Abstract
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 hbar by introducing local spatial smoothing. In the case of standard quantum hydrodynamics, the hbar ne 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.
Highlights
1.1 Factorized Wave Functions in Quantum Molecular DynamicsQuantum molecular dynamics deals with the problem of solving the molecular Schrödinger equation i ∂t Ψ = (Tn + Te + Vn + Ve + VI )Ψ =: H Ψ, (1)which governs the quantum evolution for a set of nuclei interacting with a set of electrons
This section illustrates the geometry of the hydrodynamic setting of quantum mechanics, which has its foundations in the Madelung transform [54, 55]
We have exploited momentum maps to collectivize a sequence of molecular quantum chemistry models for factorized nuclear and electronic wave functions, thereby obtaining a sequence of quantum fluid models with shared semidirect-product Lie–Poisson structures
Summary
A great deal of work has been devoted to formulating mathematical models for nonadiabatic dynamics [7, 20] Some of these approaches still exploit the Born–Huang expansion, while others introduce a fully time-dependent ansatz for the molecular wave function Ψ. This paper aims to investigate and compare the hydrodynamic approaches for both the mean-field models and the exact factorization ansatz in the context of geometric mechanics Within this framework, separating out the nuclear kinetic energy in the molecular Hamiltonian (3) corresponds in the hydrodynamic approach to transforming into a Lagrangian coordinate frame moving with the nuclei. This Poisson bracket yields the corresponding Hamiltonian equation (11) via the expected relation f = {f, h} Both the DF variational principle and the Hamiltonian structure presented above will be used again and again throughout this paper to illuminate the geometric features of current models in nonadiabatic molecular dynamics. Other important examples are given by quantum expectation values [12] and covariance matrices of Gaussian wavepackets [62]
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