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Two-dimensional materials provide a rich platform demonstrating quantum effects, and the process of electron-hole recombination occurring in them has significant applications in the fields of the photocatalytic and optoelectronic community. Here, we present nonadiabatic coupling-induced quantum coherence and quantum beats in Al-doped blue phosphorene. The work improves our understanding and utilization of nonadiabatic coupling in low-dimensional materials from a new perspective. In addition, our investigations provide meaningful guidance for manipulating quantum coherence in low-dimensional materials and promoting their novel optoelectronic properties.

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The fewest switches surface hopping (FSSH) method proposed by Tully in 1990 [Tully, J. Chem. Phys. 93, 1061 (1990)]-along with its many later variations-forms the basis for most practical simulations of molecular dynamics with electronic transitions in realistic systems. Despite its popularity, a rigorous formal derivation of the algorithm has yet to be achieved. In this paper, we derive the energy-conserving momentum jumps employed by FSSH from the perspective of quantum trajectory surface hopping (QTSH) [Martens, J. Phys. Chem. A 123, 1110 (2019)]. In the limit of localized nonadiabatic transitions, simple mathematical and physical arguments allow the FSSH algorithm to be derived from first principles. For general processes, the quantum forces characterizing the QTSH method provide accurate results for nonadiabatic dynamics with rigorous energy conservation, at the ensemble level, within the consistency of the underlying stochastic surface hopping without resorting to the artificial momentum rescaling of FSSH.

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In this paper, we investigate the time evolution of quantum coherence-the off-diagonal elements of the density matrix of a multistate quantum system-from the perspective of the Wigner-Moyal formalism. This approach provides an exact phase space representation of quantum mechanics. We consider the coherent evolution of nuclear wavepackets in a molecule with two electronic states. For harmonic potentials, the problem is analytically soluble for both a fully quantum mechanical description and a semiclassical description. We highlight the serious deficiencies of the semiclassical treatment of coherence for general systems and illustrate how even qualitative accuracy requires higher order terms in the Moyal expansion to be included. The model provides an experimentally relevant example of a molecular Schrödinger's cat state. The alive and dead cats of the exact two-state quantum evolution collapse into a "zombie" cat in the semiclassical limit-an averaged behavior, neither alive nor dead, leading to significant errors. The inclusion of the Moyal correction restores a faithful simultaneously alive and dead representation of the cat that is experimentally observable.

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In this paper, we analyze the detailed quantum-classical behavior of two alternative approaches to simulating molecular dynamics with electronic transitions: the popular fewest switches surface hopping (FSSH) method, introduced by Tully in 1990 [Tully, J. Chem. Phys., 1990, 93, 1061] and our recently developed quantum trajectory surface hopping (QTSH) method [Martens, J. Phys. Chem. A, 2019, 123, 1110]. Both approaches employ an independent ensemble of trajectories that undergo stochastic transitions between electronic surfaces. The methods differ in their treatment of energy conservation, with FSSH imposing conservation of the classical kinetic plus potential energy by rescaling the classical momentum when a surface hop occurs while QTSH incorporates a quantum force throughout the dynamics which leads naturally to the conservation of the full quantum-classical energy. We investigate the population transfer and energy budget of the surface hopping methods for several simple model systems and compare with exact quantum results. In addition, the detailed dynamics of the trajectory ensembles in phase space are compared with the quantum evolution in the Wigner representation. Conclusions are drawn.

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We describe a new method for simulating nonadiabatic dynamics using stochastic trajectories. The method, which we call quantum trajectory surface hopping (QTSH), is a variant of the popular fewest-switches surface-hopping (FSSH) approach, but with important differences. We briefly review and significantly extend our recently described consensus surface-hopping (CSH) formalism, which captures quantum effects such as coherence and decoherence via a collective representation of the quantum dynamics at the ensemble level. Using well-controlled further approximations, we derive an independent trajectory limit of CSH that recovers the FSSH stochastic algorithm but rejects the ad hoc momentum rescaling of FSSH in favor of quantum forces that couple classical and quantum degrees of freedom and lead to nonclassical trajectory dynamics. The approach is well-defined in both the diabatic and adiabatic representations. In the adiabatic representation, the classical dynamics are modified by a quantum-state-dependent vector potential, introducing geometric phase effects into the dynamics of multidimensional systems. Unlike FSSH, our method obeys energy conservation without any artificial momentum rescaling, eliminating undesirable features of the former such as forbidden hops and breakdown of the internal consistency of quantum and ensemble-based state probabilities. Corrections emerge naturally in the formalism that allow approximate incorporation of decoherence without the computational expense of the full CSH approach. The method is tested on several model systems. QTSH provides a surface-hopping methodology that has a rigorous foundation and broader applicability than FSSH while retaining the low computational cost of an independent trajectory framework.

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We present a new stochastic surface hopping method for modeling molecular dynamics with electronic transitions. The approach, consensus surface hopping (CSH), is a numerical framework for solving the semiclassical limit Liouville equation describing nuclear dynamics on coupled electronic surfaces using ensembles of trajectories. In contrast to existing techniques based on propagating independent classical trajectories that undergo stochastic hops between the electronic states, the present method determines the probabilities of transition of each trajectory collectively with input from the entire ensemble. The full coherent dynamics of the coupled system arise naturally at the ensemble level and ad hoc corrections, such as momentum rescaling to impose strict trajectory energy conservation and artificial decoherence to avoid the overcoherence of the quantum states associated with independent trajectories, are avoided.

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In this paper, we describe a new and fully coherent stochastic surface hopping method for simulating mixed quantum-classical systems. We illustrate the approach on the simple but unforgiving problem of quantum evolution of a two-state quantum system in the limit of unperturbed pure state dynamics and for dissipative evolution in the presence of both stationary and nonstationary random environments. We formulate our approach in the Liouville representation and describe the density matrix elements by ensembles of trajectories. Population dynamics are represented by stochastic surface hops for trajectories representing diagonal density matrix elements. These are combined with an unconventional coherent stochastic hopping algorithm for trajectories representing off-diagonal quantum coherences. The latter generalizes the binary (0,1) "probability" of a trajectory to be associated with a given state to allow integers that can be negative or greater than unity in magnitude. Unlike existing surface hopping methods, the dynamics of the ensembles are fully entangled, correctly capturing the coherent and nonlocal structure of quantum mechanics.

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In this Letter, we describe theoretical modeling of an experimentally realized nanoscale system that exhibits the general universal behavior of a nonlinear dynamical system. In particular, we consider the description of voltage-induced current fluctuations through a single nanopore from the perspective of nonlinear dynamics. We briefly review the experimental system and its behavior observed and then present a simple phenomenological nonlinear model that reproduces the qualitative behavior of the experimental data. The model consists of a two-dimensional deterministic nonlinear bistable oscillator experiencing both dissipation and random noise. The multidimensionality of the model and the interplay between deterministic and stochastic forces are both required to obtain a qualitatively accurate description of the physical system.

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In this paper, we investigate coherent quantum dynamics in a nonequilibrium environment. We focus on a two-state quantum system strongly coupled to a single classical environmental oscillator, and explore the effect of nonstationary statistical properties of the oscillator on the quantum evolution. A simple nonequilibrium model, consisting of an oscillator with a well-defined initial phase which undergoes subsequent diffusion, is introduced and studied. Approximate but accurate analytic expressions for the evolution of the off-diagonal density matrix element of the quantum system are derived in the second-order cumulant approximation. The effect of the initial phase choice on the subsequent quantum evolution is quantified. It is observed that the initial phase can have a significant effect on the preservation of coherence on short time scales, suggesting this variable as a control parameter for optimizing coherence in many-body quantum systems.

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We investigate H(2)O photodissociation in its first absorption band using entangled trajectory molecular dynamics method. We compare our results of entangled trajectories with exact quantum mechanical calculations, the overall agreement with the exact results is reasonable. To help understanding we show the photodissociation process with our entangled trajectories and the effect of the entangled trajectories in the system.

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In this paper, we extend the entangled trajectory molecular dynamics (ETMD) method to multidimensional systems. The integrodifferential form of the evolution equation for the Wigner function is employed, allowing general potentials not represented as a polynomial to be treated. As the example, the method is applied to a two-dimensional model of scattering from an Eckart barrier. The results of ETMD are in good agreement with quantum hydrodynamics and exact quantum simulations. By comparing the quantum and classical trajectory in phase space, the quantum tunneling phenomenon is interpreted vividly.

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We describe an analytically solvable model of quantum decoherence in a nonequilibrium environment. The model considers the effect of a bath driven from equilibrium by, for example, an ultrafast excitation of a quantum chromophore. The nonequilibrium response of the environment is represented by a nonstationary random function corresponding to the fluctuating transition frequency between two quantum states coupled to the surroundings. The nonstationary random function is characterized by a Fourier series with the phase of each term starting initially with a definite value across the ensemble but undergoing random diffusion with time. The decay of the off-diagonal density matrix element is shown to depend significantly on the particular pattern of initial phases of the terms in the Fourier series, or equivalently, the initial phases of bath modes coupled to the quantum subsystem. This suggests the possibility of control of quantum decoherence by the detailed properties of an environment that is driven from thermal equilibrium.

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This paper describes nonequilibrium molecular dynamics simulations of pressure induced transport of liquid water through model nanopores. We consider a simple model for a porous membrane consisting of a slab of water molecules held in a rigid ice structure and penetrated by a pore of nanometer scale dimensions. Both hydrophilic membranes composed of conventional TIP3P water and hydrophobic membranes consisting of modified water with the model partial charges set to zero are treated. Molecular dynamics simulation is employed to investigate the rate of water flow through the pore induced by a pressure difference across the membrane. The results are compared with the predictions of continuum hydrodynamics. We find that the flow rate of water through hydrophilic pores is much less than the continuum predictions, while the flux through hydrophobic pores can significantly exceed the continuum theory. Finally, we show asymmetric behavior in the flux vs. pressure difference for a conical nanopore, which thus acts as a Brownian ratchet.

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In this paper, we investigate the effects of boundary structure on the properties of water in nanometer scale environments. We use molecular dynamic simulations to study water enclosed in model nanocavities with rigid boundaries of ice I(h) structure and compare its behavior to that of water in cavities with smooth structureless boundaries. We show the dependence of quantities such as velocity autocorrelation function and hydrogen-bond lifetimes on the size and surface characteristics of the cavity. The boundary structure greatly influences the structure and dynamics of the water. In the smallest systems considered, with dimensions of 3-8 A, the dynamics are slowed significantly, and the velocity autocorrelation function resembles that of solid ice.

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In this paper, we develop the formalism of entangled trajectory molecular dynamics (ETMD), introduced by Donoso and Martens [Phys. Rev. Lett. 2001, 87, 223202] in a form applicable to the treatment of general (i.e., nonpolynomial) potentials. We formulate our approach directly in terms of the integrodifferential equation obeyed by the Wigner function, without assuming a Taylor series expansion of the potential in powers of the coordinate. This alternative formalism has distinct advantages for propagating distribution functions represented by finite trajectory ensembles and for nonpolynomial potentials. We use a numerical implementation of the new approach to calculate the reaction probabilities for three model systems: the cubic polynomial potential, symmetric Eckart barrier and asymmetric Eckart barrier. Our results are in excellent agreement with the results of exact quantum calculation.

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Nanoscale pores exhibit transport properties that are not seen in micrometre-scale pores, such as increased ionic concentrations inside the pore relative to the bulk solution, ionic selectivity and ionic rectification. These nanoscale effects are all caused by the presence of permanent surface charges on the walls of the pore. Here we report a new phenomenon in which the addition of small amounts of divalent cations to a buffered monovalent ionic solution results in an oscillating ionic current through a conical nanopore. This behaviour is caused by the transient formation and redissolution of nanoprecipitates, which temporarily block the ionic current through the pore. The frequency and character of ionic current instabilities are regulated by the potential across the membrane and the chemistry of the precipitate. We discuss how oscillating nanopores could be used as model systems for studying nonlinear electrochemical processes and the early stages of crystallization in sub-femtolitre volumes. Such nanopore systems might also form the basis for a stochastic sensor.

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We describe an independent trajectory implementation of semiclassical Liouville method for simulating quantum processes using classical trajectories. In this approach, a single ensemble of trajectories describes all semiclassical density matrix elements of a coupled electronic state problem, with the ensemble evolving classically under a single reference Hamiltonian chosen on the basis of physical grounds. In this paper, we introduce an additional uncoupled trajectory approximation, allowing the members of the ensemble to evolve independently of one another and eliminating the major computational costs of our previous coupled trajectory implementation. The accuracy of the method is demonstrated for model one-dimensional problems. In addition, the approach is applied to the chemical reaction dynamics of a collinear triatomic system, yielding excellent agreement with exact calculations. This method allows molecular dynamics involving coupled electronic surfaces to be modeled with essentially the same effort as classical molecular dynamics and ensemble averaging.

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We solve quantum dynamical equations of simple systems by propagating ensembles of interacting trajectories. A scheme is proposed which uses adaptive kernel density estimation for representing probability distribution functions and their derivatives. The formulation is carried on in the Husimi representation to ensure the positiveness of the distribution functions. By comparing to previous work, the effect of changing representations is studied as well as the advantage of using adaptive kernels for the estimation of probability distributions. We found significant improvement in the accuracy of the results.

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In this paper, we present simulations of the decay of quantum coherence between vibrational states of I(2) in its ground (X) electronic state embedded in a cryogenic Kr matrix. We employ a numerical method based on the semiclassical limit of the quantum Liouville equation, which allows the simulation of the evolution and decay of quantum vibrational coherence using classical trajectories and ensemble averaging. The vibrational level-dependent interaction of the I(2)(X) oscillator with the rare-gas environment is modeled using a recently developed method for constructing state-dependent many-body potentials for quantum vibrations in a many-body classical environment [J. M. Riga, E. Fredj, and C. C. Martens, J. Chem. Phys. 122, 174107 (2005)]. The vibrational dephasing rates gamma(0n) for coherences prepared between the ground vibrational state mid R:0 and excited vibrational state mid R:n are calculated as a function of n and lattice temperature T. Excellent agreement with recent experiments performed by Karavitis et al. [Phys. Chem. Chem. Phys. 7, 791 (2005)] is obtained.

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