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Title: Semiclassical initial value series solution of the spin boson problem

Abstract

A numerical solution for the quantum dynamics of the spin boson problem is obtained using the semiclassical initial value series representation approach to the quantum dynamics. The zeroth order term of the series is computed using the new forward-backward representation for correlation functions presented in the preceding adjacent paper. This leads to a rapid convergence of the Monte Carlo sampling, as compared to previous attempts. The zeroth order results are already quite accurate. The first order term of the series is small, demonstrating the rapid convergence of the semiclassical initial value representation series. This is the first time that the first order term in the semiclassical initial value representation series has been converged for systems with the order of 50 degrees of freedom.

Authors:
;  [1]
  1. Chemical Physics Department, Weizmann Institute of Science, 76100 Rehovoth (Israel)
Publication Date:
OSTI Identifier:
20991254
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Chemical Physics; Journal Volume: 126; Journal Issue: 16; Other Information: DOI: 10.1063/1.2714520; (c) 2007 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BOSONS; CONVERGENCE; CORRELATION FUNCTIONS; DEGREES OF FREEDOM; MONTE CARLO METHOD; NUMERICAL SOLUTION; SEMICLASSICAL APPROXIMATION; SPIN

Citation Formats

Martin-Fierro, Eva, and Pollak, Eli. Semiclassical initial value series solution of the spin boson problem. United States: N. p., 2007. Web. doi:10.1063/1.2714520.
Martin-Fierro, Eva, & Pollak, Eli. Semiclassical initial value series solution of the spin boson problem. United States. doi:10.1063/1.2714520.
Martin-Fierro, Eva, and Pollak, Eli. Sat . "Semiclassical initial value series solution of the spin boson problem". United States. doi:10.1063/1.2714520.
@article{osti_20991254,
title = {Semiclassical initial value series solution of the spin boson problem},
author = {Martin-Fierro, Eva and Pollak, Eli},
abstractNote = {A numerical solution for the quantum dynamics of the spin boson problem is obtained using the semiclassical initial value series representation approach to the quantum dynamics. The zeroth order term of the series is computed using the new forward-backward representation for correlation functions presented in the preceding adjacent paper. This leads to a rapid convergence of the Monte Carlo sampling, as compared to previous attempts. The zeroth order results are already quite accurate. The first order term of the series is small, demonstrating the rapid convergence of the semiclassical initial value representation series. This is the first time that the first order term in the semiclassical initial value representation series has been converged for systems with the order of 50 degrees of freedom.},
doi = {10.1063/1.2714520},
journal = {Journal of Chemical Physics},
number = 16,
volume = 126,
place = {United States},
year = {Sat Apr 28 00:00:00 EDT 2007},
month = {Sat Apr 28 00:00:00 EDT 2007}
}
  • A recently formulated continuum limit semiclassical initial value series representation (SCIVR) of the quantum dynamics of dissipative systems is applied to the study of vibrational relaxation of model harmonic and anharmonic oscillator systems. As is well known, the classical dynamics of dissipative systems may be described in terms of a generalized Langevin equation. The continuum limit SCIVR uses the Langevin trajectories as input, albeit with a quantum noise rather than a classical noise. Combining this development with the forward-backward form of the prefactor-free propagator leads to a tractable scheme for computing quantum thermal correlation functions. Here we present the firstmore » implementation of this continuum limit SCIVR series method to study two model problems of vibrational relaxation. Simulations of the dissipative harmonic oscillator system over a wide range of parameters demonstrate that at most only the first two terms in the SCIVR series are needed for convergence of the correlation function. The methodology is then applied to the vibrational relaxation of a dissipative Morse oscillator. Here, too, the SCIVR series converges rapidly as the first two terms are sufficient to provide the quantum mechanical relaxation with an estimated accuracy on the order of a few percent. The results in this case are compared with computations obtained using the classical Wigner approximation for the relaxation dynamics.« less
  • No abstract prepared.
  • The linearized semiclassical initial value representation (LSC-IVR) [H. Wang, X. Sun and W. H. Miller, J. Chem. Phys. {bold 108}, 9726 (1998)] is used to study the nonadiabatic dynamics of the spin-boson problem, a system of two electronic states linearly coupled to an infinite bath of harmonic oscillators. The spectral density of the bath is chosen to be of the Debye form, which is often used to model the solution environment of a charge transfer reaction. The simulation provides a rather complete understanding of the electronically nonadiabatic dynamics in a broad parameter space, including coherent to incoherent transitions along allmore » three axes (the {ital T}-axis, the {eta}-axis, and the {omega}{sub c}-axis) in the complete phase diagram and the determination of rate constants in several physically interesting regimes. Approximate analytic theories are used to compare with the simulation results, and good agreement is found in the appropriate physical limits. {copyright} {ital 1999 American Institute of Physics.}« less
  • The uniqueness of a nonnegative solution of the second initial boundary-value problem for the equation.