Theoretical Chemistry Accounts

, Volume 110, Issue 2, pp 49–58 | Cite as

Simulating quantum dynamics in classical environments

  • Alessandro Sergi
  • Donal Mac Kernan
  • Giovanni Ciccotti
  • Raymond KapralEmail author


Methods 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.


Nonadiabatic dynamics Quantum–classical dynamics Surface hopping Open quantum systems 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



This work was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada. Acknowledgement is made to the donors of The Petroleum Research Fund, administered by the ACS, for partial support

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Alessandro Sergi
    • 1
  • Donal Mac Kernan
    • 2
  • Giovanni Ciccotti
    • 3
  • Raymond Kapral
    • 1
    Email author
  1. 1.Chemical Physics Theory Group, Department of ChemistryUniversity of TorontoTorontoCanada
  2. 2.Department of Physics and Centre for Scientific ComputationTrinity CollegeIreland
  3. 3.INFM and Dipartimento di FisicaUniversità “La Sapienza”RomeItaly

Personalised recommendations