This thesis presents a comprehensive treatment of the mathematical foundations of the quantum mechanics of molecules. In particular it treats the molecular system H3, which consists of three protons and three electrons. The mathematical formalism of quantum mechanics is developed from first principles, and elementary examples of quantum systems, such as spin systems, the hydrogen atom and molecule are worked out as illustrations of the formalism. Molecular symmetry groups and their representation theory are also treated. The coordinate systems in the shape space of three nuclei are studied. Vector bundles and connections are developed and applied to the Berry-Simon connection in the complex vector bundle over the shape space whose fibres are the energy ground state eigenspaces. ``Berry phase'' is then interpreted as holonomy associated to parallel translation in the usual sense of differential geometry.

Since this thesis was defended (November, 2003) several errors or omissions have been detected in it and if they are not too extensive they will be corrected by Dan Dix. These (planned) amendations with their dates of completion are as follows.

- Corrected and expanded discussion of simultaneous eigenfunctions of S^2 and S_3 in the two and three spin systems (completed Feb. 4, 2004).
- Corrected and expanded discussion of tensor products of Hilbert spaces (completed Feb. 13, 2004).
- Hyperspherical coordinates on shape space of H3. Discussion of the placement of equilateral triangles, collinear shapes, and various types of isoceles triangles within shape space. Symmetries of shape space induced by permutation of the nuclei.
- Berry Phase in the Molecular System H3, amended version (Feb. 13, 2004).

The following topics would have been nice to have been discussed for the sake of completeness, and maybe in the future short accounts will be written by Dan Dix to summarize them.

- Joint spectral decomposition of H, J^2, and J_3 in the scattering sector for the Hydrogen-like atom.
- Discussion of low energy electronic states of H2, including or excluding internuclear repulsion term. Separation of space and spin variables. Discussion of spatial symmetry and electron interchange symmetry.
- Extended discussion of GVB (SCVB) and HF ansatze for H2 wavefunctions.
- More detailed discussion of known facts about the low energy electronic states of H3.

Discussion of mathematical issues left unresolved in the thesis.

- The shape space is the union of sets C_k, k=1,2,..., where C_k contains those shapes of the nuclei where the energy ground state eigenspace has (complex) dimension 2k. Mead claims (indirectly) that C_1 is an open subset of the shape space. Exactly which shapes are in C_1?
- Over C_1 we should then have a complex line bundle, where the fibre over each point (shape) is the one dimensional complex vector space of energy ground state eigenfunctions which are also eigenfunctions of the operator S_3 for the total z-component of spin with eigenvalue hbar/2. This requires one to prove the existence of an atlas of smooth local trivializations of this bundle.
- C_2 appears to contain part of the one parameter family of equilateral triangle shapes, as well as certain one parameter families of isosceles triangle shapes (near to the united atom limit). These curves in shape space meet at a special equilateral triangle shape which is in C_3. What happens to C_2 near the united atom limit?
- The fibre over each point (shape) of C_k is equipped with a representation of the point symmetry group of that shape. Is this representation always irreducible? If so, what is its character?
- Mead claims that the Berry-Simon connection over C_1 has vanishing curvature. The holonomy associated to a closed curve in C_1 spanned by a surface lying entirely in C_1 can be related to a surface integral of the curvature form, but if the closed curve in C_1 is not null homotopic in C_1 then the holonomy needs to be computed. Yarkony claims to have computed it (in certain cases).
- The fibre over each point of C_1 is spanned by a ground state electronic wavefunction, which is a complex-valued function of nine space variables (three variables for each electron) and three binary spin variables (each taking only the values 0 or 1). The visualization of such a complicated function is impossible, but if it is closely approximated by a wavefunction of GVB type then its approximate nature can be understood. How accurate is this GVB ansatz over the entire shape space?