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
Technicalities on the course
Course material (slides and list of suggested textbooks):
http://www.fhi­berlin.mpg.de/~luca/TCCEC.html
Where/Why do we need new materials?
and more...
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
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?
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 error­controlled links between the different simulation techniques
We need a reliable base!
If the electronic­structure theory base is not reliable, everything that follows may be wrong, and we may miss the key issue.
Current state of the art in atomistic modeling Full
CI
Accuracy,
Reliability,
and Predictive
Power
Beyond independent electrons (MP2, RPA, CCSD(T),...)
Computational Cost
Density­functional theory with (semi)­local and hybrid functionals
Semi­empirical methods (AM1, PM6, CNDO, tight­binding)
Empirical potentials (“force fields”)
(no explicit electrons)
Realistic predictions: bulk oxide vs surface oxide, Pd (100)
in-situ SXRD
Theory
metal
E. Lundgren et al., Phys. Rev. Lett. 92, 046101 (2004)
)
×2
)
p(2
2
×
p(2
√5
)R
27
°
(√5 × √5)R27 °
(√5
×
bu
lk
ox
id
e
bulk oxide
metal
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, 046103 (2003); Phys. Rev. B 68, 045407 (2003)
Realistic predictions: CO oxidation on RuO2 (110), kinetic
Realistic predictions: CO oxidation on RuO2 (110), kinetic
Ab initio iron melting line: Earth core
Alfè et al. Nature (1999)
Ab initio diamond melting line
Wang et al. PRL 95, 185701 (2005)
Diamond nucleation on … white dwarfs
Ghiringhelli et al., PRL (2007).
Discovery of V886 Centauri (BPM 37093), later PSR J1719­1438 b, 55 Cancri e Ab initio crystal structure trasformations in SiO2
Martonak et al. Nature Materials 5, 623 (2006)
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: Born­Oppenheimer molecular dynamics and beyond (state­of­the­art)
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.
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.
Recapitulation of useful concepts from Thermodynamics
and
Statistical Mechanics
Thermodynamics, 0th and 1st principle
Thermodynamics, 2nd principle
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. 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
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.
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
10-15
{Ri}
d
Macroscopic
regime
ils
a
et
10-9
10-3
o
m
1
r
p
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). http://www.fhi­berlin.mpg.de/th/paper.html
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
Statistical mechanics, properties of high­dimensional 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: harmonic oscillator:
E.g., D­dimensional hypersphere
Statistical mechanics, definition of temperature
Two systems, E1 + E2 = E
Probability that 1 is in state i : At equilibrium:
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
Statistical mechanics, energy distribution
Statistical mechanics, energy distribution
Statistical mechanics, the canonical ensemble
Continuum:
Continuum:
Partition function Z
Statistical mechanics, quantities derived from Z
Average energy:
Heat capacity:
Statistical mechanics, generalized forces
Statistical mechanics: thermodynamic meaning of Z
Statistical mechanics: free energy, alternative derivation
Statistical mechanics: free energy, alternative derivation
Important derivative of F:
Configurational partition function: factorizing momenta
Thermal length:
Partition function of ideal gas:
Statistical mechanics: free energy as a probabilistic concept
Energy: mapping from 3N coordinates into one scalar
so that:
Formally:
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 thermodynamic­integration methods.
Free energy, one quantity, many definitions
●
Fundamental statistical mechanics ↔ thermodynamics link
Classical statistics (for nuclei):
●
Ab initio
Probabilistic interpretation of free energy
Ab initio
Statistical mechanics: constant pressure ensemble
R for “reservoir”
Evaluation of pressure
Statistical mechanics: NPT and grand canonical ensembles
NPT
Grand­canonical
Statistical mechanics: which ensemble?
Langmuir adsorption: N particles in M sites (M is like a volume)
NMT
:
μMT