The earth orbiting the sun, the electron bonded to a proton in the hydrogen atom are both manifestations of particles in motion bound by an inverse-square force and both are governed by the principle of least action (of all the possible paths the particles may take between two points in space and time, they take those paths for which the time integral of the Lagrangian or the difference, kinetic energy-potential energy, is the least) and shaped by the same Hamiltonian (or total energy) structure. For both types of motion, the invariants (or properties that are conserved) are the energy and angular momentum of the relative motion, and the symmetry is that of the rotational motion. Differences arise because the electric force bounding the electron to the proton is forty two orders of magnitude stronger than the gravitational force, and the smallness of the hydrogen atom brings about ?quantum effects?: the mechanics of the microscopic particles which constitute the atom is wave-like. Yet the central concepts of mechanics are preserved in integrity: least action, invariants or conservation laws, symmetries, and the Hamiltonian structure. The discussion in the sections that follow on the quantum-mechanical treatment of molecular structures is based for the most part on the books by Pople and Murrell [2,3], and is by no means comprehensive but hopefully will elucidate the most relevant concepts for performing the estimation of thermochemical and kinetic properties of elementary reactions. The aim of quantum chemistry is to provide a qualitative and quantitative description of molecular structure and the chemical properties of molecules. The principal theories considered in quantum chemistry are valence bond theory and molecular orbital theory. Valence bond theory has been proven to be more difficult to apply and is seldom used, thus this discussion will deal only with the application of molecular orbital theory to molecular structures. In molecular orbital theory, the electrons belonging to the molecule are placed in orbitals that extend all the different nuclei making-up the molecule (the simplest approximation of a molecular orbital being a simple sum of the atomic orbitals with appropiate linear weighting coefficients, Figure 1 below for carbon Monoxide as example), in contrast to the approach of valence bond theory in which the orbitals are associated with the constituent atoms. The full analytical calculation of the molecular orbitals for most systems of interest may be reduced to a purely mathematical problem, the central feature of which is the calculation and diagonalization of an effective interaction energy matrix for the system. In ab initio molecular orbital calculations, the parameters that appear in such an energy matrix are exactly evaluated from theoretical considerations, while in semi-empirical methods experimental data on atoms and prototype molecular systems are used to approximate the atomic and molecular integrals entering the expression for the elements of the energy interaction matrix. Ab initio methods can be made as accurate as experiment for many purposes Zeener , the principal drawback to ?high level? ab initio work is the cost in terms of computer resources which restrict it to systems of ten or fewer atoms even for the most experienced users. This is what draws the chemist to semi-empirical methods that can be easily applied to complex systems consisting of hundreds of atoms. Presently, useful semi-empirical methods are limited in execution by matrix multiplication and diagonalization, both requiring computer time proportional to N3 where N is the number of atomic orbitals considered in the calculation or basis set.