Bridging scales:

Ab initio atomistic thermodynamics

Karsten Reuter

Fritz-Haber-Institut, Berlin

General idea

Approach:- separate system into sub-systems (exploit idea of reservoirs!)- calculate properties of sub-systems separately (cheaper…)- connect by implying equilibrium between sub-systems

gasphase

surface

bulkDrawback:- no temporal information („system properties after infinite time“)- equilibrium assumption

Motivation:- extend length scale- consider finite temperature effects

Ab initio atomistic thermodynamics and statistical mechanics of surface properties and functionsK. Reuter, C. Stampfl and M. Scheffler, in: Handbook of Materials Modeling Vol. 1,

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

I. Connecting thermodynamics,

statistical mechanics and density-functional theory

Statistical Mechanics,D.A. McQuarrie, Harper Collins Publ. (1976)

Introduction to Modern Statistical Mechanics,D. Chandler, Oxford Univ. Press (1987)

M. Scheffler in Physics of Solid Surfaces 1987,J. Koukal (Ed.), Elsevier (1988)

- In its set of variables the total derivative of each potential function is simple (derive from 1st law of ThD: dEtot = dQ + dW, dW = -pdV, dQ = TdS)

dE = TdS – pdVdH = TdS + VdpdF = -SdT – pdVdG = -SdT + Vdp

Thermodynamics in a nutshell

Internal energy (U) Etot(S,V)Enthalpy H(S,p) = Etot + pV(Helmholtz) free energy F(T,V) = Etot - TSGibbs free energy G(T,p) = Etot - TS + pV

- Equilibrium state of system minimizes corresponding potential function

Potential functions

These expressions open the gate to a whole set of general relations like: S = - (F/T)V , p = - (F/V)T Etot = - T 2 (/T)V (F/T) Gibbs-Helmholtz eq.

(T/V)S = - (p/S)V etc. Maxwell relations

- Chemical potential µ = (G/ n)T,p is the cost to remove a particle from the system. Homogeneous system: µ = G/N (= g)

i.e. Gibbs free energy per particle

- If groups of degrees of freedom are decoupled from each other (i.e. if the energetic states of one group do not depend on the state within the other group), then

Ztotal = ( i exp(-EiA / kBT) ) ( i exp(-Ei

B / kBT) ) = ZA ZB

Ftotal = FA + FB e.g. electronic nuclear (Born-Oppenheimer)

rotational vibrational

Link to statistical mechanics

A many-particle system will flow through its huge phase space, fluctuating through all microscopic states consistent with the constraints imposed on the system. For an isolated system with fixed energy E and fixed size V,N (microcanonic ensemble) these microscopic states are all equally likely at thermodynamic equilibrium (i.e. equilibrium is the most random situation).

- Partition function Z = Z(T,V) = i exp(-Ei / kBT) Boltzmann-weighted sum

over all possible system states

F = - kBT ln( Z )

- N indistinguishable, independent particles: Ztotal = 1/N! (Zone particle)N

Computation of free energies: ideal gas I

µ(T,p) = G / N = (F + pV) / N = ( - kBT ln( Z ) + pV ) / N

Z = 1/N! ( Znucl Zel Ztrans Zrot Zvib )NX

i) Electr. free energy Zel = i exp(-Eiel / kBT) Typical excitation energies eV >> kBT,

only (possibly degenerate) ground state Fel Etot – kBT ln( Ispin ) contributes significantly

Required input: Internal energy Etot

Ground state spin degeneracy Ispin

ii) Transl. free energy Ztrans = k exp(-ħk2 / 2mkBT) Particle in a box of length L = V1/3

(L) Ztrans V ( 2 mkBT / ħ2 )3/2

Required input: Particle mass m

Computation of free energies: ideal gas II

iii) Rotational free energy Zrot = J (2J+1)exp(-J(J+1)Bo / kBT) Rigid rotator

(Diatomic molecule) Zrot - kBT ln(kBT/ Bo ) = 2 (homonucl.), = 1 (heteronucl.) Bo ~ md2 (d = bond length)

Required input: Rotational constant Bo

(exp: tabulated microwave data)

iv) Vibrational free energy Zvib = i=1 n exp(-(n + ½)ħi / kBT) Harmonic oscillator

µvib(T) = i=1 ½ ħi + kBT ln( 1 - exp(-ħi/kBT) )

Required input: M fundamental vibr. modes i

M

M

Calculate dynamic matrix Dij = (mimj)-½ (2Etot/rirj)req

Solve eigenvalue problem det(D – 1 i2)

Computation of free energies: ideal gas III

O2 CO

m (amu) 32 28stretch (meV) 196 269Bo (meV) 0.18 0.24 2 1Ispin 3 1

µ = µ(T,p) = Etot + Δμ(T,p)

Δµ

(T, p

= 1

atm

) (

eV)

Temperature (K) Temperature (K)

vibr.+ electr.

+ rotat. + rotat.

+ transl. + transl.O2 CO

Alternatively: Δ(T, p) = Δ(T, po) + kT ln(p/po)and Δ(T, po = 1 atm) tabulated in thermochem. tables (e.g. JANAF)

Computation of free energies: solids

Ftrans Translational free energy Frot Rotational free energy

pV V = V(T,p) from equation of state, varies little

Fconf Configurational free energy

Etot Internal energy

Fvib Vibrational free energy

G(T,p) = Etot + Ftrans + Frot + Fvib + Fconf + pV

Etot, Fvib use differencesuse simple models to approx. Fvib (Debye, Einstein)

Solids (low T): G(T,p) ~ Etot + Fconf

1/M 0

Trouble maker…

0 for p < 100 atm

DFT

phonon band structure

II. Starting simple:Equilibrium concentration of point defects

Solid State Physics,

N.W. Ashcroft and N.D. Mermin, Holt-Saunders (1976)

Isolated point defects and bulk dissolution

On entropic grounds there willalways be a finite concentrationof defects at finite temperature,even though the creation of adefect costs energy (ED > 0).

How large is it?

Internal energy: Etot = n ED

N sites, n defects (n <<N)

Config. entropy: Fconf = kBT ln Z(n)

with Z = = N (N-1) … (N-n-1) N!

1·2· …· n (N-n)!n!

Minimize free energy: (G/n)T,p = /nT,p (Etot – Fconf + pV) = 0

Forget pV, use Stirling: ln N! N(lnN-1) n/N = exp(-ED/kBT)

III. Slightly more involved:Effect of a surrounding gas phase

on the surface structure and composition

E. Kaxiras et al., Phys. Rev. B 35, 9625 (1987)

X.-G. Wang et al., Phys. Rev. Lett. 81, 1038 (1998)

K. Reuter and M. Scheffler, Phys. Rev. B 65, 035406 (2002)

Surface thermodynamics

A surface can never be alone: there are always

“two sides” to it !!!

solid – gassolid – liquidsolid – solid (“interface”)…

Phase I

Phase II

Phase I / phase II alone (bulk):

GI = NI I

GII = NII II

Total system (with surface):

GI+II = GI + GII + Gsurf

(T,p)

A

= 1/A ( GI+II - i Ni i ) Surface tension(free energy per area)

Example: Surface in contact with oxygen gas phase

O2 gas

surface

bulk

surf. = 1/A [ Gsurf.(NO, NM) – NO O - NM M ]

ii) M = gMbulkii)

i)

i) O from ideal gas

Use reservoirs:

(T,p) ( Esurf. – NM EM )/A – NO O(T,p) /A (slab) bulk

Forget about Fvib and Fconf for the moment:

Oxide formation on Pd(100)

M. Todorova et al., Surf. Sci. 541, 101 (2003);K. Reuter and M. Scheffler, Appl. Phys. A 78, 793 (2004)

( Esurf. – NM EM )/A – NO O/A (slab) bulk

- c

lean

(m

eV/Å

2 )

p(2x2) O/Pd(100)

(√5 x √5)R27° PdO(101)/Pd(100)

Vibrational contributions to the surface free energy

Fvib(T,V) = d Fvib(T,) ()

vib = Fvib/A =

= 1/A d Fvib(T,) [ surf.( ) - NRu bulk ( ) ]

Only the vibrational changes at the surfacecontribute to the surface free energy

Use simple models for order of magnitude estimate

e.g. Einstein model: ( ) = ( - )

0

20

10

30

(meV

)

Pd bulk

Pd(bulk) ~ 25 meV

Surface induced variations of substrate modes

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

Temperature (K)

vib (

meV

/Å2 ) + 50%

- 50%

20

-20

10

-10

0

0 200 400 600 800 1000

Surface functional groups

vi

b.

(meV

/Å2 )

Temperature (K)

H2O

OH

40

30

20

10

00 200 400 600 800 1000

Q. Sun, K. Reuter and M. Scheffler, Phys. Rev. B 67, 205424 (2003)

-0.5-1.0-1.5O (eV)

(

meV

/Å2 )

clean surface O(1x1)

Configurational entropy smears out phase transitions

Configurational entropy and phase transitions

0.0

1.0

0.5

<

(O)

>

No lateral interactions:

Langmuir adsorption-isotherm

<(Ocus)> =

1 + exp((Ebind - O)/kBT)1

Fconf = kBT ln (N+n)! / (N!n!)

T = 300 KT = 600 K

IV. Exploration of configuration space:Monte Carlo simulations

and lattice gas Hamiltonians

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

A Guide to Monte Carlo Simulations in Statistical Physics,D.P. Landau and K. Binder, Cambridge Univ. Press

(2000)

- In general, the configuration space is spanned by all possible (continuous) positions rN of the N atoms in the sample:

Z = ∫ drN exp(- E(r1,r2,…,rN) / kBT)

- The average value of any observable A at temperature T in this ensemble is then

<A> = 1/Z ∫ drN A(r1,r2,…,rN) exp( -E(r1,r2,…,rN) / kBT)

Configuration space and configurational free energy

Canonic ensemble (constant temperature):

Partition function Z = Z(T,V) = i exp(-Ei / kBT) Boltzmann-weighted sum

over all possible system states

Fconf = - kBT ln( Zconf )

Evaluating high-dimensional integrals: Monte Carlo techniques

Problem: - numerical quadrature (on a grid) rapidly unfeasible- scales with: (no. of grid points)N - e.g.: 10 atoms in 3D, 5 grid points: 530 ~ 1021 evaluations

<A> = 1/Z ∫ drN A(r1,r2,…,rN) exp( -E(r1,r2,…,rN) / kBT)

Alternative: - random sampling (Monte Carlo)

Example for integrationby simple sampling

Finding a needle in a haystack: Importance sampling

<A> = 1/Z ∫ drN A(r1,r2,…,rN) exp( -E(r1,r2,…,rN) / kBT)

- Many evaluations where integrand vanishes- Need extremely fine grid (very inefficient)

- Do random walk, and reject all moves that bring you out of the water- Provides average depth of Nile, but NOT the total volume!!

Measuring the depth of the Nile

Thank you,Daan!!!

Specifying “getting out of the water”:The Metropolis algorithm

?

?

Etrial < Epresent: accept

Etrial > Epresent: accept with probability exp[- (Etrial-Epresent) / kBT ]

Some remarks: - With this definition, Metropolis fulfills „detailed balance“ and thus samples a canonic ensemble- If temperature T is steadily decreased during simulation, upward moves become less likely and one ends up with an efficient ground state search („simulated annealing“)

In short:

Modern importance sampling Monte Carlo techniques allow to

- efficiently evaluate the high-dimensional integrals needed

for evaluation of canonic averages

- properly explore the configuration space, and thus

configurational entropy is intrinsically accounted for in

MC simulations

Major limitations:

- still need easily 105 – 106 total energy evaluations

- this is presently an unsolved issue. First steps in the

direction of true „ab initio Monte Carlo“ are only

achieved using lattice models

A very simple lattice system: O / Ru(0001)

- Consider only adsorption into hcp sites (for simplicity)- Simple hexagonal lattice, one adsorption site per unit cell- Questions: which ordered phases exist ?

order-disorder transition at which temperature ?

Configuration space comprises:

disordered structuresordered structures

(arbitrary periodicity)

BUT: only periodic structures accessible to direct DFT,and supercell size quite limited

How can we then samplethe configuration space?

Lattice gas Hamiltonians / Cluster expansions

Expand total energy of arbitrary configuration in terms of lateral interactions

Elatt = i Eo + 1/2 i,j Vpair(dij)i j + 1/3 i,j,k Vtrio(dij,djk,dki)i j k + …

- Algebraic sum (very fast to evaluate)- Ising, Heisenberg models- Conceptually easily generalized to

- multiple adsorbate species- more complex lattices (different site types etc.)

…but how can we get thelateral interactions from DFT?

LGH parametrization through DFT

Since isolated clusters not compatible with supercell approach,exploit instead the interaction with supercell images in a systematic way:

- Compute many ordered structures - Write total energy as LGH expansion, e.g.

E(3x3) = 2Eo + 2V1pair + 2V3

pair

- Set up system of linear equations- „Invert“ to get lateral interactions

e.g.O / Ru(0001)

3

3

O (eV)

(

meV

/Å2 )

clean p(2x2)

p(2x1)

p(2x2) p(2x1)clean

p(2x2)p(2x1)

disorderedlattice-gas

T (

K)

clean

p(2x1)

p(2x1)clean

In short:

DFT parametrized lattice gas Hamiltonians enable

- efficient sampling of configurational space

- parameter-free prediction of phase diagrams

- first treatment of disordered structures

Major limitations:

- systematics / convergence of LGH expansion

- restricted to systems that can be mapped onto a lattice

- expansion rapidly very cumbersome for complex lattices,

multiple adsorbates, at defects/steps/etc.

allows any general thermodynamic reasoning

concentration of point defects at finite T

surface structure and composition inrealistic environments

Ab initio atomistic thermodynamics

major limitations

Vxc vs. kBT

sampling of configurational space

„only“ equilibrium

Lecture 2 tomorrow: kinetics, time scales

Use DFT in the computation of free energies Suitably exploit equilibria and concept of reservoirs