Few-body quantum and many-body classical hyperspherical approaches to reactions and to cluster dynamics

The hyperspherical method is a widely used and successful approach for the quantum treatment of elementary chemical processes. It has been mostly applied to three-atomic systems, and current progress is here outlined concerning the basic theoretical framework for the extension to four-body bound state and reactive scattering problems. Although most applications only exploit the advantages of the hyperspherical coordinate systems for the formulation of the few-body problem, the full power of the technique implies representations explicitly involving quantum hyperangular momentum operators as dynamical quantities and hyperspherical harmonics as basis functions. In terms of discrete analogues of these harmonics one has a universal representation for the kinetic energy and a diagonal representation for the potential (hyperquantization algorithm). Very recently, advances have been made on the use of the approach in classical dynamics, provided that a hyperspherical formulation is given based on “classical” definitions of the hyperangular momenta and related quantities. The aim of the present paper is to offer a retrospective and prospective view of the hyperspherical methods both in quantum and classical dynamics. Specifically, regarding the general quantum hyperspherical approaches for three- and four-body systems, we first focus on the basis set issue, and then we present developments on the classical formulation that has led to applications involving the implementations of hyperspherical techniques for classical molecular dynamics simulations of simple nanoaggregates.     

Hyperspherical coordinates in quantum mechanical collinear reactive scattering

A new hyperspherical coordinate method for performing atom—diatom quantum mechanical collinear reactive scattering calculations is described. The method is applicable at energies for which breakup channels are open. Comparison with previous results and new results at high energies for H H₂ are given. The usefulness of this approach is discussed.     

A simple classical prediction of quantal resonances in collinear reactive scattering

Quantal collinear reactive scattering computations have shown that in the vicinity of thresholds of reactant or product vibrational states, one finds resonances in the state to state reaction probability. We find that these resonances can be explained classically in terms of energy transfer between adiabatic reactant and product channels. This transfer is attributable to resonant periodic orbits, resonating between reactants and products. The classical condition for a quantal resonance is that the action of the orbit over one period be an integer (in units of h) and that the energy at which this occurs be lower than the adiabatic barrier heights of the resonating states. These conditions suffice for a prediction of the location of the quantal resonance within a 1% accuracy!     

Simulating quantum dynamics in classical environments

Methods for simulating the dynamics of composite systems, where part of the system is treated quantum mechanically and its environment is treated classically, are discussed. Such quantum–classical systems arise in many physical contexts where certain degrees of freedom have an essential quantum character while the other degrees of freedom to which they are coupled may be treated classically to a good approximation. The dynamics of these composite systems are governed by a quantum–classical Liouville equation for either the density matrix or the dynamical variables which are operators in the Hilbert space of the quantum subsystem and functions of the classical phase space variables of the classical environment. Solutions of the evolution equations may be formulated in terms of surface-hopping dynamics involving ensembles of trajectory segments interspersed with quantum transitions. The surface-hopping schemes incorporate quantum coherence and account for energy exchanges between the quantum and classical degrees of freedom. Various simulation algorithms are discussed and illustrated with calculations on simple spin-boson models but the methods described here are applicable to realistic many-body environments.     

Competition between dissociation and exchange processes in a collinear A + BC collision: Comparison of quantum and classical results

Previous calculations on a reactive/dissociative H + HD model system have been extended to higher collision energies. Exact quantum dissociation probabilities are now available in the 1–10 eV total energy range for initial vibrational quantum numbers v = 0,1 and 2. Comparison with quasiclassical results shows the absence of quantum tails at dissociation threshold, but large quantum effects at higher energies.     

氟原子与氢分子共线反应几率的量子散射计算

基于最新的6SEC势能面,用邓从豪等提出的LCAC-SW方法计算得到了共线反应F+H~2(v=0)→HF(v')+H的态-态反应几率,计算结果准确地反映出势能面的特点,进一步证明LCAC-SW方法是一成功的量子散射方法。     

Extended wave packet dynamics; exact solution for collinear atom,diatomic molecule scattering

The semiclassical wave packet dynamics method of Heller is extended to provide a formally exact theory of quantum mechanical motion for multidimensional anharmonic systems by introducing a complete, orthonormal, time-dependent basis of generalized oscillator functions. The exact wavefunction is expressed in terms of this basis and the expansions are shown to develop according to linear, coupled first-order differential equations. Application to collinear inelastic atom-diatomic molecule scattering demonstrates the feasibility and convergence of the new method.     

The direct histogram method for quasiclassical collision dynamics: application to collinear atom-diatom scattering

A method is presented for directly propagating ensembles of collision trajectories, such the histogram-type final state distributions are obtained. A substantial savings over standard quasiclassical methods is possible since only a few trajectory ensembles need to be propagated.     

Comparisons of classical chemical dynamics simulations of the unimolecular decomposition of classical and quantum microcanonical ensembles

Previous studies have shown that classical trajectory simulations often give accurate results for short-time intramolecular and unimolecular dynamics, particularly for initial non-random energy distributions. To obtain such agreement between experiment and simulation, the appropriate distributions must be sampled to choose initial coordinates and momenta for the ensemble of trajectories. If a molecule's classical phase space is sampled randomly, its initial decomposition will give the classical anharmonic microcanonical (RRKM) unimolecular rate constant for its decomposition. For the work presented here, classical trajectory simulations of the unimolecular decomposition of quantum and classical microcanonical ensembles, at the same fixed total energy, are compared. In contrast to the classical microcanonical ensemble, the quantum microcanonical ensemble does not sample the phase space randomly. The simulations were performed for CH(4), C(2)H(5), and Cl(-)---CH(3)Br using both analytic potential energy surfaces and direct dynamics methods. Previous studies identified intrinsic RRKM dynamics for CH(4) and C(2)H(5), but intrinsic non-RRKM dynamics for Cl(-)---CH(3)Br. Rate constants calculated from trajectories obtained by the time propagation of the classical and quantum microcanonical ensembles are compared with the corresponding harmonic RRKM estimates to obtain anharmonic corrections to the RRKM rate constants. The relevance and accuracy of the classical trajectory simulation of the quantum microcanonical ensemble, for obtaining the quantum anharmonic RRKM rate constant, is discussed.     

Thermodynamics and equilibrium structure of Ne38 cluster: quantum mechanics versus classical

The equilibrium properties of classical Lennard-Jones (LJ38) versus quantum Ne38 Lennard-Jones clusters are investigated. The quantum simulations use both the path-integral Monte Carlo (PIMC) and the recently developed variational-Gaussian wave packet Monte Carlo (VGW-MC) methods. The PIMC and the classical MC simulations are implemented in the parallel tempering framework. The classical heat capacity Cv(T) curve agrees well with that of Neirotti et al. [J. Chem. Phys. 112, 10340 (2000)], although a much larger confining sphere is used in the present work. The classical Cv(T) shows a peak at about 6 K, interpreted as a solid-liquid transition, and a shoulder at approximately 4 K, attributed to a solid-solid transition involving structures from the global octahedral (Oh) minimum and the main icosahedral (C5v) minimum. The VGW method is used to locate and characterize the low energy states of Ne38, which are then further refined by PIMC calculations. Unlike the classical case, the ground state of Ne38 is a liquidlike structure. Among the several liquidlike states with energies below the two symmetric states (Oh and C5v), the lowest two exhibit strong delocalization over basins associated with at least two classical local minima. Because the symmetric structures do not play an essential role in the thermodynamics of Ne38, the quantum heat capacity is a featureless curve indicative of the absence of any structural transformations. Good agreement between the two methods, VGW and PIMC, is obtained. The present results are also consistent with the predictions by Calvo et al. [J. Chem. Phys. 114, 7312 (2001)] based on the quantum superposition method within the harmonic approximation. However, because of its approximate nature, the latter method leads to an incorrect assignment of the Ne38 ground state as well as to a significant underestimation of the heat capacity.     

Finite element interpolation for combined classical/quantum mechanical molecular dynamics simulations

A method is presented to interpolate the potential energy function for a part of a system consisting of a few degrees of freedom, such as a molecule in solution. The method is based on a modified finite element (FE) interpolation scheme. The aim is to save computer time when expensive methods such as quantum-chemical calculations are used to determine the potential energy function. The expensive calculations are only carried out if the molecule explores new unknown regions of the conformation space. If the molecule resides in regions previously explored, a cheap interpolation is performed instead of an expensive calculation, using known neighboring points. We report the interpolation techniques for the energies and the forces of the molecule, the handling of the FE mesh, and an application to a simple test example in molecular dynamics (MD) simulations. Good performance of the method was obtained (especially for MD simulations with a preceding Monte Carlo mesh generation) without losing accuracy. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 : 1484–1495, 1997     

Computer simulation of quantum dynamics in a classical spin environment

In this paper, a formalism for studying the dynamics of quantum systems coupled to classical spin environments is reviewed. The theory is based on generalized antisymmetric brackets and naturally predicts open-path off-diagonal geometric phases in the evolution of the density matrix. It is shown that such geometric phases must also be considered in the quantum–classical Liouville equation for a classical bath with canonical phase space coordinates; this occurs whenever the adiabatics basis is complex (as in the case of a magnetic field coupled to the quantum subsystem). When the quantum subsystem is weakly coupled to the spin environment, non-adiabatic transitions can be neglected and one can construct an effective non-Markovian computer simulation scheme for open quantum system dynamics in classical spin environments. In order to tackle this case, integration algorithms based on the symmetric Trotter factorization of the classical-like spin propagator are derived. Such algorithms are applied to a model comprising a quantum two-level system coupled to a single classical spin in an external magnetic field. Starting from an excited state, the population difference and the coherences of this two-state model are simulated in time while the dynamics of the classical spin is monitored in detail. It is the author’s opinion that the numerical evidence provided in this paper is a first step toward developing the simulation of quantum dynamics in classical spin environments into an effective tool. In turn, the ability to simulate such a dynamics can have a positive impact on various fields, among which, for example, nanoscience.     

A classical analysis of quantum resonances in isotopic collinear H + H2 reactions

Isotopic substitution of hydrogen by muonium in the collinear H + H₂ reaction causes dramatic changes in the resonance patterns of the quantum reaction probabilities. These changes are explained by a classical model. Using the systems' resonant orbits, we show that resonances appear close to energetic thresholds of new vibrational channels.     

Ultrafast excited state dynamics of the Na3F cluster: quantum wave packet and classical trajectory calculations compared to experimental results

Short-time, excited-state dynamics of the lowest isomer of the Na(3)F cluster is studied theoretically in order to interpret the features of recent time-resolved pump-probe ionization experiments [J. M. L'Hermite, V. Blanchet, A. Le Padellec, B. Lamory, and P. Labastie, Eur. Phys. J. D 28, 361 (2004)]. In the present paper, we propose an identification of the vibrational motion responsible for the oscillations in the ion signal, on the basis of quantum mechanical wave packet propagations and classical trajectory calculations. The good agreement between experiment and theory allows for a clear interpretation of the detected dynamics.     

A comparison of semiclassical wavepacket,exact quantum and classical trajectory calculations for diffractive atom/surface scattering

Comparison of a time-dependent wavepacket approach with classical trajectory and exact coupled channel calculations is presented for He/LiF(001) diffractive scattering. The range of validity of the wavepacket method is examined in detail and attention is paid to the dependence of the results on the width of the packets. At higher energies, where the wavepacket method will be almost exact, a comparison is made with the results of classical trajectories. It is shown that, aside from a correct incorporation of quantum effects, the wavepacket approach can have a considerable computational advantage over a pure classical calculation.     

The classical Wigner method with an effective quantum force: application to the collinear H + H2 reaction

To improve the classical Wigner (CW) model, we recently proposed the classical Wigner model with an effective quantum force (CWEQF). The results of the CWEQF model are more accurate than those of the CW model. Still the simplicity of the CW model is retained. The quantum force was obtained by defining a characteristic distance η(0) between two Feynman paths that enter the expression for the flux-flux correlation function. η(0) was considered independent of the position along the reaction path. The CWEQF leads to a lowering of the effective potential barrier. Here we develop the method to use position dependent η(0) values. The method is also generalized to two dimensions. Applications are carried out on one-dimensional model problems and the two-dimensional H + H(2) collinear reaction.     

Time-dependent quantum dynamics of reactive scattering and the calculation of product quantum state distributions — A study of the collinear F+H2(v=0) → HF(v′)+H reaction

Summary A new method for the calculation of partial cross sections in the time-dependent quantum theory of molecular reactive scattering processes is discussed. Preliminary calculations are presented which clearly illustrate the power of the method. They show how all the partial cross sections associated with a single initial quantum state may be computed over a very wide energy range from a single propagation of a prepared wavepacket. The resonance behaviour is obtained without difficulty and the energies of the reactive scattering resonances are exactly reproduced.     

Hybrid quantum/classical molecular dynamics for a proton transfer reaction coupled to a dissipative bath

A hybrid quantum/classical molecular dynamics approach is applied to a proton transfer reaction represented by a symmetric double well system coupled to a dissipative bath. In this approach, the proton is treated quantum mechanically and all bath modes are treated classically. The transition state theory rate constant is obtained from the potential of mean force, which is generated along a collective reaction coordinate with umbrella sampling techniques. The transmission coefficient, which accounts for dynamical recrossings of the dividing surface, is calculated with a reactive flux approach combined with the molecular dynamics with quantum transitions surface hopping method. The hybrid quantum/classical results agree well with numerically exact results in the spatial-diffusion-controlled regime, which is most relevant for proton transfer in proteins. This hybrid quantum/classical approach has already been shown to be computationally practical for studying proton transfer in large biological systems. These results have important implications for future applications to hydrogen transfer reactions in solution and proteins.