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A trivial flaw in the utilization of artificial neural networks in interpolating chemical potential energy surfaces (PES) whose descriptors are Cartesian coordinates is their dependence on simple translations and rotations of the molecule under consideration. A different set of descriptors can be chosen to circumvent this problem, internuclear distances, inverse internuclear distances or z-matrix coordinates are three such descriptors. The objective is to use an interpolated PES in instanton rate constant calculations, hence information on the energy, gradient and Hessian is required at coordinates in the vicinity of the tunneling path. Instanton theory relies on smoothly fitted Hessians, therefore we use energy, gradients and Hessians in the training procedure. A major challenge is presented in the proper back-transformation of the output gradients and Hessians from internal coordinates to Cartesian coordinates. We perform comparisons between our method, a previous approach and on-the-fly rate constant calcuations on the hydrogen abstraction from methanol and on the hydrogen addition to isocyanic acid.

The work contained within this thesis is concerned with the explanation and usage of a set of theoretical procedures for the study of static and dynamic two–centre problems in the relativistic framework of Dirac’s equation. Two distinctly different theories for handling time–dependent atomic interactions are reviewed, namely semi–classical perturbation theory and a non–perturbative numerical technique based on the coupled channel equation to directly solve the time–dependent, two–centre Dirac equation. The non–perturbative numerical technique has been developed independently and the calculations performed with it are entirely new. Calculations for ionisation cross sections and state occupancies are conducted for both these methods. The non–perturbative technique for relativistic two–centre problems is extensively explained and, given its novelty, a probity test is conducted between this technique and that of the well established perturbation theory in calculating K-and L-shell ionisation cross sections for the alpha decay of initially Hydrogen–like Polonium. To that end, an in depth outline of the perturbative technique is also made for both collision and decay processes. As well as the comparison test mentioned, this technique is also applied to the analysis of cross sections of the promotion of a single electron into the positive continuum from either a K- or L-shell due to the alpha decay of heavy, neutral nuclei (Gadolinium, Polonium and Thorium). Dirac-Coulomb eigenfunctions centred on the parent nucleus of the decay pair are taken as the basis for use in the cross section calculations utilising first order, semi-classical pertubation theory. The excellent congruence between both techniques justifies the usage of the non-perturbative algorithms in the subsequent analysis of collisions between very heavy, highly charged ions. As such, a set of calculations are performed examining the bound and continuum state occupancy of the electronic levels during a collision between U92+ -U91+, at both over–critical and non-critical projectile velocities. Overall, the non-perturbative method developed and implemented here, is shown to be reliable, compares well with available experimental data, and most importantly is flexible enough to find continued use in studies on more extreme/exotic atomic systems.