first-principles statistical...
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First-principles statistical mechanics
Sergey V. Levchenko
FHI Theory department
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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
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Outline
1) Ab initio atomistic thermodynamics
2) Molecular dynamics: reaction barriers and config urationalsampling
3) First-principles kinetic Monte Carlo simulations
4) Concluding remarks
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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
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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
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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
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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
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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
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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
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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
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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 ∆−
−−≈
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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)
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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
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Surface-induced variations of substrate modes
< 10 meV/Å2 for T = 600 K – in this case!!!
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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)
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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∆
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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
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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
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( )]!)!/[(!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
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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)
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Interaction between the Li Mg defects
Formation of Li Mg in the top layer of MgO (001)
top view
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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)-
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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)-
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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 ∆<∆
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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
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O defects at 50% Li Mg at the surface
12.5% O vacancies
25% O vacanciesat the surface
δ/2 O vacanciesin the bulk
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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
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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)
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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)
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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
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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
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• 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
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T3
T4
Ene
rgy
Robust sampling of PES: Parallel tempering
T1
T2
some coordinates
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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.
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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
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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
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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
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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
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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)
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)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
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(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)
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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)
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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
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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)