arXiv subjects

Lauren E. Cook

Publications and source records attributed to Lauren E. Cook.

4 recordsLinked to original sources

arXiv2026

On the Symplectic Propagation of the Spin-MInt Algorithm for Non-Adiabatic Quantum Dynamics

Mapping methods are often used for the numerical simulation of nonadiabatic systems by propagating classical mapping variable trajectories. A recently popularised mapping method is spin-mapping, whose mapping variables arise from quantum mechanical operators with symmetries described by a Lie-Poisson algebra. Simulating the classical-like dynamics of spin-mapping systems accurately is generally challenging, with many methods unable to preserve the underlying geometric structure of the symplectic form. The Spin-MInt algorithm is a recently proposed algorithm propagating spin-mapping variables, with a direct proof of symplecticity existing only for 2 electronic states. Here, we directly prove the symplecticity of the Spin-MInt algorithm for a general $K$ electronic states. A review of the symplectic nature of coadjoint orbits of the $\mathfrak{su}(K)$ Lie-Poisson algebra provides the framework needed to understand symplecticity of the Spin-MInt algorithm in this general case. The symplecticity of the method on the associated coadjoint orbit is then shown for what we believe to be the first time via an explicit verification of the symplecticity condition $\mathbf{MJ}\mathbf{M}^\textrm{T}=\mathbf{J}$ exploiting the Lie-Poisson structure of the system. To our knowledge, this is the first time the monodromy matrix for the Spin-MInt algorithm has been explicitly stated using canonical coordinates on the coherent state manifold for a general number of states. We hope that this will assist the development of classical-like spin-mapping methods which might utilise elements of the monodromy matrix, and inform future work on similar symplectic algorithms for coupled and uncoupled Lie-Poisson systems.

physics.chem-ph↗

arXiv2026

The Spin-MInt Algorithm: an Accurate and Symplectic Propagator for the Spin-Mapping Representation of Nonadiabatic Dynamics

Mapping methods, including the Meyer-Miller-Stock-Thoss (MMST) mapping and spin-mapping, are commonly utilised to simulate nonadiabatic dynamics by propagating classical mapping variable trajectories. Recent work confirmed the Momentum Integral (MInt) algorithm is the only known symplectic algorithm for the MMST Hamiltonian. To our knowledge, no symplectic algorithm has been published for the spin-mapping representation without obtaining Cartesian variables and utilising the MInt algorithm. Here, we present the Spin-MInt algorithm which directly propagates the spin-mapping variables. First, we consider a two-level system which maps onto a spin-vector on a Bloch sphere. Despite the spin-variables being non-canonical, we rigorously prove the Spin-MInt is symplectic using a canonical variable transformation. We determine that the Spin-MInt is a symmetrical, second-order, time-reversible, angle invariant and geometric structure preserving algorithm. Computationally, for a one-dimensional spin-boson model, the Spin-MInt and MInt algorithms are symplectic, satisfy Liouville's theorem, provide second-order energy conservation and are more accurate than a previously-published angle-based algorithm. We additionally present accurate correlation functions for a multi-dimensional spin-boson model. We also extend this methodology to a general number of electronic states and present accurate population results for a three-state Morse potential. The Spin-MInt is faster than the MInt algorithm for all tested models, particularly so for large nuclear degrees of freedom. We believe this to be the first known symplectic algorithm for propagating the nonadiabatic spin-mapping Hamiltonian and one of the first rigorously symplectic algorithms in the case of non-trivial coupling between canonical and spin systems. These results should guide and improve future simulations.

physics.chem-ph↗

arXiv2025

On the electronic path integral normal modes of the Meyer-Miller-Stock-Thoss representation of nonadiabatic dynamics

Accurate and efficient simulation of nonadiabatic dynamics is highly desirable for understanding charge and energy transfer in complex systems. A key criterion for obtaining an accurate method is conservation of the Quantum Boltzmann Distribution (QBD). For a single surface, Matsubara dynamics is known to conserve the QBD, as a consequence of truncating the dynamics in the higher normal modes of the imaginary-time path integral. Recently, a nonadiabatic Matsubara (NA-Mats) dynamics has been proposed (J. Chem. Phys., 2021, 154, 124124) which truncates in the normal modes of the nuclear variables but not in the electronic variables, which are described with the Meyer-Miller-Stock-Thoss (MMST) representation. Surprisingly, this NA-Mats method does not appear to conserve the QBD for a general system. This poses the question of the effect of truncating the higher path integral normal modes of the electronic variables in the MMST representation. In this article, we present what we believe is the first study of electronic normal modes of the MMST representation. We find that observables are not usually a function of a finite number of normal modes and that the higher normal modes are not constrained by the distribution, unlike in conventional nuclear normal modes. Furthermore, truncating the dynamics in MMST normal modes leads to inaccurate correlation functions and while the QBD appears conserved for an ensemble of trajectories, it is not for a single trajectory. Overall, this suggests that MMST path integral normal modes are not optimal for obtaining an accurate, QBD conserving nonadiabatic dynamics method.

physics.chem-ph↗

arXiv2023

Which Algorithm Best Propagates the Meyer-Miller-Stock-Thoss Mapping Hamiltonian for Non-Adiabatic Dynamics?

A common strategy to simulate mixed quantum-classical dynamics is by propagating classical trajectories with mapping variables, often using the Meyer-Miller-Stock-Thoss (MMST) Hamiltonian or the related spin-mapping approach. When mapping the quantum subsystem, the coupled dynamics reduce to a set of equations of motion to integrate. Several numerical algorithms have been proposed, but a thorough performance comparison appears to be lacking. Here, we compare three time-propagation algorithms for the MMST Hamiltonian: the Momentum Integral (MInt) (arXiv:1709.07474), the Split-Liouvillian (SL) (arXiv:1609.00644), and the algorithm in arXiv:1201.1042 that we refer to as the Degenerate Eigenvalue (DE) algorithm due to the approximation required during derivation. We analyse the accuracy of individual trajectories, correlation functions, energy conservation, symplecticity, Liouville's theorem and the computational cost. We find that the MInt algorithm is the only rigorously symplectic algorithm. However, comparable accuracy at a lower computational cost can be obtained with the SL algorithm. The approximation implicitly made within the DE algorithm conserves energy poorly, even for small timesteps, and thus leads to slightly different results. These results should guide future mapping-variable simulations.

physics.chem-ph↗