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

Area of Science:

• Quantum Chemistry
• Computational Chemistry
• Chemical Physics

Background:

• Mixed quantum-classical dynamics simulations are crucial for understanding chemical processes.
• The Meyer-Miller-Stock-Thoss (MMST) Hamiltonian and spin-mapping approaches are common for these simulations.
• Efficient and accurate numerical algorithms are needed for integrating the equations of motion.

Purpose of the Study:

• To compare the performance of three time-propagation algorithms for the MMST Hamiltonian: Momentum Integral (MInt), Split-Liouvillian (SL), and Degenerate Eigenvalue (DE).
• To analyze accuracy, energy conservation, symplecticity, Liouville's theorem adherence, and computational cost.
• To provide guidance for selecting appropriate algorithms in future mapping-variable simulations.

Main Methods:

• Implementation and comparison of MInt, SL, and DE algorithms for the MMST Hamiltonian.
• Evaluation of individual trajectory accuracy and correlation functions.
• Assessment of conserved quantities (energy, symplecticity, Liouville's theorem) and computational expense.

Main Results:

• The MInt algorithm is the only rigorously symplectic method among those tested.
• The SL algorithm achieves comparable accuracy to MInt but with reduced computational cost.
• The DE algorithm exhibits poor energy conservation, even at small timesteps, due to its inherent approximation.

Conclusions:

• The choice of algorithm significantly impacts the accuracy, conservation properties, and efficiency of mixed quantum-classical simulations.
• The SL algorithm presents a favorable balance of accuracy and computational cost.
• The DE algorithm's limitations in energy conservation warrant caution in its application.