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Published on October 23, 2007

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Bridging scales: Ab initio atomistic thermodynamics :  Bridging scales: Ab initio atomistic thermodynamics Karsten Reuter Fritz-Haber-Institut, Berlin Slide2:  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 Drawback: - 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 functions K. Reuter, C. Stampfl and M. Scheffler, in: Handbook of Materials Modeling Vol. 1, (Ed.) S. Yip, Springer (Berlin, 2005). Slide3:  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:  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 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 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 Link to statistical mechanics:  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 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:  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 )N X 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:  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) Computation of free energies: ideal gas III:  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 + Δμ(T,p) 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:  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 differences use simple models to approx. Fvib (Debye, Einstein)  Solids (low T): G(T,p) ~ Etot + Fconf Slide10:  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:  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). How large is it? Internal energy: Etot = n ED N sites, n defects (n <<N) Minimize free energy: (G/n)T,p = /nT,p (Etot – Fconf + pV) = 0 Slide12:  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:  Surface thermodynamics A surface can never be alone: there are always “two sides” to it !!! solid – gas solid – liquid solid – 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:  Example: Surface in contact with oxygen gas phase O2 gas surface bulk  surf. = 1/A [ Gsurf.(NO, NM) – NO O - NM M ] Slide15:  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) p(2x2) O/Pd(100) (√5 x √5)R27° PdO(101)/Pd(100) Vibrational contributions to the surface free energy:  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:  ( ) =  ( - ) Surface induced variations of substrate modes:  Surface induced variations of substrate modes < 10 meV/Å2 for T < 600 K - in this case!!! Surface functional groups:  Surface functional groups Q. Sun, K. Reuter and M. Scheffler, Phys. Rev. B 67, 205424 (2003) Configurational entropy and phase transitions:  clean surface O(1x1) Configurational entropy and phase transitions Slide20:  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:  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 ) Slide22:  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) Slide23:  Finding a needle in a haystack: Importance sampling <A> = 1/Z ∫ drN A(r1,r2,…,rN) exp( -E(r1,r2,…,rN) / kBT) Slide24:  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::  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 Slide26:  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) BUT: only periodic structures accessible to direct DFT, and supercell size quite limited How can we then sample the configuration space? Slide27:  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? Slide28:  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 + 2V3pair Set up system of linear equations „Invert“ to get lateral interactions 3 3 Slide29:  O (eV)  (meV/Å2) In short::  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. Slide31:   allows any general thermodynamic reasoning concentration of point defects at finite T surface structure and composition in realistic 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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