Theoretical Concepts for Chemical Energy Conversion

Transcription

1 Theoretical Concepts for Chemical Energy Conversion processes and functional materials Luca M. Ghiringhelli Fritz Haber Institute, Theory group held during summer semester 2014 at Technische Universität Berlin

2 Technicalities on the course Course material (slides and list of suggested textbooks): berlin.mpg.de/~luca/tccec.html

3 Where/Why do we need new materials? and more...

4 Chemical energy conversion: catalysis Reactant(s) Non-catalytic free-energy barrier Free energy ΔFnon-cat Reaction Product(s) ΔFcat Adsorption Desorption Reaction coordinate Issues: Reaction rate: proportional to exp ( ΔF / kt) Selectivity: eliminate or at least reduce the undesired products

5 Veracity and reliability The example of heterogeneous catalysis: A catalyst usually gets active after a macroscopic induction time. Thus: We introduce a material, but the material that exists at the steady state of catalysis may be different from the one that was introduced. Which material is formed and active at reactive conditions?

6 Veracity: Some Words about Theory We don t (necessarily) want to identify THE key novel material, but just the most promising ones. And we want to identify anomalies in materials properties and functions that fall outside the established trends. How good is our theory? We don t want to miss the best candidate. Calculation of the function of materials typically demands multiscale modeling. We need error controlled links between the different simulation techniques We need a reliable base! If the electronic structure theory base is not reliable, everything that follows may be wrong, and we may miss the key issue.

7 Current state of the art in atomistic modeling Full CI Accuracy, Reliability, and Predictive Power Beyond independent electrons (MP2, RPA, CCSD(T),...) Computational Cost Density functional theory with (semi) local and hybrid functionals Semi empirical methods (AM1, PM6, CNDO, tight binding) Empirical potentials ( force fields ) (no explicit electrons)

8 Realistic predictions: bulk oxide vs surface oxide, Pd (100) in-situ SXRD Theory metal E. Lundgren et al., Phys. Rev. Lett. 92, (2004) ) 2 ) p(2 2 p(2 5 )R 27 ( 5 5)R27 ( 5 bu lk ox id e bulk oxide metal

9 Realistic predictions: CO oxidation on RuO2 (110) Obr / - Obr / COcus ΔμCO (ev) Obr / Ocus CObr / COcus ΔμO (ev) K. Reuter and M. Scheffler, Phys. Rev. Lett. 90, (2003); Phys. Rev. B 68, (2003)

10 Realistic predictions: CO oxidation on RuO2 (110), kinetic

11 Realistic predictions: CO oxidation on RuO2 (110), kinetic

12 Ab initio iron melting line: Earth core Alfè et al. Nature (1999)

13 Ab initio diamond melting line Wang et al. PRL 95, (2005)

14 Diamond nucleation on white dwarfs Ghiringhelli et al., PRL (2007). Discovery of V886 Centauri (BPM 37093), later PSR J b, 55 Cancri e

15 Ab initio crystal structure trasformations in SiO2 Martonak et al. Nature Materials 5, 623 (2006)

16 Topics 1. Recapitulation of key concepts in thermodynamics and statistical mechanics. 2. Introduction to importance sampling Metropolis Monte Carlo, canonical ensemble and more. 3. Introduction to (classical) molecular dynamics, microcanonical ensemble and more. 4. Ab initio molecular dynamics: Born Oppenheimer molecular dynamics and beyond (state of the art) 5. Ab initio atomistic thermodynamics: phase diagrams. Application to surface/cluster corrosion and reactivity of realistic materials. Neutral and charged defects in semiconductors. 6. Stochastic sampling of the Schrödinger equation: quantum Monte Carlo. Theory and application to realistic materials.

17 Topics 7. Sampling free energy I: (ab initio) phase diagrams. Thermodynamics integration, smart advanced techniques, and applications to realistic materials. 8. Sampling free energy II: enhanced sampling (biased sampling, metadynamics, and more). Theory and application to realistic materials. 9. Sampling free energy III: replica exchange, the problem of the choice of order parameters and reaction coordinates (from many to few relevant dimensions). Theory and application to realistic materials. 10. Chemical reactions as rare events: transition state theory and beyond. Methods (transition path sampling, transition interface sampling, and more) and application to realistic materials. 11. Stochastic sampling beyond equilibrium: ab initio kinetic Monte Carlo. Theory and application to realistic materials. 12. Multiscale approaches. QM/MM, adaptive schemes and beyond. Theory and application to realistic materials. 13. Materials discovery. The quest for descriptors.

18 Recapitulation of useful concepts from Thermodynamics and Statistical Mechanics

19 Thermodynamics, 0th and 1st principle

20 Thermodynamics, 2nd principle

21 Thermodynamics, reversible engine, entropy Another equivalent formulation of the 2nd principle tells us that Any spontaneous change in a closed system (i.e. a system exchanging neither heat nor particles with its environment) can never lead to a decrease of entropy.

22 Thermodynamics, entropy, generalized 1st principle No work done on the system: Entropy is extensive (weakly coupled systems) 1st principle, rewritten: Everything we do not know: lack of information

23 Thermodynamics, free energy Combining 1st and 2nd principles: Let's define: It means that, at constant N and T, the maximum amount of work that the system can do ( δw ) equals (the negative of) the change in free energy F. Hence the name free energy: the part of internal energy that is actually available to produce work.

24 Extending the scale Thermodynamics: p, T, V, N Length (m) 1 10 e or m -3 Potential Energy Surface: {Ri} 10-6 (3N+1) dimensional 10-9 E continuum average over all processes many atoms Mesoscopic regime few atoms many processes Microscopic regime few processes {Ri} d Macroscopic regime ils a et o m 1 pr e r s se s e oc Time (s) Essentials of computational chemistry: theories and models. 2nd edition. C. J. Cramer, JohnWiley and Sons Ltd (West Sussex, 2004). 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). berlin.mpg.de/th/paper.html

25 Statistical mechanics, microcanonical ensemble System at constant energy U, consisting of N particles in a box of volume V. If known, we can solve the equations of motion. This is useless: we are more interested in average properties of the system than detailed properties. Hypothesis: U, assuming a particular value E, is all we need to know about the system (together with N and V ) to describe the equilibrium state. We call the set of all state at energy E, the microcanonical ensemble. Defining Ω(E): the number of states between E and E + δe: Ensemble average: All state with a fixed energy E are equally probable

26 Statistical mechanics, properties of high dimensional spaces Properties of Ω(E): Number of states between 0 and E Energy per degree of freedom Number of states between 0 and ε α of order 1, e.g. free particle: E.g., D dimensional hypersphere harmonic oscillator:

27 Statistical mechanics, definition of temperature Two systems, E1 + E2 = E Probability that 1 is in state i : At equilibrium:

28 Statistical mechanics, definition of temperature Definition of entropy in statistical mechanics: S is maximum at equilibrium and extensive Invoking thermodynamics: Systems 1 and 2 at equilibrium have the same temperature

29 Statistical mechanics, energy distribution

30 Statistical mechanics, energy distribution

31 Statistical mechanics, the canonical ensemble Continuum: Continuum: Partition function Z

32 Statistical mechanics, quantities derived from Z Average energy: Heat capacity:

33 Statistical mechanics, generalized forces

34 Statistical mechanics: thermodynamic meaning of Z

35 Statistical mechanics: free energy, alternative derivation

36 Statistical mechanics: free energy, alternative derivation Important derivative of F: Configurational partition function: factorizing momenta Thermal length: Partition function of ideal gas:

37 Statistical mechanics: free energy as a probabilistic concept Energy: mapping from 3N coordinates into one scalar so that: Formally:

38 Free energy, one quantity, many definitions (in this page, Helmholtz free energy, F(N,V,T)) Thermodynamics Ab initio if we can calculate E and write analytically on approximation for S for our system, we use this expression. Example: ab initio atomistic thermodynamics. Thermodynamic Integration Ab initio or similar derivatives that yield measurable quantities (in a computer simulation): one can estimate the free energy by integrating such relations. This is the class of the so called thermodynamic integration methods.

39 Free energy, one quantity, many definitions Fundamental statistical mechanics thermodynamics link Classical statistics (for nuclei): Ab initio Probabilistic interpretation of free energy Ab initio

40 Statistical mechanics: constant pressure ensemble R for reservoir Evaluation of pressure

41 Statistical mechanics: NPT and grand canonical ensembles NPT Grand canonical

42 Statistical mechanics: which ensemble? Langmuir adsorption: N particles in M sites (M is like a volume) NMT : μmt