Quantum evolution speed and its limit in the driven double-well system

We investigate quantum evolution speed in the driven double-well system using the entangled trajectory molecular dynamics method. We emphasize not only the evolution speed of the quantum state but also its limit according to different definitions. The Wasserstein 1-distance is used to quantify the distance between distinguishable quantum states in the phase space, the quantum speed limit based on the geometry has been shown to be the strictest one. The single trajectory's contribution to the quantum speed limit is discussed, which is related to both the time evolution of the trajectory and its position in the total Wigner function. The resonance and chaos strongly enhance the evolution speed and its limit in the driven double-well system. The resonance effect makes a large proportion of representative points pass through the well as a whole, nevertheless, the chaos makes the Wigner function disperse in the phase space.     

Vibrational spectrum of a 1D oscillator: The quantum,the Wigner,and the classical ways

Infrared spectra have been used in many chemical applications, and theoretical calculations have been useful for analyzing these experimental results. While quantum mechanics is used for calculating the spectra for small molecules, classical mechanics is used for larger systems. However, a systematic understanding of the similarities and differences between the two approaches is not clear. Previous studies focused on peak position and relative intensities of the spectra obtained by various quantum and classical methods, but here, we included “absolute” intensities in the evaluation. The infrared spectrum of a one-dimensional (1D) harmonic oscillator (HO) and Morse oscillator were examined using four treatments: quantum, Wigner, truncated Wigner, and classical microcanonical treatments. For a 1D HO with a linear dipole moment function (DMF), the quantum and Wigner treatments give nearly the same spectra. On the other hand, the truncated Wigner underestimates the fundamental transition's intensity by half. In the case of cubic DMF, the truncated Wigner and classical methods fail to reproduce the relative intensity between the fundamental and second overtone transitions. Unfortunately, all the Wigner and classical methods fail to agree with the quantum results for a Morse oscillator with just 1% anharmonicity.     

A new quasiclassical method for modeling the high-resolution spectra of polyatomic systems

A new quasiclassical method for quantum autocorrelation functions based on the semiclassical limit in Wigner phase space has been derived. Unlike the existing quasiclassical method, the new method enables long-time simulations, thus making it possible to locate quantum spectral lines very precisely. The new method has been tested for a one-dimensional anharmonic oscillator fitted to the H(2) molecule and for a six-dimensional calculation of the Ar(2)I van der Waals cluster in adiabatic approximation. The obtained results compare well with the benchmark quantum-mechanical calculations and are also roughly comparable to the experimental Ar(2)I(-) zero-kinetic-energy photoelectron spectrum, which is available in the literature.     

Wigner function approach to the quantum Brownian motion of a particle in a potential

Recent progress in our understanding of quantum effects on the Brownian motion in an external potential is reviewed. This problem is ubiquitous in physics and chemistry, particularly in the context of decay of metastable states, for example, the reversal of the magnetization of a single domain ferromagnetic particle, kinetics of a superconducting tunnelling junction, etc. Emphasis is laid on the establishment of master equations describing the diffusion process in phase space analogous to the classical Fokker-Planck equation. In particular, it is shown how Wigner's [E. P. Wigner, Phys. Rev., 1932, 40, 749] method of obtaining quantum corrections to the classical equilibrium Maxwell-Boltzmann distribution may be extended to the dissipative non-equilibrium dynamics governing the quantum Brownian motion in an external potential V(x), yielding a master equation for the Wigner distribution function W(x,p,t) in phase space (x,p). The explicit form of the master equation so obtained contains quantum correction terms up to o(h(4)) and in the classical limit, h --> 0, reduces to the classical Klein-Kramers equation. For a quantum oscillator, the method yields an evolution equation coinciding in all respects with that of Agarwal [G. S. Agarwal, Phys. Rev. A, 1971, 4, 739]. In the high dissipation limit, the master equation reduces to a semi-classical Smoluchowski equation describing non-inertial quantum diffusion in configuration space. The Wigner function formulation of quantum Brownian motion is further illustrated by finding quantum corrections to the Kramers escape rate, which, in appropriate limits, reduce to those yielded via quantum generalizations of reaction rate theory.     

Vibrational energy relaxation rates via the linearized semiclassical approximation: applications to neat diatomic liquids and atomic-diatomic liquid mixtures

We report the results obtained from the application of our previously proposed linearized semiclassical method for computing vibrational energy relaxation (VER) rates (J. Phys. Chem. A 2003, 107, 9059, 9070) to neat liquid oxygen, neat liquid nitrogen, and liquid mixtures of oxygen and argon. Our calculations are based on a semiclassical approximation for the quantum-mechanical force-force correlation function, which puts it in terms of the Wigner transforms of the force and the product of the Boltzmann operator and the force. The calculation of the multidimensional Wigner integrals is made feasible by the introduction of a local harmonic approximation. A systematic analysis has been performed of the temperature and mole-fraction dependences of the VER rate constant, as well as the relative contributions of centrifugal and potential forces, and of different types of quantum effects. The results were found to be in very good quantitative agreement with experiment, and they suggest that this semiclassical approximation can capture the quantum enhancement, by many orders of magnitude, of the experimentally observed VER rate constants over the corresponding classical predictions.     

Quantum ergodicity and the wigner distribution

A quantum ergodic analysis is developed based on comparison of Wigner phase-space densities of quantum eigenstates to classical densities. The technique is applied to the two-dimensional Henon-Heiles system. Analysis of quantum densities as a function of energy is found to clearly display the regular to ergodic transition.     

Husimi representation for stationary states

This paper considers a Husimi representation of quantum mechanics in which the (stationary) state of a system or ensemble is described by a Husimi function and an observable is described by a phase space function or distribution such that the expectation value of the observable is given by an integral over phase space of the product of that function or distribution and the Husimi function. The density matrix, Wigner function, and Husimi function are considered to be alternative ways of describing the state of a system or ensemble, and methods of recovering the Wigner function or density matrix from the Husimi function are discussed. The classical limits of the Wigner and Husimi functions and of the relationship between them are considered. © 1993 John Wiley & Sons, Inc.     

Photofragment angular momentum distribution beyond the axial recoil approximation: predissociation

We present the quantum mechanical expressions for the angular momentum distribution of the photofragments produced in slow predissociation. The paper is based on our recent theoretical treatment [J. Chem. Phys. 123, 034307 (2005)] of the recoil angle dependence of the photofragment multipole moments which explicitly treat the role of molecular axis rotation on the electronic angular momentum polarization of the fragments. The electronic wave function of the molecule was used in the adiabatic body frame representation. The rigorous expressions for the fragment state multipoles which have been explicitly derived from the scattering wave function formalism have been used for the case of slow predissociation where a molecule lives in the excited quasibound state much longer than a rotation period. Possible radial nonadiabatic interactions were taken into consideration. The optical excitation of a single rotational branch and the broadband incoherent excitation of all possible rotational branches have been analyzed in detail. The angular momentum polarization of the photofragments has been treated in the high-J limit. The polarization of the photofragment angular momenta predicted by the theory depends on photodissociation mechanism and can in many cases be significant.     

Chemical reaction rates using the semiclassical Van Vleck initial value representation

A semiclassical initial value representation formulation using the Van Vleck [Proc. Natl. Acad. Sci. U.S.A. 14, 178 (1928)] propagator has been used to calculate the flux correlation function and thereby reaction rate constants. This Van Vleck formulation of the flux-flux correlation function is computationally as simple as the classical Wigner [Trans. Faraday Soc. 34, 29 (1938)] model. However, unlike the latter, it has the ability to capture quantum interference/coherence effects. Classical trajectories are evolved starting from the dividing surface that separates reactants and products, and are evolved negatively in time. This formulation has been tested on model problems ranging from the Eckart barrier, double well to the collinear H+H2.     

A linearized path integral description of the collision process between a water molecule and a graphite surface

Quantum effects in the scattering and desorption process of a water molecule from a graphite surface are investigated using the linearized path integral model. The graphite surface is quantized rigorously using the fully quantum many-body Wigner transform of the surface Boltzmann operator, while the water molecule is treated as rigid. Classical dynamics with these quantized initial conditions show that quantizing the surface at 100 and 300 K results in markedly different results, compared to a fully classical analysis. The trapping probability (defined as the probability of multiple encounters with the surface) is not sensitive to the choice of dynamical treatment, but the residence time on the surface is much shorter in the quantum case. At 300 K the transiently trapped molecules desorb from the surface with a rate constant which is 60-70% larger than the corresponding classical value. Lowering the surface temperature to 100 K decreases the quantum rate constant by approximately a factor of 3 while all trapped molecules stick to the surface in the classical case. The stability of the quantum initial state for the highly anisotropic graphite crystal is discussed in detail as well as the dynamical consequences of energy redistribution during the scattering process. The graphite surface application demonstrates that the Boltzmann operator Wigner transform for a system with 900 degrees of freedom can be obtained by the so-called gradient implementation [Poulsen et al. J. Chem. Theory Comput. 2006, 2, 1482] of the underlying Feynman-Kleinert effective frequency theory, an implementation that only requires a force and potential routine for the system at hand, and hence is applicable to any molecule-surface collision problem.     

An approach for generating trajectory-based dynamics which conserves the canonical distribution in the phase space formulation of quantum mechanics. II. Thermal correlation functions

We show the exact expression of the quantum mechanical time correlation function in the phase space formulation of quantum mechanics. The trajectory-based dynamics that conserves the quantum canonical distribution-equilibrium Liouville dynamics (ELD) proposed in Paper I is then used to approximately evaluate the exact expression. It gives exact thermal correlation functions (of even nonlinear operators, i.e., nonlinear functions of position or momentum operators) in the classical, high temperature, and harmonic limits. Various methods have been presented for the implementation of ELD. Numerical tests of the ELD approach in the Wigner or Husimi phase space have been made for a harmonic oscillator and two strongly anharmonic model problems, for each potential autocorrelation functions of both linear and nonlinear operators have been calculated. It suggests ELD can be a potentially useful approach for describing quantum effects for complex systems in condense phase.     

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.     

The eigenvalue problem in phase space

We formulate the standard quantum mechanical eigenvalue problem in quantum phase space. The equation obtained involves the c‐function that corresponds to the quantum operator. We use the Wigner distribution for the phase space function. We argue that the phase space eigenvalue equation obtained has, in addition to the proper solutions, improper solutions. That is, solutions for which no wave function exists which could generate the distribution. We discuss the conditions for ascertaining whether a position momentum function is a proper phase space distribution. We call these conditions psi‐representability conditions, and show that if these conditions are imposed, one extracts the correct phase space eigenfunctions. We also derive the phase space eigenvalue equation for arbitrary phase space distributions functions. © 2017 Wiley Periodicals, Inc.     

Solution of the master equation for Wigner's quasiprobability distribution in phase space for the Brownian motion of a particle in a double well potential

Quantum effects in the Brownian motion of a particle in the symmetric double well potential V(x)=ax(2)2+bx(4)4 are treated using the semiclassical master equation for the time evolution of the Wigner distribution function W(x,p,t) in phase space (x,p). The equilibrium position autocorrelation function, dynamic susceptibility, and escape rate are evaluated via matrix continued fractions in the manner customarily used for the classical Fokker-Planck equation. The escape rate so yielded has a quantum correction depending strongly on the barrier height and is compared with that given analytically by the quantum mechanical reaction rate solution of the Kramers turnover problem. The matrix continued fraction solution substantially agrees with the analytic solution. Moreover, the low-frequency part of the spectrum associated with noise assisted Kramers transitions across the potential barrier may be accurately described by a single Lorentzian with characteristic frequency given by the quantum mechanical reaction rate.     

Entangled trajectory molecular dynamics in multidimensional systems: two-dimensional quantum tunneling through the Eckart barrier

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.     

Real time correlation function in a single phase space integral beyond the linearized semiclassical initial value representation

It is shown how quantum mechanical time correlation functions [defined, e.g., in Eq. (1.1)] can be expressed, without approximation, in the same form as the linearized approximation of the semiclassical initial value representation (LSC-IVR), or classical Wigner model, for the correlation function [cf. Eq. (2.1)], i.e., as a phase space average (over initial conditions for trajectories) of the Wigner functions corresponding to the two operators. The difference is that the trajectories involved in the LSC-IVR evolve classically, i.e., according to the classical equations of motion, while in the exact theory they evolve according to generalized equations of motion that are derived here. Approximations to the exact equations of motion are then introduced to achieve practical methods that are applicable to complex (i.e., large) molecular systems. Four such methods are proposed in the paper--the full Wigner dynamics (full WD) and the second order WD based on "Wigner trajectories" [H. W. Lee and M. D. Scully, J. Chem. Phys. 77, 4604 (1982)] and the full Donoso-Martens dynamics (full DMD) and the second order DMD based on "Donoso-Martens trajectories" [A. Donoso and C. C. Martens, Phys. Rev. Lett. 8722, 223202 (2001)]--all of which can be viewed as generalizations of the original LSC-IVR method. Numerical tests of the four versions of this new approach are made for two anharmonic model problems, and for each the momentum autocorrelation function (i.e., operators linear in coordinate or momentum operators) and the force autocorrelation function (nonlinear operators) have been calculated. These four new approximate treatments are indeed seen to be significant improvements to the original LSC-IVR approximation.     

Dominant channels of vibronic transitions in molecules with several identical modes

Weak-coupling radiationless transitions (internal conversion or inter system crossing) are studied assuming separability and symmetry over N identical modes. Franck–Condon factors control the branching ratios between exciting just one of the equivalent modes, or equally distributing the available energy. The dominant process can be predicted by an exact quantum mechanical solution if the wavefunctions are known (Gaussian initial distributions and accepting Morse or Poeschl-Teller oscillators, for example); or more generally by a Wigner phase space surface-jumping analysis based on a classical limit of the Wigner function, using only the donor distribution and the acceptor potential surface.     

Solution of quantum Langevin equation: approximations, theoretical and numerical aspects

Based on a coherent state representation of noise operator and an ensemble averaging procedure using Wigner canonical thermal distribution for harmonic oscillators, a generalized quantum Langevin equation has been recently developed [Phys. Rev. E 65, 021109 (2002); 66, 051106 (2002)] to derive the equations of motion for probability distribution functions in c-number phase-space. We extend the treatment to explore several systematic approximation schemes for the solutions of the Langevin equation for nonlinear potentials for a wide range of noise correlation, strength and temperature down to the vacuum limit. The method is exemplified by an analytic application to harmonic oscillator for arbitrary memory kernel and with the help of a numerical calculation of barrier crossing, in a cubic potential to demonstrate the quantum Kramers' turnover and the quantum Arrhenius plot.