Rare events: classical and quantum

Croucher ASI, Hong Kong, Dec. 9 2005

Roberto Car, Princeton University

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 [ ] 0V

A longitudinal constraint, requiring only uniform stretching, is imposed by Lagrange multipliers:

ˆ( ( )) ( ) ( ) 0V

This is easily solved by Damped Molecular Dynamics using the SHAKE procedure for the Lagrange multipliers

ˆ( ; ) ( ( ; )) ( ; ) ( ; ) ( ; )t V t t t t

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)

Damped Molecular Dynamics of a String

First Principles String Molecular Dynamics

,ˆ( ) ( )

,

( )

I I II

n e n nm mmn

E RMR MR

R

E R

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

Takeuchi, Kanai, Selloni JACS (2004)A surface chain reaction

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

► 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)

Intra Inter 1,2 1,1

PBE 0.40 0.24 0.98 0.50

TPSS 0.56 0.40 1.27 0.67

B3LYP _ _ 1.31 0.60

QMC 0.66

± 0.15

0.54

± 0.09

_ _

H2+Si(100) H2 + Si2H 4

PBE 1.94 2.11

TPSS 2.18 2.37

B3LYP __ 2.25

QMC 2.40 ± 0.15 _

Reaction Energy (eV)

Reaction Barriers (eV)

H2+Si(100) H2+Si2H4

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.

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

exp( / )i i Br E k T

ii

r

1, 1 1,1

j jj i j i

r r

2ln

t

1 2 and are random numbers (0,1)

2ln 1t

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 Åoxygen

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 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 cations is employed

• Repeat over a range of Y and oxygen vacancy concentrations

Oxygen vacancy in Cation lattice

Calculated Results: Oxygen Diffusivity

Co

nd

uct

ivit

y (

-1cm

-1)

(Strickler and Carlson, 1964)

Y2O3 (mole %)

Activation Energy

(Oishi and Ando, 1985)

Simulation

Experiment

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

field collisions

df f f

dt t t

Steady State:

field collisions

f f

t t

( , ; )f f x p t 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

f S

,dS

i H S Sdt

C

,i H S S=CSteady State

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

,dS

i H S Sdt C

The single-particle Kohn-Sham approach is generalized to dissipative quantum system (Burke, Gebauer, RC, PRL 2005)

, E x E

,A

c t

E A c t E

The v-gauge corresponds to a ring geometry in which an electric current is induced by a magnetic flux

x-gauge

v-gauge

The electrons are subject to a steady electromotive force: coupling to a heat bath prevent them from accelerating indefinitely

( )

( ) ( ) ( )( )

, , , , ,

, , , , , , , , ,

( ) ( )

+ 1

n m n p p m n p p mp

n m n m n p m p p p n m p n p m p pp p

S i H t S S H t

S S S S

E E

d

=- -

- G +G - G +G -

å

å å

&

The Liouville-Master equation

Here: 2

0( ) ( ) [ ]2 HXC

p tH t U x V n

E E

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 1eV

Gebauer, Piccinin, RC ChemPhysChem 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

dIG

dV

2

ii

eG T

h

Landauer formula

Carbon nanotube suspended between two gold electrodes

A self-consistent tight binding calculation

I-V characteristics: CNT on gold

Experiment: from Tao, Kane, and Dekker PRL 84, 2941 (2000)

Tight-binding calculations using self-consistent master equation, including nanotube, contacts and gold electrodes