Bridging scales:

Ab initio molecular dynamics and

kinetic Monte Carlo simulations

Karsten Reuter

Fritz-Haber-Institut, Berlin

Multiscale modeling

Ab initio atomistic thermodynamics and statistical mechanics of surface properties and functions

K. Reuter, C. Stampfl and M. Scheffler, in: Handbook of Materials Modeling, Part A. Methods,

(Ed.) Sidney Yip, Springer (Berlin, 2005).http://www.fhi-berlin.mpg.de/th/paper.html

I. Straightforward time-evolution:Ab initio molecular dynamics

Understanding Molecular Simulation,D. Frenkel and B. Smit, Academic Press (2002)

)())](()1

[()()(2)( 42 terrorttrVm

ttrtrttr

MD: Numerical integration of Newton’s equation

),,,( 21

..

Nrii rrrVrmi

e.g. Verlet algorithm

Alternative algorithms: - velocity Verlet- leapfrog- predictor-corrector

- Runge-Kutta…

Solid-state: Δt ~ 10-15 s

Keeping everything: expensive, limited time scale

Example:

O2 dissociation at Al(111)

Total time of trajectory: 0.5 psTime step: 2.5 fs (200 steps)

CPU cost: 45 days on 1 Compaq ES45 processor

J. Behler et al., Phys. Rev. Lett. 94, 036104 (2005)

II. First-principles kinetic Monte Carlo simulations:rare events and long time scales

IPAM tutorial by Kristen Fichthornhttp://www.ipam.ucla.edu/publications/matut/matut_5907.ppt

First-principles kinetic Monte Carlo simulations

A

BMolecularDynamics

ΔEA ΔEBrA→B

rB→A

kineticMonte CarloN

t

B

A

equilibriumMonte Carlo

A

B

< N >

EB

EA

TS

Molecular Dynamics:the whole trajectory

Kinetic Monte Carlo:coarse-grained hops

ab initio MD:up to 50 ps

ab initio kMC:up to minutes

Kinetic Monte Carlo: essentially coarse-grained MD

The crucial ingredients to a kMC simulation

i) Elementary processes

Fixed process list vs. „on-the-fly“ kMC

Lattice vs. off-lattice kMC

ii) Process rates

PES from density-functional theory

Transition state theory

x

xOcus

CObr

Flowchart of a kinetic Monte-Carlo simulation

determine all possible processes i for given configuration of your

system and build a list.

Get all rates (i)

Get two random numbers 1 , 2 [0,1[

Calculate R = i (i)

and find process “k”:

k k-1 (i) 1 R (i)

i=1 i =1

Execute process number “k”, i.e. update configuration

update clock

t t – ln(2)/R

START

END

0

1

1 Rk

III. Getting the rates…

Chemical kinetics,J.K. Laidler, Harper & Row 3rd ed. (New York, 1987)

Methods for finding saddle points and minimum energy paths,G. Henkelman, G. Jóhannesson, and H. Jónsson,in “Progress on theoretical chemistry & physics”,S.D. Schwartz (Ed.), Kluwer (Amsterdam, 2000)

Rare event dynamics

E

IS

TS

FS

Brute force approach to rate constants:

i) Have accurate potential energy surface (forces)ii) Run MD trajectory so long, that it establishes

equilibrium, crossing the barrier many, manytimes back and forth:

k =no. of crossings IS FS per unit time

fraction of time system has spent in IS

require approximate theories to obtain process rates!

Yet: - Relevant time step in MD run is fs (vibrations) - Typical barrier E for surface reactions ~ 1 eV 10-2 - 102 reactions per second (TOF!) - Requires to run trajectory over about 1015 – 1020 time steps unfeasable…

…and essentially 99,9999% of the time, the system willjust vibrate around IS basin (short time dynamics)

Transition state (activated complex) theory I

Assumptions:

i) Reaction system passes the barrier only once (no re-crossings)

ii) Energy distribution of reactant DOF is Boltzmann-like (many collisions compared to reaction eventsyield equilibrium between activated complex and IS,except with respect to the reaction coordinate)

iii) Passage over barrier is the motion of only one DOF,the reaction coordinate, which is independent of allother motions of the activated complex (no concertedmotions)

iv) Passage over barrier is a classical event (no tunneling)

( Eyring, Evans, Polanyi, ~1935 )

IS FS

eq.

X

X

kTST = [ ( ) e S/k ] e-E/kTkThISFS

Derivation: see e.g.K.J. Laidler, Chemical kinetics,

Harper & Row, New York (1987)

Transition state (activated complex) theory II

problem reduces to locating transition states,i.e. saddle points in high dimensional PES

- Attempt frequency/preexponential factor ko

TST = e S/k

~ 1013 sec-1 ~ 1

- In harmonic TST, koTST is given by the harmonic normal modes at the IS and TS

(as explained in talk by Christian Ratsch on Tuesday)

kTh

- If assumptions i)-iv) are fulfilled, kTST is exact. In general, kTST is an upper limit to the real rate

k = fdyn kTST

In principle, one can compute so-called dynamical corrections. In contrast to liquid & gas phase, fdyn ~ 1 for solid-state processes ( TST is a rather good approximation)

i) Grid method:

- Compute PES on a (regular) grid- Scales like: (no. of points) dim

- e.g. 59 ~ 2 million grid points

often unfeasable

ii) Drag method:

- Choose appropriate reaction coordinate q- Constrain q and relax all other DOF- Move from IS to FS

highly dependent on good reaction coordinate hysteresis!

IS

FS

TSx

x

xx

x

x

x

x

x

x

x

x

x

Transition state search algorithms I: grid and drag

Transition state search algorithms II: ridge

- Initialize with straight line interpolation and choose max-energy point Ro

- Create two replicas slightly displaced from Ro on either side of the ridge (side-step)- Displace replicas along gradient (downhill- step)- Find max-energy point Ri along connecting line between two replicas

- Sequentially decrease displacements in downhill- and side-steps when approaching TS

works nicely on well-defined ridges difficult to optimize the displacements

for a given system then poor performance (many force

evaluations required)

I.V. Ionova and E.A. Carter,J. Chem. Phys. 98, 6377 (1993)

IS

FS

TSx

x

x

x

x

x

xx

xRo

xx

xxxxxx

R1

R2

- Initialize with several images {Ri} along a straight-line interpolation- Minimize

S(R1, …, RN) = i E(Ri) + i k/2 (Ri+1 - Ri )2

- Problem:- elastic band cuts corners- images tend to slide down towards low-energy IS/FS regions, leaving few images for relevant TS region

- Solution:- only spring force component parallel to path (no corner cutting)- only true force component perpendicular to path (no down-sliding)

widely applied workhorse has problems, if energy varies largely along path,

but very little perpendicular to it (kinky PES)

G. Mills and H. Jónsson,Phys. Rev. Lett. 72, 1124 (1994)

Transition state search algorithms III: nudged elastic band

IS

FS

TSx

x

xx x

xx

x

x

x

F true

F ||spring

x

Transition state search algorithms IV: dimer

- Initialize by putting dimer at an extremum of a high temperature MD-run- Rotate dimer to minimize energy ( direction of lowest frequency normal mode)- Move dimer along projected gradient perpen- dicular to dimer axis works without any information about FS

G. Henkelman and H. Jónsson,J. Chem. Phys. 111, 7010 (1999)

IS

FS

TSx

x

x

Generally:

- performance scaling with DOF not really known- not good for rough PES- high CPU cost - no algorithm is fool-proof (still lots of room for new ideas)

F

F R

IV. Identifying the processes…

Extending the time scale in atomistic simulation of materials, A.F. Voter, F. Montalenti and T.C. Germann,

Annu. Rev. Mater. Res. 32, 321 (2002)

Diffusion at metal surfaces: surprises…

Hopping mechanism

Ag(100) E = 0.45 eVAu(100) E = 0.83 eV

B.D. Yu and M. Scheffler, Phys. Rev. B 56, R15569 (1997)

Exchange mechanism

Ag(100) E = 0.73 eVAu(100) E = 0.65 eV

Hyperdynamics

Automatized process identification

TAD

Accelerated molecular dynamics:

Other approaches: - metadynamics- dimer method…

V. If it works, it works:First-principles kMC simulations for oxidation catalysis

cus site bridge site

RuO

A materials gap resolved: CO oxidation at Ru(0001) vs. RuO2(110)

H. Over and M. Muhler, Prog. Surf. Sci. 72, 3 (2004)

K. Reuter et al., Chem. Phys. Lett. 352, 311 (2002)

kMC events for CO oxidation over RuO2(110)

Adsorption: CO - unimolecular, O2 – dissociativeno barrierrate given by impingement r ~ So p/(2mkT)

Desorption: CO – 1st order, O2 – 2nd orderout of DFT adsorption well (= barrier)prefactor from detailed balance

Diffusion: hops to nearest neighbor sitessite and element specificbarrier from DFT (TST)prefactor 1012 s-1 (generic)

Reaction: site specificimmediate desorption, no readsorptionbarrier from DFT (TST)prefactor from detailed balance

26 elementary processes considered

The steady-state of heterogeneous catalysis

T = 600 K, pO2 = 1 atm, pCO = 20 atm

K. Reuter and M. Scheffler, Phys. Rev. B (submitted)

Explicit information about fluctuations, correlations and spatial distribution of chemicals at the catalyst surface

A ( pCO , pO2 )-map of catalytic activity

COCObrbr/-/-

600 K pO (atm)2

1

COCObrbr/CO/COcuscus

OObrbr/O/Ocuscus

p CO (

atm

)

1

10-5

105

10-5 10+510-15

(CO/O

)br / (

CO/O)cus

10-10

OObrbr/ -/ -

COCObrbr/CO/COcuscus

OObrbr/O/Ocuscus

4.03.02.0

1.0

-1.0-2.0-3.0

0.0

pO (atm)2

10-10 110-5 10+5

log(TOF)

p CO (

atm

)

1

10-5

105

K. Reuter, D. Frenkel and M. Scheffler, Phys. Rev. Lett. 93, 116105 (2004)

pC

O (

atm

)

10-30 110-20 10-10

10-20

10-10

1 COCObrbr/CO/COcuscus

COCObrbr/-/-

- / -- / - OObrbr/O/Ocuscus

T = 350 KpCO = 10-10 atmT = 350 KpO2

= 10-10 atm

…and how about experiment?

J. Wang, C.Y. Fan, K. Jacobi, and G. Ertl,J. Phys. Chem. B 106, 3422 (2002)

pO (atm)2

-9.0

-4.0-5.0-6.0-7.0

-10.0-11.0

-8.0

log(TOF)

350 K

Ab initio MD and kMC simulations

Ab initio kinetic Monte Carlo simulations:

- coarse-grained time evolution (rare event dynamics)

- efficient treatment of statistical interplay of a larger

number of elementary processes

- time scales given by process rates, often seconds or longer

- process list (process identification, lattice models)

- accuracy of rates (DFT-TST), high CPU cost

- low speed-up, if very fast processes present

Ab initio molecular dynamics:

- fully dynamics of the system- straightforward, easy to implement

- times scales up to ~ns- acceleration techniques under development