Eur. Phys. J. Special Topics 224, 2389–2407 (2015) © EDP Sciences, Springer-Verlag 2015 DOI: 10.1140/epjst/e2015-02418-7 T HE EUROPEAN P HYSICAL JOURNAL SPECIAL TOPICS Review Free energies for rare events: Temperature accelerated MD and MC S. Meloni 1, a and G. Ciccotti 2, b 1 Laboratory of Computational Chemistry and Biochemistry, Institut des Sciences et Ing´ enierie Chimiques, ´ Ecole Polytechnique F´ ed´ erale de Lausanne, 1015 Lausanne, Switzerland 2 Dipartimento di Fisica and CNISM, Universit` a La Sapienza, P. le A. Moro 5, 00185 Rome, Italy; School of Physics, University College Dublin, Belfield, Dublin, Ireland Received 20 March 2015 / Received in final form 5 May 2015 Published online 22 June 2015 Abstract. In this article we review a set of methods for exploring the space of a set of collective variables, and to reconstruct the associ- ated Landau free energy in presence of metastabilities: Temperature Accelerated Molecular Dynamics (TAMD), its extension, Temperature Accelerate Monte Carlo (TAMC), and the Single Sweep Method (SSM). TAMD and TAMC can be used for both exploring and reconstructing the Landau free energy landscape. However, SSM is more efficient at accomplishing this last task. We illustrate the use of these methods by presenting their application to the nucleation of a Lennard-Jones crys- tal from its melt, and the H-vacancy migration in an NaAlH 6 crystal. 1 Introduction In this article we review a set of methods for studying rare events by atomistic simu- lations. Rare events, which we will define more precisely below, are infrequent transi- tions between high probable states of a system. From the point of view of statistical mechanics rare events are like any other (more frequent) event. The difference stands in the simulation techniques that must be used to investigate them: because of their low frequency we cannot use standard (brute force) molecular dynamics (MD) or Monte Carlo (MC). To put rare events in their more general theoretical context, and to introduce quantities that are relevant for their description, in this introduction we start revising the basic ideas behind statistical mechanics. Let us consider a classical system made of N point particles. A microscopic state of this system is described by a point in the 6N -dimensional phase space Γ =(r, p). r and p are the 3N -dimensional vectors of the positions and momenta of the parti- cles. Consider an observable O(Γ). The corresponding macroscopic value is the time average of the observable: ¯ O = lim t→∞ 1 t  t 0 dτ O(Γ(τ )), (1) a e-mail: simone.meloni@epfl.ch b e-mail: giovanni.ciccotti@roma1.infn.it