Computing free energy: Replica exchange

Transcription

1 Computing free energy: Replica exchange

2 Extending the scale Length (m) 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 {Ri} 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).

3 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.

4 Free energy, one quantity, many definitions Fundamental statistical mechanics thermodynamics link Classical statistics (for nuclei): Ab initio Probabilistic interpretation of free energy Ab initio

5 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

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

7 Parallel tempering: concept Parallel the tempering: the concept

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

9 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

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

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

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

13 Replica exchange: cluster size

14 Replica exchange: temperature weighted histogram analysis such that

15 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:

16 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]

17 Path collective variables

18 Path collective variables

19 The quest for the right variable(s) : Sketch maps (thanks to Michele Ceriotti, Oxford, for providing figures)

20 Describing structural complexity

21 Dimensionality reduction

22 Dimensionality reduction

23 Dimensionality reduction

24 Dimensionality reduction

25 Dimensionality reduction

26 (Non linear) dimensionality reduction

27 (Non linear) dimensionality reduction

28 (Non linear) dimensionality reduction

29 (Non linear) dimensionality reduction

30 (Non linear) dimensionality reduction

31 Proximity matching

32 Proximity matching

33 Proximity matching

34 Proximity matching

35 Sketch map algorithm (multidimensional scaling)

36 Sketch map algorithm

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

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

39 A simplified model for ala12 landscape

40 Projecting the model FES

41 Projecting the model FES

42 Projecting the model FES

43 Sketch map of folded lanscape of ala12

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

45 Sketch map of folded lanscape of ala12

46 Accelerating rare events

47 Accelerating rare events

48 Accelerating rare events

49 Accelerating rare events

50 Accelerating rare events

51 Accelerating rare events

52 Sketch map based metadynamics

53 Sketch map based metadynamics

54 Sketch map based metadynamics

55 Discontinuous trajectories?

56 Discontinuous trajectories?

57 Discontinuous trajectories?

58 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.

59 Field representation

60 Field representation

61 Field overlap metadynamics

62 Field overlap metadynamics

63 Field overlap metadynamics

64 From clusters to defects in bulk

65 From clusters to defects in bulk

66 From clusters to defects in bulk