Bridging scales: Ab initio atomistic thermodynamics

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Bridging scalesAb initio atomistic
thermodynamics

• Karsten Reuter
• Fritz-Haber-Institut, Berlin

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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
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
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• 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)
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Thermodynamics in a nutshell
Internal energy (U) Etot(S,V) Enthalpy H(S,p)
Etot pV (Helmholtz) free energy F(T,V) Etot
- TS Gibbs free energy 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
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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 )

• 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(-EiB /
  kBT) ) ZA ZB
• ? Ftotal FA FB e.g. electronic ? nuclear
  (Born-Oppenheimer)
• rotational ? vibrational

• N indistinguishable, independent
  particles Ztotal 1/N! (Zone particle)N

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Computation of free energies ideal gas I
X
Z 1/N! ( Znucl 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) Typical excitation energies eV gtgt
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(-hk2
/ 2mkBT) Particle in a box of length L V1/3
(L??) ? Ztrans ? V ( 2? mkBT / h2
)3/2 Required input Particle mass m
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Computation of free energies ideal gas II
iii) Rotational free energy Zrot ?J
(2J1)exp(-J(J1)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)
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Computation of free energies ideal gas III

• O2 CO
• m (amu) 32 28
• ?stretch (meV) 196 269
• Bo (meV) 0.18 0.24
• ? 2 1
• Ispin 3 1

? ? ?
µ µ(T,p) Etot ?µ(T,p)
Alternatively ??(T, p) ??(T, po) kT
ln(p/po) and ??(T, po 1 atm) tabulated in
thermochem. tables (e.g. JANAF)
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Computation of free energies solids
G(T,p) Etot Ftrans Frot Fvib
Fconf pV
Ftrans Translational free energy
Frot Rotational free energy pV V V(T,p) from
equation of state, varies little Fconf Configurat
ional free energy Etot Internal
energy Fvib Vibrational free energy
Etot, Fvib use differences use simple models
to approx. Fvib (Debye, Einstein) ? Solids (low
T) G(T,p) Etot Fconf
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II. Starting simple Equilibrium concentration of
point defects
Solid State Physics, N.W. Ashcroft and N.D.
Mermin, Holt-Saunders (1976)
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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 gt 0). How large is it?
N sites, n defects (n ltltN)
Internal energy Etot n ED
Minimize free energy (?G/?n)T,p ?/?nT,p (Etot
Fconf pV) 0
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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)
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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 Total system (with
surface) GIII GI GII ?Gsurf
Phase II
(T,p)
Phase I
?A
? 1/A ( GIII - ?i Ni ?i )
Surface tension (free energy per area)
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Example Surface in contact with oxygen gas phase
? surf. 1/A Gsurf.(NO, NM) NO ?O - NM ?M
O2 gas
surface
bulk
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Oxide formation on Pd(100)
p(2x2) O/Pd(100)
(v5 x v5)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)
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Vibrational contributions to the surface free
energy
Fvib(T,V) ? d? Fvib(T,?) ? (?)

• Use simple models for order of magnitude
  estimate
• e.g. Einstein model ? (? ) ? (? - ?)

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Surface induced variations of substrate modes
lt 10 meV/Å2 for T lt 600 K - in this case!!!
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Surface functional groups
Q. Sun, K. Reuter and M. Scheffler, Phys. Rev. B
67, 205424 (2003)
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Configurational entropy and phase transitions
clean surface
O(1x1)
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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)
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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
• ltAgt 1/Z ? drN A(r1,r2,,rN) exp(
  -E(r1,r2,,rN) / kBT)

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Evaluating high-dimensional integrals Monte
Carlo techniques
ltAgt 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

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Finding a needle in a haystack Importance
sampling
ltAgt 1/Z ? drN A(r1,r2,,rN) exp(
-E(r1,r2,,rN) / kBT)
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Specifying getting out of the waterThe
Metropolis algorithm
Etrial lt Epresent accept Etrial gt
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)
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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

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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
ordered structures (arbitrary periodicity)
disordered structures
How can we then sample the configuration space?
BUT only periodic structures accessible to
direct DFT, and supercell size quite limited
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Lattice gas Hamiltonians / Cluster expansions
Expand total energy of arbitrary configuration in
terms of lateral interactions
Elatt åi Eo 1/2 åi,j Vpair(dij) si sj 1/3
åi,j,k Vtrio(dij,djk,dki) si sj sk

• 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?
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LGH parametrization through DFT
Since isolated clusters not compatible with
supercell approach, exploit instead the
interaction with supercell images in a
systematic way
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• Compute many ordered structures
• Write total energy as LGH expansion,
• e.g.
• E(3x3) 2Eo 2V1pair 2V3pair
• Set up system of linear equations
• Invert to get lateral interactions

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? (meV/Å2)
??O (eV)
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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.

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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