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Novel experimental methods have allowed for the routine production of ultracold (sub-Kelvin) atoms and small molecules. This has facilitated the study of chemical reactions involving only a small number of partial waves, allowing for unprecedented control over ultracold chemical reactions. This thesis describes work towards a new set of theories, based on Wigner's R-matrix methodology, which are adapted for so-called heavy particle scattering, and in particular atom-atom scattering. From these new theories a new set of methods are constructed to accurately simulate scattering observables such as scattering lengths, cross-sections, and resonances for atom-atom scattering events at ultracold temperatures by producing high resolution plots of these observables. The methods utilise software built for high-accuracy diatomic spectra, such as Duo, to provide molecular eigenenergies and wavefunctions of the bound system at short internuclear distances (in a region known as the inner region), only requiring as input a matrix of diatomic internuclear potential energy curves and couplings. These methods then act as 'harnesses', allowing this information to be used to perform an R-matrix propagation at long internuclear distances (in a region known as the outer region) using R-matrix propagation codes such as PFARM. The result of this propagation is then used to produce the aforementioned scattering observables. In this work these new R-matrix methods are applied to the case of a particle scattering off a Morse potential, to elastic argon-argon collisions, and to the intramultiplet mixing of oxygen when impacted by helium. This work also serves as a basis for the future simulation of more complex scattering events, such as atom-diatom collisions and higher polyatomic collisions.

This thesis presents research carried out as part of the ExoMol project, towards calculating theoretical spectra for the main isotopologue of acetylene, C2H2, for use in characterising hot exoplanet or cool stellar atmospheres. A large component of this work was in the development of numerical methods for treating linear polyatomic molecules such that these calculations could be carried out in an efficient and feasible way; ro-vibrational calculations of linear molecules are very non-trivial and require a unique treatment in order to avoid singularities in the Hamiltonian. A novel approach was employed in variational nuclear motion programme TROVE, which involves the use of a finite Dnh symmetry group and classification of ro-vibrational states using the vibrational angular momentum operator, L̂z. This has been used in nuclear-motion calculations to compute an ab initio linelist of ¹²C₂H₂ covering 13.9 million transitions between 2.7 million states, up to a rotational excitation of J=58. In order to facilitate an accurate calculated spectra, available experimental data of ¹²C₂H₂ were collated and analysed to obtain an accurate set of 11,213 empirical energy levels using the MARVEL procedure. As demonstrated, these can be used to produce a high-accuracy potential energy surface and subsequent semi-empirical model for the ro-vibrational energies and intensities of acetylene, which can be computed up to high ro-vibrational excitations. Calculations using this semi-empirical model are in progress for an accurate high-temperature linelist, expected to be valid up to 1000-1200 K. This will be published in due course and will be appropriate for characterising exoplanet and cool stellar atmospheres; these ongoing calculations are discussed.

Variational calculations of rotation-vibration spectra are presented for a range of four- and five-atom molecules of atmospheric and astrophysical importance. Using state-of-the-art electronic structure methods, new nine-dimensional potential energy and dipole moment surfaces are constructed for methyl chloride (CH3Cl), silane (SiH4), and methane (CH4). The respective surfaces are rigorously evaluated against high-resolution spectroscopic data from a variety of experimental sources. The ab initio potential energy surfaces represent some of the most accurate to date, whilst intensity simulations utilizing the dipole moment surfaces show good agreement with experiment. A novel application of rotation-vibration computations is introduced to investigate the sensitivity of spectral lines to a possible space-time variation of the proton-to-electron mass ratio μ. The approach relies on finding the mass dependence of the computed energy levels and is only possible because of the remarkable accuracy of variational calculations. Highly sensitive transitions are uncovered for ammonia (NH3) and the hydronium cation (H3O+) which could lead to a tighter constraint on a varying μ. An advantage of the variational approach is that Einstein A coefficients can be determined to help guide future laboratory and astronomical observations. This thesis demonstrates the current capabilities of variational calculations of rotation-vibration spectra and highlights the challenges faced by the field.

The major part of this work is construction of 54 room-temperature infrared absorption line lists for isotopologues of carbon dioxide. In accurate nuclear motion calculations an exact nuclear kinetic energy operator is used in the Born-Oppenheimer approximation and three ab initio and semi-empirical potential energy surfaces for generation of rotational-vibrational wavefunctions and energy levels. Transition intensities are calculated with two different high quality ab initio dipole moment surfaces. The generated line lists are comprehensively compared to state-of-the-art measurements, spectroscopic databases and other theoretical studies. As a result, uncertainties in calculated transition intensities in several vibrational CO2 bands are shown below 1%, which is sufficient for use in remote sensing measurements of carbon dioxide in the Earth’s atmosphere. Results of the present calculations set a new state-of-the-art and have been included in the 2016 release of the HITRAN database. A theoretical procedure for estimating reliability of computed transition intensities is presented and applied to CO 2 line lists. As a result, each transition intensity received a reliability factor, a particularly useful descriptor for detecting resonance interactions between rotational-vibrational energy levels, as well as a good measure quantifying the strength of such interactions. The theoretical procedure used for CO 2 is extended to electronic transitions in the Born-Oppenheimer approximation. In this extended framework rotational- vibrational-electronic line lists for SO2 and CaOCa molecules are generated. For this purpose appropriate ab initio potential energy surfaces and a transition dipole moment surface are generated. Absolute transition intensities are then calculated both in the Franck-Condon approximation and with a full transition dipole moment surface. Resulting line lists are compared with available experimental and theoretical data. The unprecedented accuracy of the model used in these calculations and the rotational resolution of transition lines renders the present approach as promising for future uses in atmospheric science. Finally a theoretical framework for fully non-adiabatically coupled Hamiltonian is derived and discussed. A proposition for computer implementation of this theoretical scheme is also given.

New linelists (list of wavelengths with associated frequencies) for the open-shell transition metal diatomics Chromium Hydride (CrH) and Manganese Hydride (MnH) have been calculated. These linelists have been calculated from first principles making use of the Born-Oppenheimer approximation which decomposes the Schr\"{o}dinger equation for a molecule into an electronic equation and rovibronic equation. To solve the electronic Schr\"{o}dinger equation and thus produce Potential Energy Curves (PECs), Dipole Moment Curves (DMCs) and all relevant couplings (transition dipole, electronic angular momentum, spin-orbit) the MRCI (Multi-Reference Configuration Interaction) method as implemented in the \textit{ab initio} \texttt{MOLPRO} package was used. The electronic states considered were those up to $20\;000$ cm$^{-1}$ as this region is of most importance to astronomers to whom we are creating the linelists for within the ExoMol project. The construction of linelists for transition metal diatomics is very much at a developmental stage due to the myriad of low-lying electronic states of high multiplicity which couple together. Hence during the calculation of \textit{ab initio} curves a systematic study of the CASSCF orbitals used for the proceeding MRCI calculations was undertaken for both CrH and MnH. Also the effect of changing the configuration space of electrons was found to have profound effects on the behaviour of the PECs obtained. Additionally, the variation in properties obtained by changing the number of states calculated within a single MRCI calculation was investigated. A choice selection of these \textit{ab initio} curves were then implemented into the in-house programme Duo to produce linelists for CrH and MnH. These linelists were refined using available experimental data. At present, the existing linelist available for CrH in literature has 2 electronic states. Our new linelist is composed from 8 low-lying electronic states of CrH. CrH is used to classify L type dwarfs under the widely accepted classification scheme of Kirkpatrick. Additionally it has been shown theoretically that CrH could be used as a sensitive probe of magnetic fields of stars and also in the so-called ``deuterium test'' to probe the evolutionary history of sub-stellar objects. Due to the limited coverage of the existing linelist, a new linelist is sought after by astronomers. At present MnH has not been discovered in outer space but due to the favourable abundance of manganese it has been speculated that it could be present in the ISM. Hence the creation of a linelist opens up the possibility of the first astronomical detection of this molecule. Since January 2016 I have been working as a teaching fellow at Aberystwyth University. Hence preliminary work has been done on creating projects for undergraduate students for the molecules of BeH, MgH and CaH. A summary of literature has also been created for FeH with a view to in future creating a new linelist for this molecule which is of considerable interest both from an astronomical and theoretical perspective.

During the end of the 1950's Alexander Grothendieck observed the importance of the coefficient groups in cohomology. Three decades later, he presented his ``Esquisse d'un Programme" to the main french funding body. This program also included the use of different coefficient groups in the definition of various (co)homologies. His proposal was rejected. Another three decades later, in the 21st century, his research proposal is considered one of the most inspiring and important collection of ideas in pure mathematics. His ideas brought together algebraic topology, geometry, Galois theory, etc. becoming the origin for several new branches of mathematics. Today, less than one year after his death, Grothendieck is considered one of the most influential mathematicians worldwide. His ideas were important for the proofs of some of the most remarkable mathematical problems like the Weil Conjectures, Mordell Conjectures and the solution of Fermat's last theorem. Grothendieck's dessins d'enfant have been used in mathematical physics in various domains. Seiberg-Witten curves, N=1 and N=2 gauge theories and matrix models are a few examples where his insights are relevant. In this thesis I try to connect the idea of cohomology with coefficients in various sheaves to some areas of modern research in physics. The applications are manifold: the universal coefficient theorem presents connections to the topological genus expansion invented by 't Hooft and applied to quantum chromodynamics (QCD) and string theory, but also to strongly coupled electronic systems or condensed matter physics. It also appears to give a more intuitive explanation for topological recursion formulas and the holomorphic anomaly equations. The counting of BPS states may also profit from this new perspective. Indeed, the merging of cohomology classes when a change in coefficient groups is implemented may be related to the wall-crossing formulas and the phenomenon of decay or coupling of BPS states while crossing stability walls. The $Ext$ groups appearing in universal coefficient theorems may be regarded as obstructions characterizing the phenomena occurring when BPS stability walls are being crossed. Another important aspect is the existence of dualities. These are the non-perturbative analogue of symmetry transformations. Until now, they were discovered more by accident or by educated guesswork. I show in this thesis that there exists an underlying structure to the dualities, a structure that connects them the number fields used as coefficients in (co)homologies. This observation makes a nontrivial connection between number theory and physics.

Molecular nitrogen, N₂, is the most abundant molecule in the terrestrial atmosphere. Its cation N₂⁺ is therefore prevalent in the earth's ionosphere as well as in nitrogen plasmas produced for reasons varying from lightning strikes to combustion. Any model which seeks to describe plasmas in air must contain a description of nitrogen ion chemistry. Despite this, there is a distinct paucity of data describing electron-N₂⁺ interactions and the resultant bound and quasi-bound electronic structure of N₂. The characterisation of these states is essential for describing dissociative recombination which is the main destroyer of molecular ions in a plasma. This thesis aims to alleviate this problem by performing extensive ab initio R-matrix calculations to create a comprehensive map of the highly-excited electronic structure of N₂ which can the be used to perform a dissociative recombination cross-section calculation. Potential energy curves were found by performing resonant and bound state calculations for all singlet and triplet molecular symmetries of N₂ up to l ≤ 4. The use of a dense grid meant that highly-excited electronic states could be found with an unprecedented level of detail. Many of the states were previously unknown. A new fitting method was developed for the characterisation of resonant states using the time-delay method. It was shown that whilst the R-matrix method is not competitive with conventional quantum chemistry techniques for low lying valence states, it is particularly appropriate for highly-excited states, such as Rydberg states. The data gained from these calculations was then used as an input for a multichannel quantum defect theory calculation of a dissociative recombination cross-section. A description is given of how to prepare the data from the R-matrix calculation for input into a multichannel quantum defect theory dissociative recombination cross-section calculation. Cross-sections were found for v=0-3 including three ionic cores. Whilst previous studies of dissociative recombination using R-matrix data required some empirical intervention, the cross-section found in this thesis is completely ab initio and is in good agreement with experiment.

The main aim of this study is to calculate a high temperature line list for H_2^32S. The results will form an important addition to the databases used for space applications, as well as laboratory investigations and pollution studies. The Dvr3d program suite is used to calculate the bound ro-vibrational energy levels, and dipole moment transition intensities. The most accurate available potential energy surface is empirically determined. This surface is used in our calculations after refining it by fitting to the up-to-date experimental data. For accurate line intensities, an accurate dipole moment surface (DMS) is needed. Constructing an accurate DMS for H_2S is well known to be difficult. A systematic ab initio study for the DMS has been performed. Different methods were tested in conjunctions with different basis sets taking into account the relativistic corrections and core-valence effects. The resulting (ATY2013) line list should be valid from 0 to 9000 cm^−1 and for temperature up to 2000 K. ATY2013 with cut off intensity of order 10^−31 cm^−1/(molecule×cm^−2) contains ∼36×10^6 transitions at 2000 K. In addition, the pure rotational transition frequencies of H_2S in natural abundance in its ground and first excited vibrational states have been recorded at room temperature at 0.005 cm^−1 resolution in the region 45 to 360 cm^−1 with a globar continuum source at SOLEIL synchrotron. 2400 rotational transitions are assigned to ground vibrational state of the four isotopologues H_2^32S, H_2^33S, H_2^34S and H_2^36S where 65% of them are new. 91 rotational transitions of H_2^36S were identified for the first time, as well as 406 rotational lines of H_2^32S and H_2^34S in their first excited bending vibrational state were recorded and analysed for the first time.

In the first section of this thesis, we present the atomic part of our investigation. A C2+ atomic target was prepared and used to generate theoretical data required in the investigation of recombination lines that arise from collisions between electrons and ions in thin plasma found in planetary nebulae and other astrophysical objects. The R-matrix method of electron scattering theory was used to describe the C2+ plus electron system. Theoretical data concerning bound and autoionizing states were generated in the intermediate-coupling approximation by R-matrix and Autostructure codes and compared to experimental data. The comparison revealed very good agreement. These theoretical data were then used to generate dielectronic recombination data for C+ which include transition lines, oscillator strengths, radiative transition probabilities, as well as emissivity and dielectronic recombination coefficients. The data were cast in the form of a line list, called SS1, containing 6187 optically-allowed transitions which include many C II lines observed in astronomical spectra. The data were validated by comparison to C+ recombination data obtained from a number of sources in the literature. This line list was used to analyze the spectra from a number of astronomical objects, mainly planetary nebulae, and identify their electron temperature where the observational data were obtained from the literature. The electron temperature investigation was also extended to include free electron energy analysis which uses observational and theoretical data of FF and FB transitions to investigate the long-standing problem of discrepancy between the results of recombination and forbidden lines analysis and its possible connection to the electron distribution (Maxwellian or non-Maxwellian). In the course of this investigation two elaborate methods, one for finding and analyzing resonances (K-matrix method) and the other for analyzing and identifying electron temperature from astronomical spectra (least squares minimization), were employed. A computer program for atomic transition analysis was also developed and used as the main tool for obtaining the line list and analyzing the observational spectra. In the second section of the thesis we present the results of our molecular investigation; the generation of a comprehensive, calculated line list of frequencies and transition probabilities for the singly deuterated isotopologue of H3+, H2D+. The line list, which is the most comprehensive one of its kind in existence, contains over 22 million rotational-vibrational transitions occurring between more than 33 thousand energy levels and covers frequencies up to 18500 cm-1. All energy levels with rotational quantum number, J, up to 20 are considered, making the line list useful for temperatures up to at least 3000 K. About 15% of these levels are fully assigned with approximate rotational and vibrational quantum numbers. The list is calculated using a previously proposed, highly accurate, ab initio model implemented in a high-accuracy computer code based on a two-stage discrete variable representation (DVR) approach. Various consistency checks were carried out to test and validate the results. All these checks confirmed the accuracy of the list. A temperature-dependent partition function, valid over a more extended temperature range than those previously published, and cooling function are presented. Temperature-dependent synthetic spectra for the temperatures T=100, 500, 1000 and 2000 K in the frequency range 0-10000 cm-1 were also generated and presented graphically.

Here the ab-initio R-Matrix method has been used to carry out electron-molecule collision calculations on the the molecules of interstellar interest C3N, C2H & CN, and molecules found in industrial plasma applications SiBr, SiBr2 and NaI. These were carried out using the UK Molecular R-Matrix codes, along with the Quantemol expert system for running these codes. Calculations have also been carried out on electron collisions with atomic oxygen using these codes, with details included about the problems faced in running an atomic calculation with the molecular codes. Calculations on each species include comparison of different models, including staticexchange and close-coupling models (with different size CAS tried), various basis sets, and for some species different initial orbitals. These different initial orbitals were either taken from the codes themselves (for SCF orbitals), or the quantum chemistry program MOLPRO (for natural orbitals), for the latter numerous state averaged orbitals were tried with different weightings in order to produce good target energies for carrying into the scattering calculation. Results for all calculations include scattering observables such as eigenphase sums, elastic and excitation cross-sections, bound anionic states and resonance positions and widths. Also a new theory has been developed for calculating rotational cross-sections which includes the spin angular momentum of the incoming scattering electron, this has been implemented into the already existing code ROTLIN, which can calculate rotational cross-sections using the scattering data from an R-Matrix calculation.