Abstract
We recently derived a spin-mapping approach for treating the nonadiabatic dynamics of a two-level system in a classical environment [J. E. Runeson and J. O. Richardson, J. Chem. Phys. 151, 044119 (2019)] based on the well-known quantum equivalence between a two-level system and a spin-1/2 particle. In the present paper, we generalize this method to describe the dynamics of N-level systems. This is done via a mapping to a classical phase space that preserves the SU(N)-symmetry of the original quantum problem. The theory reproduces the standard Meyer-Miller-Stock-Thoss Hamiltonian without invoking an extended phase space, and we thus avoid leakage from the physical subspace. In contrast to the standard derivation of this Hamiltonian, the generalized spin mapping leads to an N-dependent value of the zero-point energy parameter that is uniquely determined by the Casimir invariant of the N-level system. Based on this mapping, we derive a simple way to approximate correlation functions in complex nonadiabatic molecular systems via classical trajectories and present benchmark calculations on the seven-state Fenna-Matthews-Olson light-harvesting complex. The results are significantly more accurate than conventional Ehrenfest dynamics, at a comparable computational cost, and can compete in accuracy with other state-of-the-art mapping approaches.
Highlights
The full quantum dynamics of complex systems is, in general, far too complicated to be simulated in practice
Each diabatic state represents an exciton localized on one of the sites. This is a challenging benchmark problem for quasiclassical nonadiabatic dynamics and allows our method to be compared with other mapping approaches[16,20,67,69] as well as to numerically exact results obtained via the hierarchical equations of motion (HEOM) approach.[70–74]
We have generalized the spin mapping of a twolevel system in Ref. 27 to N-level systems, which is a problem that had not been satisfactorily solved since it was first posed by the seminal works of Meyer and Miller in 1979.28 The general idea is to make the classical phase space inherit the SU(N)-symmetry properties of the quantum system
Summary
The full quantum dynamics of complex systems is, in general, far too complicated to be simulated in practice. Other options of comparable cost include surface hopping[3] and mapping-based techniques.[4] In particular, the Meyer–Miller–Stock–Thoss (MMST) mapping[5,6] has recently regained attention.[7,8,9,10,11,12,13,14,15,16] As a generalization of the Schwinger bosonization to N-level systems, its basic principle is to represent the N electronic states by N coupled harmonic oscillators that share a single excitation This mapping is formally exact and has inspired a number of methods for calculating correlation functions, such as the linearized semiclassical initial-value representation (LSC-IVR),[17] the Poisson-bracket mapping equation (PBME),[18,19] the symmetrical quasiclassical windowing approach (SQC),[7,11] partially linearized density matrix dynamics (PLDM),[20,21] and the forward–backward trajectory solution (FBTS)[22,23] of the quantum-classical Liouville equation (QCLE).[24] These quasiclassical approaches all use a classical description of the nuclear dynamics. The results can compete in accuracy with other state-of-the-art methods in the mapping community and are far superior to conventional Ehrenfest dynamics with comparable cost
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