The Born-Oppenheimer (BO) approximation and its extension, the Born-Huang expansion, is at the core of molecular physics but only in the past two decades has the full richness of the underlying theory been explored. From the traditional perspective, molecular forces are expressed in terms of a mean scalar field generated by the collective behavior of the fast degrees of freedom (i.e. electrons). This mean field, the Born-Oppenheimer potential, acts upon the slow degrees of freedom (vibronic coordinates) and it determines the vibrational structure of molecules and the collision properties of atoms. The modern expression of the BO theory, and its multichannel extension, allows for the presence of effective vector, gauge, potentials. These potentials can generate velocity dependent forces, as if from an effective magnetic field, acting upon the slow degrees of freedom. M. V. Berry coined the term geometric magnetism to describe the latter. They can also lead to the appearance of geometric phases. In this research, a Rydberg electron in orbit around a complex core is considered for investigation of the non-Abelian geometric phase in atomic systems. If the core has non-vanishing angular momentum the Rydberg electron experiences an effective, non-Abelian, gauge potential. The potential has magnetic monopole-like structure and there are interesting analogues with phenomena familiar in particle theory. An effective spin-orbit interaction is derived and is related to the screened charge of the monopole. The effect of that interaction on the level structure will be calculated and compared with experimental measurements. Rydberg atoms can also be manipulated to form coherent wave packets, so called designer atoms, and a study of numerical wavepackets in the field of the monopole is proposed. This work will advance the fundamental understanding of Rydberg systems that have possible applications to quantum information and computing. In addition to Rydberg systems, similar gauge structures arise when describing the collision of atoms. In the scattering problem, serious difficulties in application of the Born-Huang method arise. The mathematical framework of gauge theory promises to shed new light on some of these long standing difficulties. There is lots of interest, generated by advances in the laboratory realization of cold molecules, to develop advanced capabilities for tests of parity violation and the variation of fundamental constants. Full understanding of molecular dynamics is required in order to account for all systemic effects, including those of induced vector potentials, that might influence high precision measurements.
