Rare events classical and quantum
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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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Rare events classical and quantum

Rare events: classical and quantum

Roberto Car, Princeton University

Croucher ASI, Hong Kong, Dec. 9 2005


Reaction pathways

Reaction Pathways

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


String method at t 0

String Method (at T=0)

A Minimum Energy Path

connecting two end points

satisfies

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


Rare events classical and quantum

Damped Molecular Dynamics of a String

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)


First principles string molecular dynamics

First Principles String Molecular Dynamics

Y. Kanai, A. Tilocca, A. Selloni and R.C., JPC (2004)


Rare events classical and quantum

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)


Influence of xc functional pbe gga vs tpss 2 meta gga

Influence of xc functional : PBE (GGA) vs. TPSS2 (meta-GGA)

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)

H2+Si(100)

H2+Si2H4


Long time evolution due to activated processes coarse grained dynamics by kmc

Long time evolution due to activated processes: coarse grained dynamics by kMC

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.


Rare events classical and quantum

kinetic Monte Carlo

Continuous atomic dynamics is replaced by a Markov process consisting of a succession of hops with rates ri, which must be known in advance


Rare events classical and quantum

oxygen

yttrium or zirconium

Example: Oxygen Diffusion in YSZ

  • YSZ has a fluorite structure with oxygen in tetrahedral sites

  • Oxygen diffuses primarily in <100> directions across <110> cation edges

  • Molecular Dynamics (MD) and Monte Carlo simulations suggest that the cations on the <110> edge determine the oxygen ion diffusion barrier

  • Oxygen diffusivity determined by set of <110> edges traversed (can be Zr-Zr, Zr-Y, Y-Y)

a = 5.629 Å


Rare events classical and quantum

Kinetic Monte Carlo Simulation

  • Random (frozen) fcc cation lattice with Y and Zr according to bulk concentration

  • Oxygen ions and vacancies distributed on tetrahedral sites according to Y2O3 concentration

  • Oxygen vacancies hop to new sites using rates determined from first-principles calculations – (repeat 109 times)

  • A periodic cell with 1,000,000 oxygen ions and 500,000 cationsis employed

  • Repeat over a range of Y and oxygen vacancy concentrations

Oxygen vacancy in

Cation lattice


Rare events classical and quantum

Activation Energy

Conductivity (W-1cm-1)

Simulation

Experiment

(Oishi and Ando, 1985)

Y2O3 (mole %)

(Strickler and

Carlson, 1964)

Calculated Results: Oxygen Diffusivity


What can we do if we only know the starting point but not the end point of a reaction

What can we do if we only know the starting point but not the end point of a reaction?

  • Metadynamics (Laio and Parrinello (2002)) gives a viable strategy, provided we know the important reaction coordinates (collective variables)

  • In this approach the microscopic dynamics is biased by a coarse grained (in the space of the order parameters) history dependent dynamics


Rare events classical and quantum

Cope Rearrangement

?

?

?

1,5-hexadiene


Rare events classical and quantum

cope rearrangement of 1,5-hexadiene


Modeling quantum systems in non equilibrium situations molecular electronics

Modeling quantum systems in non-equilibrium situations: Molecular Electronics:

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.


Boltzmann s equation the standard approach for bulk transport includes kinetics and dissipation

Boltzmann’s equation, the standard approach for bulk transport, includes kinetics and dissipation

Steady State:

is a classical probability distribution


Quantum formulation

Quantum formulation

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

Steady State


Rare events classical and quantum

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)


Rare events classical and quantum

x-gauge

v-gauge

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


Rare events classical and quantum

The Liouville-Master equation

Here:

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


Rare events classical and quantum

Benzene dithiol between gold electrodes

Atomic point contact (Gold on gold)


Results for an applied bias of 1ev

Results for an applied bias of 1eV

Gebauer, Piccinin, RC ChemPhysChem 2005


I v characteristics

I-V characteristics

Quantitatively similar results to S. Ke, H.U. Baranger, W. Yang, JACS (2004)


Rare events classical and quantum

Steady state electron current flux through an atomic point contact (S. Piccinin, R. Gebauer, R.C., to be published)


Rare events classical and quantum

Quantum tunneling through a molecular contact

Landauer formula


Carbon nanotube suspended between two gold electrodes

Carbon nanotube suspended between two gold electrodes

A self-consistent tight binding calculation


Rare events classical and quantum

I-V 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)


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