Computing free energy: Replica exchange

Extending the scale Length (m) 1 10 3 Potential Energy Surface: {Ri} 10 6 (3N+1) dimensional 10 9 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 or m Microscopic regime few processes 10 15 {Ri} 10 9 10 3 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 thermodynamic integration 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

Outline Free energy evaluation: Harmonic approximation (solids) Thermodynamic integration. Phase diagrams Thermodynamics perturbation (overlap, umbrella sampling) Accelerated sampling, metadynamics. Replica Exchange MD: finding the optimal dimensionality reduction

Energy Parallel tempering: concept Parallel the tempering: the concept T4 T3 T2 T1 Configurational coordinate Exchange rule, ensuring canonical sampling at all temperatures:

Parallel tempering: concept Parallel the tempering: the concept

Parallel tempering: concept Parallel the tempering: the concept Overlap necessary: the smaller (the system size) the better

T4 T3 T2 T1 Swap 2 MD or MC run 3 Swap 1 MD or MC run 2 T5 MD or MC run 1 Parallel tempering: the implementation Parallel Parallel tempering: tempering: the implementation the concept

Parallel tempering: monitoring Parallel Parallel tempering: tempering: the implementation the monitoring concept

Au4: coexistence of several isomers Parallel Tempering 0.00 ev 0.04 ev 0.36 ev Potential Energy (ev) 100 620 K 100 K PT (100 ps) 100 K serial (100 ps) (degree)

Replica exchange: other than temperature biasing potential means sampling according to distribution:

Replica exchange: cluster size

Replica exchange: temperature weighted histogram analysis such that

Replica exchange: temperature weighted histogram analysis Assuming each count in each histogram as independent, then likelihood of observing the ith histrogram: If all histograms are independent: Maximum likelihood estimate:

Au4, relative population Free Energy (Landau): F/kBT = ln [ P (Q) ] 0.04 ev Probability Partition Function (integrated probability) Free Energy [ev] 0.00 ev Angle [degree] Angle [degree]

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

(Non linear) dimensionality reduction

Proximity matching

Sketch map algorithm (multidimensional scaling)

Sketch map algorithm

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

Ala12 landscape 24D Gaussian of stdev 0.5 Distribution between pair of ala2 configurations 24D uniform distribution

A simplified model for ala12 landscape

Projecting the model FES

Sketch map of folded lanscape of ala12

Sketch map of folded lanscape of ala12 Is point wise (relative) free energy invariant upon dimensionality reduction? No, only for regions:

Sketch map of folded lanscape of ala12

Accelerating rare events

Sketch map based metadynamics

Discontinuous trajectories?

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

Field representation

Field overlap metadynamics

From clusters to defects in bulk