Computing free energy: thermodynamic integration(s) · (in this page, Helmholtz free energy,...

Computingfree energy:

thermodynamic integration(s)

Extending the scale

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 functionsK. 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

{Ri}

E

Potential Energy Surface: {Ri}

(3N+1)dimensional

109

106

103

1

1015 109 103 1

Length(m)

Time (s)

Microscopicregime

Mesoscopicregime

Macroscopicregime

few processes

few atoms

many atoms

many processes

continuum

average overall processes

more deta

ils

more proce

sses

Thermodynamics:p, T, V, N

(in this page, Helmholtz free energy, F(N,V,T))

if we can calculate E and write analytically on approximation for S for our system, we use this expression. Example: ab initio atomistic thermodynamics.

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.

Thermodynamics

Thermodynamic Integration

Ab initio

Free energy, one quantity, many definitions

Classical statistics (for nuclei):

● Fundamental statistical mechanics thermodynamics link↔

Ab initioAb initio

● Probabilistic interpretation of free energy

Free energy, one quantity, many definitions

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

Phase diagram of ZrO2

The harmonic solid

The harmonic solid: low temperature phases of ZrO2

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

The problem of free energy sampling

Solutions:

1. “physical”-path thermodynamics integration

2. “unphysical”-path thermodynamics integration (after Kirkwood)

The problem of free energy sampling

Free energy “physical”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

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

Reference phasesLiquid: Lennard Jones

Considered phases: diamond, graphite, and liquid(s)

σ, ε ? Maximum resemblance of LJ liquid and “real”: alignment radial distribution function peaks


α? Maximum resemblance of harmonic and “real” potential


Reference phasesSolid(s): Einstein solid

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

Pressure [GPa]

Difference in slopes: difference in specific volumes

Case study: equating Gibbs free energies

Alterative method for finding phase coexistence via F(V)

Notable cases (at 0 K): Silicon (1980)

Yin and Cohen, PRL 1980DFT 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

De Koning et al. PRL 83: 3973 (1999)

time ?!? reversible?

Sparing CPU time: adiabatic switch

Silicon (classical potential)

Sparing CPU time: adiabatic switch

Too fast switch:Not reversible

Wang et al. PRL 95, 185701 (2005)

Ab initio diamond melting line

Beyond equilibrium: Jarzynski theorem

Clausius inequality:

Jarzynski equality (1997!)

Microscopic version of I principle

Proof:

Jarzynski theorem

Work (second term) depends on the trajectory, ID EST, on the initial point

Average over ensemble of initial points.

Taking:

Ensemble average of :

If Hamiltonian dynamics:

Change of variable (Hamiltonian trajectory) →

Conservation of the phase space volume!

By noting:

Jarzynski theorem

Jarzynski theorem: steered dynamics

Inefficient because:

Better estimated with cumulant:

● Harmonic free energy (analytic)

● When non harmonic: thermodynamic integration, along “physical” or “unphysical” paths

● Construction of accurate phase diagrams

● Speeding up: adiabatic switch

● Faster, non equilibrium: Jarzynski equality

Summary

Computing free energy: thermodynamic integration(s) · (in this page, Helmholtz free energy,...

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