Download Computing free energy: thermodynamic integration(s)

Computing
free energy:
thermodynamic integration(s)
Extending the scale
Length
(m)
1
10­3
Potential Energy Surface: {Ri}
10­6
(3N+1)­dimensional
10­9
E
Thermodynamics:
p, T, V, N
continuum
ls
Macroscopic
i
a
t
e
regime
d
e
average over
or
m
all processes
many atoms
es
Mesoscopic
s
s
e
c
regime
ro
p
many processes
e few atoms
r
o
m
Microscopic
regime few processes
10­15
{Ri}
10­9
10­3
1
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.fhi-berlin.mpg.de/th/paper.html
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):
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Ab initio
Probabilistic interpretation of free energy
Ab initio
Outline
Free­energy evaluation: ­ Harmonic approximation (solids)
­ Thermodynamic integration. Phase diagrams
­ Thermodynamics perturbation (overlap, umbrella sampling)
­ Accelerated sampling, metadynamics.
­ Replica Exchange MD, free energy from probability
Outline
Free­energy evaluation: ­ Harmonic approximation (solids)
­ Thermodynamic integration. Phase diagrams
­ Thermodynamics perturbation (overlap, umbrella sampling)
­ Accelerated sampling, metadynamics.
­ Replica Exchange MD, free energy from probability
Phase diagram of ZrO2
Phase diagram of ZrO2
Phase diagram of ZrO2
The harmonic solid
The harmonic solid
The harmonic solid
The harmonic solid: low temperature phases of ZrO2
Phase diagram of ZrO2
Phase diagram of ZrO2
Phase diagram of ZrO2
Phase diagram of ZrO2
Outline
Free­energy evaluation: ­ Harmonic approximation (solids)
­ Thermodynamic integration. Phase diagrams
­ Thermodynamics perturbation (overlap, umbrella sampling)
­ Accelerated sampling, metadynamics.
­ Replica Exchange MD, free energy from probability
The problem of free energy sampling
The problem of free energy sampling
Solutions:
1. “physical”-path thermodynamics integration
2. “unphysical”-path thermodynamics integration
(after Kirkwood)
Free energy “physical”­path thermodynamics integration
Free energy “physical”­path thermodynamics integration
Free energy “unphysical”­path thermodynamics integration
Free energy “unphysical”­path thermodynamics integration
Free energy “unphysical”­path thermodynamics integration
Example of application: ­ ZrO2 phase diagram (only temperature)
­ Carbon (temperature, pressure) phase diagram
Phase diagram of ZrO2
Phase diagram of ZrO2
Case study: phase diagram of pure carbon
Road map:
●
●
●
Calculation of change of Helmoltz free energy from chosen reference state to a particular (T,p) point, for each involved phase (overlooked phases?), by means of thermodynamics integration.
Search for of all coexistence points at a given T between all pairs of phases, via integration of equations of state P(ρ) and evaluation of crossing points (alternative: common tangent construction). Prolongation of coexistence line by Gibbs­Duhem integration
Case study: phase diagram of pure carbon
Considered phases: diamond, graphite, and liquid(s)
Reference phases
Liquid: Lennard Jones
σ, ε ? Maximum resemblance of LJ liquid and
“real”: alignment radial distribution function
peaks
Case study: phase diagram of pure carbon
Considered phases: diamond, graphite, and liquid(s)
Reference phases
Liquid: Lennard Jones
σ, ε ? Maximum resemblance of LJ liquid and
“real”: alignment radial distribution function
peaks
Case study: phase diagram of pure carbon
Reference phases
Solid(s): Einstein solid
α? Maximum resemblance of harmonic and “real” potential
Case study: λ–ensemble sampling and integration
Some gymnastic: Gibbs free energy as function of density
Dependence of a thermodynamic potential (Gibbs free energy density, or chemical potential) from a thermodynamic variable? look at the derivatives
By integrating:
With the ansatz:
Case study: P(ρ) equations of state
Case study: equating Gibbs free energies
Difference in slopes: difference in specific volumes
Pressure [GPa]
Alterative method for finding phase coexistence via F(V)
Notable cases (at 0 K): Silicon (1980)
Yin and Cohen, PRL 1980
DFT with LDA functional
Notable cases (at 0 K): Cerium (2013)
Casadei et al. PRL (2013)
From one coexistence point to coexistence line
Gibbs­Duhem (Clausius­Clapeyron) integration :
Latent heat
Clausius­Clapeyron equation? (I)
specific entropy
specific volume
two phases
?
(I principle)
Clausius­Clapeyron equation? (II)
Gibbs­Duhem relation
From one coexistence point to coexistence line
Sparing CPU time: adiabatic switch
time ?!?
reversible?
De Koning et al. PRL 83: 3973 (1999)
Sparing CPU time: adiabatic switch
Too fast switch:
Not reversible
Silicon (classical potential)
Ab initio diamond melting line
Wang et al. PRL 95, 185701 (2005)
Beyond equilibrium: Jarzynski theorem
Clausius inequality:
Jarzynski equality (1997!)
Proof:
Microscopic version of I principle
Jarzynski theorem
Work (second term) depends on the trajectory, ID EST, on the initial point If Hamiltonian dynamics:
Average over ensemble of initial points. Taking:
Ensemble average of
:
Jarzynski theorem
Change of variable (Hamiltonian trajectory)
Conservation of the phase space volume!
By noting:
→ Jarzynski theorem: steered dynamics
Inefficient because:
Better estimated with cumulant:
Summary
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Harmonic free energy (analytic)
When non harmonic: thermodynamic integration, along “physical” or “unphysical” paths
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Construction of accurate phase diagrams
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Speeding up: adiabatic switch
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Faster, non equilibrium: Jarzynski equality