First-principles statistical mechanics

Sergey V. Levchenko

FHI Theory department

space

Continuum Equations,Rate Equations

and Finite ElementModeling

ab initio

ab initiokinetic Monte Carlo

m

mm

µm

density-functional

theory

Towards Realistic Time Scales

time

f s p s n s µ s m s s hours years

ElectronicStructureTheory

ab initioMolecularDynamics

kinetic Monte Carlo

We need robust, error-controlled links with knowledge of uncertainty between the various simulation methodologies

nm

Outline

1) Ab initio atomistic thermodynamics

2) Molecular dynamics: reaction barriers and config urationalsampling

3) First-principles kinetic Monte Carlo simulations

4) Concluding remarks

Real versus model materials and interfaces (materials and pressure gaps)

1) A straightforward DFT calculation corresponds to T = 0 K

2) Real materials may be full of defects (vacancies , dislocations, steps on surfaces, etc.)

3) Material’s structure and composition can strongl y depend on preparation

4) Impurities and adsorbates can influence the surfa ce morphology

Special care must be taken in experiment to produce a defect-free pure surface of known termination. But even then…

5) A surface cannot be separated from a gas (or liq uid) above it

mkT

p

πν

2=

For = 300 K, = 1 atm � ~ 108 site -1 s-1ν

Requires 10 -12 atm to keep a “clean” surface clean; surface can also lose atomsT p

≤p

DFT (internal) versus free energy

At constant T a system minimizes its free energy (- TS), not internal energy U

If also volume V is constant, the energy minimized is Helmholtz free e nergy F

TSUF −=If (T,p) are constant, the energy minimized is Gibbs free e nergy G

NTSpVUG ∑=−+= µ ii

i NTSpVUG ∑=−+= µ

Chemical potential of the i-th atom type is the change in free energy as the number of atoms of that type in the system incr eases by 1

iµ

In thermodynamic equilibrium, is the same in the whole system (surface, bulk, gas)

iµ

Statistics plays a crucial role due to a macroscopi cally large number of particles in the system

Bridging the pressure gap

Nanometer and sub-nanometer thin oxide films at surfaces of late transition metals,K. Reuter, in: Nanocatalysis: Principles, Methods, Case Studies,(Eds.) U. Heiz, H. Hakkinen, and U. Landman, Springe r (Berlin, 2006)http://www.fhi-berlin.mpg.de/th/paper.html

Theory: Development and use of first-pr inciple statisticalmechanics approaches

Experiment: Awareness and diligent experiments…Development and use of „ in situ“ techniques

First-principles atomistic thermodynamics

equilibrium

+−+= confvibtot

),(22 OO pTµ

pVTSFEpTG +−+= confvibtot),(

DFT

C.M. Weinert and M. Scheffler, Mater. Sci. Forum 10 -12, 25 (1986); E. Kaxiras et al., Phys. Rev. B 35, 9625 (1987)

Surface free fromation energy:

−= ∑i

iii NNGA

pT µγ })({1

),( sufrsurf

Defect free formation energy:

−∆−∆+=∆ ∑ })({})({1

),( perfdef ii

iiii NGNNNGA

pTG µ

Computation of free energies: Ideal gas

NpVZTkNGpT /)ln(/),( B +−==µ

( )NZZZZZN

Z vibrottranselnucl!

1= – partition function (canonical)

– chemical potential

For an ideal gas of N particles in a box:

1) Electronic free energy: ∑ −= TkEZ )/exp( el1) Electronic free energy: ∑ −=i

i TkEZ )/exp( Bel

el

Typical excitation energies eV >> kBT , only (possibly degenerate) ground statecontributes significantly

)12ln(),( B0el +−≈ ITkEpTµ

Required input: ,0E I

2) Translational free energy (particle in a box): ∑ −=k

Bk )2/exp( 22transl TmkZ h

Box length of length ∞→L 2/32Btransl )/2( hTmkVZ π≈

Required input: particle mass m

Computation of free energies: Ideal gas

3) Rotational energy (rigid rotator): ∑ +−+=J

TkBJJJZ )/)1(exp()12( B0rot

Diatomic molecule: )/ln( 0BBrot BTkTk σµ −≈σ = 2 (homonucl.), = 1 (heteronucl.)B0 ~ md2 (d – bond length)

Required input: rotational constant B0 (calculations, microwave spectroscopy)

∑∑M 1

4) Vibrational free energy (harmonic approx.): ∑∑=

+−=M

i ni TknZ

1Bvib )/)

2

1(exp( ωh

∑=

−−+=M

iii TkTkpT

1BBvib ))/exp(1ln(

2

1),( ωωµ hh

Required input: M fundamental vibrational modes iω

Calculate dynamic matrix , eq

)/()( 022/1

rjijiij rrEmmD ∂∂∂= −

0)det( 2 =− ω1Dsolve eigenvalue problem

Computation of free energies: Ideal gas

It is convenient to define a reference for :),( pTµ ),(),( 0 pTEpT µµ ∆+=

Alternatively: )/ln(),(),( Boo ppTkpTpT +∆=∆ µµ

and from thermoch emical tables (e.g., JANAF)),( atm1=∆ opTµ

Example I: Metal surface in contact with O 2 gas

[ ]MMOOMOsufrsurf ),(1

),( µµγ NNNNGA

pT −−=

surface

(1) Reservoirs:1) from ideal gas),( 2OO pTµ

bulk metal

surface

(2) 2)

2

bulkMM g=µ

Neglect for now Fvib and TSconf :

ApTNAENENEpT /),(/2

1),( OOOO

bulkMM

slabsurf

2µγ ∆−

−−≈

Example I: Metal surface in contact with O 2 gas

AN

AENENE

/

/2

1

OO

OObulkPdPd

slabsurf

2

µ

γ

∆−

−−≈

p(2x2) O/Pd(100)

(√5x√5)R27o PdO(101)/Pd(100)M. Todorova et al., Surf. Sci. 541, 101 (2003); K. Reuter and M. Scheffler, Appl. Phys. A 78, 793 (2004)

Vibrational contributions to the surface free energ y

∫= ωωσω dTFVTF )(),(),( vibvib

[ ]∫ −

=∆=

ωωσωσω

γ

dNTFA

AF

)()(),(1

/

bulkPdsurfvib

vibvib

)(ωσ

Only changes in vibrational free energy contribute to the surface free energy

Make estimate from simple models

e.g., Einstein model: )()( ωωδωσ −=

Pdω (bulk) ~ 25 meV

Surface-induced variations of substrate modes

< 10 meV/Å2 for T = 600 K – in this case!!!

First-principles atomistic thermodynamics: constrained equilibria

constrainedequilibrium

X),(22 OO pTµ ),( COCO pTµ

pVTSFEpTG +−+= confvibtot),(

DFT

C.M. Weinert and M. Scheffler, Mater. Sci. Forum 10-12, 25 (1986); E. Kaxiras et al., Phys. Rev. B 35, 9625 (1987);

K. Reuter and M. Scheffler, Phys. Rev. B 65, 035406 (2001); Phys. Rev. B 68. 045407 (2003)

Surface phase diagrams)atm(

2Op10-10 1 1010

10-10 110-2010-3010-40

1

105

105

300 K

600 K

CO oxidation on RuO2(110)

K. Reuter and M. Scheffler,Phys. Rev. Lett. 90, 046103 (2003)

10-10

10-5

1

10-15

10-20

105

1

10-5

p CO(a

tm)

)eV(Oµ∆

Example II: defects on oxide surfacesLi-induced defects at MgO (001) surface

(for oxidative coupling of methane)

1) LiMg: Li substitutional defects

MgO (001) surface with subsurface Li Mg defects

top layer

2nd layerLi

CH

2CH4 + O2 → C2H4 + 2H2O

1) LiMg: Li substitutional defects (rLi+ ≈ rMg2+)

2) LiMg with adjacent O or Mg vacancies or interstitials

3) LiMg with adjacent O or Mg ad-atoms

Electronic structure, total energies + ab initio atomistic thermodynamics

proposed model of OCM (H abstraction at “Li +O-”)

2nd layer

3rd layer O

Mg

Defect formation energies: methodologyAb initio atomistic thermodynamics

fixed concentration δof Li Mg in the bulk

),(2O pTµ

),(Li pTµ

DFT

constant from DFT

∑ ∆−∆+−∆+∆=∆i

ii NVpTSFEpTG µconfvibtot),(

)/ln(2

1),(

2

1),( BOO 2

oo ppTkpTpT +∆=∆ µµ

X X

),(/ln(2

1),( OBLi ) pTppTkpT E µµ −+= o

( )]!)!/[(!ln defdefBconf NNNNkS −=

)]ln()ln()(ln[ defdefdefdefBconf NNNNNNNNkS −−−−≈

Configurational entropy and defect concentration

For a very large supercell, when N >> 1, Ndef >>1, and (N - Ndef) >> 1 (Stirling formula):

[ ] 0confdefconfdef

perf

defdef

def

=−∆+=+

TSGNGdN

d

dN

dGTS

In equilibrium: (assumes no interaction between defects)

1)/exp(

1

Bdef

conf

def

+∆=

+TkGN

N

TS

Formation energies (PBE) of the Li Mg defect in the first and second atomic layer (1/8 concentration) relative to the

bulk value

Li energetically prefers to be in the top layer of MgO (001) (GGA and GGA+ U#)

#M. Nolan and G.W. Watson, Surface Science 586, 25 (2005)

Interaction between the Li Mg defects

Formation of Li Mg in the top layer of MgO (001)

top view

Stabilization of O vacancies

F0 top + Li MgF+ + (LiMg)-

VO under LiVO to the side of Li

F0 top + 2Li Mg

F2+ + 2(LiMg)-

Isolated V OIsolated Li MgF+ + (LiMg)-

Equilibrium defect concentrations in Li/MgO

F2++2(LiMg)-

Concentration of Li Mg and VO at the surface increases under reducing conditions

5% LiMg in the bulkUHV (pO2 = 10-10 atm)F++(LiMg)-

LiMg island formation at the surface: methodology

loss in configurational entropy ↔ gain in formation energy

island size as a function of T

formation energy at high coverage

Model:

( )

−

−

−−−∆≈∆n

N

n

N

n

NN

n

NNNNkTENG LiLiLi

sitesLi

sitessitessiteshcLiisland lnlnln

versus

( ) ( ) ( ) ( )( )LiLiLisitesLisitessitessiteslcLiseparate lnlnln NNNNNNNNkTENG −−−−−∆≈∆

formation energy at high coverage

formation energy at low coverage

island size

Find such thatn separateisland GG ∆<∆

LiMg island formation at the surfaceloss in configurational entropy ↔ gain in formation energy

5% LiMg in the bulk

25

30

35

40

45

50M

gin

the

isla

nd

50% coverage

The Li Mg defects form islands at the surface

0

5

10

15

20

25

Num

ber

of L

i Mg

700 750 800 850 900 950 1000 1050 1100

Temperature, K

O defects at 50% Li Mg at the surface

12.5% O vacancies

25% O vacanciesat the surface

δ/2 O vacanciesin the bulk

Surface transformation scenario based on calculated surface free energies

1) segregation of Li Mg

2) sintering

3) loss of oxygen

4) further sintering and loss of oxygen

Experiment: STM images (80x80 nm) of Li-doped MgOfilm in 5x10 -9 atm O2 on Mo(001)

as-deposited annealed at 800K annealed at 1050K

1) Li in the MgO matrix, high concentration of Li Mg at the surface, Li xO at the surface

2) Strong segregation of Li Mg to the surface and continued formation of Li xO at the surface, Li Mg defects form islands rich in O vacancies

3) Material loses more O, the Li-rich islands and r emaining Li xO desorb due to high temperature

P. Myrach, N. Nilius, S.V. Levchenko et al., ChemCatChem 2, p. 854 (2010)

conductance image (6.8 V)Mg deposition (shows the pattern characteristic of Mg on pristine MgO (001)

When vibrations do matter eV

)

µO (eV)ZnO (0001) surface phase diagram in H 2O-O2 atmosphere –no vibrations

No structure with (2x2)periodicity as

µ H(e

V

M. Valtiner, M. Todorova, G. Grundmeier, and J. Neugebauer, PRL 103, 065502 (2009)

periodicity as seen at the ZnO(0001) surfaceannealed in a dry oxygen atmosphere (containing at maximum2 ppm water)

When vibrations do matter H

2(b

ar)

A (2x2)-O adlayerstructure is stabilized by vibrationalentropy effects

Observed at

M. Valtiner, M. Todorova, G. Grundmeier, and J. Neugebauer, PRL 103, 065502 (2009)

pH

2

pO2 (bar)

Observed at “humid” conditions

Explicit atom motion: Molecular dynamics Goal: Integrate equations of motion, add heat bath; input – forces

p

qpH

dt

dq

q

qpH

dt

dp

∂∂=

∂∂−= ),(

,),(

Not trivial to integrate!

Hamiltonian is by construction energy conserving

How to model a system at constant temperature?

H

bathsys HHH +=conserves energy yields Maxwell-Boltzmann distribution

of velocities for given T

Nosé-Hoover thermostat (fictitious degrees of freed om):

βηπηπ N

QU

mH i

i i

iii 3

2

1})({

2),},{},({ 2

2+++=∑ r

prp

i

ii

mdt

d pr =Q

U

dt

d i

i

i πpr

p −∂∂−=

Qdt

d πη = βπ N

mdt

d

i i

i 3−=∑p

Thermostat takes and gives energy to the atomic sub system

• T-dependent rotational-vibrational spectra with anha rmonic contributions

• Rates of reactions (or conformational changes, or phase transformations)

• Out-of-equilibrium quantities, e.g. reaction of t he system to a time dependent external field

What can we use it for?

Explicit atom motion: Molecular dynamics

• Car-Parrinello MD: coupled equations of motion fo r both ions and electrons (versus Born-Oppenheimer dynamics)• Wave function extrapolation (Kühne-Parrinello, Nikl asson)

Including electronic degrees of freedom

T3

T4

Ene

rgy

Robust sampling of PES: Parallel tempering

T1

T2

some coordinates

Parallel tempering : the scheme

Swap 2Swap 1

MD

run

1

MD

run

2

T5

T4

T3

MD

run

3

T3

T2

T1

- Swap move accepted with prob = exp {( ββββc- ββββh)(Ec-Eh)}, where ββββ =1/kBT. This ensures canonical ensemble at each temperature

- Swap acceptance is high when temperatures are so close thatenergy distributions overlap.

First-principles kinetic Monte Carlo simulations

B

TS

∆EA→B ∆EB→A

equilibriumMonte Carlo

EB

kA→B

kB→A

kineticMonte Carlo

N

t

B

A

A

B

MolecularDynamics

<N>

EB

EA

t

∑∑ →→ +−=j

jijj

ijii tPktPkdt

tdP)()(

)(

∆−Γ=

=

→

→→

Tk

E

Z

Z

h

Tkk

ji

i

jiji

B

)(TSB

exp

o

Transition State Theory

Master equation

Kinetic Monte Carlo – essentially coarse-grained MD

Molecular Dynamics:the whole trajectory

Kinetic Monte Carlo:coarse-grained hops

ab initio MD:up to 50 ps

ab initio kMC:up to minutes

Crucial ingredients of a kMC simulation

1) Elementary processes

Fixed process list vs. „on-the-fly“ kMCLattice vs. off -lattice kMCOcus

∑∑ →→ +−=j

jijj

ijii tPktPkdt

tdP)()(

)(

Lattice vs. off -lattice kMC

2) Process rates

PES accuracyReaction rate theory

x

xOcus

CObr

kMC events for CO oxidation over RuO 2(110)

Adsorption: CO - unimolecular, O 2 – dissociativeno barrierrate given by impingement

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

Diffusion: hops to nearest neighbor sites

)2/( B0 TmkpSk π≈

K. Reuter and M. Scheffler, Phys. Rev. B 73, 045433 (2006)

Diffusion: hops to nearest neighbor sitessite and element specificbarrier from DFT (TST)prefactor from DFT (hTST)

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 = 7 atm

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

K. Reuter, C. Stampfl, and M. Scheffler, Handbook o f materials modeling, part A. Methods, p. 149, Springer, Berlin (2005)

)atm(2Op

10-10 1 1010

105

10-15 10-10 10-5 1 105

)atm(2Op

atm

)

105

Thermodynmic versus kinetic phase diagram

1

10-5

p CO(a

tm)

p CO(a

tm

1

10-5

(pO2, pCO)-map of catalytic activity

10-15 10-10 10-5 1 105

105

pO2 (atm)

(atm

)

10-10 10-5 1 105

4.03.02.0

pO2 (atm)600 K

1

10-5

p CO

(atm

)

2.01.00.0

-1.0-2.0-3.0

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

Comparison to experiment

10-30 10-20 10-10 1

10-10

1

pO2 (atm)p C

O(a

tm)

350 K

10-20

p

-4.0-5.0-6.0-7.0-8.0-9.0

-10.0-11.0

J. Wang et al., J. Phys. Chem. B 106, 3422 (2002)

→ cf. NH3/Ru(0001):K. Honkala et al., Science 307, 555 (2005)

Concluding remarks

1) DFT is only the beginning

2) Standard DFT accuracy may be enough for rough estimate of reaction energies, but going beyond is necessary for predictive statistical calculations

3) To predict catalytic activity, statistical effec ts must be taken into account

Is there anything left to do in methodology?

1) Beyond DFT (electronic correlation)

2) Heat and mass transfer in catalytic systems

3) Excited states (photocatalysis)