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The Journal of Chemical Physics

Ehrenfest dynamics with spontaneous localization

A. A. Tomaz, Rafael Souza Mattos, Saikat Mukherjee, Mario Barbatti

Published Dec 4, 2025• 1 citation
Publisher

Abstract

We propose Ehrenfest Dynamics with Spontaneous Localization (SLED), a decoherence-corrected extension of Ehrenfest dynamics based on the Gisin-Percival quantum-state diffusion equation. In SLED, the electronic wave function evolves stochastically in the adiabatic energy basis, producing trajectory-level localization. The trajectory ensemble reproduces a Lindblad-type propagation of the reduced electronic density matrix. This approach ensures linearity, trace preservation, and complete positivity, providing a physically consistent alternative to ad hoc decoherence corrections commonly adopted in mixed quantum-classical methods. Benchmark simulations on one-dimensional Tully models and multidimensional spin-boson Hamiltonians demonstrate that SLED reproduces electronic populations and captures the essential features of coherence decay. The tests, however, also reveal that accurate treatment will require generalizing the localization kernel controlling the electron-nucleus coupling strength from a constant into a function of time and phase space coordinates. SLED is implemented in the newly developed Skitten program and will be integrated into Newton-X. While the present study serves as a proof of concept, SLED establishes a rigorous and extensible framework that bridges mixed quantum-classical dynamics with open quantum system theory.

DOI
10.1063/5.0297596

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We propose Ehrenfest Dynamics with Spontaneous Localization (SLED), a decoherence-corrected extension of Ehrenfest dynamics based on the Gisin-Percival quantum-state diffusion equation. In SLED, the electronic wave function evolves stochastically in the adiabatic energy basis, producing trajectory-level localization. The trajectory ensemble reproduces a Lindblad-type propagation of the reduced electronic density matrix. This approach ensures linearity, trace preservation, and complete positivity, providing a physically consistent alternative to ad hoc decoherence corrections commonly adopted in mixed quantum-classical methods. Benchmark simulations on one-dimensional Tully models and multidimensional spin-boson Hamiltonians demonstrate that SLED reproduces electronic populations and captures the essential features of coherence decay. The tests, however, also reveal that accurate treatment will require generalizing the localization kernel controlling the electron-nucleus coupling strength from a constant into a function of time and phase space coordinates. SLED is implemented in the newly developed Skitten program and will be integrated into Newton-X. While the present study serves as a proof of concept, SLED establishes a rigorous and extensible framework that bridges mixed quantum-classical dynamics with open quantum system theory.
We derive the L-mean-field Ehrenfest (MFE) method to incorporate Lindblad jump operator dynamics into the MFE approach. We map the density matrix evolution of Lindblad dynamics onto pure state coefficients using trajectory averages. We use simple assumptions to construct the L-MFE method that satisfies this exact mapping. This establishes a method that uses independent trajectories that exactly reproduce Lindblad decay dynamics using a wavefunction description, with deterministic changes of the magnitudes of the quantum expansion coefficients, while only adding on a stochastic phase. We further demonstrate that when including nuclei in the Ehrenfest dynamics, the L-MFE method gives semi-quantitatively accurate results, with the accuracy limited by the accuracy of the approximations present in the semiclassical MFE approach. This work provides a general framework to incorporate Lindblad dynamics into semiclassical or mixed quantum-classical simulations.
We present a semiclassical approach for nonadiabatic molecular dynamics based on the Ehrenfest method with corrections for decoherence and detailed balance. Decoherence is described via a coherence penalty functional that drives dynamics away from regions in Hilbert space characterized by large values of coherences. Detailed balance is incorporated by modification of the off-diagonal matrix elements with a quantum correction factor used in semiclassical approximations to quantum time-correlation functions. Both decoherence and detailed balance corrections introduce nonlinear terms to the Schrödinger equation. At the same time, the simplicity of fully deterministic dynamics and a single trajectory for each initial condition is preserved. In contrast, surface hopping is stochastic and requires averaging over multiple realization of the stochastic process for each initial condition. The Ehrenfest-decoherence-detailed-balance (Ehrenfest-DDB) method is adapted to the classical path approximation and ab initio time-dependent density functional theory and applied to an experimentally studied nanoscale system consisting of a fluorophore molecule and an scanning tunneling microscopy tip and undergoing current-induced charge injection, cooling, and recombination. Ehrenfest-DDB produces time scales that are similar to those obtained with decoherence induced surface hopping, which is a popular nonadiabatic molecular dynamics technique applied to condensed matter. At long times, Ehrenfest-DDB dynamics slows down considerably because the detailed balance correction makes off-diagonal elements go to zero on approach to Boltzmann equilibrium. The Ehrenfest-DDB technique provides efficient means to study quantum dynamics in large systems.
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.
Electronic energy flow in an isolated molecular system involves coupling between the electronic and nuclear subsystems, and the coupled system evolves to a statistical mixture of pure states. In semiclassical theories, nuclear motion is treated using classical mechanics, and electronic motion is treated as an open quantal system coupled to a "bath" of nuclear coordinates. We have previously shown how this can be simulated by a time-dependent Schrödinger equation with coherent switching and decay of mixing, where the decay of mixing terms model the dissipative effect of the environment on the electronic subdynamics (i.e., on the reduced dynamics of the electronic subsystem). In the present paper we reformulate the problem as a Liouville-von Neumann equation of motion (i.e., we propagate the reduced density matrix of the electronic subsystem), and we introduce the assumption of first-order linear decay. We specifically examine the cases of equal relaxation times for both longitudinal (i.e., population) decay and transverse decay (i.e., dephasing) and of longitudinal relaxation only, yielding the linear decay of mixing (LDM) and the population-driven decay of mixing (PDDM) schemes, respectively. Because we do not generally know the basis in which coherence decays, that is, the pointer basis, we judge the semiclassical methods in part by their ability to give good results in both the adiabatic and diabatic bases. The accuracy in the prediction of physical observables is shown to be robust not only with respect to basis but also with respect to the way in which demixing is incorporated into the master equation for the density matrix. The success of the PDDM scheme is particularly interesting because it incorporates the least amount of decoherence (i.e., the PDDM scheme is the most similar of the methods discussed to the fully coherent semiclassical Ehrenfest method). For both the new and previous decay of mixing schemes, four kinds of decoherent state switching algorithms are analyzed and compared to one another: natural switching (NS), self-consistent switching (SCS), coherent switching (CS), and globally coherent switching (GCS). The CS formulations are examples of a non-Markovian method, in which the system retains some memory of its history, whereas the GCS, SCS, and NS schemes are Markovian (time local). These methods are tested against accurate quantum mechanical results using 17 multidimensional atom-diatom test cases. The test cases include avoided crossings, conical interactions, and systems with noncrossing diabatic potential energy surfaces. The CS switching algorithm, in which the state populations are controlled by a coherent stochastic algorithm for each complete passage through a strong interaction region, but successive strong-interaction regions are not mutually coherent, is shown to be the most accurate of the switching algorithms tested for the LDM and PDDM methods as well as for the previous decay of mixing methods, which are reformulated here as Liouville-von Neumann equations with nonlinear decay of mixing (NLDM). We also demonstrate that one variant of the PDDM method with CS performs almost equally well in the adiabatic and diabatic representations, which is a difficult objective for semiclassical methods. Thus decay of mixing methods provides powerful mixed quantum-classical methods for modeling non-Born-Oppenheimer polyatomic dynamics including photochemistry, charge-transfer, and other electronically nonadiabatic processes.
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 that 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 wave packet 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.
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.