Rare events classical and quantum Roberto Car Princeton

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 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) 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

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 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)

Influence of xc functional : PBE (GGA) vs. TPSS 2 (meta-GGA) Reaction Energy (e. V) ► DFT-GGA underestimates the barriers for these reactions 3, 4. H 2+Si(100) H 2 + Si 2 H 4 PBE 1. 94 2. 11 TPSS 2. 18 2. 37 __ 2. 25 2. 40 ± 0. 15 _ B 3 LYP ► Barriers as well as reaction energies improve using meta. GGA (TPSS). QMC Reaction Barriers (e. V) ► There are, however, situations where neither B 3 LYP nor TPPS work well (e. g. a proton transfer reaction in a H-bond) H 2+Si(100) H 2+Si 2 H 4 Intra Inter 1, 2 1, 1 PBE 0. 40 0. 24 0. 98 0. 50 TPSS 0. 56 0. 40 1. 27 0. 67 _ _ 1. 31 0. 60 _ _ B 3 LYP QMC 0. 66 ± 0. 15 0. 54 ± 0. 09

Long time evolution due to activated processes: coarse grained dynamics by k. MC Activation energies an reaction pathways identified by the string method provide the input data for kinetic Monte Carlo simulations (k. MC). 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 Continuous atomic dynamics is replaced by a Markov process consisting of a succession of hops with rates ri, which must be known in advance

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 oxygen yttrium or zirconium

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 Y 2 O 3 concentration • Oxygen vacancies hop to new sites using rates determined from first-principles calculations – (repeat 109 times) • A periodic cell with 1, 000 oxygen ions and 500, 000 cations is employed • Repeat over a range of Y and oxygen vacancy concentrations Oxygen vacancy in Cation lattice

Simulation Experiment (Oishi and Ando, 1985) Conductivity (W-1 cm-1) Calculated Results: Oxygen Diffusivity Activation Energy Y 2 O 3 (mole %) (Strickler and Carlson, 1964)

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

Cope Rearrangement ? ? ? 1, 5 -hexadiene

cope rearrangement of 1, 5 -hexadiene

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 Steady State: is a classical probability distribution

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

A scheme introduced by R. Gebauer and RC allows to deal with an electron flux in a close circuit. (PRL 2004, PRB 2004) Kinetic approach: master equation The single-particle Kohn-Sham approach is generalized to dissipative quantum system (Burke, Gebauer, RC, PRL 2005)

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

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

Benzene dithiol between gold electrodes Atomic point contact (Gold on gold)

Results for an applied bias of 1 e. V Gebauer, Piccinin, RC Chem. Phys. Chem 2005

I-V characteristics 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 Landauer formula

Carbon nanotube suspended between two gold electrodes A self-consistent tight binding calculation

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)