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