Acta Appl Math (2019) 162:63–103 https://doi.org/10.1007/s10440-019-00257-1
Geometry of Nonadiabatic Quantum Hydrodynamics
Michael S. Foskett1,2 · Darryl D. Holm3 · Cesare Tronci1,2
Received: 26 December 2018 / Accepted: 8 April 2019 / Published online: 7 May 2019 © The Author(s) 2019
Abstract The Hamiltonian action of a Lie group on a symplectic manifold induces a mo- mentum map generalizing Noether’s conserved quantity occurring in the case of a symme- try 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 com- position 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 stan- dard 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.
Keywords Lie group action · Momentum map · Quantum hydrodynamics · Nonadiabatic molecular dynamics
B C. Tronci [email protected] M.S. Foskett [email protected] D.D. Holm [email protected]
1 Department of Mathematics, University of Surrey, Guildford, UK 2 Mathematical Sciences Research Institute, Berkeley, CA, USA 3 Department of Mathematics, Imperial College London, London, UK 64 M.S.Foskettetal.
1 Introduction
1.1 Factorized Wave Functions in Quantum Molecular Dynamics
Quantum molecular dynamics deals with the problem of solving the molecular Schrödinger equation i ∂t Ψ = (Tn + Te + Vn + Ve + VI )Ψ =: HΨ, (1) which governs the quantum evolution for a set of nuclei interacting with a set of electrons. In the equation above, Ψ({rk}, {xl},t)is called the molecular wave function. The notation is such that {rk}={rk : k = 1,...,Nn} and {xl}={xl : l = 1,...,Ne} denote, respectively, Nn nuclear and Ne electronic coordinates. Each rk corresponds to a nucleus of mass Mk whilst all electrons have the same mass m. The notation T and V in (1) refers to the kinetic energy and potential energy operators, while the subscripts n and e denote nuclear and electronic energies, respectively. The subscript I refers to the interaction potential between nuclei and electrons. More explicitly, one has