The aim of the summer school is to give thorough knowledge of ab initio and classical methods for molecular simulation. The first week will be dedicated to classical molecular simulation. The basic principles will be tackled both for the physical (Michel Mareschal’s lectures) and the numerical (Gabriel Stoltz’s lectures) point of view: various ensembles, numerical schemes, stochastic methods, … For each theme various applied exercises will be proposed. The second week will be dedicated to ab-initio molecular dynamics with an introduction to the density functional theory (DFT). Implementation of ab initio molecular dynamics within the framework of plane wave pseudopotential density functional theory will be given in detail.
Specific topics include : Car-Parinello molecular dynamics (CPMD), Path Integral molecular dynamics (PIMD), Tigh-Binding methods, Lean on the fly (LOTF)… Emphasis will be given to problems that can only be tackled using ab-initio molecular dynamics as matter in extreme conditions, or properties related to electronic excitations.
The participants will use the Abinit program to perform molecular dynamics : parallelism, calculations of melting temperature, phonons density of state, transport properties, PIMD...

Accommodation and courses take place in a castle depending of the CEA (French AlternativeEnergies and Atomic Energy Commission) at Cadarache, South of France, from june 10 to june 21, 2013.

These courses can interest a public of industrial engineers as well as academics (in particular researchers from CEA, EDF and INRIA), confirmed or not, in particular the PhDand the post-doctorate students.

* Lecture 1:

**Introduction to Molecular Dynamics and Monte Carlo for classical models**

*Michel MARESCHAL (ZCAM, Zaragoza)*

The lectures will introduce the basics of the Molecular Dynamics and Monte Carlo methods for systems which can be modeled by classical mechanics. The techniques allowing to compute static and transport properties will be introduced, based on equilibrium statistical mechanics. Applications in various ensembles will be presented, with an emphasis on thermostating the dynamics. The last part of the course will be dedicated to direct non-equilibrium modeling.

* Lecture 2:

**Numerical Methods in Computational Statistical Physics **

*Gabriel STOLTZ (CERMICS, Ecole des Ponts & MICMAC, INRIA Rocquencourt)*

The aim of this set of lectures is to present the mathematical underpinnings of the most common numerical approaches to integrate dynamics in molecular simulation. The first part will be devoted to the numerical integration of Hamiltonian dynamics, with an emphasis on symplectic algorithms (such as the celebrated Verlet method). The second part will focus on stochastic methods, namely stochastic differential equations such as the Langevin dynamics, and Markov chain approaches such as the Metropolis algorithm. We will give rigorous results on the convergence of time averages along trajectories of the system, and present numerical schemes to approximately integrate the corresponding dynamics. An important point will be the estimation of numerical errors.

* Lecture 3:

**Density functional theory: formalism, implementation, dynamical properties.**

*Xavier GONZE (UC Louvain)*

http://www.uclouvain.be/xavier.gonze

Density functional theory (DFT) is nowadays the workhorse for the first-principle prediction of properties of condensed matter, i.e. without parameterization relying on experimental data. Any type of system can be examined, provided the computing ressources are available.

In this lecture, I will first present the basic theorems of density functional formalism, and the approximations leading to realistic calculations. I will then detail how DFT equations can be implemented, using plane wave and pseudopotentials.

I will present the computation of forces acting on the nuclei, when these are treated using classical mechanics.

This opens the way to the use of Born-Oppenheimer molecular dynamics.

Finally, I will also focus on the computation of vibrational properties of solids (and related thermodynamical properties), based on density functional perturbation theory.

DFT

1.A. Formal DFT (1h30) (Theorems, XC functionals, the band gap problem)

1.B. DFT mapped into a computer (1h) (Planewave, pseudopotentials, k point sampling) 1. C. Algorithms (1h) (Computation of forces, conjugate gradient, self-consistency, optimization of geometry)

DFPT

2. A. Formal DFPT (1h30) (Perturbations, Sternheimer equation, 2n+1 theorem) 2. B. Phonon and dielectric properties (1h) (incl. phonon band structure, DOS, thermodynamical properties)

* Lecture 4:

**Ab initio Molecular Dynamics**

*Gilles ZERAH (CEA)*

Coupling molecular dynamics and ab initio calculation has made possible a qualitative investigation of the properties of many states of matter, from quantum solids to high temperature plasmas.

This set of lectures will present some of the methodological developments and applications which occurred in this respect during the past twenty years.

A special emphasis will be given to the interplay between these theoretical developments and experimental results.