Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1805.03534 · Journal of Chemical Theory and Computation

Coupled-Trajectory Mixed Quantum-Classical Algorithm: A Deconstruction

G. H. Gossel, Federica Agostini, Neepa T. Maitra

Published Jul 31, 2018• 1 citation
arXivHTMLPDF

Abstract

We analyze a mixed quantum-classical algorithm recently derived from the exact factorization equations [Min, Agostini, Gross, PRL 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 nonadiabatic terms in these equations, including how they are computed in practice. 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 nonadiabatic Tully model systems to illustrate our analysis.

DOI
10.1021/acs.jctc.8b00449

Related papers

Coupled-Trajectory Mixed Quantum-Classical Algorithm: A Deconstruction

Gossel Graeme H., Federica Agostini, Neepa T. Maitra

We analyze a mixed quantum-classical algorithm recently derived from the exact factorization equations [Min, Agostini, Gross, PRL 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 nonadiabatic terms in these equations, including how they are computed in practice. 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 nonadiabatic Tully model systems to illustrate our analysis.

•Jan 1, 2018•45 citations

Coupled-Trajectory Quantum-Classical Approach to Electronic Decoherence in Nonadiabatic Processes

Seung Kyu Min, Federica Agostini, Eberhard K. U. Gross

We present a novel quantum-classical approach to nonadiabatic dynamics, deduced from the coupled electronic and nuclear equations in the framework of the exact factorization of the electron-nuclear wave function. The method is based on the quasiclassical interpretation of the nuclear wave function, whose phase is related to the classical momentum and whose density is represented in terms of classical trajectories. In this approximation, electronic decoherence is naturally induced as an effect of the coupling to the nuclei and correctly reproduces the expected quantum behavior. Moreover, the splitting of the nuclear wave packet is captured as a consequence of the correct approximation of the time-dependent potential of the theory. This new approach offers a clear improvement over Ehrenfest-like dynamics. The theoretical derivation presented in this Letter is supported by numerical results that are compared to quantum mechanical calculations.

arXiv:1504.00250•Physical Review Letters•Aug 10, 2015•180 citations

Nonadiabatic dynamics via the classical limit Schrödinger equation

John C. Burant, John C. Tully

The coupled Schrödinger equations that describe nonadiabatic dynamics are recast using the Bohm formulation of quantum mechanics. The resulting coupled Bohm equations are solved numerically for two scattering models, giving results that are essentially identical to wave-packet solution of the original coupled Schrödinger equations. The classical limit of the set of coupled Bohm equations is then described, producing a mixed quantum-classical theory incorporating classical-like motion on each potential energy surface accompanied by quantum transitions between the quantum states. Numerical tests of the mixed quantum-classical method are in excellent agreement with the accurate full-quantum results for the model problems. The method is contrasted with the related surface-hopping approach. It is shown that computing the dynamics of a distribution of classical particles is more consistent and more accurate than computing the motion of independent point particles as with surface hopping.

The Journal of Chemical Physics•Apr 8, 2000•104 citations

Stabilised Coupled Trajectory Mixed Quantum Classical Algorithm with Improved Energy Conservation: CTMQC-EDI

Aaron Dines, Matthew Ellis, Jochen Blumberger

Coupled trajectory mixed quantum classical (CTMQC) dynamics is a rigorous approach to trajectory-based non-adiabatic dynamics, which has recently seen an improvement to energy conservation via the introduction of the CTMQC-E algorithm. Despite this, the method's two key quantities distinguishing it from Ehrenfest dynamics, the modified Born-Oppenheimer momentum and the quantum momentum, require regularisation procedures in certain circumstances. Such procedures in the latter can cause instabilities leading to undesirable effects such as energy drift and spurious population transfer, which is expected to become increasingly prevalent the larger the system as such events would happen more frequently. We propose a further modification to CTMQC-E which includes a redefinition of the quantum momentum, CTMQC-EDI (Double Intercept), such that it has no formal divergences. We then show for Tully models I-IV that the algorithm has greatly improved total energy conservation and negligible spurious population transfer at all times, in particular in regions of strong non-adiabatic coupling. CTMQC-EDI therefore shows promise as a numerically robust non-adiabatic dynamics technique that accounts for decoherence from first principles and that is scalable to large molecular systems and materials.

arXiv:2310.06915•Oct 10, 2023•0 citations

Quantum entanglement from classical trajectories

Johan E. Runeson, Jeremy O. Richardson

A long-standing challenge in mixed quantum-classical trajectory simulations is the treatment of entanglement between the classical and quantal degrees of freedom. We present a novel approach which describes the emergence of entangled states entirely in terms of independent and deterministic Ehrenfest-like classical trajectories. For a two-level quantum system in a classical environment, this is derived by mapping the quantum system onto a path-integral representation of a spin-1/2. We demonstrate that the method correctly accounts for coherence and decoherence and thus reproduces the splitting of a wavepacket in a nonadiabatic scattering problem. This discovery opens up a new class of simulations as an alternative to stochastic surface-hopping, coupled-trajectory or semiclassical approaches.

•May 5, 2021•0 citations

Decoherence and Its Role in Electronically Nonadiabatic Dynamics

Yinan Shu, Donald G. Truhlar

Decoherence is the tendency of a time-evolved reduced density matrix for a subsystem to assume a form corresponding to a statistical ensemble of states rather than a coherent combination of pure-state wave functions. When a molecular process involves changes in the electronic state and the coordinates of the nuclei, as in ultraviolet or visible light photochemistry or electronically inelastic collisions, the reduced density matrix of the electronic subsystem suffers decoherence, due to its interaction with the nuclear subsystem. We present the background necessary to conceptualize this decoherence; in particular, we discuss the density matrix description of pure states and mixed states, and we discuss pointer states and decoherence time. We then discuss how decoherence is treated in the coherent switching with decay of mixing algorithm and the trajectory surface hopping method for semiclassical calculations of electronically nonadiabatic processes.

Journal of Chemical Theory and Computation•Jan 9, 2023•54 citations
We analyze a mixed quantum-classical algorithm recently derived from the exact factorization equations [Min, Agostini, Gross, PRL 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 nonadiabatic terms in these equations, including how they are computed in practice. 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 nonadiabatic Tully model systems to illustrate our analysis.
We analyze a mixed quantum-classical algorithm recently derived from the exact factorization equations [Min, Agostini, Gross, PRL 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 nonadiabatic terms in these equations, including how they are computed in practice. 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 nonadiabatic Tully model systems to illustrate our analysis.
We present a novel quantum-classical approach to nonadiabatic dynamics, deduced from the coupled electronic and nuclear equations in the framework of the exact factorization of the electron-nuclear wave function. The method is based on the quasiclassical interpretation of the nuclear wave function, whose phase is related to the classical momentum and whose density is represented in terms of classical trajectories. In this approximation, electronic decoherence is naturally induced as an effect of the coupling to the nuclei and correctly reproduces the expected quantum behavior. Moreover, the splitting of the nuclear wave packet is captured as a consequence of the correct approximation of the time-dependent potential of the theory. This new approach offers a clear improvement over Ehrenfest-like dynamics. The theoretical derivation presented in this Letter is supported by numerical results that are compared to quantum mechanical calculations.
The coupled Schrödinger equations that describe nonadiabatic dynamics are recast using the Bohm formulation of quantum mechanics. The resulting coupled Bohm equations are solved numerically for two scattering models, giving results that are essentially identical to wave-packet solution of the original coupled Schrödinger equations. The classical limit of the set of coupled Bohm equations is then described, producing a mixed quantum-classical theory incorporating classical-like motion on each potential energy surface accompanied by quantum transitions between the quantum states. Numerical tests of the mixed quantum-classical method are in excellent agreement with the accurate full-quantum results for the model problems. The method is contrasted with the related surface-hopping approach. It is shown that computing the dynamics of a distribution of classical particles is more consistent and more accurate than computing the motion of independent point particles as with surface hopping.
Coupled trajectory mixed quantum classical (CTMQC) dynamics is a rigorous approach to trajectory-based non-adiabatic dynamics, which has recently seen an improvement to energy conservation via the introduction of the CTMQC-E algorithm. Despite this, the method's two key quantities distinguishing it from Ehrenfest dynamics, the modified Born-Oppenheimer momentum and the quantum momentum, require regularisation procedures in certain circumstances. Such procedures in the latter can cause instabilities leading to undesirable effects such as energy drift and spurious population transfer, which is expected to become increasingly prevalent the larger the system as such events would happen more frequently. We propose a further modification to CTMQC-E which includes a redefinition of the quantum momentum, CTMQC-EDI (Double Intercept), such that it has no formal divergences. We then show for Tully models I-IV that the algorithm has greatly improved total energy conservation and negligible spurious population transfer at all times, in particular in regions of strong non-adiabatic coupling. CTMQC-EDI therefore shows promise as a numerically robust non-adiabatic dynamics technique that accounts for decoherence from first principles and that is scalable to large molecular systems and materials.
A long-standing challenge in mixed quantum-classical trajectory simulations is the treatment of entanglement between the classical and quantal degrees of freedom. We present a novel approach which describes the emergence of entangled states entirely in terms of independent and deterministic Ehrenfest-like classical trajectories. For a two-level quantum system in a classical environment, this is derived by mapping the quantum system onto a path-integral representation of a spin-1/2. We demonstrate that the method correctly accounts for coherence and decoherence and thus reproduces the splitting of a wavepacket in a nonadiabatic scattering problem. This discovery opens up a new class of simulations as an alternative to stochastic surface-hopping, coupled-trajectory or semiclassical approaches.
Decoherence is the tendency of a time-evolved reduced density matrix for a subsystem to assume a form corresponding to a statistical ensemble of states rather than a coherent combination of pure-state wave functions. When a molecular process involves changes in the electronic state and the coordinates of the nuclei, as in ultraviolet or visible light photochemistry or electronically inelastic collisions, the reduced density matrix of the electronic subsystem suffers decoherence, due to its interaction with the nuclear subsystem. We present the background necessary to conceptualize this decoherence; in particular, we discuss the density matrix description of pure states and mixed states, and we discuss pointer states and decoherence time. We then discuss how decoherence is treated in the coherent switching with decay of mixing algorithm and the trajectory surface hopping method for semiclassical calculations of electronically nonadiabatic processes.