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Register a new userCommutator Matrix in Phase Space Mapping Models for Nonadiabatic Quantum Dynamics
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We show that a novel, general phase space mapping Hamiltonian for nonadiabatic systems, which is reminiscent of the renowned Meyer-Miller mapping Hamiltonian, involves a commutator variable matrix rather than the conventional zero-point-energy parameter. In the exact mapping formulation on constraint space for phase space approaches for nonadiabatic dynamics, the general mapping Hamiltonian with commutator variables can be employed to generate approximate trajectory-based dynamics. Various benchmark model tests, which range from gas phase to condensed phase systems, suggest that the overall performance of the general mapping Hamiltonian is better than that of the conventional Meyer-Miller Hamiltonian.
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We propose a trajectory-based method for simulating nonadiabatic dynamics in molecular systems with two coupled electronic states. Employing a quantum-mechanically exact mapping of the two-level problem to a spin-1/2 coherent state, we construct a classical phase space of a spin vector constrained to a spherical surface with a radius consistent with the quantum magnitude of the spin. In contrast with the singly-excited harmonic oscillator basis used in Meyer-Miller-Stock-Thoss (MMST) mapping, the theory requires no additional projection operators onto the space of physical states. When treated under a quasiclassical approximation, we show that the resulting dynamics is equivalent to that generated by the MMST Hamiltonian. What differs is the value of the zero-point energy parameter as well as the initial distribution and the measurement operators. For various spin-boson models the results of our method are seen to be a significant improvement compared to both standard Ehrenfest dynamics and linearized semiclassical MMST mapping, without adding any computational complexity.
The mapping approach addresses the mismatch between the continuous nuclear phase space and discrete electronic states by creating an extended, fully continuous phase space using a set of harmonic oscillators to encode the populations and coherences of the electronic states. Existing quasiclassical dynamics methods based on mapping, such as the linearised semiclassical initial value representation (LSC-IVR) and Poisson bracket mapping equation (PBME) approaches, have been shown to fail in predicting the correct relaxation of electronic-state populations following an initial excitation. Here we generalise our recently published modification to the standard quasiclassical approximation for simulating quantum correlation functions. We show that the electronic-state population operator in any system can be exactly rewritten as a sum of a traceless operator and the identity operator. We show that by treating the latter at a quantum level instead of using the mapping approach, the accuracy of traditional quasiclassical dynamics methods can be drastically improved, without changes to their underlying equations of motion. We demonstrate this approach for the seven-state Frenkel-Exciton model of the Fenna-Matthews-Olson light harvesting complex, showing that our modification significantly improves the accuracy of traditional mapping approaches when compared to numerically exact quantum results.
We derive an exact quantum propagator for nonadiabatic dynamics in multi-state systems using the mapping variable representation, where classical-like Cartesian variables are used to represent both continuous nuclear degrees of freedom and discrete electronic states. The resulting expression is a Moyal series that, when suitably approximated, can allow for the use of classical dynamics to efficiently model large systems. We demonstrate that different truncations of the exact propagator lead to existing approximate semiclassical and mixed quantum-classical methods and we derive an associated error term for each method. Furthermore, by combining the imaginary-time path-integral representation of the Boltzmann operator with the exact propagator, we obtain an analytic expression for thermal quantum real-time correlation functions. These results provide a rigorous theoretical foundation for the development of accurate and efficient classical-like dynamics to compute observables such as electron transfer reaction rates in complex quantized systems.
We present a new partially linearized mapping-based approach for approximating real-time quantum correlation functions in condensed-phase nonadiabatic systems, called spin-PLDM. Within a classical trajectory picture, partially linearized methods treat the electronic dynamics along forward and backward paths separately by explicitly evolving two sets of mapping variables. Unlike previously derived partially linearized methods based on the Meyer-Miller-Stock-Thoss mapping, spin-PLDM uses the Stratonovich-Weyl transform to describe the electronic dynamics for each path within the spin-mapping space; this automatically restricts the Cartesian mapping variables to lie on a hypersphere and means that the classical equations of motion can no longer propagate the mapping variables out of the physical subspace. The presence of a rigorously derived zero-point energy parameter also distinguishes spin-PLDM from other partially linearized approaches. These new features appear to give the method superior accuracy for computing dynamical observables of interest, when compared with other methods within the same class. The superior accuracy of spin-PLDM is demonstrated within this paper through application of the method to a wide range of spin-boson models, as well as to the Fenna-Matthews-Olsen complex.
In the previous paper [J. R. Mannouch and J. O. Richardson, J.~Chem.~Phys.~xxx, xxxxx (xxxx)] we derived a new partially linearized mapping-based classical-trajectory technique, called spin-PLDM. This method describes the dynamics associated with the forward and backward electronic path integrals, using a Stratonovich-Weyl approach within the spin-mapping space. While this is the first example of a partially linearized spin mapping method, fully linearized spin mapping is already known to be capable of reproducing dynamical observables for a range of nonadiabatic model systems reasonably accurately. Here we present a thorough comparison of the terms in the underlying expressions for the real-time quantum correlation functions for spin-PLDM and fully linearized spin mapping in order to ascertain the relative accuracy of the two methods. In particular, we show that spin-PLDM contains an additional term within the definition of its real-time correlation function, which diminishes many of the known errors that are ubiquitous for fully linearized approaches. One advantage of partially linearized methods over their fully linearized counterparts is that the results can be systematically improved by re-sampling the mapping variables at intermediate times. We derive such a scheme for spin-PLDM and show that for systems for which the approximation of classical nuclei is valid, numerically exact results can be obtained using only a few `jumps. Additionally, we implement focused initial conditions for the spin-PLDM method, which reduces the number of classical trajectories that are needed in order to reach convergence of dynamical quantities, with seemingly little difference to the accuracy of the result.
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