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
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