1,721,194 research outputs found
Mixed quantum-classical dynamics
No full textMixed quantum-classical equations of motion are derived for a quantum subsystem of light (mass m) particles coupled to a classical bath of massive (mass M) particles. The equation of motion follows from a partial Wigner transform over the bath degrees of freedom of the Liouville equation for the full quantum system, followed by an expansion in the small parameter mu(m/M)(1/2) in analogy with the theory of Brownian motion. The resulting mixed quantum-classical Liouville equation accounts for the coupled evolution of the subsystem and bath. The quantum subsystem is represented in an adiabatic (or other) basis and the series solution of the Liouville equation leads to a representation of the dynamics in an ensemble of surface-hopping trajectories. A generalized Pauli master equation for the evolution of the diagonal elements of the density matrix is derived by projection operator methods and its structure is analyzed in terms of surface-hopping trajectories. (C) 1999 American Institute of Physics. [S0021-9606(99)51117-3]
Linearization approximations and Liouville quantum-classical dynamics
No full textWe show that linearization methods, commonly used to approximate the evolution of the density operator in mixed quantum-classical systems, can be justified when a small parameter, the ratio of masses of the quantum subsystem and bath, is introduced. The same parameter enters in the derivation of the quantum-classical Liouville equation. Although its original derivation followed from a different formalism, here we show that the basis-free form of the quantum-classical Liouville equation for the density operator can also be obtained by linearization of the exact time evolution of this operator. These results show the equivalence among various quantum-classical schemes. (C) 2009 Elsevier B.V. All rights reserved
Analysis of the quantum-classical Liouville equation in the mapping basis
No full textThe quantum-classical Liouville equation provides a description of the dynamics of a quantum subsystem coupled to a classical environment. Representing this equation in the mapping basis leads to a continuous description of discrete quantum states of the subsystem and may provide an alternate route to the construction of simulation schemes. In the mapping basis the quantum-classical Liouville equation consists of a Poisson bracket contribution and a more complex term. By transforming the evolution equation, term-by-term, back to the subsystem basis, the complex term (excess coupling term) is identified as being due to a fraction of the back reaction of the quantum subsystem on its environment. A simple approximation to quantum-classical Liouville dynamics in the mapping basis is obtained by retaining only the Poisson bracket contribution. This approximate mapping form of the quantum-classical Liouville equation can be simulated easily by Newtonian trajectories. We provide an analysis of the effects of neglecting the presence of the excess coupling term on the expectation values of various types of observables. Calculations are carried out on nonadiabatic population and quantum coherence dynamics for curve crossing models. For these observables, the effects of the excess coupling term enter indirectly in the computation and good estimates are obtained with the simplified propagation. (C) 2010 American Institute of Physics. [doi:10.1063/1.3480018
Statistical mechanics of quantum-classical systems
No full textThe statistical mechanics of systems whose evolution is governed by mixed quantum-classical dynamics is investigated. The algebraic properties of the quantum-classical time evolution of operators and of the density matrix are examined and compared to those of full quantum mechanics. The equilibrium density matrix that appears in this formulation is stationary under the dynamics and a method for its calculation is presented. The response of a quantum-classical system to an external force which is applied from the distant past when the system is in equilibrium is determined. The structure of the resulting equilibrium time correlation function is examined and the quantum-classical limits of equivalent quantum time correlation functions are derived. The results provide a framework for the computation of equilibrium time correlation functions for mixed quantum-classical systems. (C) 2001 American Institute of Physics
Mixed quantum-classical surface hopping dynamics
No full textAn algorithm is presented for the exact solution of the evolution of the density matrix of a mixed quantum-classical system in terms of an ensemble of surface hopping trajectories. The system comprises a quantum subsystem coupled to a classical bath whose evolution is governed by a mixed quantum-classical Liouville equation. The integral solution of the evolution equation is formulated in terms of a concatenation of classical evolution segments for the bath phase space coordinates separated by operators that change the quantum state and bath momenta. A hybrid Molecular Dynamics-Monte Carlo scheme which follows a branching tree of trajectories arising from the action of momentum derivatives is constructed to solve the integral equation. We also consider a simpler scheme where changes in the bath momenta are approximated by momentum jumps. These schemes are illustrated by considering the computation of the evolution of the density matrix for a two-level system coupled to a low dimensional classical bath. (C) 2000 American Institute of Physics. [S0021-9606(00)50215-3]
Surface-hopping dynamics of a spin-boson system
No full textThe spin-boson model is solved within the framework of quantum-classical dynamics using our recently-developed surface-hopping scheme. The quantum-classical equation of motion is expressed in an adiabatic basis and its solution is constructed from an ensemble of trajectories which undergo nonadiabatic transitions and evolve coherently on the adiabatic surfaces. Details of the algorithm for the simulation of the dynamics are presented and the method of simple Monte Carlo sampling used to evaluate the expectation values of observables is discussed. The simulation method is applied to a spin-boson system with a harmonic bath composed of ten oscillators with an Ohmic spectral density. For the spin-boson model the present implementation of quantum-classical dynamics is exact and the results of our surface-hopping simulations are in accord with previous numerically exact results for this model. (C) 2002 American Institute of Physics
Activation free energy for proton transfer in solution
No full textPath integral molecular dynamics methods are employed to compute the free energy for proton transfer reactions for strongly hydrogen bonded systems in a polar solvent. The free energy profile is calculated using several different techniques, including: integration of the mean force acting on the proton path with its centroid constrained at different values, the integral form of the free energy calculation in the constrained-reaction-coordinate-dynamics ensemble and direct simulation of the unconstrained dynamics. The results show that estimates of the free energy barrier obtained by harmonic extrapolation are likely to be in error. Both quantum and classical results for the free energy are obtained and compared with simulations using adiabatic quantum dynamics. Comparison of the quantum and classical results show that there are quantum corrections to the solvent contributions to the free energy
Simulating quantum dynamics in classical environments
No full textMethods 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
Quantum effects on the solvent contribution to the activation free energy
No full textQuantum effects on the activation free energy for the transfer of a quantum particle between two ions immersed in a classical dipolar solvent are investigated. The bare potential for the quantum particle-ion complex has an intrinsic barrier that is negligible compared to k(B)T and corresponds to a strongly hydrogen bonded system if the quantum particle is a proton. In this circumstance the activation free energy is dominated by solvent contributions. We compare the solvent free energies along the reaction coordinate for a classical particle with charge +1, a proton and a muon. The quantum calculations are carried out using Feynman's path integral methods. The smaller mass of the muon gives rise to enhanced quantum effects and allows one to probe the effects of the dispersion of the charge density of the particle undergoing transfer on the solvent contribution to the free energy
Activation energies by molecular dynamics with constraints
No full textRecently Tobias and Broods introduced a method to compute by molecular dynamics with constraints the probability density P(≈r) = ; associated with rate values≈r of a spatial coordinate r. In this Letter we extend their approach to the case of a general reaction coordinate ξ(r), an arbitrary function of the configuration-space coordinates. The generalized version is shown to be the integral form of the free energy calculation in the constrained-reaction-coordinate ensemble where the mean force is computed as an average in a ξ-constrained ensemble. The two approaches are shown to be of equal computational efficiency for a very simple Lennard-Jones test case
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