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Journal of Chemical Physics / Volume 136 / Issue 8 / ARTICLES / Theoretical Methods and Algorithms

J. Chem. Phys. 136, 084101 (2012); http://dx.doi.org/10.1063/1.3685420 (14 pages)

Mapping quantum-classical Liouville equation: Projectors and trajectories

Aaron Kelly1,2, Ramses van Zon1,3, Jeremy Schofield1, and Raymond Kapral1

1Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada
2Department of Chemistry, Stanford University, 333 Campus Drive, Stanford, California 94305, USA
3SciNet HPC Consortium, University of Toronto, 256 McCaul St, Toronto, Ontario M5T 1W5, Canada

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(Received 14 December 2011; accepted 27 January 2012; published online 22 February 2012)

The evolution of a mixed quantum-classical system is expressed in the mapping formalism where discrete quantum states are mapped onto oscillator states, resulting in a phase space description of the quantum degrees of freedom. By defining projection operators onto the mapping states corresponding to the physical quantum states, it is shown that the mapping quantum-classical Liouville operator commutes with the projection operator so that the dynamics is confined to the physical space. It is also shown that a trajectory-based solution of this equation can be constructed that requires the simulation of an ensemble of entangled trajectories. An approximation to this evolution equation which retains only the Poisson bracket contribution to the evolution operator does admit a solution in an ensemble of independent trajectories but it is shown that this operator does not commute with the projection operators and the dynamics may take the system outside the physical space. The dynamical instabilities, utility, and domain of validity of this approximate dynamics are discussed. The effects are illustrated by simulations on several quantum systems.

© 2012 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. QUANTUM-CLASSICAL LIOUVILLE EQUATION: MAPPING, PROJECTORS, AND EXPECTATION VALUES
    1. Representation in mapping basis and projection operators
    2. Forms of operators in the mapping subspace
    3. Expectation values
    4. Equations of motion
  3. TRAJECTORY DESCRIPTION OF DYNAMICS
    1. Ensemble of entangled trajectories
    2. Ensemble of independent trajectories
  4. DYNAMICAL INSTABILITIES IN APPROXIMATE EVOLUTION EQUATIONS
    1. Simulations of the dynamics
      1. Curve crossing dynamics: Nuclear momentum distributions
      2. Conical intersection model
      3. Collinear reactive collision model
  5. SUMMARY AND COMMENTS

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KEYWORDS and PACS

Keywords

Liouville equation, oscillators, quantum entanglement

PACS

  • 03.65.Ta

    Foundations of quantum mechanics; measurement theory

  • 03.65.Ud

    Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)

  • 05.45.Xt

    Synchronization; coupled oscillators

  • 02.30.Rz

    Integral equations

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3685420

PUBLICATION DATA

ISSN

0021-9606 (print)  
1089-7690 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

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