom and the bond angle is investigated, using the canonical ground state wavefunctions derived in the CI calculations. The non-numerical results are independent of the radial parts of the wavefunctions and inserting the CI wavefunctions leads to excellent qualitative and reasonably accurate quantitative numerical predictions.Accepted versio