[Paper Review] The Coupled-Trajectory Mixed Quantum-Classical Algorithm: A Deconstruction

G. H. Gossel, Federica Agostini+1 more|arXiv (Cornell University)|May 9, 2018

Spectroscopy and Quantum Chemical Studies74 references45 citations

TL;DR

The paper analyzes the CT-MQC method from exact factorization, detailing how coupled-trajectory terms drive decoherence and wavepacket branching, and comparing electronic vs nuclear contributions.

ABSTRACT

We analyze a mixed quantum-classical algorithm recently derived from the exact factorization equations [Min, Agostini, Gross, PRL {\\bf 115}, 073001 (2015)] to show the role of the different terms in the algorithm in bringing about decoherence and wavepacket branching. The algorithm has the structure of Ehrenfest equations plus a "coupled-trajectory" term for both the electronic and nuclear equations, and we analyze the relative roles played by the different non-adiabatic terms in these equations, including how they are computed in practise. In particular, we show that while the coupled-trajectory term in the electronic equation is essential in yielding accurate dynamics, that in the nuclear equation has a much smaller effect. A decoherence time is extracted from the electronic equations and compared with that of augmented fewest-switches surface-hopping. We revisit a series of non-adiabatic Tully model systems to illustrate our analysis.

Motivation & Objective

• Explain the role of different non-adiabatic terms in CT-MQC and how they induce decoherence and wavepacket branching.
• Determine the relative importance of coupled-trajectory terms in electronic vs nuclear equations.
• Illustrate the mechanism across Tully model systems and compare decoherence times with augmented fewest-switches surface-hopping.

Proposed method

• Derive and analyze CT-MQC equations from the exact factorization framework.
• Decompose dynamics into Ehrenfest-like terms plus coupled-trajectory corrections.
• Use trajectory-resolved (spatially resolved) electronic populations and quantum momentum to study coupling effects.
• Reconstruct quantum momentum from a swarm of nuclear trajectories and apply a trajectory-specific shift to enforce physical population transfer.
• Compare full CT-MQC with CTe-MQC (nuclear coupling only) and pure Ehrenfest dynamics across multiple model systems.

Experimental results

Research questions

• RQ1 How do coupled-trajectory terms in the electronic and nuclear equations contribute to wavepacket branching and decoherence in CT-MQC?
• RQ2 What is the relative impact of the coupled-trajectory term in the electronic equation versus the nuclear equation on correct non-adiabatic dynamics?
• RQ3 Can CT-MQC reproduce features like branching and decoherence observed in exact dynamics and how does its decoherence time compare with augmented FSSH?
• RQ4 How do CT-MQC dynamics manifest across different non-adiabatic model systems (extended coupling, single-avoided crossing, double-arch, dual avoided-crossing)?

Key findings

• The coupled-trajectory term in the electronic equation is essential for accurate branching and decoherence.
• The coupled-trajectory term in the nuclear equation has a smaller effect and Ehrenfest nuclei coupled to CT-MQC electronic dynamics reproduce much of the full behavior.
• Ehrenfest-only nuclei with CT-MQC electronic coupling (CTe-MQC) capture nuclear wavepacket splitting similarly to the full CT-MQC in many cases.
• Nuclear wavepacket splitting can occur even with only weighted-BO forces, due to trajectory-dependent populations and quantum momentum.
• A decoherence time can be defined within CT-MQC and compared qualitatively with augmented fewest-switches surface-hoping results, showing similar onset but quantitative differences.
• The study uses four Tully models to illustrate the roles of terms and the mechanisms of branching and decoherence.