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Scientific Report 2007-2009
Condensed matter physics and biophysics
C21. Mixed quantum-classical dynamics for condensed matter
simulations
The computational cost of quantum dynamic simulations
scales exponentially with the number of degrees of
freedom (DOF). This prevents a brute force solution of
the problem and fosters research to find alternative, viable
simulation methods. Large systems in which the
quantum nature of just some DOFs is relevant can be
studied with mixed quantum-classical methods. These
methods start from a full quantum description of all
DOFs and then partition them into two subsets: the
quantum subsystem and the bath. A classical limit for
the evolution of the bath alone is taken to substantially
reduce the cost of the calculation while preserving, approximately,
the quantum evolution of the subsystem.
Taking this limit is non-trivial and part of our research
explores using different approaches to analyze the formal
properties of mixed quantum classical schemes [1].
We also study a specific kind of mixed quantum-classical
problems: non-adiabatic dynamics. In non-adiabatic situations,
the coupling between nuclear (the bath) and
electronic (quantum subsystem) motions in a molecular
system, or the interactions with the environment, can induce
transitions among the eigenstates of the electronic
Hamiltonian. A Born-Oppenheimer description of the
nuclear dynamics is thus invalid and advanced simulation
methods are necessary. Non-adiabatic transitions
can affect the energy and charge distribution of a system,
change the products of a chemical reaction by opening up
different reaction channels, modify the relaxation path
and the final state of a molecule excited by light and
influence the time scale for its dissociation or recombination
in the presence of solvent. Non-adiabatic simulations
then open the possibility to control a wide range
of interesting processes by suggesting how to modify the
nature of the transitions, for example via coupling with
a controlled environment or an appropriate pattern of
excitations.
In collaboration with Ray Kapral (University of
Toronto) and David Coker (Boston University) our
group developed two approaches for simulating nonadiabatic
mixed quantum-classical dynamics: the
quantum-classical Liouville equation and the iterative
linearized density matrix propagation. Both methods
derive from well-defined approximations for propagating
the density operator; the first exploits the Wigner-
Liouville representation of quantum mechanics, the
second the path integral formalism by Feynman. The
approximation in both dynamics is controlled by the
mass ratio of quantum and classical DOFs, and the
coupling among the different dynamics arises naturally
from a Taylor series expansion of the propagator in
this parameter. The solution of the mixed-quantum
classical equations can be expressed in an iterative form
and solved by means of hybrid molecular dynamics
- Monte Carlo algorithms whose accuracy increases
with the order of the iteration. Tests on standard
Figure 1: Simulation of a photodissociation experiment for
a diatomic molecule [3]. Upper panel: schematic representation
of 3 molecular electronic states and of the couplings
among them plotted as a function of the internuclear distance.
Bottom panel: time evolution of the probability to
find the system on the different states (i.e. of the population
of the states) after photoexcitation of the molecule on
state 1 (solid line in the upper panel). The changes in the
populations reflect the non-adiabatic transitions among the
states.
benchmark models (such as the spin-boson system)
have proved that our methods are indeed capable
of describing non-adiabatic processes [2,3]. Current
research is focused on two technical fronts: improvements
the algorithmic properties of the methods and
further theoretical analysis to clarify their relationship
and relative accuracy [4]. Progress in these areas is
crucial for our goal of non-adiabatic applications to
systems as complex as those that we have studied in the
past with Born-Oppenheimer mixed quantum-classical
methods (e.g. the diffusion of an excess electron in a
metal-molten salt solution).
References
1. F. Agostini et al., Europhys. Letts. 78, 30001 (2007).
2. D. Mackernan et al., J. Chem. Phys. 112, 424 (2008).
3. E. Dunkel et al., J. Chem. Phys. 129, 1141106 (2008).
4. S. Bonella et al., in Energy Transfer Dynamics in Bio
material Systems, Eds. E. R. Bittner et al., Springer
(2009).
Authors
G. Ciccotti, S. Bonella, S. Caprara, F. Agostini
http://abaddon.phys.uniroma1.it/
Sapienza Università di Roma 74 Dipartimento di Fisica