Rare events: classical and quantum. Roberto Car, Princeton University. Croucher ASI, Hong Kong, Dec. 9 2005. Reaction Pathways. FPMD simulations are currently limited to time scales of tens of ps
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Roberto Car, Princeton University
Croucher ASI, Hong Kong, Dec. 9 2005
FPMD simulations are currently limited to time scales of tens of ps
Most chemical reactions are activated processes that occur on longer time scales and are not accessible in direct FPMD simulations (and would not be accessible even in classical MD simulations).
Identifying reaction pathways is central to the study of chemical reactions. The string method for reaction pathways (W. E et al (PRB 66 (2002))) can be easily combined with FPMD
A Minimum Energy Path
connecting two end points
A longitudinal constraint, requiring only uniform stretching, is imposed by Lagrange multipliers:
This is easily solved by Damped Molecular Dynamics using the SHAKE procedure for the Lagrange multipliers
In this way an initial trial path is locally optimized to get a MEP
This is closely related to the NEB method by H. Jonsson and co.: the latter can be seen as a string method in which a constraint is imposed by a penalty function (rather than a Lagrange multiplier)
Y. Kanai, A. Tilocca, A. Selloni and R.C., JPC (2004)
Acetylene interacting with a partially hydrogenated Si(111) surface: reaction pathways from string damped molecular dynamics
A surface chain reaction
Takeuchi, Kanai, Selloni JACS (2004)
Reaction Energy (eV)
► DFT-GGA underestimates the
barriers for these reactions 3,4.
► Barriers as well as reaction energies improve using meta-GGA (TPSS).
► There are, however, situations where neither B3LYP nor TPPS work well (e.g. a proton transfer reaction in a H-bond)
Reaction Barriers (eV)
Activation energies an reaction pathways identified by the string method provide the input data for kinetic Monte Carlo simulations (kMC). This multi-scale approach allows us to model long-time micro-structural evolution (i.e. processes that occur on time scales of minutes or even hours and are completely outside the realm of MD simulations.
kinetic Monte Carlo grained dynamics by kMC
Continuous atomic dynamics is replaced by a Markov process consisting of a succession of hops with rates ri, which must be known in advance
oxygen grained dynamics by kMC
yttrium or zirconium
Example: Oxygen Diffusion in YSZ
a = 5.629 Å
Kinetic Monte Carlo Simulation grained dynamics by kMC
Oxygen vacancy in
Activation Energy grained dynamics by kMC
(Oishi and Ando, 1985)
Y2O3 (mole %)
Calculated Results: Oxygen Diffusivity
Cope Rearrangement the end point of a reaction?
cope rearrangement of 1,5-hexadiene the end point of a reaction?
We are interested in the steady state current. The relaxation time to achieve stationary conditions is large compared to the timescales of both electron dynamics and lattice dynamics. This makes a kinetic approach possible.
is a classical probability distribution
When the dimensions of a device are comparable to the electron wavelength, the semi-classical Boltzmann equation should be replaced by a quantum-mechanical Liouville-Master equation for the reduced density operator describing a quantum system coupled to a heat bath
A scheme introduced by R. Gebauer and RC allows to deal with an electron flux in a close circuit. (PRL 2004, PRB2004)
Kinetic approach: master equation
The single-particle Kohn-Sham approach is generalized to dissipative quantum system (Burke, Gebauer, RC, PRL 2005)
x-gauge an electron flux in a close circuit. (
The v-gauge corresponds to a ring geometry in which an electric current is induced by a magnetic flux
The electrons are subject to a steady electromotive force: coupling to a heat bath prevent them from accelerating indefinitely
The Liouville-Master equation an electron flux in a close circuit. (
The collision term gives a Fermi-Dirac distribution to the electrons in absence of applied electromotive force
In the numerical implementation the electric field is systematically “gauged” away to avoid indefinite “growth” of the Hamiltonian with time
Benzene dithiol between gold electrodes an electron flux in a close circuit. (
Atomic point contact (Gold on gold)
Gebauer, Piccinin, RC ChemPhysChem 2005
Quantitatively similar results to S. Ke, H.U. Baranger, W. Yang, JACS (2004)
Steady state electron current flux through an atomic point contact (S. Piccinin, R. Gebauer, R.C., to be published)
Quantum tunneling through a molecular contact contact (S. Piccinin, R. Gebauer, R.C., to be published)
A self-consistent tight binding calculation
I-V contact (S. Piccinin, R. Gebauer, R.C., to be published)characteristics: CNT on gold
Tight-binding calculations using self-consistent master equation, including nanotube, contacts and gold electrodes
Experiment: from Tao, Kane, and Dekker PRL 84, 2941 (2000)