Physics

Quantum dynamics

Degrees of freedom (physics and chemistry)

Mathematics

Statistical physics

Statistics

Mathematical optimization

Curse of dimensionality

Phase space

Hamiltonian (control theory)

Hamiltonian mechanics

Quantum mechanics

Classical mechanics

Quantum

Configuration space

Accounts of Chemical Research

Nonadiabatic dynamical processes are one of the most important quantum mechanical phenomena in chemical, materials, biological, and environmental molecular systems, where the coupling between different electronic states is either inherent in the molecular structure or induced by the (intense) external field. The curse of dimensionality indicates the intractable exponential scaling of calculation effort with system size and restricts the implementation of "numerically exact" approaches for realistic large systems. The phase space formulation of quantum mechanics offers an important theoretical framework for constructing practical approximate trajectory-based methods for quantum dynamics. This Account reviews our recent progress in phase space mapping theory: a unified framework for constructing the mapping Hamiltonian on phase space for coupled F-state systems where the renowned Meyer-Miller Hamiltonian model is a special case, a general phase space formulation of quantum mechanics for nonadiabatic systems where the electronic degrees of freedom are mapped onto constraint space and the nuclear degrees of freedom are mapped onto infinite space, and an isomorphism between the mapping phase space approach for nonadiabatic systems and that for nonequilibrium electron transport processes. While the zero-point-energy parameter is conventionally assumed to be positive, we show that the constraint implied in the conventional Meyer-Miller mapping Hamiltonian requires that such a parameter can be negative as well and lies in (-1/F, +∞) for each electronic degree of freedom. More importantly, the zero-point-energy parameter should be interpreted as a special case of a commutator matrix in the comprehensive phase space mapping Hamiltonian for nonadiabatic systems. From the rigorous formulation of mapping phase space, we propose approximate but practical trajectory-based nonadiabatic dynamics methods. The applications to both gas phase and condensed phase problems include the spin-boson model for condensed phase dissipative two-state systems, the three-state photodissociation models, the seven-site model of the Fenna-Matthews-Olson monomer in photosynthesis of green sulfur bacteria, the strongly coupled molecular/atomic matter-optical cavity systems designed for controlling and manipulating chemical dynamical processes, and the Landauer model for a quantum dot state coupled with two electrodes. In these applications the overall performance of our phase space mapping dynamics approach is superior to two prevailing trajectory-based methods, Ehrenfest dynamics and fewest switches surface hopping.

Unified Formulation of Phase Space Mapping Approaches for Nonadiabatic Quantum Dynamics