Bridging scales:
Ab initio atomistic thermodynamics
Karsten Reuter
Fritz-Haber-Institut, Berlin
General idea
Motivation:
- extend length scale
- consider finite temperature effects
Approach:
- separate system into sub-systems
(exploit idea of reservoirs!)
- calculate properties of sub-systems
separately (cheaper…)
- connect by implying equilibrium
between sub-systems
Drawback:
- no temporal information
(„system properties after infinite time“)
- equilibrium assumption
gas
phase
surface
bulk
Ab initio atomistic thermodynamics and statistical mechanics of surface properties and functions
K. 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)
Thermodynamics in a nutshell
Internal energy (U)
Enthalpy
(Helmholtz) free energy
Gibbs free energy
Etot(S,V)
H(S,p) = Etot + pV
F(T,V) = Etot - TS
G(T,p) = Etot - TS + pV
Potential functions
- Equilibrium state of system minimizes corresponding potential function
- 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 – pdV
dH = TdS + Vdp
dF = -SdT – pdV
dG = -SdT + Vdp
 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
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).
Z = Z(T,V) = i exp(-Ei / kBT)
- Partition function
 Boltzmann-weighted sum
over all possible system states
 F = - kBT ln( Z )
- 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 =
(  exp(-E
i
 Ftotal = FA + FB
i
A
/ kBT)
) (  exp(-E
i
i
B
/ kBT)
) =Z
A
ZB
e.g. electronic  nuclear (Born-Oppenheimer)
rotational  vibrational
- N indistinguishable, independent particles:
(
Ztotal = 1/N! Zone particle
)
N
Computation of free energies: ideal gas I
Z = 1/N!
Z
(X
nucl
Zel Ztrans Zrot Zvib
)
N
 µ(T,p) = G / N = (F + pV) / N = ( - kBT ln( Z ) + pV ) / N
i) Electr. free energy
Zel = i exp(-Eiel / kBT)
 Fel  Etot – kBT ln( Ispin )
Required input:
ii) Transl. free energy
Ztrans = k exp(-ħk2 / 2mkBT)
Typical excitation energies eV >> kBT,
only (possibly degenerate) ground state
contributes significantly
Internal energy Etot
Ground state spin degeneracy Ispin
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)
M
iv) Vibrational free energy
Zvib = i=1 n exp(-(n + ½)ħi / kBT)
Harmonic oscillator
M
 µvib(T) = i=1 ½ ħi + kBT ln( 1 - exp(-ħi/kBT) )
Required input:
M fundamental vibr. modes i
Calculate dynamic matrix Dij = (mimj)-½ (2Etot/rirj)req
Solve eigenvalue problem det(D – 1 i2)
Computation of free energies: ideal gas III
m
stretch
Bo

Δµ(T, p = 1 atm) (eV)
Ispin
(amu)
(meV)
(meV)
O2
CO
32
196
0.18
2
3
28
269
0.24
1
1
O2
Alternatively:
µ = µ(T,p)
= Etot + Δμ(T,p)
+ electr.
vibr.
+ rotat.
+ rotat.
+ transl.
Temperature (K)



CO
+ transl.
Temperature (K)
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
G(T,p) = Etot + Ftrans + Frot + Fvib + Fconf + pV



Ftrans
Frot
Translational free energy
Rotational free energy
pV
V = V(T,p) from equation of state, varies little
 0 for p < 100 atm
Fconf
Configurational free energy
 Trouble maker…
Etot
Internal energy
 DFT
Fvib
Vibrational free energy
 phonon
band structure
Etot, Fvib
 1/M  0
use differences
use simple models to approx. Fvib (Debye, Einstein)
 Solids (low T):
G(T,p) ~ Etot + Fconf
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 will
always be a finite concentration
of defects at finite temperature,
even though the creation of a
defect costs energy (ED > 0).
N sites,
n defects (n <<N)
How large is it?
Internal energy:
Config. entropy:
Minimize free energy:
Etot = n ED
Fconf = kBT ln Z(n)
N (N-1) … (N-n-1)
with
Z=
1·2· …· n
N!
=
(N-n)!n!
(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
solid – gas
solid – liquid
solid – solid (“interface”)
…
A surface can never be alone:
there are always
“two sides” to it !!!
Phase I / phase II alone (bulk):
GI = NI I
GII = NII II
Phase II
(T,p)
Phase I
Total system (with surface):
GI+II = GI + GII + Gsurf
A
 = 1/A ( GI+II - i Ni i )
Surface tension
(free energy per area)
Example: Surface in contact with oxygen gas phase
 surf. = 1/A [ Gsurf.(NO, NM) – NO O - NM M ]
O2 gas
i)
Use reservoirs:
surface
bulk
i) O from ideal gas
ii)
bulk
ii) M = gM
Forget about Fvib and Fconf for the moment:
(slab) – N E bulk )/A – N  (T,p) /A
(T,p)  ( Esurf.
M M
O O
Oxide formation on Pd(100)
p(2x2) O/Pd(100)
(√5 x √5)R27° PdO(101)/Pd(100)
M. Todorova et al., Surf. Sci. 541, 101 (2003);
K. Reuter and M. Scheffler,
Appl. Phys. A 78, 793 (2004)
 - clean (meV/Å2)
(slab) – N E bulk )/A – N  /A
  ( Esurf.
M M
O O
Vibrational contributions to the surface free energy
Fvib(T,V) =  d Fvib(T,)  ()
30
 vib = Fvib/A =
 (meV)
= 1/A  d Fvib(T,) [ surf.( ) - NRu bulk ( ) ]
20
Only the vibrational changes at the surface
contribute to the surface free energy
10
0
Pd bulk
 Use simple models for order of magnitude estimate
e.g. Einstein model:  ( ) =  ( - )
Pd(bulk) ~ 25 meV
Surface induced variations of substrate modes
20
vib (meV/Å2)
10
+ 50%
0
- 50%
-10
-20
0
200
400
600
800
Temperature (K)
< 10 meV/Å2 for T < 600 K - in this case!!!
1000
Surface functional groups
H2O
 vib. (meV/Å2)
40
30
20
OH
10
0
0
200
400
600
800
1000
Temperature (K)
Q. Sun, K. Reuter and M. Scheffler,
Phys. Rev. B 67, 205424 (2003)
 (meV/Å2)
Configurational entropy and phase transitions
clean surface
No lateral interactions:
Fconf = kBT ln (N+n)! / (N!n!)
< (O) >
1.0
T = 300 K
T = 600 K
Langmuir adsorption-isotherm
0.5
<(Ocus)> =
0.0
-1.5
1
-1.0
O (eV)
-0.5
1 + exp((Ebind - O)/kBT)
Configurational entropy smears out phase transitions
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)
Configuration space and configurational free energy
Fconf = - kBT ln( Zconf )
Canonic ensemble (constant temperature):
Partition function
Z = Z(T,V) = i exp(-Ei / kBT)  Boltzmann-weighted sum
over all possible system states
- 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)
Evaluating high-dimensional integrals:
Monte Carlo techniques
<A> = 1/Z ∫ drN A(r1,r2,…,rN) exp( -E(r1,r2,…,rN) / kBT)
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
Alternative:
- random sampling
(Monte Carlo)
Example for integration
by simple sampling
Finding a needle in a haystack: Importance sampling
<A> = 1/Z ∫ drN A(r1,r2,…,rN) exp( -E(r1,r2,…,rN) / kBT)
Measuring the depth of the Nile
Thank you,
Daan!!!
- 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!!
Specifying “getting out of the water”:
The Metropolis algorithm
?
?
Etrial < Epresent:
Etrial > Epresent:
Some remarks:
accept
accept with probability exp[- (Etrial-Epresent) / kBT ]
- 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 structures
ordered structures
(arbitrary periodicity)
How can we then sample
the configuration space?
BUT: only periodic structures accessible to direct DFT,
and supercell size quite limited
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 the
lateral 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:
3
- Compute many ordered structures
3
- Write total energy as LGH expansion,
e.g.
E(3x3) = 2Eo + 2V1pair + 2V3pair
- Set up system of linear equations
- „Invert“ to get lateral interactions
e.g.
O / Ru(0001)
 (meV/Å2)
clean
cleanclean
p(2x2)
p(2x1)
p(2x1)
T (K)
disordered
lattice-gas
p(2x1)
p(2x2)
O (eV)
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.
Ab initio atomistic thermodynamics
Use DFT in the computation of free energies
Suitably exploit equilibria and concept of reservoirs
 allows any general thermodynamic reasoning
concentration of point defects at finite T
surface structure and composition in
realistic environments
 major limitations
Vxc vs. kBT
sampling of configurational space
„only“ equilibrium
Lecture 2 tomorrow: kinetics, time scales