Computing free energy: Thermodynamic perturbation and beyond

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

Thermodynamic perturbation If poor overlap: sequence of systems F

Chemical potential

(Widom's) test particle insertion

Hard spheres

Overlapping distribution

Non Boltzmann sampling

Umbrella sampling

Statistical mechanics: free energy as a probabilistic concept Energy: mapping from 3N coordinates into one scalar so that: Formally:

Free energy à la Landau

Intermezzo: collective variables Distance Angle Torsion Coordination number

Intermezzo: collective variables Hydrogen bonds Radius of gyration Cell parameters (in a while) From A to B: Path collective variables... next time

Umbrella sampling and Landau Free energy

Reconstructing the free energy profile

Umbrella sampling: estimating the probability distribution Let's estimate from a series of observations If we observe i entries in the interval Probability of observing i out of n entries (Poisson distribution, uncorrelated entries)

Umbrella sampling: estimating the probability distribution Average: Estimated error: In real life : mind correlations between observations

Umbrella sampling: Weighted Histogram Analysis Method Biasing potential in run i: During run i statistics collected near si (Bad) estimate for the unbiased probability distribution. Assume the best estimate as linear combination of the With normalized weights:

Umbrella sampling: Weighted Histogram Analysis Method

Time dependent hamiltonian: Metadynamics Alessandro Laio & Michele Parrinello, PNAS (2002) A method to drive chemical reactions using collective variables Add a small, repulsive potential at the present value of the reaction coordinate Free energy surface can be reconstructed after the simulation

Time dependent hamiltonian: Metadynamics Algorithm Choose a set of collective variables, e. g. distances, coordination number, simulation cell parametres,... Constraint these collective variables at a given point in s Perform metadynamics in space of collective coordinates... History dependent potential: either in steps: coarse grained dynamics continuously: smooth metadynamics

Metadynamics: the movie

Metadynamics: reconstruction of free energy profile The free energy surface can be reconstructed afterwards! Slowly all the local minima are filled and

Metadynamics: pros and cons Advantages: General Can cope with high dimensionality Predictive, wide exploration of free energy surface (with lower resolution) Disadvantages: Careful choice of the collective variables Inaccurate if a slow variable is forgotten (can be checked a posteriori) Choice of good (optimal) parametres (masses, coupling constants,... ) not straightforward What happenes if a single reaction coordinate is not enough? The low energy path might not be captured

Application of metadynamics: dissociation of carbonic acid

Application of metadynamics: fluxionality of Au7

Application of metadynamics: Parrinello Rahman Lagrangian Imposed pressure

Application of metadynamics: Structural phase transition Gibbs free energy Helmholtz free energy Eliminating rotations: h becomes upper triangular

Structural change of MgSiO3 Perovskite Post perovskite

Crystal structure transformations in SiO2 Structural change in SiO2 Martonak et al. Nature Materials 5, 623 (2006)

Summary Thermodynamic perturbation Test particle insertion Umbrella Sampling Weighted Histogram Analysis Method Metadynamics