Computing free energy:Replica exchange

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: finding the optimal dimensionality reduction

Outline

T1

T2

T3

T4

Configurational coordinate

Ene

rgy

Parallel tempering: the concept

Exchange rule, ensuring canonical sampling at all temperatures:

Parallel tempering: the concept

Parallel tempering: the conceptParallel tempering: the conceptParallel tempering: the concept

Parallel tempering: the conceptParallel tempering: the concept

Overlap necessary: the smaller (the system size) the better

Swap 2Swap 1M

D o

r M

C

run

1

MD

or

MC

ru

n 2

T3

T1

MD

or

MC

ru

n 3

T2

T4

T5

Parallel tempering: the implementationParallel tempering: the conceptParallel tempering: the implementation

Parallel tempering: monitoringParallel tempering: the implementationParallel tempering: the conceptParallel tempering: monitoring

Pote

ntia

l Ene

rgy 

(eV

)

 (degree)

  100—620 K  100 K PT (100 ps)  100 K serial (100 ps)

Au4: coexistence of several isomers

0.00 eV

0.04 eV

0.36 eV

Parallel Tempering

biasing potential

means sampling  according to distribution:

Replica exchange: other than temperature

Replica exchange: cluster size

Replica exchange: temperature – weighted histogram analysis

such that

Assuming each count in each histogram as independent, then likelihood of observing the ith histrogram:

Replica exchange: temperature – weighted histogram analysis

If all histograms are independent:

Maximum likelihood estimate:

Angle [degree]Angle [degree]

Free

 Ene

rgy 

[eV

]Pr

obab

ility Pa

rtit

ion 

Func

tion

 (i

nteg

rate

d pr

obab

ility

)

0.00 eV 0.04 eV

Au4, relative population

Free Energy (Landau): F/kBT = ­ ln [ P (Q) ] 

Path collective variables

Path collective variables

The quest for the “right variable(s)”:Sketch maps

(thanks to Michele Ceriotti, Oxford, for providing figures)

Describing structural complexity

Dimensionality reduction

Dimensionality reduction

Dimensionality reduction

Dimensionality reduction

Dimensionality reduction

(Non linear) dimensionality reduction

(Non linear) dimensionality reduction

(Non linear) dimensionality reduction

(Non linear) dimensionality reduction

(Non linear) dimensionality reduction

Proximity matching

Proximity matching

Proximity matching

Proximity matching

Sketch­map algorithm (multidimensional scaling)

Sketch­map algorithm

Sketch­map algorithm

Minimization of the stress function (for a set of landmarks points)

Ala12 landscape

24D Gaussian of stdev 0.5

24D uniform distribution Distribution between pair of ala2 configurations

A simplified model for ala12 landscape

Projecting the model FES

Projecting the model FES

Projecting the model FES

Sketch map of folded lanscape of ala12

Sketch map of folded lanscape of ala12

Is point­wise (relative) free energy invariant upon dimensionality reduction?No, only for regions:

Sketch map of folded lanscape of ala12

Accelerating rare events

Accelerating rare events

Accelerating rare events

Accelerating rare events

Accelerating rare events

Accelerating rare events

Sketch­map based metadynamics

Sketch­map based metadynamics

Sketch­map based metadynamics

Discontinuous trajectories?

Discontinuous trajectories?

Discontinuous trajectories?

Field representation

This field replaces the usual representation based on d­dimensional points x. The overlap between fields, which measures their similarity, replaces the distance.

Field representation

Field representation

Field overlap metadynamics

Field overlap metadynamics

Field overlap metadynamics

From clusters to defects in bulk

From clusters to defects in bulk

From clusters to defects in bulk