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Title: Communication: Wigner functions in action-angle variables, Bohr-Sommerfeld quantization, the Heisenberg correspondence principle, and a symmetrical quasi-classical approach to the full electronic density matrix

Abstract

It is pointed out that the classical phase space distribution in action-angle (a-a) variables obtained from a Wigner function depends on how the calculation is carried out: if one computes the standard Wigner function in Cartesian variables (p, x), and then replaces p and x by their expressions in terms of a-a variables, one obtains a different result than if the Wigner function is computed directly in terms of the a-a variables. Furthermore, the latter procedure gives a result more consistent with classical and semiclassical theory - e.g., by incorporating the Bohr-Sommerfeld quantization condition (quantum states defined by integer values of the action variable) as well as the Heisenberg correspondence principle for matrix elements of an operator between such states - and has also been shown to be more accurate when applied to electronically non-adiabatic applications as implemented within the recently developed symmetrical quasi-classical (SQC) Meyer-Miller (MM) approach. Moreover, use of the Wigner function (obtained directly) in a-a variables shows how our standard SQC/MM approach can be used to obtain off-diagonal elements of the electronic density matrix by processing in a different way the same set of trajectories already used (in the SQC/MM methodology) to obtain the diagonal elements.

Authors:
 [1];  [2]
+ Show Author Affiliations
  1. Univ. of California, Berkeley, CA (United States). Dept. of Chemistry, Kenneth S. Pitzer Center for Theoretical Chemistry
  2. Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Chemcial Sciences Division
Publication Date:
Research Org.:
Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Basic Energy Sciences (BES)
OSTI Identifier:
1379579
Alternate Identifier(s):
OSTI ID: 1328640
Grant/Contract Number:  
AC02-05CH11231; CHE-1148645
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Chemical Physics
Additional Journal Information:
Journal Volume: 145; Journal Issue: 8; Journal ID: ISSN 0021-9606
Publisher:
American Institute of Physics (AIP)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

Citation Formats

Miller, William H., and Cotton, Stephen J. Communication: Wigner functions in action-angle variables, Bohr-Sommerfeld quantization, the Heisenberg correspondence principle, and a symmetrical quasi-classical approach to the full electronic density matrix. United States: N. p., 2016. Web. doi:10.1063/1.4961551.
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Miller, William H., & Cotton, Stephen J. Communication: Wigner functions in action-angle variables, Bohr-Sommerfeld quantization, the Heisenberg correspondence principle, and a symmetrical quasi-classical approach to the full electronic density matrix. United States. https://doi.org/10.1063/1.4961551
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Miller, William H., and Cotton, Stephen J. Sun . "Communication: Wigner functions in action-angle variables, Bohr-Sommerfeld quantization, the Heisenberg correspondence principle, and a symmetrical quasi-classical approach to the full electronic density matrix". United States. https://doi.org/10.1063/1.4961551. https://www.osti.gov/servlets/purl/1379579.
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@article{osti_1379579,
title = {Communication: Wigner functions in action-angle variables, Bohr-Sommerfeld quantization, the Heisenberg correspondence principle, and a symmetrical quasi-classical approach to the full electronic density matrix},
author = {Miller, William H. and Cotton, Stephen J.},
abstractNote = {It is pointed out that the classical phase space distribution in action-angle (a-a) variables obtained from a Wigner function depends on how the calculation is carried out: if one computes the standard Wigner function in Cartesian variables (p, x), and then replaces p and x by their expressions in terms of a-a variables, one obtains a different result than if the Wigner function is computed directly in terms of the a-a variables. Furthermore, the latter procedure gives a result more consistent with classical and semiclassical theory - e.g., by incorporating the Bohr-Sommerfeld quantization condition (quantum states defined by integer values of the action variable) as well as the Heisenberg correspondence principle for matrix elements of an operator between such states - and has also been shown to be more accurate when applied to electronically non-adiabatic applications as implemented within the recently developed symmetrical quasi-classical (SQC) Meyer-Miller (MM) approach. Moreover, use of the Wigner function (obtained directly) in a-a variables shows how our standard SQC/MM approach can be used to obtain off-diagonal elements of the electronic density matrix by processing in a different way the same set of trajectories already used (in the SQC/MM methodology) to obtain the diagonal elements.},
doi = {10.1063/1.4961551},
journal = {Journal of Chemical Physics},
number = 8,
volume = 145,
place = {United States},
year = {Sun Aug 28 00:00:00 EDT 2016},
month = {Sun Aug 28 00:00:00 EDT 2016}
}
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https://doi.org/10.1063/1.4961551
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Works referenced in this record:

The Symmetrical Quasi-Classical Model for Electronically Non-Adiabatic Processes Applied to Energy Transfer Dynamics in Site-Exciton Models of Light-Harvesting Complexes
journal, February 2016

  • Cotton, Stephen J.; Miller, William H.
  • Journal of Chemical Theory and Computation, Vol. 12, Issue 3
  • DOI: 10.1021/acs.jctc.5b01178

A classical analog for electronic degrees of freedom in nonadiabatic collision processes
journal, April 1979

  • Meyera), Hans‐Dieter; Miller, William H.
  • The Journal of Chemical Physics, Vol. 70, Issue 7
  • DOI: 10.1063/1.437910

A Symmetrical Quasi-Classical Spin-Mapping Model for the Electronic Degrees of Freedom in Non-Adiabatic Processes
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  • Cotton, Stephen J.; Miller, William H.
  • The Journal of Physical Chemistry A, Vol. 119, Issue 50
  • DOI: 10.1021/acs.jpca.5b05906

Symmetrical Windowing for Quantum States in Quasi-Classical Trajectory Simulations
journal, March 2013

  • Cotton, Stephen J.; Miller, William H.
  • The Journal of Physical Chemistry A, Vol. 117, Issue 32
  • DOI: 10.1021/jp401078u

�ber quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen.
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  • The Journal of Chemical Physics, Vol. 141, Issue 8
  • DOI: 10.1063/1.4893345

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journal, April 2015

  • Miller, William H.; Cotton, Stephen J.
  • The Journal of Chemical Physics, Vol. 142, Issue 13
  • DOI: 10.1063/1.4916945

Symmetrical windowing for quantum states in quasi-classical trajectory simulations: Application to electronically non-adiabatic processes
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  • Cotton, Stephen J.; Miller, William H.
  • The Journal of Chemical Physics, Vol. 139, Issue 23
  • DOI: 10.1063/1.4845235

Unified treatment of quantum coherent and incoherent hopping dynamics in electronic energy transfer: Reduced hierarchy equation approach
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  • Ishizaki, Akihito; Fleming, Graham R.
  • The Journal of Chemical Physics, Vol. 130, Issue 23
  • DOI: 10.1063/1.3155372
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    Works referencing / citing this record:

    Classical molecular dynamics simulation of electronically non-adiabatic processes
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    • DOI: 10.1039/c6fd00181e

    A new symmetrical quasi-classical model for electronically non-adiabatic processes: Application to the case of weak non-adiabatic coupling
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    • DOI: 10.1063/1.5031788

    The symmetrical quasi-classical approach to electronically nonadiabatic dynamics applied to ultrafast exciton migration processes in semiconducting polymers
    journal, July 2018

    • Liang, Ruibin; Cotton, Stephen J.; Binder, Robert
    • The Journal of Chemical Physics, Vol. 149, Issue 4
    • DOI: 10.1063/1.5037815

    Performance evaluation of the symmetrical quasi-classical dynamics method based on Meyer-Miller mapping Hamiltonian in the treatment of site-exciton models
    journal, November 2018

    • Xie, Yu; Zheng, Jie; Lan, Zhenggang
    • The Journal of Chemical Physics, Vol. 149, Issue 17
    • DOI: 10.1063/1.5047002

    A symmetrical quasi-classical windowing model for the molecular dynamics treatment of non-adiabatic processes involving many electronic states
    journal, March 2019

    • Cotton, Stephen J.; Miller, William H.
    • The Journal of Chemical Physics, Vol. 150, Issue 10
    • DOI: 10.1063/1.5087160

    Two-dimensional vibronic spectra from classical trajectories
    journal, April 2019

    • Polley, Kritanjan; Loring, Roger F.
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    Trajectory-adjusted electronic zero point energy in classical Meyer-Miller vibronic dynamics: Symmetrical quasiclassical application to photodissociation
    journal, May 2019

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