The Hamiltonian action of a Lie group on a symplectic manifold induces a momentum map generalizing Noether's conserved quantity occurring in the case of a symmetry group. Then, when a Hamiltonian function can be written in terms of this momentum map, the Hamiltonian is called 'collective'. Here, we derive collective Hamiltonians for a series of models in quantum molecular dynamics for which the Lie group is the composition of smooth invertible maps and unitary transformations. In this process, different fluid descriptions emerge from different factorization schemes for either the wavefunction or the density operator. After deriving this series of quantum fluid models, we regularize their Hamiltonians for finite by introducing local spatial smoothing. In the case of standard quantum hydrodynamics, the = 0 dynamics of the Lagrangian path can be derived as a finite-dimensional canonical Hamiltonian system for the evolution of singular solutions called 'Bohmions', which follow Bohmian trajectories in configuration space. For molecular dynamics models, application of the smoothing process to a new factorization of the density operator leads to a finite-dimensional Hamiltonian system for the interaction of multiple (nuclear) Bohmions and a sequence of electronic quantum states.

In this paper we consider a new geometric approach to Madelung's quantum hydrodynamics (QHD) based on the theory of a gauge connection. Unlike previous similar approaches this connection is no longer an exact differential, instead allowing for a constant curvature and endowing QHD with intrinsic non-zero holonomy through the use of only single-valued phase-factors. In the hydrodynamic context, this connection corresponds to a fluid velocity which no longer is constrained to be irrotational, thus possessing a non-trivial circulation theorem and can allow for solutions corresponding to vortex filaments. After exploiting the Rasetti-Regge method to couple the Schrödinger equation to vortex filament dynamics, the latter is then considered as a source of molecular geometric phase in the context of Born-Oppenheimer molecular dynamics. Similarly, we consider the Pauli equation for the motion of spin particles in electromagnetic fields and we exploit its underlying hydrodynamic picture to include filament dynamics.

This thesis investigates geometric approaches to quantum hydrodynamics (QHD) in order to develop applications in theoretical quantum chemistry.Based upon the momentum map geometric structure of QHD and the associated Lie-Poisson and Euler-Poincaré equations, alternative geometric approaches to the classical limit in QHD are presented. Firstly, a new regularised Lagrangian is introduced, allowing for singular solutions called 'Bohmions' for which the associated trajectory equations are finite-dimensional and depend on a smoothened quantum potential. Secondly, the classical limit is considered for quantum mixed states. By applying a cold fluid closure to the density matrix the quantum potential term is eliminated from the Hamiltonian entirely.The momentum map approach to QHD is then applied to the nuclear dynamics in a chemistry model known as exact factorization. A variational derivation of the coupled electron-nuclear dynamics is presented, comprising an Euler-Poincaré structure for the nuclear motion. The QHD equations for the nuclei possess a Kelvin-Noether circulation theorem which returns a new equation for the evolution of the electronic Berry phase. The geometric treatment is then extended to include unitary electronic evolution in the frame of the nuclear flow, with the resulting dynamics carrying both Euler-Poincaré and Lie-Poisson structures. A new mixed quantumclassical model is then derived by applying both the QHD regularisation and cold fluid closure to a generalised factorisation ansatz at the level of the molecular density matrix.A new alternative geometric formulation of QHD is then constructed. Introducing a u(1) connection as the new fundamental variable provides a new method for incorporating holonomy in QHD, which follows from its constant non-zero curvature. The associated fluid flow is no longer constrained to be irrotational, thus possessing a non-trivial circulation theorem and allows for vortex filament solutions. This approach is naturally extended to include the coupling of vortex filament dynamics to the Schrödinger equation. This formulation of QHD is then applied to Born-Oppenheimer molecular dynamics suggesting new insights into the role of Berry phases in adiabatic phenomena.Finally, non-Abelian connections are then considered in quantum mechanics. The dynamics of the spin vector in the Pauli equation allows for the introduction of an so(3) connection whilst a more general u(H ) connection can be introduced from the unitary evolution of a quantum system. This is used to provide a new picture for the Berry connection and quantum geometric tensor and well as derive more general systems of equations which feature explicit dependence on the curvature of the connection. Relevant applications to quantum chemistry are then considered.