Ibargüen César

Cedric Becerra

PhD Pascal Crespos

PhD. Larregaray

Influence of van der Waals forces in the Dynamics of Nitrogen and Hydrogen in Interaction with the W(100) Surface

Keywords: Science des surfaces, catalyse heterogéne, dynamique réactionnelle, simulations numériques, surfaces d'énergie potentielle Surface science, heterogeneous catalysis, reaction dynamics, numerical simulations, potential energy surfaces

An important part of scientific literature is devoted to the heterogeneous elementary processes occurring at gas-solid interface due to their key role in many fundamental domains and applications. Interaction of gas atoms/molecules and surfaces are, for instance, of primary importance in the study of: heterogeneous catalysis, combustion of solid fuel and coal gasification, processes of corrosion, hydrogen storage in solid material, automotive and oil industry, plasma-wall interactions in the context of thermonuclear experimental reactor (ITER), atmospheric re-entries technologies and astrochemistry, to cite a few examples. When an atom or molecule impinges on a surface many different elementary processes can take place, that depend on factors such as: the collision energy of the projectile, the angle of incidence to the surface, the surface temperature, the internal state of the molecules, etc... All these factors determine the mechanisms and dynamics of the processes. Experimental molecular beams (MB) and other experimental techniques are able to accurately control the initial state of the reactant and characterize the products of gas-surface reactions.

However, in most cases experimental techniques do not provide enough details about the mechanisms through which elementary processes occur. Therefore, the development of theoretical models becomes essential to rationalise the description. The main goal of this thesis work is to propose an analysis of the dynamics of several benchmark elementary processes occurring at the W(100) surface, such as: the inelastic scattering of N 2 and H 2 , the dissociative and non-dissociative adsorption of H 2 and the adsorption and absorption of H and N. The originality of the work stems from the inclusion of long range interactions of van der Waals type, which prove essential to reach a good agreement between theoretical and experiment results, especially in the low collision energy regime. To rationalize the non-adiabatic effects, the energy dissipation to lattice vibrations and electronic excitations are taken in to account by means of effective models.

Titre: Effet des Forces de van der Waals sur la Dynamique de L'azote et de L'hydrogène en Interaction avec la Surface de W [START_REF] Kr Geethalakshmi | Non-reactive scattering of N 2 from the W(110) surface studied with different exchange-correlation functionals[END_REF] Résumé: Une littérature scientifique nourrie est consacrée aux processus élémentaires hétérogènes se produisant à l'interface gaz-solide en raison de leur rôle clé dans de nombreux domaines. Ainsi, l'interaction d'atomes et de molécules avec les surfaces revêt une importance primordiale dans l'étude de la catalyse hétérogène, la combustion, la corrosion, le stockage de l'hydrogène, l'industrie automobile et pétrolière, les interactions plasma/paroi dans le contexte du réacteur expérimental thermonucléaire (ITER), les technologies du spatial, la chimie atmosphérique et l'astrochimie, pour ne citer que quelques exemples. Lorsqu'un atome ou une molécule entre en collision avec une surface, de nombreux processus élémentaires peuvent avoir lieu. Ils dépendent de nombreux facteurs tels que : l'énergie de collision du projectile, l'angle d'incidence sur la surface, la température de surface, l'état interne des molécules, etc... Tous ces facteurs influencent fortement les mécanismes réactionnels et la dynamique de ces processus. Les expériences de faisceaux moléculaires permettent un contrôle toujours plus précis de l'état initial des réactifs associé à un caractérisation fine des produits de réactions.

Cependant, dans la plupart des cas, ces observations expérimentales ne fournissent pas tous les détails permettant de décrire finement les mécanismes gouvernant les processus élémentaires étudiés. Par conséquent, l'élaboration de modèles théoriques devient essentielle pour en rationaliser la description. L'objectif principal de ce travail de thèse est de proposer une analyse théorique de la dynamique de plusieurs processus élémentaires pouvant se produire sur une surface du Tungsène W(100) en contact avec de l'hydrogène et de l'azote (diffusion inélastique de N 2 et H 2 , l'adsorption dissociative et non dissociative de H 2 et l'adsorption et l'absorption de H et N). L'originalité de ce travail réside dans la prise en compte des interactions à longue portée de type van der Waals, essentielles pour atteindre un bon accord théorie expérience dans le régime des faibles énergies de collision. La dissipation de l'énergie aux vibrations du réseau et aux excitations électroniques est prise en compte au moyen de modèles effectifs. Les résultats sont comparés aux données expérimentales disponibles et aux résultats théoriques antérieurs.

Chapter 1

General Introduction 1.1 Motivations

An important scientific literature is devoted to the heterogeneous elementary processes occurring at gas-solid interface [START_REF] Nilsson | Chemical bonding at surfaces and interfaces[END_REF] due to their great interest and key role in many different domains and applications. Several examples supporting that the interaction of gas atoms/molecules with surfaces is of primary importance can be listed: i) In the chemical industry, heterogeneous catalysis has been used for a long time, and still today catalytic steps are involved in most manufacturing processes. The development of new catalysts, which exhibit enhanced performance, has led to an unquestionable improvement in many fields of application [START_REF] Libuda | A molecular beam/surface spectroscopy apparatus for the study of reactions on complex model catalysts[END_REF], from the extraction of metal to the combustion of solid fuels and coal gasification. [START_REF] Szekely | Gas-solid Reactions[END_REF] Another outstanding example is the synthesis of ammonia (NH 3 ), in which the rate-limiting step in the reaction is the dissociation on N 2 molecules, the elucidation of which has led to the 2007 Nobel Prize (Gerhard Ertl) [START_REF] Honkala | Ammonia synthesis from first-principles calculations[END_REF][START_REF] Rayment | Structure of the ammonia synthesis catalyst[END_REF][START_REF] Gabor | Introduction to surface chemistry and catalysis[END_REF]. ii) The hydrogen storage in solid material [START_REF] Sakintuna | Metal hydride materials for solid hydrogen storage: A review[END_REF][START_REF] Dogan | Hydrogen storage tank systems and materials selection for transport applications[END_REF]. iii) Automotive industry, in which platinum (rhodium as well) have been used in catalytic reactions for controlling and reducing the emission of carbon monoxide (CO) or nitrogen oxides (NO x ) [START_REF] Twigg | Progress and future challenges in controlling automotive exhaust gas emissions[END_REF]. iv) In the oil industry, the interaction of gas atoms/molecules with surfaces is of interest for the development of polymetallic cluster catalysts, which have found extensive applications in petroleum refining for the production of high "antiknock" components for gasoline [START_REF] John | Structure of metal catalysts[END_REF][START_REF] John | Highly dispersed catalytic materials[END_REF]. v) For Space shuttle Thermal Protection System (TPS) during the hypersonic re-entry into the earth's atmosphere, the reactions on the refractory material exposed to N 2 and O 2 , and in particular the N-atom recombination,

ELEMENTARY PROCESSES

can contribute up to a third of the heat flux on TPS [START_REF] Halpern | Chemical energy accommodation at catalyst surfaces. Flow reactor studies of the association of nitrogen atoms on metals at high temperatures[END_REF][START_REF] Martin | Contribution théorique à l'étude de la réactivité élémentaire gaz/surface d'intérêt en rentrée atmosphérique[END_REF][START_REF] Wilbur | Re-entry Aerodynamics[END_REF]. vi) The N 2 molecule is also involved in many processes of interest in astrochemical research: N 2 may be the most abundant nitrogen-bearing molecule in the interstellar medium [START_REF] David C Knauth | The interstellar N 2 abundance towards hd 124314 from far-ultraviolet observations[END_REF], but the amount of nitrogen present as N 2 depends on the competition between its gas-phase formation and the depletion of atomic nitrogen onto grains [START_REF] Daranlot | Elemental nitrogen partitioning in dense interstellar clouds[END_REF]. vii) In the framework of the International Thermonuclear Experimental Reactor (ITER) project, the description of plasma-wall interactions is a key issue, especially for choosing the Plasma-Facing Materials (PFMs) of prime importance for the design of the first fusion device to produce net energy [17][START_REF] Federici | Plasma-Wall interactions in ITER[END_REF][START_REF] Shimada | Overview of goals and performance of ITER and strategy for plasma-wall interaction investigation[END_REF].

In view of all these examples, it is difficult to overstate the significance of the gas/surface reactions. In this thesis, we focus on the scattering dynamics of hydrogen and nitrogen on Tungsten. The interactions of hydrogen (and its isotopes) and tungsten is of practical interest within the framework of nuclear fusion and has been intensively studied in the last ten years. The study of nitrogen is more of fundamental interest, in particular to compare the scattering of "light" and "heavy" atoms/molecules when considering dissipation of energy to the solid. A special focus is here made on the influence of the van der Waals forces on the scattering dynamics observables for these species.

Elementary Processes

When an atom or a diatomic molecule impinges a surface, different processes can take place, which depend on factors such as: i) the collision energy of the projectile, ii) the angle of incidence, iii) surface temperature, iv) the initial internal state of the molecule, v) the dissipation of energy to the surface, among others. Some of the possible elementary mechanisms are schematically sketched in Fig 1 .1.

• Reflection occurs when after collision, the projectile is sent back to vacuum.

Reflection can be elastic and inelastic. If the projectile (could be a single atom or a molecule) conserves its initial energy after the collision it is referred to as elastic reflection. On the contrary, if upon reflection the impinging projectile looses or gains energy, it is said to be an inelastic reflection. Reflected molecule may 1.2. ELEMENTARY PROCESSES experience rotational and vibrational states transitions.

• Adsorption: The atomic adsorption takes place when the atom equilibrates with the lattice and gets stuck on the surface upon collision. In the case of molecules, it is possible to distinguish two types of adsorption: -Dissociative adsorption happens when the internuclear bond breaks and the two atoms stick separately on the surface.

• Absorption takes place when atoms (or molecules) collide the metal and penetrates into the subsurface finding a stable site inside the bulk and remains stuck on it.

• Abstraction or Recombination: The Eley-Rideal (ER) [START_REF] Eley | Parahydrogen conversion on tungsten[END_REF] process takes place when adsorbed atoms are removed from the surface by combining with atoms coming from the gas phase. If the impinging atom diffuses on the surface before

EXPERIMENTAL TECHNIQUES

colliding with the adsorbed one, the process is defined as Hot-Atom (HA) [START_REF] Harris | On precursor mechanisms for surface reactions[END_REF] recombination. The Langmuir-Hinshelwood (LH) [START_REF] Langmuir | The mechanism of the catalytic action of platinum in the reactions 2CO+O 2 →2CO 2 and 2H 2 +O 2 = 2H 2 O[END_REF] process occurs when the two atoms that form the new molecule are originally pre-adsorbed, in thermal equilibrium with the surface.

Experimental Techniques

From experimental point of view, a number of techniques allowing accurate measurement for atoms/molecules reacting on surface, have been developed [START_REF] Gabor | Introduction to surface chemistry and catalysis[END_REF]. When one wants to determine the adsorption energy of atomic/molecular species on surfaces the Temperature Programmed Desorption (TPD) technique [START_REF] Rj Cvetanović | A Temperature-Programmed Desorption technique for investigation of practical catalysts[END_REF][START_REF] John | Temperature-Programmed Desorption and reaction: applications to supported catalysts[END_REF] is applied. TPD is a powerful tool for both the evaluation of active sites on catalyst surfaces and the understanding of the mechanisms of catalytic reactions [START_REF] Ishii | Temperature programmed desorption[END_REF]. The topography properties of surfaces (e.g. metallic ones) can be measured by using a scanning technique that has sub-nanometer scale resolution as the Scanning Tunneling Microscope (STM) [START_REF] Binnig | Surface studies by scanning tunneling microscopy[END_REF].

Similar studies on surface topography with an unprecedented accuracy can be performed throughout the Atomic Force Microscopy [START_REF] Hudlet | Evaluation of the capacitive force between an atomic force microscopy tip and a metallic surface[END_REF]. By means of Energy Electron Loss Spectroscopy (EELS) it is possible to determine the vibrational modes, frequencies and the structure of the surface and adsorbed entities on the crystal [START_REF] Ibach | Electron energy loss spectroscopy and surface vibrations[END_REF]. Other experimental technique commonly used for determining and characterizing surface geometries is the Low-Energy Electron Diffraction (LEED) [START_REF] Diehl | Low-energy electron diffraction from quasicrystal surfaces[END_REF].

All the experimental techniques listed above provide a refined description of the surfaces and their adsorbates. However, to deal thoroughly into the study of the reactive processes it is necessary to use techniques allowing to gather information on the dynamics and kinetics of the processes. The Molecular Beams approach (MB) provides a adequate way of performing detailed and quantitative experiments on the kinetics and dynamics of surface reactions [START_REF] Libuda | A molecular beam/surface spectroscopy apparatus for the study of reactions on complex model catalysts[END_REF][START_REF] Sebastiaan | Manipulation and control of molecular beams[END_REF].

A MB is a spatially well-defined, directed and collision-free flow of atoms or molecules. The beam source is produced by a supersonic expansion of a gas from a so-called stagnation state flowing through an nozzle to a chamber at Ultra High Vacuum 1.4. THEORETICAL SIMULATIONS conditions (UHV, ∼ 10 -8 Pa) to avoid the presence of unwanted species in the reaction, and avoiding also possible multi collisions reactions. One of the main advantages of the MB experiments is the precise control of the initial states of the incoming atoms or molecules as well as the collision energy. The translational energy, the temperature of the surface, the internal states of the molecules, the initial angle of incidence influence significantly the dynamical processes at surfaces after the first encounter. Precise control of such parameters enables, in some cases, to manipulate surface chemical reactions [START_REF] Okada | Surface chemical reactions induced by well-controlled molecular beams: translational energy and molecular orientation control[END_REF][START_REF] Pirani | A simple and compact mechanical velocity selector of use to analyze/select molecular alignment in supersonic seeded beams[END_REF]. MB techniques allow to measure the Sticking Probability (combination of non-dissociative and dissociative probabilities). The coupling of MB techniques quadrupole mass spectrometry or Resonance Enhanced Multiphoton Ionization (REMPI) Spectroscopy [START_REF] Mnr Ashfold | Multiphoton spectroscopy of molecular species[END_REF], allows to gather information about the scattered molecules or atoms. An important number of MB setups of different complexity have been described in the reference [START_REF] Libuda | A molecular beam/surface spectroscopy apparatus for the study of reactions on complex model catalysts[END_REF]. More details about Molecular Beams experiments will be opportunely addressed and put into context of this work in the Chapter 4. Others techniques such as the femtosecond transition-state spectroscopy (FTS) [START_REF] Dantus | Real-time femtosecond probing of "transition states" in chemical reactions[END_REF] and laser induce fluorescence (LIF) [START_REF] James | Laser-induced fluorescence[END_REF] can also be used to detect transition states during the reaction, thus, bond breaking and bond forming processes.

Theoretical Simulations

If molecular beams (MB) experiments are able to accurately prepare reactants and characterize products of gas-surface reactions, in most of the case, they do not provide enough details about the mechanisms through which elementary processes (adsorption, reflection, absorption, recombination, i.e.) occur. Consequently, developping models and theoretical approaches become very important since they can complement the description that experimental techniques can not reach.

Huge theoretical advances on both the description and analysis of dynamics involved in elementary processes have been made since the beginning of this century, boosted by the improvement in the computational resources. Nowadays, refined calculations at the quantum level, allow to tackle an important amount of open questions about gas-surface reactions [START_REF] Díez | Dynamics of Gas-Surface Interactions[END_REF]. But, in spite of all this achievements, in many occasions, the simulations

THEORETICAL SIMULATIONS

can not reproduce the data obtained by experimental techniques. An exact theoretical treatment would have to deal with the entire system in a quantum manner, what is increasingly difficult as the size of the system grows. In other words, solving the time-independent Schrödinger Equation (TDSE) for all nuclei and electron can be done for a very limited number of atoms. For this reason, in this area of research it is necessary to constantly test new theoretical approximations that reproduce experimental observables and are able to predict and rationalize phenomena. This is the aim of this thesis, contributing in the ongoing theoretical research in gas-surface reactions.

The theoretical studies of reactive or non-reactive processes at surface imply the use of some approximations. Separating the movement of nuclei and electrons, as a result of the Born-Oppenheimer Approximation (BOA), is one of those. Usually the electronic part of the problem is tackled by performing calculation based on the Density Functional Theory, which has proven to be a reliable tool to periodic systems, as solid surfaces. Another approximation is to consider the surface as a rigid with atoms fixed at their equilibrium position. By interpolating the energies obtained from electronic structure calculations, the construction of a continuous representation of the global Potential Energy Surface (PES) may be achieved. It is then feasible to simulate the dynamics of the nuclei by solving classically the equations of motion. Doing so, the modelling of molecular beams experiments may be achieved. The construction of a PES is usually not a straightforward process as the fitting or interpolation becomes more and more complicated as the number of degree of freedom of the system increases.

Other parameters of the dynamics simulation as the surface temperature and the eventual transfer of energy to the vibrations of the lattice can be approximated by using the Generalized Langiven Oscillator, GLO model. The dissipation due to the electronic excitations can be taken into account by using the Local Density Friction Approximation (LDFA) (see Chapter 2).

In this work, the results for the dynamics of several atomic and molecular elementary processes on a surface of tungsten W(100) are reported: i) Inelastic scattering of N 2 and H 2 , ii) non reactive adsorption and dissociative adsorption of H 2 , iii) adsorption and

THEORETICAL SIMULATIONS

absorption of H and N. The first step of H diffusion into the bulk is also studied. An especial attention on the influence of van der Waals forces in dynamics of these processes is paid. The results are compared with available experimental data and previous theoretical results. The effect of including energy dissipation channels is also investigated.

The structure of the manuscript is as follows:

• In Chapter 2 an overview of the theoretical methods is presented. This chapter starts with the basics of the atoms/molecules-surface interactions, reviewing the electronic structure problem, within the framework of the Born-Oppenheimer Approximation.

DFT main concepts are described paying special attention to the van der Waals functionals. Details about the construction of the potential energy surface, its most important features and the interpolation method (Corrugation Reducing Procedure) are presented. The effective models for energy dissipation are also reviewed.

• such as the preferred sites for absorption and adsorption, the number of rebounds of the atoms before reaching exit channels, the minimum approach to the surface, the diffusion path into the bulk and role played for the energy dissipation channels.

• Chapter 6 summarizes the most important results and proposes an outlook for future research resulting from this work.

Chapter 2

ATOM-MOLECULE/SURFACE INTERACTION

Atom-Molecule/Surface interaction

Computer simulations have become an indispensable tool to describe properties of atomic and molecular systems, most of the time providing essential information otherwise difficult to obtain, even through sophisticated experimental techniques.

However, both the quality and predicting power of a computer simulation are tightly related to the accuracy with which the atomic or molecular interactions and their dynamics is described. For that reason, an intensive work is dedicated to the development of accurate methods for the calculation of electronic structure and dynamics for systems of interest [START_REF] Díez | Dynamics of Gas-Surface Interactions[END_REF].

When it comes to the dynamics of gas-surface reactions, the complete description of the total number of nuclei and electron is often computationally unaffordable. Nevertheless, by means of adequate approximations, dynamics simulation of nuclear motion is made possible. This chapter aims at presenting the theoretical methods used to simulate the atom-molecule/surface interactions.

The Born-Oppenheimer Approximation (BO)

A system as the one studied in this thesis, composed of atoms or molecules interacting with a surface, constitutes a many-body problem. Let us considers a system of M nuclei and N electrons, which are described by coordinates r 1 , . . .,r k ≡ r for electrons (of mass m e ) and R 1 , . . .,R k ≡ R for nuclei (of masses m 1 , • • • , m M ). For a stationary state, the wavefunction containing all the information about the system depends of the coordinates of nuclei and electrons: Ψ(R, r). This function satisfies the time independent Schröedinger equation

ĤΨ(R, r) = EΨ(R, r) (2.1)
in which Ĥ is the Hamiltonian operator and E the total energy of the system. Eq. 2.1 describes the simultaneous motion of the electrons and nuclei and the interaction between them. The non-relativistic Hamiltonian operator contains kinetic (T) and potential (V)

ATOM-MOLECULE/SURFACE INTERACTION

energies for all particles. It is the sum of five terms

H ≡ T e (r) + T n (R) + V ne (R, r) + V ee (r) + V nn (R). (2.2)
In the right side of the equation shown above, the first term corresponds to the operator for kinetic energy of electrons, the second one is the operator of kinetic energy for nuclei.

The Coulomb interactions between the electrons and the nuclei is accounted for in third term. Fourth and fifth term represents the repulsive potential between electrons and nuclei, respectively. The terms of the Hamiltonian in Eq. 2.2 can be rewritten as, atomics units,

T e (r) = - 1 2 N i=1 ∇ 2 i (2.3) T n (R) = - 1 2 M A=1 1 m A ∇ 2 A (2.4) V ne (R, r) = - M A=1 N i=1 Z A |R A -r i | (2.5) V ee (r) = N i=1 N j>i 1 |r i -r j | (2.6) V nn (R) = M A=1 M B>A Z A Z B |R A -R B | (2.7)
In this Hamiltonian (Eq. 2.2), nuclei and electrons are considered as point masses and spin and relativistic effects have been neglected. The Schrödinger Equation is a 2nd order non-separable differential equation that can only be fully solved for systems with a small number of atoms [START_REF] Frank | Introduction to Computational Chemistry[END_REF], which is not the case of gas/surface systems. Therefore, approximations are required to solve this equation.

Very commonly, the first approximation consist of splitting the problem into electronic and nuclear degrees of freedom using the so-called Born-Oppenheimer approximation (BO), (also known as Adiabatic Approximation) [START_REF] Born | Zur quantentheorie der molekeln[END_REF]. Since nuclei are ≈ 2000 times heavier than electrons, it is possible to neglect the coupling between the electronic and nuclear motions as nuclei are considered static from electrons point of view. Their kinetic energy (T n ) may be considered negligible and the repulsive potential between them (V nn ) can be seen as a constant, when solving the electronic problem. A consequence of this approximation is that electrons adapt instantaneously to any motion of nuclei.

From this, one can write the time-independent Schröedinger equation to describe the motion of electrons around the static nuclei at the position R as

H el Ψ el (r, R) = E el Ψ el (r, R) (2.8)
where the so-called Electronic Hamiltonian (H el ) arises from grouping the first and the last three of the Eq. 2.2 and has the form

H el = T e + V ne + V ee + V nn . Note that V nn
does not depend on r and is included in the Eq. 2.8 just as a constant. E el refers to the electronic energy state of the system. Under the Born-Oppenheimer approximation E el (R) is used as a potential acting on the nuclei, in other words, E el (R), is defined as the PES on which the atoms moves. The corresponding Schröedinger equation of motion

for nuclei is i ∂ ∂t Ψ n (R,t) = (T n + E el (R))Ψ n (R,t) (2.9)
In practice, the use of the BO approximation in atoms/molecule-surface problems consists in obtaining the electronic ground state PES by using Eq. 2.8 and carry out nuclear dynamics on that PES.

The BO approximation has been used successfully to describe many ground state material properties [START_REF] Frank | Introduction to Computational Chemistry[END_REF]. Many of theoretical chemistry and physics studies relies on this approximation, which allows splitting the theoretical description of the gas-surface reaction problem in two steps. In the first one, we deal with electronic problem by solving the Schrödinger equation for a set of nuclear positions using density functional theory (DFT), that is an electronic structure calculation method particularly well adapted to periodic systems, leading to a good balance between chemical accuracy and computational efficiency. This procedure allows to obtain a PES that contain all the information about the interaction of the system. In the second step, the motion of nuclei

DENSITY FUNCTIONAL THEORY (DFT)

governing the gas-surface reactions is simulated, here within the classical framework.

Density Functional Theory (DFT)

The description of the structure and dynamics of many-electron systems, in particular, the electron-electron interaction is a complicated issue. In the last four decades, DFT has emerged as one of the most used electronic structure methods. The DFT approach is fundamentally based upon a strategy of modeling the electron correlation through the determination of the electron density. A wave function for an N electron system contains 4N variables, three spatial and one spin coordinates for each electron [START_REF] Frank | Introduction to Computational Chemistry[END_REF]. As soon as the size of the system increases, the complexity of the wave function increases exponentially and the solution becomes intractable using the wave function-based methods. In contrast, the electron density, which is the square of the wavefunction integrated over N -1 electron coordinates, depends on 3 spatial coordinates (X,Y,Z), and not on the number of electrons. Unlike the wave function, which does not have a physical meaning, the electron density is a physical observable.

The DFT method has gained popularity because it takes into account electron correlation while being less demanding computationally than, for instance, Configuration-Interaction (CI) or Second Order Møller-Plesset Perturbation Theory (MP2) Methods. In the same line, DFT methods offers an important increase in computational accuracy without paying the additional increase in computing time.

The electron density ρ is defined as the integral of the square of the n-electron wave function over all except one arbitrary electron coordinate r 1 ,

ρ(r 1 ) = n |Ψ(r 1 ,r 2 ,r 3 , . . . r n )| 2 dr 2 , . . . , dr n (2.10)
The electron density can be also expressed as a sum over one electron orbitals φ i , of an independent particle system, in which the wave function is given by a Slater Determinant, as

ρ(r) = n i=1 φ * i (r)φ i (r).
(2.11)

The Hohenberg-Kohn Theorems

The starting point of any discussion of DFT is the Hohenberg-Kohn [START_REF] Hohenberg | Inhomogeneous electron gas[END_REF] (HK) theorems published in 1964. Such theorem states that the ground state electronic energy and other ground state electronic properties are fully determined by the electron density ρ. In other words, there exists a one-to-one correspondence between the electron density of a system and the energy. [START_REF] Peter | Molecular Quantum Mechanics[END_REF] Thus, the energy E is said to be a functional of the electron density.

It is denoted as E[ρ(r)

]. The first Hohenberg-Kohn theorem proves the existence of a unique functional but does not provides with its functional form. The electronic energy can bel written as follow:

E[ρ(r)] = T e [ρ(r)] + V ee [ρ(r)] + V eN [ρ(r)] = V eN [ρ(r)] + F HK [ρ(r)], (2.12)
where T e is the kinetic energy term, V ee refers to the electron-electron repulsion term and V eN includes terms describing potential energy of the nuclei-electrons attraction and nuclei-nuclei repulsion. The last term in Eq. 2.12 is the Hohenberg-Kohn functional

F HK [ρ(r)] and contains T e and V ee . Theses last two terms are "universal" for a given Coulomb interaction and does not depend on the external potential.

The second Hohenberg-Kohn theorem states that it is possible to obtain the ground state energy variationally: the density that minimizes the total energy is the exact ground state density. The latter implies that any density different from the ground state will yield a higher energy than the one obtained with the ground state density. The term V ee can be written as:

Ψ|V ee |Ψ = 1 2 ρ(r) • ρ(r ) |r -r | drdr + G XC [ρ(r)] = E H [ρ(r)] + G XC [ρ(r)] = V ee [ρ(r)], (2.13)
i.e., a sum of the electrostatic repulsion energy E H and a correction G XC , which contains the quantum mechanical many-body effects, so-called as exchange-correlation functional.

With this, the F HK [ρ(r)] can be rewritten as follow:

F HK [ρ(r)] = T e [ρ(r)] + E H [ρ(r)] + G XC [ρ(r)]. (2.14)

The Kohn-Sham Equations

Equation 2.14 looks nice, but it still has an unknown term. To face the problem of not knowing the analytical form of the functional, a practical scheme was developed by Kohn-Sham in 1965. The key to Kohn-Sham theory is to calculate the kinetic energy of a reference system under the assumption of non-interacting electrons with a corresponding Slater determinant as wave function [START_REF] Frank | Introduction to Computational Chemistry[END_REF]. This Slater Determinant should be built from a set of orbitals φ i , those are the Kohn-Sham (KS) orbitals. Within the Kohn-Sham approach it is possible to find a system subject to an effective potential that yield the electron density of the "real" system,

ρ(r) = n i=1 φ i |φ i . (2.15)
Having now a reference system, Kohn and Sham thought of separating the kinetic energy in two terms: T S for the non-interacting contribution and an unknown component T C that recovers the correctionsof the electronic interaction, as depicted in Eqs. 2.16 and 2.17

T S [ ρ(r)] = - 1 2 n i=1 φ i |∇ 2 i |φ i (2.16
)

T e [ρ(r)] = T S [ ρ(r)] + T C [ρ(r)] = - 1 2 n i=1 φ i |∇ 2 i |φ i + T C [ρ(r)]
(2.17)

T C [ρ(r)] gather an unspecified part ot the true kinetic energy and the additional contribution of the electron-electron interaction is contained into G XC , both are combined together into one exchange-correlation functional E X C[ρ(r)]. This latter includes all the many-body effects. With that in mind, Kohn and Sham showed that the exact ground-state electronic energy E of an n-electron system can be written as the total energy functional presented in the Eq. 2.12 including on it the Eq 2.17, as follow

E[ρ(r)] = T S [ ρ(r)] + V eN [ρ(r)] + E H [ρ(r)] + E XC [ρ(r)] (2.18)
The Kohn-Sham method allows to turn the problem of n interacting particles in an external field v ext into a simpler problem of n non-interacting particles in an effective field v ef f . The solution of this type of problem is a single Slater-Determinant. Thus,

EXCHANGE-CORRELATION FUNCTIONALS

KS equations are derived by applying the variational principle to the electronic energy E[ρ(r)] and them correspond to one-electron Schrödinger-like equations separated into individual ones:

- 1 2 ∇ 2 + v ef f φ i (r) = ε i φ i (r) (2.19)
where ε i are the KS orbitals energies. On the other hand, v ef f can be expanded as

v ef f = v eN + v H + v XC .
All these potentials are functional derivatives with respect to the density of the corresponding energy functionals. v eN and v H are known analytically.

v XC is the exchange-correlation potential, and can be represented as follow:

v XC = δE XC [ρ(r)] δρ(r) (2.20)
The KS equations have to be solved in a self-consistent fashion. Usually superposition of atomic densities are used as a initial guess for the density to obtain the derivatives. By using some approximate form (which remains fixed during all iterations) for the functional

E XC [ρ], to next compute v XC .
The set of KS equations is then solved to obtain an initial set of KS orbitals. This set of orbitals is then used to compute an improved density. That process is repeated until the density and exchange-correlation energy have converged to within a given tolerance [START_REF] Peter | Molecular Quantum Mechanics[END_REF].

Exchange-Correlation Functionals

The difference between various DFT methods is the choice of functional form for the exchange-correlation energy. Numerous schemes have been developed in order to get approximate forms of the functional for the exchange-correlations energy E XC [ρ(r)].

The tireless search of new and more accurate functionals is nowadays a line of research for itself. This latter makes as, most of the time, the main source of errors (or successes) in DFT stems from the approximate nature of the E XC , which is often separated into an exchange (for exchange energy) functional and a correlation functional.

Local Density Approximation (LDA)

One of the approximations (maybe the simplest one) to E XC [ρ(r)], proposed initially by Kohn-Sham [START_REF] Kohn | Self-consistent equations including exchange and correlation effects[END_REF] is the Local Density Approximation (LDA). Within LDA, it is assumed that the density is a slowly varying function which can be locally treated as a uniform electron gas. Under this approximation the exchange-correlation function for the density ρ(r) is given by

E LDA XC [ρ(r)] = ρ(r)ε XC [ρ(r)]dr, (2.21)
with ε XC [ρ(r)] being the exchange correlation energy per electron in the homogeneous electron gas. ε xc is usually divided into separate parts:

ε xc = ε x + ε c , referred to as
exchange and correlation parts. The contribution associated with the exchange is known analytically as

ε x (ρ) = - 3 4 3ρ π 1 3 .
(

The analytical form for the correlation part ε c of a "uniform" electron gas, which consists of purely dynamical correlations, has been derived in the high and low density limits [START_REF] Wj | Ground-state energy of a high-density electron gas[END_REF]. For intermediate densities, the correlation energy has been determined to a high precision by quantum Monte Carlo methods [START_REF] David | Ground state of the electron gas by a stochastic method[END_REF], and subsequently parametrized to provide an analytic form [START_REF] Seymour | Accurate spin-dependent electron liquid correlation energies for local spin density calculations: A critical analysis[END_REF].

Although the Eq. 2.21 is certainly an approximation (neither positive nor electronic charges are uniformly distributed in actual molecules), LDA is still computationally convenient and besides surprisingly accurate, especially for predicting structural properties. But, LDA usually overestimates binding energies. However, it is expected to perform well for systems with slowly varying density, e.g. simple metals.

Generalized Gradient Approximation (GGA)

Under the Local Density Approximation, the exchange and correlation functionals depend only on ρ but not on any derivative of ρ, therefore its accuracy is questioned with varying electron density, as occurs in molecules. In order to account for the inhomogeneity of the electron density, a non-local correction involving the gradient of ρ is often added to

VAN DER WAALS FUNCTIONALS

the exchange correlation energy. The LDA with gradient corrections yields the so-called Generalized Gradient Approximation (GGA) functionals. Within GGA-XC functionals approach, the first derivative of electron density is included as a variable, (see Eq. 2.23).

E GGA XC [ρ(r)] = f (ρ(r), ∇ρ(r))dr (2.23)
GGA-XC yields ground-state bond distances accurate to within 0.03 Å and binding energies accurate around 20 kJmol -1 [START_REF] Frank | Introduction to Computational Chemistry[END_REF]. GGA functionals gives quite accurate results

for the chemical bonds and reaction barrier heights. Nowadays it exists a variety of exchange-correlation functionals, some of them commonly used in computational chemistry, such as Lee, Yang and Parr (LYP) [START_REF] Lee | Development of the colle-salvetti correlation-energy formula into a functional of the electron density[END_REF] and B3LYP. This latter represent a particular type of "hybrid" functional that combines Hartree-Fock corrections with density functional correlation exchange.

For atoms or diatoms interacting with transition metal surfaces, the PBE (Perdew-Burke-Ernzerhof ) [START_REF] John P Perdew | Generalized Gradient Approximation made simple[END_REF], the PW91 (Perdew-Wang 1991 ) [START_REF] John P Perdew | Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation[END_REF] and the RPBE (Revised-PBE) [START_REF] Hammer | Improved adsorption energetics within density-functional theory using revised Perdew-Burke-Ernzerhof functionals[END_REF] functionals can be mentioned among the most used ones. It has been reported that RPBE functional improves significantly the accuracy of calculated adsorption energies [START_REF] Hammer | Improved adsorption energetics within density-functional theory using revised Perdew-Burke-Ernzerhof functionals[END_REF], whereas PW91 reproduces in a better way the features of the PES far from the surface, compared to RPBE. The PW91 functional has been tuned to improve the performance for weak interactions, producing the mPW91 acronym. More recently, van der Waals functionals (vdW) [START_REF] Lee | Higher-accuracy van der waals density functional[END_REF][START_REF] Dion | Van der Waals density functional for general geometries[END_REF] have been introduced to account for long range interactions. Such interactions can have a strong effect in the dynamics of molecular systems [START_REF] Migliorini | Application of van der waals functionals to the calculation of dissociative adsorption of N 2 on W(110) for static and dynamic systems[END_REF][START_REF] Loncarić | Strong anisotropic interaction controls unusual sticking and scattering of CO at Ru(0001)[END_REF][START_REF] Peña-Torres | Dynamics of N 2 sticking on W(100): the decisive role of van der Waals interactions[END_REF][START_REF] Peña-Torres | Energy dissipation effects on the adsorption dynamics of N 2 on W(100)[END_REF][START_REF] Ibargüen | Inelastic scattering of N 2 off W(001): Reconciling experiment and theory at low collision energies[END_REF][START_REF] Busnengo | Representation of the 6D potential energy surface for a diatomic molecule near a solid surface[END_REF]. One of these vdW functionals, more specifically the vdW-DF2 functional, have been used to carry out the necessary DFT calculation in this work.

Van der Waals Functionals

Some authors [START_REF] Lee | Higher-accuracy van der waals density functional[END_REF] have defined the van der Waals (vdW) interaction as "an attraction in quantum mechanical phenomenon with charge fluctuations in one part of an atomic system that are electrodynamically correlated with charge fluctuations in another".

These vdW interactions are considered the weakest of all intermolecular or interatomic attractions. However, despite how weak they are, the long range vdW interaction has

shown to have an effect on the gas-surface reaction dynamics [START_REF] Loncarić | Strong anisotropic interaction controls unusual sticking and scattering of CO at Ru(0001)[END_REF][START_REF] Peña-Torres | Dynamics of N 2 sticking on W(100): the decisive role of van der Waals interactions[END_REF][START_REF] Peña-Torres | Energy dissipation effects on the adsorption dynamics of N 2 on W(100)[END_REF][START_REF] Ibargüen | Inelastic scattering of N 2 off W(001): Reconciling experiment and theory at low collision energies[END_REF][START_REF] Busnengo | Representation of the 6D potential energy surface for a diatomic molecule near a solid surface[END_REF]. So, a full description of the gas-surface elementary processes, should include the interaction at sparse electron densities.

The exact density functional contains the vdW non-local forces. Unfortunately, there is no direct access to it. But approximate versions are abundant, such as Meta-GGA (a higher order gradient method) or Hybrid Functionals, to name some examples. vdW interactions have been known for a long time, but it has not been long since they are studied within the DFT framework. Some of the first studies do not go beyond 30 years ago. The first attempts to include the vdW forces were made for Rydberg et al. [START_REF] Rydberg | Tractable nonlocal correlation density functionals for flat surfaces and slabs[END_REF][START_REF] Kohn | van der Waals energies in Density Functional Theory[END_REF][START_REF] Andersson | Van der Waals interactions in Density Functional Theory[END_REF] as asymptotic interactions between fragments. The advances have been continued and applied in sparse systems, including soft matter, van der Waals complexes, and biomolecules [START_REF] Lee | Higher-accuracy van der waals density functional[END_REF]. The advances have resulted in the recently published vdW-DF. This functional has been successfully tested for layered structures and some geometries [START_REF] Dion | Van der Waals density functional for general geometries[END_REF][START_REF] Thonhauser | Van der Waals density functional: Self-consistent potential and the nature of the van der Waals bond[END_REF]. However, it has been found that vdW-DF underestimates hydrogen bonds strength in comparison to other ab-initio calculations [START_REF] Gulans | Linear-scaling selfconsistent implementation of the van der Waals density functional[END_REF]. On the other hand, an overestimation of distances of equilibrium is observed in many cases [START_REF] Thonhauser | Van der Waals density functional: Self-consistent potential and the nature of the van der Waals bond[END_REF][START_REF] Dc Langreth | A density functional for sparse matter[END_REF].

In 2010 Lee and coworker [START_REF] Lee | Higher-accuracy van der waals density functional[END_REF] proposed a second version of the van der Waals density functional (vdW-DF2). To build vdW-DF2 functional they used a more accurate semilocal exchange functional PW86 [START_REF] Perdew | Density-functional approximation for the correlation energy of the inhomogeneous electron gas[END_REF][START_REF] Eamonn D Murray | Investigation of exchange energy density functional accuracy for interacting molecules[END_REF] and large-N asymptote gradient corrections [START_REF] Elliott | Non-empirical derivation of the parameter in the B88 exchange functional[END_REF] in determining the vdW kernel. Authors report significant improvements regarding previous functionals which correspond to: i) equilibrium separations distances, (ii) the hydrogen bond strengths, and (iii) vdW attractions at intermediate separations longer than the equilibrium ones.

For the reasons briefly exposed, we have decided to use vdW-DF2 functional, although it should be noted the predictive power of a given functional may vary from systems to systems. Nevertheless, several studies have made use of functionals that account for vdW corrections in the past few years [START_REF] Loncarić | Strong anisotropic interaction controls unusual sticking and scattering of CO at Ru(0001)[END_REF][START_REF] Busnengo | Representation of the 6D potential energy surface for a diatomic molecule near a solid surface[END_REF] and in some cases, the description of dynamical observables in gas-surface systems have been improved. The works and reviews presented in references [START_REF] Lee | Higher-accuracy van der waals density functional[END_REF][START_REF] Dion | Van der Waals density functional for general geometries[END_REF][START_REF] Hyldgaard | Interpretation of van der Waals density functionals[END_REF][START_REF] Berland | van der Waals forces in density functional theory: A review of the vdW-DF method[END_REF] are recommended for a more detailed study of the vdW 2.5. DFT FOR PERIODIC SYSTEMS functionals.

DFT for Periodic Systems

A general picture on DFT has been so far presented based on the resolution of the Kohn-Sham equations (Eq. 2.19). We now discuss how these equation are solved for the particular case of a periodic system such as atoms/molecules in interaction with a perfect slab. A deeper overview can be found in several references, e.g, Ref [START_REF] Richard | Electronic Structure: Basic theory and practical methods[END_REF].

One of the first important decision is to choose the way the electronic wave function is represented. The electronic wave functions are eigenfunctions of the free-electron [START_REF] Smith | Phase analysis of image states and surface states associated with nearlyfree-electron band gaps[END_REF],

and therefore, they are the natural basis to describe bands in the nearly-free-electron approximation (NFEA). A number of different basis function are nowadays in use.

Gaussian and Slater-type orbitals, Numerical Atomic Orbitals, are some of those basis functions. However, for periodic systems with translational symmetries, the so-called Bloch Function are the most suitable basis functions, defined as the product of a periodic function times a plane wave [START_REF] Frank | Introduction to Computational Chemistry[END_REF]. Furthermore, these basis set provides a intuitive understanding as well as simple algorithms for practical calculations [START_REF] Richard | Electronic Structure: Basic theory and practical methods[END_REF]. For this reason, the DFT calculations in this thesis were conducted out with this type of basis functions.

The periodicity of the nuclei and potential in the system involve that the square of the wave function must display the same periodicity. This periodicity is, in fact, inherent in the Bloch Theorem (Eq. 2.24), which states that: the wave function value at equivalent positions in different cells are related by a complex phase factor involving the lattice vector t and a vector in the reciprocal space [START_REF] Frank | Introduction to Computational Chemistry[END_REF][START_REF] Bloch | About the quantum mechanics of electrons in crystal lattices[END_REF].

φ(r + t) = e ik•t φ(r) (2.24)
It is possible to use the Bloch theorem to write the eigenstates (in this case KS orbitals)

and obtaining the Bloch functions,

φ i,k (r) = u i,k (r)e ik•r (2.25)
with k as a wavevector within the first Brillouin zone, and u i,k is a function with the same periodicity of the lattice. The KS orbitals can be rewritten after expanding u i,k in the plane wave basis set (Fourier series) to obtain

φ i,k (r) = 1 √ Ω G C i,k (G)e i(k+G)•r , (2.26)
where G are the reciprocal lattice vectors and 1

√ Ω corresponds to a normalization factor. If the Eq. 2.26 is introduced into the KS equations (Eq. 2.19), the matrix eigenvalue equation is obtained:

G 1 2 (k + G) 2 + v ef f (G -G ) c i,k+G = i (k)c i,k+G (2.27)
where the kinetic energy operator is diagonal and v ef f is the Fourier transform of the effective potential. An advantage of using plane waves is that Eq. 2.27 can be efficiently solved by means of Fast Fourier Transformations (FFT). To get a very accurate calculation, it is necessary that the dimension of the plane-wave basis set is infinite.

Fortunately the plane-waves at the lowest kinetic energiesare the most important. Thus, a reasonable solution of can be obtained by truncating the basis set to a finite number of plane-waves. This number is conveniently expressed by a cut-off kinetic energy,

1 2 (k + G) 2 E cut-off . (2.28)
Often, a series of calculations are performed increasing the value of E cut-off , keeping the energy value for which the property of interest converges.

The plane waves or Bloch functions are ideal for describing delocalized slowly varying electron densities, for instance, the valence bands in a metal. Nevertheless drawbacks arise close to the nuclei, where the electrons are strongly localized due to the strong attractive potential, and the valence orbitals exhibit rapid oscillations to maintain orthogonality. Large number of plane waves are needed to properly describe this region.

Given that core electrons do not play a crucial role in the interaction with other atoms, those can be explicitly neglected and replaced through the use of a pseudopotential. In this approach, the valence electrons are still explicitly considered, and a pseudopotential replaces the strong Coulomb potential of a nucleus and the effects of the tightly bound core electrons with a softer potential of a pseudo-atom, which contains the nucleus, the core electrons, and the interaction between them.

The associated pseudo-wavefunction is built in a way that is as smooth inside a cut-off radius r c and reproduces the true wavefunction outside r c . There are several types of pseudopotentials. The first were called norm-conserving since the norm of each pseudo wavefunction outside r c had to be exactly the same than the corresponding electron wavefunction [START_REF] Dr Hamann | Norm-conserving pseudopotentials[END_REF]. This kind of pseudopotentials was a simple approach that it is still used, nevertheless, the E cut-off needed was still too large. In order to minimize the value for E cut-off , ultrasoft (US) pseudopotentials [START_REF] Vanderbilt | Soft self-consistent pseudopotentials in a generalized eigenvalue formalism[END_REF][START_REF] Laasonen | Implementation of ultrasoft pseudopotentials in ab initio molecular dynamics[END_REF][START_REF] Laasonen | Car-parrinello molecular dynamics with Vanderbilt ultrasoft pseudopotentials[END_REF] and projector-augmented waves (PAW) were later introduced [START_REF] Peter | Projector Augmented-Wave method[END_REF][START_REF] Kresse | From ultrasoft pseudopotentials to the Projector Augmented-Wave method[END_REF].

It has been reported that PAW pseudopotentials are better in terms of accuracy with respect to US pseudopotentials due to the PAW method retains the entire set of all-electron core functions along with smooth parts of the valence functions [START_REF] Richard | Electronic Structure: Basic theory and practical methods[END_REF]. However, the requirement of using a pseudopotential to describe the core region means that there is a fundamental limitation in how accurate the results can be.

Continuous sampling of the Brillouin zones is another computational bottleneck related with these kind of calculations since we have to compute a finite number of eigenstates at an infinite number of k-points in a single unit cell. Nevertheless, it is possible to carry out efficiently and with decent accuracy this sampling by discrete (and finite) k-point grids. Monkhorst and Pack [START_REF] Hendrik | Special points for brillouin-zone integrations[END_REF] proposed a useful method to generate the k-point grid which was used in this work.

As was aforementioned, the DFT plane waves are intrinsically periodic in the three dimensions. The same does not occur with the surface, which is just laterally periodic.

Periodicity is thus broken in the perpendicular direction to the surface, (i.e., Z), in which the system goes from the metallic bulk structure to the vacuum. Therefore, it is necessary to devise a reliable way to represent such system, i.e. the supercell approach. In this approach, the surface is represented as a set of finite atomic planes, called slab, which are separated by a layer of vacuum in the Z direction. Then, the atom or molecule is bulk are well simulated by the slabs, and the cell must be large enough so that spurious interaction with the periodic image molecules does not take place. In this way, and taking into account that the vacuum region separating must guarantee there is no interaction between the slab, it only remains to balance the size of the system and the computational cost.

Construction of an Adiabatic PES

In the precedent sections, specially Sec. 2.5, an overview about how to determine the total energy for given molecular configurations by means of electronic structure methods has been presented. In order to know the time evolution of the system, i.e. to simulate the dynamics of nuclei, it is necessary to know the energy (and forces) for any configuration. To that end, a continuous representation of the multi-dimensional PES should be built. The crucial step is to calculate the electronic energies (DFT data) for selected configurations on selected symmetry sites of the surface. Then, the task is to fit them to an analytical representation or interpolate them, considering all the relevant degrees of freedom, to finally obtain the global PES.

To do so, fitting methods represent a first scheme. They usually rely on physically justified analytical expressions involving adjustable parameters. The London-Eyring-Polanyi-Sato (LEPS) [START_REF] Martin | Contribution théorique à l'étude de la réactivité élémentaire gaz/surface d'intérêt en rentrée atmosphérique[END_REF][START_REF] Varandas | A leps potential for H 3 from force field data[END_REF][START_REF] Sato | On a new method of drawing the potential energy surface[END_REF][START_REF] Karplus | Exchange reactions with activation energy. I. Simple barrier potential for (H, H 2 )[END_REF][START_REF] Martin-Gondre | Detailed description of the flexible periodic London-Eyring-Polanyi-Sato potential energy function[END_REF] and the Embedded Atom Method (EAM) [START_REF] Murray S Daw | The embedded-atom method: A review of theory and applications[END_REF][START_REF] Michael | Modified embedded-atom potentials for cubic materials and impurities[END_REF] potentials are famous examples. However, their precision and flexibility is often limited. A second scheme relies on interpolating the data by mathematical function which do not always have a physical insight. Neural Networks (NN) [START_REF] Lorenz | Representing high-dimensional potential-energy surfaces for reactions at surfaces by neural networks[END_REF][START_REF] Behler | Neural network potential energy surfaces in chemistry: A tool for large-scale simulations[END_REF][START_REF] Behler | Representing molecule-surface interactions with symmetry-adapted neural networks[END_REF],

Spline-based methods [START_REF] William H Press | Numerical Recipes 3rd Edition: The art of Scientific Computing[END_REF], Interpolating Moving Least Squares (IMLS) [START_REF] Gia G Maisuradze | Interpolating moving least-squares methods for fitting potential energy surfaces: Detailed analysis of one-dimensional applications[END_REF][START_REF] Guo | Interpolating moving least-squares methods for fitting potential energy surfaces: Applications to classical dynamics calculations[END_REF], the Modified Sheppard Interpolation (MSI) [START_REF] Ischtwan | Molecular potential energy surfaces by interpolation[END_REF][START_REF] Meredith Jt Jordan | Convergence of molecular potential energy surfaces by interpolation: Application to the OH+H 2 →H 2 O+H reaction[END_REF], Genetic Programming [START_REF] Dmitrii | Fitting potential-energy surfaces: A search in the function space by directed genetic programming[END_REF] and the corrugation reducing procedure (CRP) [START_REF] Busnengo | Representation of the 6D potential energy surface for a diatomic molecule near a solid surface[END_REF] have, for example, been proposed.

Interpolation provides good results given its flexibility, but the number of DFT data points needed to accurately describe the PES, which clearly increases with the number of degrees of freedom may be prohibitive. To avoid (at least partially) this bottleneck, the surface atoms are usually considered fixed in their equilibrium positions. The so-called "static surface" approximation [START_REF] Richard S Keiffer | The role of the frozen surface approximation in small wave-height perturbation theory for moving surfaces[END_REF] is used in this work for determining the PES. This approximation has been successfully used to simulate the reactivity of a number of diatomic molecules, such as N 2 , H 2 , O 2 , on metal surfaces [START_REF] Behler | Dissociation of O 2 at Al(111): The role of spin selection rules[END_REF][START_REF] Díaz | Chemically accurate simulation of a prototypical surface reaction: H 2 dissociation on Cu(111)[END_REF][START_REF] Pijper | Six-dimensional quantum dynamics of scattering of (ν=0, J=0) H 2 from Pt(111): Comparison to experiment and to classical dynamics results[END_REF][START_REF] Alducin | Why N 2 molecules with thermal energy abundantly dissociate on W(100) and not on W(110)[END_REF][START_REF] Salin | Theoretical study of hydrogen dissociative adsorption on the Cu(110) surface[END_REF]. Here we treat atomic and diatomic systems (three and six degrees of freedom), so the PESs under the static surface approximation are 3D and 6D. A schematic representation of the coordinate system used in the mapping of the six-dimensional (6D) PES is depicted in Fig. 2.2. The position of the molecule is defined in terms of the X,Y and Z center-of-mass coordinates and the interactomic distance r.

Z

The molecular orientation relative to the surface is given by the polar (θ) and azimuthal angles (φ).

The construction of a PES starts by calculating a grid of ab initio points as dense as possible. Commonly, the DFT calculations are performed on a grid of r and Z values for several configuration, varying the positions (X,Y) and orientations of the molecule (θ, φ). The following step is the interpolation of the set of DFT points by any technique mentioned above. In this work, the Corrugation Reducing Procedure (CRP) was used.

Some details about its implementation to build 3D and 6D PES are presented in the next chapter.

The Corrugation Reduction Procedure (CRP)

During the last three decades, many improvements have been made in determining the PES of molecule-surface interactions from ab initio calculations. In the case of diatomic molecules interaction with a static surface, it is possible to build a 6D PES from ab initio data using a mixture of analytical and numerical interpolation techniques. This approach is the Corrugation Reducing Procedure (CRP). It was first proposed by [START_REF] Busnengo | Representation of the 6D potential energy surface for a diatomic molecule near a solid surface[END_REF], and since then, the CRP has successfully become a widely used method to represent in an accurately way the interaction of different gas-metal systems [52-55, 95-97, 99-104].

The CRP method overcomes one of the major problems when dealing with the parametrization of a molecular potential, which is the strong variation of the repulsive potential at the vicinities of the surface for different configuration of the molecules. The idea at the heart of the CRP is that variation of the potential, referred as corrugation, can be reduced. The strongest variations of the 6D PES originate from the individual atomic interaction with the surface when the molecule approaches it. Thus, if the 3D atom-surface interactions V 3D A and V 3D B are subtracted from the full PES V 6D , a smoother function I 6D arises. It is then possible to interpolate separately the highly corrugated terms V 3D and the smoother "interpolation function" by standard numerical

THE CORRUGATION REDUCTION PROCEDURE (CRP)

interpolation techniques. This can be expressed as follows:

I 6D (X, Y , Z, r, θ, φ) =V 6D (X, Y , Z, r, θ, φ)- V 3D (X A , Y A , Z A ) -V 3D (X B , Y B , Z B ) (2.29)
where the coordinates (X, Y , Z, r, θ, φ) represent the six DOF of the molecule over the surface, (X A,B , Y A,B , Z A,B ) as it is defined in Fig. 2.2. The I 6D (X, Y , Z, r, θ, φ) function can be interpolated following this series of steps.

• For each given configuration (X, Y , θ, φ) the dependence of the I 6D function with r and Z in the grid is interpolated using 2D cubic splines.

• The interpolation over the azimuthal (φ) and polar (θ) angles is performed making use of trigonometric functions. Those should fulfill the required symmetry as basis functions.

• In the last step, the remaining degrees of freedom (X, Y ) are interpolated by means of 2D cubic splines or low order Fourier expansion.

Once the interpolation on I 6D has been done, V 3D

A and V 3D

A must be added to the interpolation function in order to obtain the full V 6D (X cm , Y cm , Z cm , r, θ, φ). From this, it becomes clear that in the CRP scheme, the atomic potentials V 3D A,B have to be calculated as well, suggesting that in addition to the initial set of molecular energies data over the surface, DFT energy calculation of an atom-surface grid are needed.

V 3D are highly corrugated due to the strong potential variations. An analogous procedure to that depicted in Eq. 2.29 is followed to address this corrugation,

I 3D (R) = V 3D (R) - n i=1 V 1D (|R -R i |).
(2.30)

In the Eq. 2.30 V 1D s is a "pair" potential describing the interaction between one isolated atom and the i th surface atom located at positions

R ≡ (X, Y , Z) and R ≡ (X i , Y i , Z i ),
respectively. Reference 1D potential (V 1D ) is chosen as the one obtained over a surface atom (Top site). In order to interpolate I 3D , 3D splines over X, Y and Z can be easily computed. The sum on equation above runs over all slab atoms that give a non zero

2.8. MOLECULAR DYNAMICS contribution to V 1D .
Once the task of building the PES is completed, it remains only an important step:

to test its accuracy. This test can be done by comparing the interpolated values with calculated ab initio data not included in the interpolation procedure. The CRP method is general in the sense that it is not restricted to a particular surface symmetry or a particular diatomic molecule. Moreover, it preserves the full symmetry of the system. It also has the advantage that its precision can be systematically improved by adding more data from ab initio calculations.

The steps involved in the CRP construction for a 6D-PES are here summarized:

1. Performing DFT Calculation in order to obtain V 1D , V 3D and V 6D

Obtaining

I 3D = V 3D -V 1D and interpolate it 3. Retrieve the V 3D , V 3D = I 3D + V 1D
4. Construct the 6D interpolation function,

I 6D = V 6D -V 3D A -V 3D
A and then interpolate it 5. Recover the final

V 6D = I 6D + V 3D A -V 3D B 6.
Test the accuracy of the PES and include more DFT point if it is necessary

Molecular Dynamics

Beyond the meaningful information that can be collected by exploring the PES, a static analysis falls short and molecular dynamics simulations become crucial to gain insights into the understanding of gas-surfaces mechanisms. In principle, the motion of all atoms in the system should be accounted for, but the computational cost that this involves is huge. Commonly, only the movement of the reactive atoms is considered through the static surface approximation (Sec. 2.6). Beyond the static surface approximation, energy exchange with the surface may be accounted for by effective models. From the determination of the PES, equation of motion for the nuclear degrees of freedom can be integrated numerically, and the dynamics can be studied. In the following we discuss adiabatic and non-adiabatic simulation within the classical framework).

Classical Dynamics on a PES

Dynamics calculations of gas-surface reactions might involve the motion of all the atoms. Unfortunately, the appropriate dimensionality [START_REF] Díez | Dynamics of Gas-Surface Interactions[END_REF][START_REF] Groß | Reactions at surfaces studied by ab initio dynamics calculations[END_REF][START_REF] Kroes | Six-dimensional quantum dynamics of dissociative chemisorption of H 2 on metal surfaces[END_REF] to use makes a quantum mechanical description of the nuclear motion computationally prohibitive. Due to their masses, the motion of the nuclei can analyzed within the framework of Classical Mechanics. To that end, the Hamilton equations of motion are solved,

dR i dt = ∂H ∂P i , dP i dt = - ∂H ∂R i (2.31)
where P i and R i are the momentum and the position of atoms. Within the dynamics codes used in this work, a integrator algorithm of Beeman type proposed by Schofield [START_REF] Schofield | Computer simulation studies of the liquid state[END_REF] is implemented. This method of integration improves the efficiency and accuracy provided by the Velocity Verlet [START_REF] Verlet | Computer experiments on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules[END_REF][START_REF] Verlet | Computer experiments on classical fluids. II. Equilibrium correlation functions[END_REF] algorithm. Others algorithm as Bulirsh-Stoer [START_REF] Stoer | bulirsch introduction to numerical analysis springer-verlag[END_REF] can be also used.

Classical equations do not take intrinsically into account the vibrational zero-point energy (ZPE) of the molecule. To correct for that, calculations are usually done in the socalled quasi-classical way, i.e. the internuclear distance and radial momenta are initially sampled in such a way that the vibrational energy corresponds to the ZPE. In this work, all calculations are performed with this quasi-classical approach, even if at the end the difference with respect to purely classical results is not significant.

Dissipation of Energy in the Classical Dynamics

Up to this point, the discussion have focused on dynamics calculations on 6D PESs within the framework of an ideal static surface. Hence, no energy transfers between atoms or molecules and the surface can be accounted for. Notwithstanding the simplicity of these model, they have proven able to describe the elementary process in many gas-surface systems. But, the ideal static surface approximation neglect (at least) the possible effects on the reaction of: i) surface temperature (the thermal displacement

MOLECULAR DYNAMICS

of the surface atoms from their equilibrium positions), ii) exchange of energy between atoms or molecules with lattice vibrations [START_REF] Ac Luntz | CH 4 dissociation on metals: A quantum dynamics model[END_REF], iii) electronic excitations. For that reason, to rationalize the non-adiabatic effects, we perform quasi-classical calculations on models incorporating the two main energy dissipation channels: i) exchange of energy between lattice vibrations and, ii) electronic excitations. Next, the Generalized Langevin Oscillator (GLO) and the Local Density Friction Approximation methods, through which energy dissipation is accounted for, are presented.

Dissipation and exchange with the lattice: Generalized Langevin Oscillator model

Recently, there have been many efforts in the implementation of models that include the temperature of the surface, the motion of this latter and the exchange of energy between the atoms or molecules and the surface. One of the model that has emerged is the Generalized Langevin Oscillator (GLO) [START_REF] Sa Adelman | Generalized langevin theory for many-body problems in chemical Dynamics: General formulation and the equivalent harmonic chain representation[END_REF][START_REF] Busnengo | Trapping, molecular adsorption, and precursors for nonactivated chemisorption[END_REF][START_REF] Busnengo | Surface temperature effects in dynamic trapping mediated adsorption of light molecules on metal surfaces: H 2 on Pd(111) and Pd (110)[END_REF][START_REF] John | Dynamics of gas-surface interactions: 3D generalized Langevin model applied to fcc and bcc surfaces[END_REF][START_REF] Polanyi | Dynamics of simple gas-surface interaction. II. rotationally inelastic collisions at rigid and moving surfaces[END_REF], which considers the surface as a 3D harmonic oscillator coupled to a thermal bath. GLO makes possible to simulate surface motion and temperature while keeping the accuracy of the 6D-PES. The surface (of mass of one surface atom, m s ) oscillates as a whole, about a position vector R s , with a 3×3 frequency matrix ωs . The energy dissipation and thermal fluctuations are accounted for by using a so-called 3D Ghost Oscillator, which is defined with a position vector R g . The mass and the associated 3×3 matrix frequency are m g and ωg , respectively. The surface and the ghost oscillator are coupled through the coupling matrix Λgs . The equation of motion for the system for a molecule and surface and ghost oscillator can be written as follows:

M i d 2 R i dt 2 = -∇ i V 6D (R i -R s , R j -R s ), (2.32
)

m s d 2 R s dt 2 = -∇ s V 6D (R i -R s , R j -R s ) -ω2 s R s + Λgs R g , (2.33) m s d 2 R g dt 2 = -ω2 g R g + Λgs R s -γg dR g dt + F r (t). (2.34)
with ∆t being the time integration step, k B the Boltzmann constant and T s is the

MOLECULAR DYNAMICS

temperature of the surface. Eq. 2.32 describes the motion of the atom or molecule due to the adiabatic force obtained from 6D-PES. The remaining two equations represents the motion of the surface and ghost oscillator. The matrices ωs and ωg are both identical and diagonals [START_REF] Busnengo | Trapping, molecular adsorption, and precursors for nonactivated chemisorption[END_REF][START_REF] Busnengo | Surface temperature effects in dynamic trapping mediated adsorption of light molecules on metal surfaces: H 2 on Pd(111) and Pd (110)[END_REF], and their used values are related to the Debye frequencies determined experimentally for the phonons of the surface under study. The 3D ghost oscillator is subjected to a friction force through a damping matrix ( γg ) and a random fluctuation force (F r (t)). This latter force, which models the heating of the surface atoms as a consequence of thermal motion of the bulk atoms, is basically a Gaussian white noise with variance

V ar(F r (t)) = 2k B T s γ g m s ∆t , ( 2.35)
In simulation including GLO model, the initial conditions of the 3D oscillators are sampled through a conventional Monte-Carlo procedure that adjusts the initial positions and velocities to surface temperature. A rather complete and rigorous description of the GLO model can be found in references [START_REF] Sa Adelman | Generalized langevin theory for many-body problems in chemical Dynamics: General formulation and the equivalent harmonic chain representation[END_REF][START_REF] John | Dynamics of gas-surface interactions: 3D generalized Langevin model applied to fcc and bcc surfaces[END_REF]. In particular, we use the implementation made by H. F. Busnengo and co-workers [START_REF] Díez | Dynamics of Gas-Surface Interactions[END_REF][START_REF] Busnengo | Surface temperature effects in dynamic trapping mediated adsorption of light molecules on metal surfaces: H 2 on Pd(111) and Pd (110)[END_REF].

Electron-hole excitations: Local Density Friction Approximation

The evidence of electronic excitations occurring during the gas-surface reactions has been well documented from experiment [START_REF] Amirav | Electron-hole pair creation by atomic scattering at surfaces[END_REF][START_REF] Gergen | Chemically induced electronic excitations at metal surfaces[END_REF][START_REF] Krix | Generation of hot charge carriers by adsorption of hydrogen and deuterium atoms on a silver surface[END_REF][START_REF] Brandbyge | Electronically driven adsorbate excitation mechanism in femtosecond-pulse laser desorption[END_REF][START_REF] Ajb Robertson | Exoelectron emission from solids[END_REF][START_REF] Nienhaus | Selective H atom sensors using ultrathin Ag/Si Schottky diodes[END_REF][START_REF] Hasselbrink | Non-adiabaticity in surface chemical reactions[END_REF][START_REF] Kasemo | Metal-gas reactions studied by surface chemiluminescence[END_REF][START_REF] Lamont | Dynamics of atomic adsorbates: Hydrogen on Cu(111)[END_REF][START_REF] David M Bird | Electronic excitation in atomic adsorption on metals: A comparison of ab initio and model calculations[END_REF]. These finding can eventually break down the applicability of the adiabatic approach [START_REF] Juaristi | Role of electron-hole pair excitations in the dissociative adsorption of diatomic molecules on metal surfaces[END_REF], which has led to development of a variety of methods tending to incorporate this non-adiabatic effect in molecular dynamics simulations [START_REF] Ac Luntz | CH 4 dissociation on metals: A quantum dynamics model[END_REF][START_REF] Juaristi | Role of electron-hole pair excitations in the dissociative adsorption of diatomic molecules on metal surfaces[END_REF][START_REF] Lindenblatt | Molecular dynamics of nonadiabatic processes at surfaces: Chemisorption of H/Al(111)[END_REF][START_REF] Shenvi | Dynamical steering and electronic excitation in NO scattering from a gold surface[END_REF][START_REF] Oliver Bünermann | Electron-hole pair excitation determines the mechanism of hydrogen atom adsorption[END_REF][START_REF] Wodtke | Electronically non-adiabatic influences in surface chemistry and dynamics[END_REF]. Among the methods that have arisen, the Local Density Friction Approximation (LDFA) [START_REF] Juaristi | Role of electron-hole pair excitations in the dissociative adsorption of diatomic molecules on metal surfaces[END_REF] has shown to be a reliable to include electronic excitations in the framework of the multidimensional PES while keeping a good compromise between accuracy, simplicity in terms of implementation and low computational cost. LDFA approach has already been used to investigate dissipation of energy in gas-surface elementary processes [53-55, 102, 130, 132-136]. The central idea of the method is to approximate that the friction coefficient η(R i ) acting on each atom at its position R i to the friction coefficient that the atom would experience when moving in an homogeneous free electron gas (FEG) with electron density n(R i ) [START_REF] Juaristi | Role of electron-hole pair excitations in the dissociative adsorption of diatomic molecules on metal surfaces[END_REF][START_REF] Novko | Ab initio molecular dynamics with simultaneous electron and phonon excitations: Application to the relaxation of hot atoms and molecules on metal surfaces[END_REF]. Therefore, applying LDFA in practice implies, first, to calculate the electron density for the bare surface at R i . This electronic density can be obtained from

MOLECULAR DYNAMICS

DFT calculations. Then, the energy losses of the molecule caused by e-h pair excitations are obtained in terms of a friction coefficient, and the dissipative force is introduced in the equations of motion via a force of friction which is proportional to the velocity v of the atom or molecule traveling through a free electron gas of density n 0 ,

F diss = -ηv = n 0 vk F σ tr (k F ). (2.36)
In the Eq. 2.36 k F and σ tr (k F ) are respectively the Fermi momentum and the transport cross section at the Fermi level. Notice that the corresponding friction coefficient η is proportional to the transport cross section evaluated at the Fermi level of the spherically symmetric static projectile-electron potential [START_REF] Bönig | Time-dependent local perturbation in a free-electron gas: Exact results[END_REF]. Thus, arises that friction coefficient can be written as η = n 0 k F σ tr (k F ). The friction coefficient so calculated has been successfully used for determining the stopping power of atoms and ions in metal surfaces [START_REF] Juaristi | Charge state dependence of the energy loss of slow ions in metals[END_REF][START_REF] Winter | Energy loss of slow ions in a nonuniform electron gas[END_REF][START_REF] Pm Echenique | Nonlinear stopping power of an electron gas for slow ions[END_REF]. A wider description of coefficient friction equation is provided in reference [START_REF] Díez | Dynamics of Gas-Surface Interactions[END_REF]. Lastly, the classical equations of motion for each atom of the specie including the dissipative force read

m i d 2 R i dt 2 = -∇ i V (R i , R j ) -η(R i ) dR i dt , (2.37)
where the first term in the right-hand side is the adiabatic force obtained from the 6D PES and the second term is the dissipative force [START_REF] Juaristi | Role of electron-hole pair excitations in the dissociative adsorption of diatomic molecules on metal surfaces[END_REF]. It is also possible to combine both GLO and LDFA models, in that case the equation of motion for an atom (Eq. 2.32)

becomes in:

M i d 2 R i dt 2 = -∇ i V 6D (R i -R s , R j -R s ) - 1 m i η(R i ) dR i dt (2.

Introduction

Within the context of surface science and heterogeneous catalysis [START_REF] Nilsson | Chemical bonding at surfaces and interfaces[END_REF][START_REF] Honkala | Ammonia synthesis from first-principles calculations[END_REF][START_REF] Gabor | Introduction to surface chemistry and catalysis[END_REF][START_REF] Chorkendorff | Concepts of Modern Catalysis and Kinetics[END_REF][START_REF] Ertl | Primary steps in catalytic synthesis of ammonia[END_REF] the scattering of diatoms on metallic surfaces has been thoroughly studied both theoretically [START_REF] Díez | Dynamics of Gas-Surface Interactions[END_REF][START_REF] Groß | Reactions at surfaces studied by ab initio dynamics calculations[END_REF][START_REF] Kroes | Six-dimensional quantum dynamics of dissociative chemisorption of H 2 on metal surfaces[END_REF][START_REF] George | The dissociation of diatomic molecules at surfaces[END_REF][START_REF] Kroes | Towards chemically accurate simulation of molecule-surface reactions[END_REF] and experimentally [START_REF] Ct Rettner | Chemical dynamics at the gassurface interface[END_REF][START_REF] Wodtke | Energy transfer and chemical dynamics at solid surfaces: The special role of charge transfer[END_REF][START_REF] Sitz | Gas surface interactions studied with state-prepared molecules[END_REF][START_REF] Aw Kleyn | Molecular beams and chemical dynamics at surfaces[END_REF][START_REF] Golibrzuch | The dynamics of molecular interactions and chemical reactions at metal surfaces: Testing the foundations of theory[END_REF][START_REF] Lbf Juurlink | State-resolved probes of methane dissociation dynamics[END_REF]. The understanding of elementary reaction dynamics and the rationalization of state-selective mechanisms, such as the scattering and sticking of H 2 on metals, has become accessible in the last decades due to joint improvements in both theoretical and experimental domains. Thus, the validation of any new theoretical model can be achieved by comparison to experimental observables, mainly from molecular beam techniques in ultra-high vacuum (UHV) conditions [START_REF] Ct Rettner | Chemical dynamics at the gassurface interface[END_REF][START_REF] Wodtke | Energy transfer and chemical dynamics at solid surfaces: The special role of charge transfer[END_REF][START_REF] Sitz | Gas surface interactions studied with state-prepared molecules[END_REF][START_REF] Aw Kleyn | Molecular beams and chemical dynamics at surfaces[END_REF][START_REF] Golibrzuch | The dynamics of molecular interactions and chemical reactions at metal surfaces: Testing the foundations of theory[END_REF][START_REF] Lbf Juurlink | State-resolved probes of methane dissociation dynamics[END_REF]. In this context, the sticking of H 2 on W(100) has been extensively studied since the early eighties with the effusive molecular beam experiments of King et al [START_REF] Alnot | Adsorption and desorption kinetics with no precursor trapping: hydrogen and deuterium on W(100)[END_REF].

More recently, several supersonic molecular beams measurements [START_REF] Berger | Adsorption of hydrogen on tungsten: a precursor path plus direct adsorption[END_REF][START_REF] Butler | Precursor dynamics in dissociative hydrogen adsorption on W(100)[END_REF][START_REF] Ct Rettner | Rotationally mediated selective adsorption, physisorption and dissociative chemisorption of HD on W(110)[END_REF][START_REF] Ba Chuikov | Adsorption of H 2 and D 2 on W(111) and W(110) surfaces at low temperatures (down to 5 K)[END_REF][START_REF] Busnengo | H 2 chemisorption on W(100) and W(110) surfaces[END_REF] have exhibited a non monotonous shape of the sticking probability (S 0 ) as a function of the initial collision energy of the molecule. S 0 is presenting a minimum at a given collision energy, dividing the sticking curve in two parts. The low collision energy regime, where S 0 decreases with collision energy, is often attributed to indirect mechanisms involving a large number of rebounds of the molecule onto the surface and total energy scaling (TES). After reaching the minimum, S 0 exhibits an opposite trend with an increase of the probability with the collision energy.

This second part of the curve is due to more direct mechanisms that tend to be characterized by a normal energy scaling (NES). The nature of the mechanism underlying the low collision energy regime remains a matter of discussion. Would it be due to dynamic trapping leading uniquely to dissociation, or is there any evidence for molecular non-dissociative adsorption states? Other authors argue for the strong influence of surface defects for stabilizing molecular species or enhancing dissociation.

Finally, whereas non-monotonous behavior for the sticking curve is well established by experimental measurements, the dynamics of H 2 on W(100) still requires clarifications.

Recent theoretical works addressed those questions by performing full dimensional (6D) quasi-classical dynamics simulations on H 2 /W(100) and H 2 /W(110) systems [START_REF] Busnengo | H 2 chemisorption on W(100) and W(110) surfaces[END_REF].

In this work, a 6D PES, based on density functional theory (DFT) [START_REF] Hohenberg | Inhomogeneous electron gas[END_REF][START_REF] Kohn | Self-consistent equations including exchange and correlation effects[END_REF] calculations using PW91 functional [START_REF] John P Perdew | Generalized Gradient Approximation made simple[END_REF][START_REF] John P Perdew | Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation[END_REF], has been used. Simulations were carried out within the In the seek for a better description of diatom-surface interactions, several exchange-correlation functionals have been used. Comparisons have been made between state of the art functionals, i.e. PW91 [START_REF] John P Perdew | Generalized Gradient Approximation made simple[END_REF][START_REF] John P Perdew | Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation[END_REF][START_REF] Ozoliņš | Full-potential calculations using the Generalized Gradient Approximation: Structural properties of transition metals[END_REF] and RPBE [START_REF] Hammer | Improved adsorption energetics within density-functional theory using revised Perdew-Burke-Ernzerhof functionals[END_REF], for systems such as Coming back to the H 2 /W(100) system under concern in this work, the question to address is: will the vdw-DF2 help to reach a better level of agreement with experiments when considering the sticking probability of H 2 ? And, as a consequence, will the vdW functional help in the description of the dynamics at low energy?.

This chapter is organized as follow: In Sec. 3.2, details on the DFT calculations and the construction of the PES are given, including some PESs accuracy considerations. In order to quantify the effect of vdW forces, several low dimension cuts of the vdW-PES are proposed in comparison to PES based on PW91 developed by Busnengo et al [START_REF] Busnengo | H 2 chemisorption on W(100) and W(110) surfaces[END_REF] in Sec. 3.3 along with the analysis of the simulation results and an interpretation of the dynamics. Final Sec. 3.4 summarizes the main conclusions of this study.

POTENTIAL ENERGY SURFACE FOR H

2 ON W(100)

Potential Energy Surface for H 2 on W(100)

H 2 scattering on W(100) is described by a 6D-PES obtained under the static surface approximation. The Corrugation Reducing Procedure (CRP) [START_REF] Busnengo | Representation of the 6D potential energy surface for a diatomic molecule near a solid surface[END_REF] (see Chapter 2 for details) has been used to build the PES. All the DFT calculations were run using vdW-DF2 functional [START_REF] Lee | Higher-accuracy van der waals density functional[END_REF][START_REF] Dion | Van der Waals density functional for general geometries[END_REF] as implemented in the Viena ab initio Simulation Package (VASP) [START_REF] Hafner | The Vienna ab-initio Simulation Program VASP: An efficient and versatile tool for studying the structural, dynamic, and electronic properties of materials[END_REF][START_REF] Kresse | Ab initio molecular dynamics for liquid metals[END_REF][START_REF] Kresse | Ab initio molecular dynamics for open-shell transition metals[END_REF][START_REF] Kresse | Efficient iterative schemes for ab initio totalenergy calculations using a plane-wave basis set[END_REF].

DFT calculations were performed using a supercell approach. Electronic wave-functions are decomposed on a plane-wave basis set, and electron-ion interactions are described through Plane Augmented Wave (PAW) pseudopotentials [START_REF] Peter | Projector Augmented-Wave method[END_REF][START_REF] Kresse | From ultrasoft pseudopotentials to the Projector Augmented-Wave method[END_REF]. A cut-off energy of 500 eV has been used and a (8 × 8 × 8) Monkhorst-Pack grid of special k-points has been set up to sample the complete k-space of the system. A lattice constant of a= 3.2390 Å has been calculated by minimization of bulk energy. The reported experimental value for this parameter a is by 4.0% shorter [START_REF] Jayant | Thermal expansion of tungsten at low temperatures[END_REF]. The supercell is modelled by a 5-layer thick slab (using a 2×2 bcc surface unit cell) and 15.5 Å of vacuum. After relaxing the top two layers, an interlayer distance of 0.437a between the top two and of 0.517a between the second and third layer was observed.

3D H/W(100) PES

The construction of the 6D-PES, under the CRP method, requires the calculation of 3D atomic contributions that basically contain the high variations of the potential. Once 3D-PES are substracted from the 6D-PES, the remaining part known as interpolation function (I6D) is a smooth function much easier to interpolate than the full 6D-PES.

This decomposition strategy leads to accurate representation of both low and high variations of the PES. For the construction of the 3D atomic contributions, grid of 61 values of various atomic altitudes above 15 sites of the surface has been sampled. 3.1 displays the 15 sites for which a total set of about 915 spin-polarized DFT single point were computed. The zero of the potential corresponds to the H-atom far from the surface, at 6 Å of altitude. We checked that at this distance the atom does not interact with the surface or its periodic images. The DFT energies were calculated between Z = 6 Å above the surface and Z = -3.34 Å into the bulk. To

POTENTIAL ENERGY SURFACE FOR H 2 ON W(100)

sample the Brillouin zone of the supercell, an 8×8×1 Monkhorst-Pack grid of special k-points was used. The interpolation of the 3D PES contributions were carried out using cubic splines.

6D H 2 /W(100) PES

About 7500 DFT energy calculations, distributed in a total of 28 different molecular configurations, were used for the construction of the full 6D-PES. For each configuration,

POTENTIAL ENERGY SURFACE FOR H 2 ON W(100)

r varies from 0.4 Å to 2.3 Å (step size of 0.146 Å), while Z ranges from 0 up to 6 Å, with a step size of 0.235 Å. The (X, Y ) positions over which the DFT energy were computed are shown in Fig. 3.2. Values of (θ, φ) selected for the interpolation depends on the configuration as listed below:

1. Top Site (X = 0, Y = 0) 3 Configurations:

θ = 0 • ; θ = 90 • with φ = 45 • ; and θ = 90 • with φ = 90 • . 2. Bridge Site (X = 0.5a, Y = 0) 3 Configurations: θ = 0 • , θ = 90 • with φ = 0 • ; θ = 90 • with φ = 90 • .
3. Hollow Site (X = 0.5a, Y = 0.5a) 3 Configurations:

θ = 0 • ; θ = 90 • with φ = 0 • ; θ = 90 • with φ = 90 • .

Bridge-to-Hollow (B2H) Site

(X = 0.5a, Y = 0.25a) 7 Configurations θ = 0 • , θ = 45 • with φ = 0 • ; θ = 45 • with φ = 90 • ; θ = 45 • with φ = 270 • ; θ = 90 • with φ = 0 • ; θ = 90 • with φ = 45 • ; θ = 90 • with φ = 90 • .

Top-to-Hollow (T2H) Site

(X = 0.25a, Y = 0.25a) 5 Configurations:

θ = 0 • , θ = 45 • with φ = 0 • ; θ = 45 • with φ = 180 • ; θ = 90 • with φ = 0 • ; θ = 90 •
with φ = 90 • .

Top-to-Bridge (T2B) Site

(X = 0.25a, Y = 0) 7 Configurations:

θ = 0 • , θ = 45 • with φ = 45 • ; θ = 45 • with φ = 90 • ; θ = 45 • with φ = 135 • ; θ = 45 • with φ = 225 • ; θ = 90 • with φ = 45 • ; θ = 90 • with φ = 135 • .
As for the 3D terms, the accuracy of the 6D-PES is checked by comparing to additional data not used for the interpolation. Other PESs built using the CRP method, that have shown similar range of errors, have successfully been used [START_REF] Peña-Torres | Energy dissipation effects on the adsorption dynamics of N 2 on W(100)[END_REF][START_REF] Busnengo | Six-dimensional quantum and classical dynamics study of H 2 (ν= 0, j= 0) scattering from Pd(111)[END_REF] for simulating the dynamics of diatomic molecules in interaction with metal surfaces. Thus, we expect that the present 6D CRP PES provides us good reliability for the H 2 /W(100) dynamics.

Analysis of the PES

In Fig. 3.4, a selection of 2D-(Z, r) cuts of the potential are shown for vdW-DF2 and PW91

based PESs [START_REF] Busnengo | H 2 chemisorption on W(100) and W(110) surfaces[END_REF]. Both PESs present similar features with a non activated pathway for H 2 approaching the surface above the vicinity of the top site in horizontal orientation with respect to the surface. As already concluded for the N 2 /W(100) in previous studies [START_REF] Peña-Torres | Dynamics of N 2 sticking on W(100): the decisive role of van der Waals interactions[END_REF][START_REF] Peña-Torres | Energy dissipation effects on the adsorption dynamics of N 2 on W(100)[END_REF][START_REF] Ibargüen | Inelastic scattering of N 2 off W(001): Reconciling experiment and theory at low collision energies[END_REF] the vdW long range forces leads to a more attractive PES far from the surface and more repulsive when getting closer. One remarkable feature for those 2D cuts is the presence of a local minimum at Z=2.1 Å, in horizontal orientation over the top site for the vdW-DF2 case. This local minimum is characterized by a binding energy of 155 meV.

RESULTS

Results

Analysis of the Dynamics

Quasi-classical trajectories have been performed on the vdW-DF2 PES within the Born-Oppenheimer approximation. To account for surface temperature effects and molecular energy transfer to lattice vibrations a generalized Langevin oscillator (GLO) model was used, as described in previous works [START_REF] Busnengo | Surface temperature effects in dynamic trapping mediated adsorption of light molecules on metal surfaces: H 2 on Pd(111) and Pd (110)[END_REF]. Energy dissipation to the electrons of the metal (electron-hole (e-h) pairs excitations) are modeled by using the Local Density Friction Approximation (LDFA) [START_REF] Juaristi | Role of electron-hole pair excitations in the dissociative adsorption of diatomic molecules on metal surfaces[END_REF]. This model has proven his efficacy in several molecule-surface theoretical studies [START_REF] Goikoetxea | The dynamics of adsorption and dissociation of N 2 in a monolayer of iron on W(110)[END_REF][START_REF] Oliver Bünermann | Electron-hole pair excitation determines the mechanism of hydrogen atom adsorption[END_REF][START_REF] Novko | Ab initio molecular dynamics with simultaneous electron and phonon excitations: Application to the relaxation of hot atoms and molecules on metal surfaces[END_REF][START_REF] Martin-Gondre | Competition between electron and phonon excitations in the scattering of nitrogen atoms and molecules off tungsten and silver metal surfaces[END_REF][START_REF] Blanco-Rey | Electronic friction dominates hydrogen hot-atom relaxation on Pd(100)[END_REF][START_REF] Galparsoro | Energy dissipation to tungsten surfaces upon Eley-Rideal recombination of N 2 and H 2[END_REF][START_REF] Martin-Gondre | Energy dissipation channels in the adsorption of N on Ag(111)[END_REF]. As detailed in previous publications [START_REF] Busnengo | Six-dimensional quantum and classical dynamics study of H 2 (ν= 0, j= 0) scattering from Pd(111)[END_REF][START_REF] Crespos | Analysis of H 2 dissociation dynamics on the Pd (111) surface[END_REF], a conventional Monte Carlo sampling is used for setting up the initial conditions of the trajectories and exit channels probabilities are calculated by averaging over a large number of trajectories (typically 100 000 trajectories).

Quantization of the vibrational and rotational states of the molecules are taken into account as initial conditions. Molecules are counted as reflected when molecule center of mass recrosses the initial altitude Z 0 with a positive velocity pointing toward the vacuum. The dissociative adsorption is supposed to occur whenever the H-H distance reaches the value of 1.9 Å (which is more than twice the internuclear equilibrium distance r eq ), with a positive radial velocity. Then, molecular adsorption corresponds to the situation where H 2 is neither dissociated nor reflected once the maximum simulation time of t max =15 ps is reached. Sticking probability is thus defined as the sum of dissociative and non-dissociative molecular adsorption probabilities. In other words, sticking probability (S 0 ) is defined as the probability that molecules remain stuck at the surface without being reflected, thus S 0 = 1 -P ref , where P ref is the probability of reflection.

Fig. 3.5 compares calculated H 2 (J i = 0) and H 2 (J i = 1) sticking probabilities, where J i denotes the initial rotational state of the molecule. As can be seen, the effect of initial rotation on S 0 is rather weak, with a conservation of the non-monotonous shape typical of non activated diatoms/surface systems, (i.e. a system for which there exist at least one reaction pathways without potential energy barriers). Dissociative adsorption can further be decomposed into direct and indirect components according to the number of rebounds N reb of the molecule before being dissociated. N reb is defined as the number of times the center of mass momentum along Z changes its sign. The number of rebounds defining indirect and direct mechanisms is somehow arbitrary, but the criteria of N reb = 5 is often used [START_REF] Busnengo | H 2 chemisorption on W(100) and W(110) surfaces[END_REF]. Results show that the majority of the molecules are undergo a direct process in the whole range of energy. As for molecular adsorption, the indirect mechanism otherwise called trapping mechanism is only observed for low collision energies, i.e. below 20 meV.

RESULTS

RESULTS

As already commented in the work of Busnengo et al. [START_REF] Busnengo | H 2 chemisorption on W(100) and W(110) surfaces[END_REF] the outcome of a given trajectory, whether it is adsorbed or reflected, is decided during the approach at rather large distances. Fig. 3.8 shows probabilities P Zstop of reaching given altitudes Z= 3, 2.5, and 2 Å as a function of the initial collision energy. Those probabilities are compared with the total dissociation probability P diss . At low collision energy, typically below 100 meV, the probability P Zstop of reaching 2.5 Å qualitatively compares with P diss , leading to the conclusion that dissociation is decided at altitudes higher than 2.5 Å. For higher energies, such an agreement between P Zstop and P diss is obtained for a slightly lower altitude of about 2 Å. Same conclusion can be drawn, emphasizing on the role of the PES in the entrance channel. An analysis of the PES in the region of high altitudes shows the crucial role of the degrees of freedom X, Y, and θ for selecting trajectories able to follow dissociation route. 2D-(X,Y) cuts of the PES are shown in Fig. 3.9 for parallel orientation of the molecule (θ = 90, φ = 0) above the top site or its vicinity. On this graph, one can realize that only way for the molecules to penetrate further and dissociate is to be directed towards the top site in parallel orientation. In comparison, bridge and hollow sites are rather inaccessible for dissociation at low collision energies.

Comparison with Experiments and previous theoretical results

In Fig. 3.10, the sticking probability, obtained at different levels of approximation, i.e. BOSS, GLO and GLO-LDFA, is compared with data obtained from supersonic MB experiments [START_REF] Alnot | Adsorption and desorption kinetics with no precursor trapping: hydrogen and deuterium on W(100)[END_REF][START_REF] Berger | Adsorption of hydrogen on tungsten: a precursor path plus direct adsorption[END_REF]. The sticking curve obtained with the PW91 based-PES, is also plotted for comparison with previous theoretical simulations not including long range interactions.

Following experimental conditions, sticking probabilities have been calculated for a normal hydrogen beam (25% of J i = 0 and 75% of J i = 1 initial rotational states). As seen in the previous section (Sec. [START_REF] Berger | Adsorption of hydrogen on tungsten: a precursor path plus direct adsorption[END_REF][START_REF] Butler | Precursor dynamics in dissociative hydrogen adsorption on W(100)[END_REF]. Results for quasi-classical simulation within GLO-LDFA, GLO, BOSS model are also included. Pw91 curve corresponds to results obtained by Busnengo el al [START_REF] Busnengo | H 2 chemisorption on W(100) and W(110) surfaces[END_REF] (in green). Results for normal incidence of H 2 impinging molecules (25% of J i = 0 and 75% of J i = 1 rotational states), at 300K . lower energy. In general terms, PW91-PES is more reactive than the vdW one leading to an overestimation of the sticking when compared to experiments. The origin of this discrepancy comes from the fact that vdW PES is more repulsive close to the surface, therefore molecular adsorption or dissociative adsorption do not occur in most cases, except when molecule approaches by the Top site.

A prospective for further analysis of the discrepancies between theory and experiments at low energy, would be the use of gaussian weighting procedure as already done with success for H 2 /Pd(111) [START_REF] Crespos | Classical molecule-surface scattering in a quantum spirit: Application to H 2 /Pd(111) nonactivated sticking[END_REF] . In this method, the final actions of reflected molecules are quantized following a Gaussian Binning procedure. Then sticking probability can be recalculated by affecting a negligible weight to trajectories that are not following quantum constraints, as for example trajectories corresponding to molecules reflected back with vibrational energies below the zero poin energy.

Conclusions

Quasi-classical trajectories have been performed to analyze the dynamics of dissociative and non-dissociative adsorption of H 2 on W(100). A special care has been applied in the representation of the PES in a region that corresponds to the approach of the molecules far from the surface, by including long range forces in our DFT calculations with the vdW-DF2 functional. It has been shown that molecules can only dissociate or being molecularly adsorbed by approaching the surface through the top site and its vicinity.

This latter result is in agreement with previous theoretical studies [START_REF] Busnengo | H 2 chemisorption on W(100) and W(110) surfaces[END_REF] based on PW91

PES. The main difference is that in the vdW-DF2 case the effect of redirection towards the top site is more pronounced because of the repulsive contribution of vdW functional when molecules get close to the surface. This involves a lower sticking probability and a better agreement with experiments especially at energies higher than 150 meV. Including energy dissipation channels, do not cause significants changes on the dynamics with respect to rigid surface, neither improves the agreement between theory and experiments.

INTRODUCTION

Introduction

The scattering dynamics of atoms/molecules off solid surfaces has been extensively investigated over the last decades by both theory and experiments [START_REF] Díez | Dynamics of Gas-Surface Interactions[END_REF]. By using supersonic beams in Ultra High Vacuum conditions (UHV), the reactive and non-reactive scattering dynamics of N 2 on a W(100) surface has been experimentally studied by Rettner and co-workers [START_REF] Ct Rettner | Dynamics of the chemisorption of N 2 on W(100): Precursor-mediated and activated dissociation[END_REF]. A schematic representation of the experimental apparatus is shown in Fig. 4.1. Briefly, inside an UHV chamber supersonic beams of N 2 are directed toward a W(100) crystal. The surface is mounted on a rotatable manipulator that leads to an accurate control of the incident angle of the beams. The temperature of the surface can be also controlled. The final velocity of the scattered molecules, in the plane defined by the direction of the beam and that of the surface normal, is determined from the time of flight from a high-speed chopper to a differentially pumped rotatable mass spectrometer (QMS). More details about the experiments can be found in Rettner-1986[170]. As previously mentionned, although many simulations have been successfully performed within the framework of the Born-Oppenheimer Static Surface (BOSS) approximation, it is sometimes crucial to account for energy dissipation to the metal for elementary processes such as trapping, adsorption, inelastic scattering, and recombination [START_REF] Díez | Dynamics of Gas-Surface Interactions[END_REF][START_REF] Bonfanti | Classical and quantum dynamics at surfaces: Basic concepts from simple models[END_REF][START_REF] Alducin | Non-adiabatic effects in elementary reaction processes at metal surfaces[END_REF][START_REF] Simon P Rittmeyer | Energy dissipation at metal surfaces[END_REF]. To that end, effective models to describe energy transfers between the impinging species and surface phonons and electrons have been developed over the last 20 years [START_REF] Díez | Dynamics of Gas-Surface Interactions[END_REF][START_REF] Bonfanti | Classical and quantum dynamics at surfaces: Basic concepts from simple models[END_REF][START_REF] Alducin | Non-adiabatic effects in elementary reaction processes at metal surfaces[END_REF][START_REF] Simon P Rittmeyer | Energy dissipation at metal surfaces[END_REF]. The quality of the simulations is closely related to the accuracy of the electronic structure calculations, which critically depends on the choice of the exchange-correlation (EXC) functional. Several theoretical works have thus analyzed the influence of the functional on dynamical observables relevant to gas-surface interactions [START_REF] Díez | Dynamics of Gas-Surface Interactions[END_REF][START_REF] Díaz | Chemically accurate simulation of a prototypical surface reaction: H 2 dissociation on Cu(111)[END_REF][START_REF] Kroes | Towards chemically accurate simulation of molecule-surface reactions[END_REF].

INTRODUCTION

The role of exchange-correlation functionals in the dynamics of N 2 dissociation on W(100) and W(110) surfaces have been studied by using RPBE [START_REF] Hammer | Improved adsorption energetics within density-functional theory using revised Perdew-Burke-Ernzerhof functionals[END_REF] and PW91 [START_REF] John P Perdew | Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation[END_REF][START_REF] Bocan | The role of exchange-correlation functionals in the potential energy surface and dynamics of N 2 dissociation on W surfaces[END_REF] functionals. The results suggest that: the drastic reduction of the dissociation probability at low energies in RPBE calculations is due to large increase in repulsion with respect to the PW91 PES far away from the surface.

Recently, Pétuya et al. [START_REF] Pétuya | Revisiting the nonreactive scattering of N 2 off W(100): On the influence of the scattering azimuth on in-plane angular distributions[END_REF] have reported results of quasi-classical simulations for the non-reactive scattering of N 2 off W(100) making us of a DFT-based (PW91) potential energy surface. They have analyzed the evolution of the widths of the scattering angular distribution as a function of normal energy in the range E n = 0.1 -2.65 eV. It is suggested that, at low energy, dynamics is largely governed by the topology of the PES upon approaching the surface and the details of the interactions in that region, which might significantly influence trapping, could be of critical importance [START_REF] Pétuya | Revisiting the nonreactive scattering of N 2 off W(100): On the influence of the scattering azimuth on in-plane angular distributions[END_REF].

The evolution of the widths of the in-plane scattering angular distributions, as a function of collision energy, have been accurately reproduced for normal collision energies greater than 0.1 eV by Pétuya and coworkers. However, simulations fail badly at reproducing those experimental data at lower collision energies (E n < 100meV ).

Very recently, a PES has been built based on the vdW-DF2 [START_REF] John P Perdew | Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation[END_REF] exchange-correlation (EXC) functional for N 2 +W(100), which has been shown to overcome some of the main shortcomings of the previously used PESs at low energies, relying on the PW91 EXC functional [START_REF] Lee | Higher-accuracy van der waals density functional[END_REF]. In particular, molecular sticking probability at low collision energy (<500 meV) is now found in very good agreement with experiments [START_REF] Peña-Torres | Dynamics of N 2 sticking on W(100): the decisive role of van der Waals interactions[END_REF]. The analysis of the dynamics shows that this stems from both the presence of van der Waals (vdW) forces, which remove repulsive potential energy structures lying above 3 Å from the surface in the PW91 PES, and the more repulsive character of the vdW-DF2 PES at short distances.

In this chapter, we study the scattering dynamics of N 2 on W(100) by means of QCT calculations performed using the newly developed vdW-DF2 PES. The results are compared with the ones from previous simulations and experiments. As there is a

INTRODUCTION

special interest in describing and analyzing the influence of possible energy transfers to the surface, the dissipation to phonons and electrons, within the framework of the generalized Langevin oscillator (GLO) model [START_REF] Busnengo | Trapping, molecular adsorption, and precursors for nonactivated chemisorption[END_REF][START_REF] John | Dynamics of gas-surface interactions: 3D generalized Langevin model applied to fcc and bcc surfaces[END_REF] and local density friction approximation (LDFA), [START_REF] Díez | Dynamics of Gas-Surface Interactions[END_REF][START_REF] Díaz | Chemically accurate simulation of a prototypical surface reaction: H 2 dissociation on Cu(111)[END_REF][START_REF] Juaristi | Role of electron-hole pair excitations in the dissociative adsorption of diatomic molecules on metal surfaces[END_REF][START_REF] Alducin | Non-adiabatic effects in elementary reaction processes at metal surfaces[END_REF][START_REF] Simon P Rittmeyer | Energy dissipation at metal surfaces[END_REF] is accounted for.

Computational Details and Methods

As mentioned in Section 4.1, the recently developed vdW-DF2 PES [START_REF] Peña-Torres | Dynamics of N 2 sticking on W(100): the decisive role of van der Waals interactions[END_REF] is used in all calculations. The dynamics results are compared with those obtained for the PW91 PES. [START_REF] Pétuya | Revisiting the nonreactive scattering of N 2 off W(100): On the influence of the scattering azimuth on in-plane angular distributions[END_REF][START_REF] Ct Rettner | Dynamics of the chemisorption of N 2 on W(100): Precursor-mediated and activated dissociation[END_REF] For the construction of the PES, the Corrugation Reducing Procedure (CRP) [START_REF] Busnengo | Representation of the 6D potential energy surface for a diatomic molecule near a solid surface[END_REF][START_REF] Ra Olsen | Constructing accurate potential energy surfaces for a diatomic molecule interacting with a solid surface: H 2 +Pt(111) and H 2 +Cu(100)[END_REF] has been applied to interpolate density functional theory (DFT) energies computed within the generalized gradient approximation. Plane wave basis sets for the electronic orbitals and ultrasoft pseudopotentials, [START_REF] Vanderbilt | Soft self-consistent pseudopotentials in a generalized eigenvalue formalism[END_REF] to approximate the interaction with atomic cores, have been used within a periodic supercell approach. All of the electronic structure calculations were performed using the VASP package [START_REF] Hafner | The Vienna ab-initio Simulation Program VASP: An efficient and versatile tool for studying the structural, dynamic, and electronic properties of materials[END_REF]. It is known that structural phase transition that leads to a c(2x2) zigzag surface atom rearrangement takes place at temperatures lower than 200 K, in this case that rearrangement [START_REF] Ernst | Helium-atom-scattering study of the structure and phonon dynamics of the W 001) surface between 200 and 1900[END_REF][START_REF] Titmuss | Reconstruction of clean and adsorbate-covered metal surfaces[END_REF] has been ignored. Here calculation are done for surface temperature higher thank 300 K.

QCT simulations have been performed within four different models: (1) the Born-Oppenheimer Static Surface (BOSS) [START_REF] Díez | Dynamics of Gas-Surface Interactions[END_REF], for which the surface is static and no energy exchange between the molecule and the surface is allowed; (2) the Static Surface including electronic friction (LDFA) [START_REF] Díaz | Chemically accurate simulation of a prototypical surface reaction: H 2 dissociation on Cu(111)[END_REF][START_REF] Juaristi | Role of electron-hole pair excitations in the dissociative adsorption of diatomic molecules on metal surfaces[END_REF][START_REF] Alducin | Non-adiabatic effects in elementary reaction processes at metal surfaces[END_REF], which only accounts for dissipation to electron-hole (e-h) pair excitations; (3) the moving surface (GLO) [START_REF] Busnengo | Trapping, molecular adsorption, and precursors for nonactivated chemisorption[END_REF][START_REF] Cynthia | Molecular electronic property density functions: The nuclear magnetic shielding density[END_REF]. In this approach the surface is treated as an oscillator coupled to a thermal bath; and (4) the moving surface with electronic friction (LDFA-GLO), which accounts for both phonons and (e-h) pair excitations. The use of the GLO and LDFA effective models has been

shown previously [START_REF] Alducin | Non-adiabatic effects in elementary reaction processes at metal surfaces[END_REF] to provide a good compromise between the accuracy of results

and simplicity of implementation. The combination of GLO and LDFA models allows analyzing the competition between the different energy exchange channels or their possible coupling.

The incident diatoms are initiated in their rovibrational ground state (E 0 = 0.143 meV). Monte Carlo samplings of the N 2 initial orientation (internal angles θ and φ) and vibrational phase are carried out. All initial conditions for calculation have been set

up following faithfully the experimental setup. The direction of the incoming beam is fixed as follow: the polar angle (with respect to the surface normal) is set to Θ i =

Effect of the DFT Functional in the in-plane Angular

Distribution at Low Energies

The evolution of the experimental FWHMs of in-plane scattering angle distributions [START_REF] Ct Rettner | Dynamics of the chemisorption of N 2 on W(100): Precursor-mediated and activated dissociation[END_REF] is displayed as a function of normal incident energy (E n ) for The origin of the significant difference between both functionals at low collision energies is illustrated in Fig 4 .5. It displays, in the middle panels, the in-plane angular distributions for Θ i =60 • and E i = 0.088 eV collision energy. For the PW91 functional (middle left panel), simulations lead to a narrow distribution centered about the specular angle, at odd with the experiment, which shows significant broadening toward small angles and backward directions. Such a feature is correctly recovered when performing simulations using the vdW-DF2 functional (middle right panel). The origin of the broadening may be understood in terms of the dynamics of two subsets of scattering events.

Θ i = 45 • (up) and Θ i = 60 • (
The lower panels of Fig. 4.5 display the distributions of the lowest rebound altitudes.

A rebound is defined here as a change in the sign of molecular center-of-mass momentum from negative to positive along the Z-axis ([001] direction). The X and Y axes lie along the [START_REF] Kr Geethalakshmi | Non-reactive scattering of N 2 from the W(110) surface studied with different exchange-correlation functionals[END_REF] and [010] surface directions, respectively. The distributions reveal two components: they correspond to molecules bouncing at low (Z<5a 0 ) and high (Z>5a 0 ) altitudes from the surface, as it is well represented in the upper panel. It is to be noticed that a 0 refers to Bohr radius. As apparent from the lower panel, the fraction of N 2 molecules getting close to the surface, where the corrugation is high, is much greater when using the vdW-DF2 (60%) than that when using the PW91 (<5%) PES. This stems from the fact that the repulsive region lying above 3 Å from the surface for the PW91 PES is significantly reduced when using the vdW-DF2 PES, which includes van der Waals interactions [START_REF] Peña-Torres | Dynamics of N 2 sticking on W(100): the decisive role of van der Waals interactions[END_REF].

RESULTS AND DISCUSSION

This is illustrated in Fig. 4.6, in which one-dimensional cuts of both PESs are displayed as a function of N 2 altitude. This corresponds to a parallel orientation of the N 2 molecule, at its gas-phase equilibrium geometry, along the [START_REF] Kr Geethalakshmi | Non-reactive scattering of N 2 from the W(110) surface studied with different exchange-correlation functionals[END_REF] direction over the top, bridge, and hollow high-symmetry sites. PESs, respectively, highlighting the more intricate dynamics created by the latter. If the average number of rebounds is similar for trajectories reflecting at altitudes above Z = 5a 0 for both PESs (about 1.2), the trajectories reflecting at altitudes below Z = 5a 0 , which amount to 60% of the scattering events for the vdW-DF2 PES, involve on average 6.7 rebounds.

Effect of the Temperature on the Trapping-Desorption Fraction

Corrugation-mediated dynamics may lead to significant trapping of the impinging N 2 molecules. The interplay between trapping-desorption and direct inelastic scattering has

RESULTS AND DISCUSSION

been experimentally investigated [START_REF] Ct Rettner | Dynamics of the chemisorption of N 2 on W(100): Precursor-mediated and activated dissociation[END_REF]. It was found that though surface temperature does not affect the low-collision-energy trapping probability, "it significantly reduces the fraction of molecules that go on to dissociate by biasing the kinetics in favor of desorption" [START_REF] Ct Rettner | Role of surface temperature in the precursor-mediated dissociative chemisorption of N 2 on W(100)[END_REF]. The fractions of scattered molecules involving trapping-desorption for The increase in the trapping-desorption fraction with temperature, which is concomitant with a decrease of dissociative sticking [START_REF] Peña-Torres | Dynamics of N 2 sticking on W(100): the decisive role of van der Waals interactions[END_REF], is significant in experiments and simulations based on the vdW-DF2 PES. Conversely, the PW91 PES is not able to reproduce this feature. This again results from the negligible amount of trapping (<5% of the scattered molecules) generated by this PES because of the repulsive potential far

E i =0.

RESULTS AND DISCUSSION

from the surface. Considering the level of approximation in the experiments to extract the cosine component [START_REF] Ct Rettner | Angular and velocity distributions characteristic of the transition between the thermal and structure regimes of gassurface scattering[END_REF], as well as the use of rather simple effective models (GLO and LDFA) in the theoretical simulations, the agreement between theory and experiment might be considered very satisfactory when using the vdW-DF2 PES. When accounting for dissipation to both phonons and electrons (LDFA-GLO), QCT simulations are found in good agreement with experiments for scattering angles ranging from 20 to 80 • . This angle range corresponds to the maximum of the scattering angle distributions at such collision energies [START_REF] Pétuya | Revisiting the nonreactive scattering of N 2 off W(100): On the influence of the scattering azimuth on in-plane angular distributions[END_REF]. The contribution of electronic friction is also minor as revealed by the slight difference between GLO and LDFA-GLO models.The energy transfer is here mainly ruled by phonons, as it was observed for other gas-surface scattering processes involving atoms or molecules heavier than hydrogen [START_REF] Galparsoro | Energy dissipation to tungsten surfaces upon Eley-Rideal recombination of N 2 and H 2[END_REF]. For grazing scattering angles (> 70 • ), simulations relying on vdW-DF2 do not involve the fast decay of final translation energy with the scattering angle that shows up when using the PW91 PES. Such a behavior appears in better agreement with experiment. These scattering events involve a larger residence time close to the surface as N 2 molecules reflect back with low normal energies and thus are largely affected by the repulsive structure lying above 3 Å from the surface in the PW91-based PES. to be independent of collision energy. About 35% of the N 2 final energy is channeled into the rovibrational motion.

The remaining translation energy is shared approximately equally between the parallel and normal motions. The average scattering polar angle is thus close to Θ ∼ 45 • . It is to be noticed that the results are similar to those found previously for the PW91 PES [START_REF] Busnengo | Representation of the 6D potential energy surface for a diatomic molecule near a solid surface[END_REF], thus suggesting the weak influence of the functional for such average observables at these collision energies.

SUMMARY

Summary

The dynamics of the nonreactive scattering of N 2 on W(001) surface have been studied.

Quasi-classical simulations of the N 2 nonreactive scattering off W(001) have been performed, making use of the newly developed vdW-DF2 functional-based potential energy surface, including dispersion interactions and energy dissipation channels to surface through electrons hole-pairs excitation (e-h) and phonons. In light of the precedent discussion several conclusions arise:

• Dynamics simulations nicely capture the scattering dynamics even at low collision energy (below E n ∼ 0.1 eV) in contrast with the ones relying on the PW91 functional.

• van der Waals forces render the interaction more attractive at distances above 3 Å from the surface. This leads to a significant fraction of trapping-desorption scattering at odd with a previously developed potential energy surface relying on the PW91 functional, which is responsible of repulsive structures at such altitudes.

• Trapping is responsible for the broadening of the in-plane scattering angle distribution at low collision energies.

In agreement with experiments, trapping-desorption is favored by surface temperature at the expense of sticking.

At higher collision energies (E n > 0.2 eV), both PESs lead to similar results.

• At high energies (E n > 0.75 eV), for which experimental results are available, dissipation to phonons was found to govern energy transfers between the N 2 molecules and the W(001) surface, electronic friction being responsible for ∼10% of the energy transfer.

Introduction

The scattering of atomic species on metal surfaces is of high interest in many domains for both fundamental and applied science, as for example heterogeneous catalysis [START_REF] Nilsson | Chemical bonding at surfaces and interfaces[END_REF][START_REF] Honkala | Ammonia synthesis from first-principles calculations[END_REF][START_REF] Gabor | Introduction to surface chemistry and catalysis[END_REF][START_REF] Langmuir | The mechanism of the catalytic action of platinum in the reactions 2CO+O 2 →2CO 2 and 2H 2 +O 2 = 2H 2 O[END_REF][START_REF] Chorkendorff | Concepts of Modern Catalysis and Kinetics[END_REF][START_REF] Ertl | Primary steps in catalytic synthesis of ammonia[END_REF][START_REF] Mp Andersson | Structure sensitivity of the methanation reaction: H 2 -induced CO dissociation on Nickel surfaces[END_REF], plasma-wall interactions in nuclear physics [START_REF] Takagi | Hydrogen-deuterium exchange on plasma-exposed W and SS surface[END_REF][START_REF] Heinola | Hydrogen interaction with point defects in tungsten[END_REF][START_REF] Terentyev | Dislocations mediate hydrogen retention in tungsten[END_REF][185] or for the aerothermochemistry of thermal proctection systems in atmospheric reentry. [START_REF] Halpern | Chemical energy accommodation at catalyst surfaces. Flow reactor studies of the association of nitrogen atoms on metals at high temperatures[END_REF][START_REF] Capitelli | Molecular physics and hypersonic flows[END_REF][START_REF] Kurotaki | Construction of catalytic model on SiO 2 -based surface and application to real trajectory[END_REF][START_REF] Barbato | Model for heterogeneous catalysis on metal surfaces with applications to hypersonic flows[END_REF][START_REF] Balat | Thermal and chemical approaches for oxygen catalytic recombination evaluation on ceramic materials at high temperature[END_REF][START_REF] Bedra | A combined md simulation and experimental investigation of the O 2 formation on β-quartz at high temperature[END_REF][START_REF] Arasa | Adsorption of atomic oxygen and nitrogen at β-cristobalite (100): A Density Functional Theory study[END_REF][START_REF] Perron | Adsorption of atomic oxygen on the Cu(100) surface[END_REF]. A key question to address is the release of energy into the surface by means of coupling between the atom translation degrees of freedom and the surface atoms motion or the electrons of the solid [START_REF] Busnengo | Surface temperature effects in dynamic trapping mediated adsorption of light molecules on metal surfaces: H 2 on Pd(111) and Pd (110)[END_REF][START_REF] Martin-Gondre | Competition between electron and phonon excitations in the scattering of nitrogen atoms and molecules off tungsten and silver metal surfaces[END_REF][START_REF] Blanco-Rey | Electronic friction dominates hydrogen hot-atom relaxation on Pd(100)[END_REF][START_REF] Martin-Gondre | Energy dissipation channels in the adsorption of N on Ag(111)[END_REF]. H-atom sticking, after single or multiple collisions with the surface, has been shown to occur mainly through loss of energy to electron-hole (e-h) pairs excitations [START_REF] Blanco-Rey | Electronic friction dominates hydrogen hot-atom relaxation on Pd(100)[END_REF][START_REF] Nørskov | Correlation between sticking probability and adsorbate-induced electron structure[END_REF][START_REF] Svenja | An accurate full-dimensional potential energy surface for H-Au(111): Importance of nonadiabatic electronic excitation in energy transfer and adsorption[END_REF][START_REF] Kroes | Ab initio molecular dynamics calculations on scattering of hyperthermal H atoms from Cu(111) and Au(111)[END_REF][START_REF] Pavanello | Adiabatic energy loss in hyperthermal H-atom collisions with Cu and Au: A basis for testing the importance of nonadiabatic energy loss[END_REF][START_REF] Dorenkamp | Hydrogen collisions with transition metal surfaces: Universal electronically nonadiabatic adsorption[END_REF][START_REF] Ahsan | Electronic stopping power in gold: The role of d electrons and the H/He anomaly[END_REF]. Because of the large mass mismatch between atoms of the metal and hydrogen, transfer of energy to the lattice vibrations is unlikely to occur. Recent experimental and theoretical works on H-atom collision with several (111) surfaces of fcc transition metals (Au, Pt, Ag, Pd, Cu, and Ni) have exhibited a universal behavior with respect to nonadiabatic adsorption processes [START_REF] Dorenkamp | Hydrogen collisions with transition metal surfaces: Universal electronically nonadiabatic adsorption[END_REF]. In this early work, authors have shown that energy loss of the atoms is almost independent of the metal and is mainly due to electronic friction. Since no influence of metals work functions is observed, authors also conclude that no charge transfer occurs between the impinging atom and the surface upon collision for an initial energy of 1.92 eV.

In this work, simulations have been performed on the H/W(100) benchmark system leading to a further analysis of the universal energy loss scheme proposed by Dorenkamp et al. in the context of a more open surface and a bcc metal. Moreover, the penetration of atomic hydrogen on W metal is of high interest for nuclear physics [START_REF] Shimada | Overview of goals and performance of ITER and strategy for plasma-wall interaction investigation[END_REF][START_REF] Heinola | Hydrogen interaction with point defects in tungsten[END_REF][START_REF] Fernandez | Hydrogen diffusion and vacancies formation in tungsten: Density Functional Theory calculations and statistical models[END_REF][START_REF] Liu | Hydrogen diffusion in tungsten: A molecular dynamics study[END_REF][START_REF] Donald | Hydrogen in tungsten: Absorption, diffusion, vacancy trapping, and decohesion[END_REF].

Consequently a discussion on the first step of H absorption and diffusion into the solid is proposed, pointing out the importance of electronic friction for the description of sub-surface atoms motion.

For heavier atoms, such as nitrogen, energy exchange with lattice vibrations are expected to be much more efficient. Scattering dynamics simulations are here also performed for N/W(100). Again, energy dissipation channels are included in the simulations addressing the question of competition between both e-h pairs and lattice

INTRODUCTION

vibration excitation upon heavy atoms sticking on a metal surface [START_REF] Galparsoro | Energy dissipation to tungsten surfaces upon Eley-Rideal recombination of N 2 and H 2[END_REF][START_REF] Galparsoro | Hydrogen abstraction from metal surfaces: When electron-hole pair excitations strongly affect hot-atom recombination[END_REF][START_REF] Oihana Galparsoro | Energy dissipation to tungsten surfaces upon hot-atom and Eley-Rideal recombination of H 2[END_REF].

Coupling between the normal and parallel motion of the N atoms mediated through corrugation has been shown to play a role [START_REF] Volpilhac | Scattering of atomic nitrogen on W(100)[END_REF][START_REF] Strömquist | The dynamics of H absorption in and adsorption on Cu(111)[END_REF] especially at low collision energy. A special focus on this issue is made in this work, by revisiting the early work of G.

Volpilhac et al. [START_REF] Volpilhac | Scattering of atomic nitrogen on W(100)[END_REF][START_REF] Strömquist | The dynamics of H absorption in and adsorption on Cu(111)[END_REF] where no energy loss channels were included in the simulations.

A new potential energy surface (PES) has been built for H/W(100) by interpolation of density functional theory (DFT) [START_REF] Hohenberg | Inhomogeneous electron gas[END_REF][START_REF] Kohn | Self-consistent equations including exchange and correlation effects[END_REF] calculations making use of the vdW-DF2 [START_REF] Lee | Higher-accuracy van der waals density functional[END_REF] exchange correlation functional. For the N/W(100), we have used a similar PES, developed and presented elsewhere [START_REF] Peña-Torres | Dynamics of N 2 sticking on W(100): the decisive role of van der Waals interactions[END_REF][START_REF] Peña-Torres | Energy dissipation effects on the adsorption dynamics of N 2 on W(100)[END_REF]. The choice of the functional was originally motivated by previous works suggesting that dispersion forces may play an important role in the dynamics of molecules in interaction with metal surfaces [START_REF] Migliorini | Application of van der waals functionals to the calculation of dissociative adsorption of N 2 on W(110) for static and dynamic systems[END_REF][START_REF] Peña-Torres | Energy dissipation effects on the adsorption dynamics of N 2 on W(100)[END_REF][START_REF] Martin-Gondre | Influence of the van der Waals interaction in the dissociation dynamics of N 2 on W(110) from first principles[END_REF].

However, comparing the vdW-DF2 based PESs with older ones calculated at PW91 level, no significant differences has been noticed for the particular cases of both N/W(100) and H/W(100). As a consequence, the dynamics is not expected to be altered by the choice of functional.

This chapter is organized as follows: In Sec. 5.2 the computational details and methodology are presented. The results are discussed in Sec. 5.3. A summary with the main conclusions is given in Sec. 5.4

Computational Details

The interaction between H and N atoms and the W(100) surface has been modeled by a 3-dimensional PES which depends on X, Y, and Z coordinates referring the position of the H/N atom with respect to one surface atom at the origin of the Cartesian frame. A periodic supercell approximation has been used. The metallic surface is represented by a 5-layer slab, and the calculations were performed using a 15×15×1

Monkhorst-Pack grid of k points for a (2×2) structure. We obtained a bulk lattice constant of a = 3.2390 Å. After relaxation, an interlayer distance of 0.437a has been found for the two topmost layers and of 0.517a between the second and the third layer.

The DFT calculations have been carried out within the Generalized Gradient Approximation (GAA) [START_REF] Perdew | Density-functional approximation for the correlation energy of the inhomogeneous electron gas[END_REF] making use of the vdW-DF2 [START_REF] Lee | Higher-accuracy van der waals density functional[END_REF] exchange-correlation

COMPUTATIONAL DETAILS

functional. Ultrasoft pseudo-potentials [START_REF] Vanderbilt | Soft self-consistent pseudopotentials in a generalized eigenvalue formalism[END_REF][START_REF] Laasonen | Implementation of ultrasoft pseudopotentials in ab initio molecular dynamics[END_REF][START_REF] Laasonen | Car-parrinello molecular dynamics with Vanderbilt ultrasoft pseudopotentials[END_REF] have been used to describe the interaction with atomic cores. All the calculations were performed with the Vienna Ab Initio Simulation Package (VASP) [START_REF] Hafner | The Vienna ab-initio Simulation Program VASP: An efficient and versatile tool for studying the structural, dynamic, and electronic properties of materials[END_REF][START_REF] Kresse | Ab initio molecular dynamics for liquid metals[END_REF][START_REF] Kresse | Ab initio molecular dynamics for open-shell transition metals[END_REF][START_REF] Kresse | Efficient iterative schemes for ab initio totalenergy calculations using a plane-wave basis set[END_REF]. The 3-dimensional PES for H/W(100) has been built within the framework of this investigation, while the N/W(100) PES has been taken from the work of Peña-Torres et al [START_REF] Peña-Torres | Dynamics of N 2 sticking on W(100): the decisive role of van der Waals interactions[END_REF].

The dynamics simulations are based on the adiabatic 3D-PES thus obtained, making use of the classical trajectories method. In order to calculate probabilities for the different exit channels such as adsorption, absorption and/or reflection, 10 000

trajectories have been performed for several initial collision energies ranging from 10 meV to 1.0 eV. The initial X and Y positions of the atoms above the W(100) unit cell were uniformly sampled. All trajectories have been started at Z 0 = 6.0 and Z 0 = 7.0 Å from the surface in the asymptotic region of the PES for H and N atoms, respectively.

Normal and off-normal incidence conditions have been analyzed. An atom is considered as reflected whenever it reaches back its initial distance from the surface Z 0 with a velocity pointing towards vacuum. Absorption occurs when atoms reach the value of Z min =-3.0 Å, which corresponds to a position slightly above the third layer of the slab.

The atoms are also considered absorbed if, after the maximum integration time of 2.0 ps, they remain between the top layer (Z=0) and the third layer with kinetic energy lower than the one required to escape from the bulk. Adsorption occurs when the atom is neither reflected nor absorbed with a kinetic energy lower than the minimum required to escape the attractive potential of the surface. The maximum integration time of 2.0 ps has been chosen such that the exit channel probabilities are all converged.

In the calculations, a surface oscillator was used to simulate energy exchange between the atom and vibrational modes of the lattice (phonons dissipation channel) via the socalled generalized Langevin oscillator (GLO) model [START_REF] Busnengo | Surface temperature effects in dynamic trapping mediated adsorption of light molecules on metal surfaces: H 2 on Pd(111) and Pd (110)[END_REF]. Then energy dissipation to the metal mediated by (e-h) pairs excitations is modelled by an effective friction term introduced in the classical equations of motion, and evaluated through the Local density friction approximation (LDFA) [START_REF] Juaristi | Role of electron-hole pair excitations in the dissociative adsorption of diatomic molecules on metal surfaces[END_REF][START_REF] Galparsoro | Energy dissipation to tungsten surfaces upon Eley-Rideal recombination of N 2 and H 2[END_REF]. Previous works on H scattering on various metals have excluded any formation of a transient H -specie since energy loss spectra have been shown to be mostly independent of the work function of metals [START_REF] Dorenkamp | Hydrogen collisions with transition metal surfaces: Universal electronically nonadiabatic adsorption[END_REF]. Thus,

RESULTS AND DISCUSSIONS

Results and Discussions

Analysis of the PES

In order to describe the main features of the PESs for the two systems under study, i.e.

H/W(100) and N/W(100), 1D cuts of the potentials for the highest symmetry sites as a function of altitude in Z are displayed in Fig. 5.2. In agreement with previous experimental and theoretical works [START_REF] Alnot | Adsorption and desorption kinetics with no precursor trapping: hydrogen and deuterium on W(100)[END_REF][START_REF] Busnengo | H 2 chemisorption on W(100) and W(110) surfaces[END_REF][START_REF] Barnes | Hydrogen-adsorption-induced reconstruction of tungsten (100): Observation of surface vibrational modes[END_REF], the bridge site is the most stable site for H adsorption on W(100) at Z=1.11 Å with 3.08 eV binding energy.

The penetration of the H atom into the slab is energetically favorable through the bridge site with barriers below the asymptotic potential energy. At large distances (between 2.0 and 3.0 Å) the top site is the most attractive site for both H and N. Unlike what is found for H/W, the most stable site is the hollow one at Z=0.63 Å for N/W with a 6.75 eV binding energy, similar to that of the B2H site, in agreement with previous results reported for Volpilhac [START_REF] Volpilhac | Dissociative adsorption of N 2 on the W(100) surface[END_REF]. The absorption process is shown to be activated with a barrier height of about 2 eV on the bridge (with respect to the origin of the potential energy that is set for N far from the surface), whereas the top site remains as the most attractive site at large distances, like in the H/W case. A comparison with a previous PES based on PW91 functional has been made and, in contrast to what was found for N 2 [START_REF] Peña-Torres | Dynamics of N 2 sticking on W(100): the decisive role of van der Waals interactions[END_REF][START_REF] Ibargüen | Inelastic scattering of N 2 off W(001): Reconciling experiment and theory at low collision energies[END_REF], for instance, no significant changes have been noticed by adding van der Waals contributions to the description of atom-surface interaction.

Sticking Probability

The total sticking probability S 0 , defined as S (GLO-LDFA). By comparing the 3 models, it is possible to infer which dissipation channel dominates energy transfer to the surface. The study of diffusion of H in tungsten has gained a lot of interest in the last decades in the context of plasma facing materials (PFMs) [START_REF] Takagi | Hydrogen-deuterium exchange on plasma-exposed W and SS surface[END_REF][START_REF] Heinola | Hydrogen interaction with point defects in tungsten[END_REF][START_REF] Terentyev | Dislocations mediate hydrogen retention in tungsten[END_REF][185], in which tungsten is a promising candidate. Hydrogen is known to induce blisters formation when tungsten is exposed to the high flux plasma irradiation, specially at low incident energy [START_REF] Shu | Blister bursting and deuterium bursting release from tungsten exposed to high fluences of high flux and low energy deuterium plasma[END_REF]. It is also known that blistering problem induced by H retention may cause reduction of the lifetime of the material. The diffusion pathway into the bulk observed in the simulations is the Bridge-T2B-Bridge, as depicted in the Fig. 5.6. In short, the atoms that are neither reflected nor adsorbed penetrates the surface through the Bridge site and find the absorption site 1 (Bridge-1) and then migrate to site 2 (T2B) overcoming a barrier of 0.73 eV. The site 2 is the most populated for absorption, as have been already discussed. Then, the H-atoms undergoes a migration to the site 3 (Bridge-3), site that has an activation barrier of 0.28 eV. Sites 1, 2, 3 shown in Figs. 5.2 and 5.6 correspond to tetrahedral interstitial sites (TIS), so we have found that motion of H-atoms occurs along a TIS-TIS-TIS path. Similar results have been reported by Liu et al [START_REF] Liu | Hydrogen diffusion in tungsten: A molecular dynamics study[END_REF]. In the present work, it was also found that atoms absorbed in any of the three sites can remain trapped. The lack of kinetic energy does not allow them to move out of their absorption site. Interestingly, absorbed atoms are weakly bounded when compares to species adsorbed on the surface (E bind =3.08 eV), which makes this system a promising candidate for sub-surface recombination reactions with atoms coming from the gas phase. Results reported by Markejl et al [START_REF] Markelj | Production of vibrationally excited hydrogen molecules by atom recombination on Cu and W materials[END_REF] suggests that such recombination reactions would be characterized by a large exothermicity (of about 4 eV), producing vibrationaly hot molecules. Moreover, if most of the exothermicity is transferred to vibration motion, one can expect producing molecules excited until v = 7 vibrational state [START_REF] Markelj | Production of vibrationally excited hydrogen molecules by atom recombination on Cu and W materials[END_REF]. Au [START_REF] Oliver Bünermann | Electron-hole pair excitation determines the mechanism of hydrogen atom adsorption[END_REF], the atomic adsorption follows a penetration/resurfacing mechanism for both H/W(100) and N/W(100) cases. If we consider penetration to happen when the atom goes below Z=0, about 60% of the H atoms adsorbed on W(100) has followed a penetration/resurfacing mechanism, against 15% for the N atoms case. In comparison, for H on Au, Bunermann et al [START_REF] Oliver Bünermann | Electron-hole pair excitation determines the mechanism of hydrogen atom adsorption[END_REF] found a penetration-resurfacing contribution of about 80%. At 0.8 eV, H-atoms can penetrate as deep as Z=-2 Å before resurfacing and being adsorbed on a bridge site of the surface. The loss of energy by electronic friction is enhanced by this penetration effect. For N-atoms, the penetration is somehow weaker since Z=-0.5 Å is only rarely reached. N/W(100) PES does not allow penetration of the atoms as easily as in the H-atoms case. The results discussed throughout this section corresponds to trajectories initiated with normal incidence to the surface.

Diffusion

Energy Loss upon Reflection

For several [START_REF] Ac Luntz | CH 4 dissociation on metals: A quantum dynamics model[END_REF] metal surfaces (i.e. Au, Pt, Ag, Pd, Cu, and Ni), comparisons between theoretical simulations and atomic beam experiments have been performed [START_REF] Svenja | An accurate full-dimensional potential energy surface for H-Au(111): Importance of nonadiabatic electronic excitation in energy transfer and adsorption[END_REF][START_REF] Dorenkamp | Hydrogen collisions with transition metal surfaces: Universal electronically nonadiabatic adsorption[END_REF] by considering energy loss spectra for H-atoms scattering back to the vacuum. In these works, H-atoms inelastic scattering has been analyzed for high collision energy of 1.92 eV and off-normal incidence (initial angle of the beam with respect to the surface normal vector equal to 45). It was found that the average energy loss is about 0.75 eV and very similar for all metals ranging from the lightest (Ni) to the heaviest surface atoms (Au). Because the loss of energy is mainly due to electronic effects (that depends only weakly on the metal under study when compared with surface motion dissipation channel, where the mass is playing a key role), there is no significant change with the nature of the metal.

For comparison, the energy loss spectra of H scattering off the W(100) surface has been displayed in Fig. 5.9, for an initial energy of 1.92 eV and two incidence angles (normal incidence and off normal incidence with a polar angle of 45 degrees taking an average over all azimuthal angles). In the off-normal case the mean energy loss is 0.8 eV in close agreement with previous works [START_REF] Svenja | An accurate full-dimensional potential energy surface for H-Au(111): Importance of nonadiabatic electronic excitation in energy transfer and adsorption[END_REF][START_REF] Dorenkamp | Hydrogen collisions with transition metal surfaces: Universal electronically nonadiabatic adsorption[END_REF] on other metals. The shape of the energy loss spectrum exhibits three peaks, that can be attributed to reflections at different values of Z, roughly 1.2 , -0.5, and -2 Å. As expected, the highest is the penetration of the atoms (i.e. the lowest altitude), the highest is the energy loss by electronic friction since atoms then travel in regions where electronic density is high. As a consequence the third peak between 1.0 and 1.5 eV of energy loss is attributed to the We have also checked that sticking probability is also weakly influenced by the incidence angle (calculations were made from normal incidence to 75 off-normal incidence). The energy dissipation to the electrons of the surface follows total energy scaling (TES) process whatever if considering reflection, or sticking.

Chapter 6

General Conclusions

The main goal of this work was to get more insights about the dynamics of the interactions of atomic and molecular Nitrogen and Hydrogen with the W(100) surface.

In particular, reactive and non-reactive adsorption of H 2 , inelastic scattering of N We have first investigated the dynamics of dissociative and molecular adsorption of H 2 . A new PES has been built with special care for the representation of the PES in the region that corresponds to the approach of the molecules far from the surface. Our results suggest that molecules can only dissociate or being molecularly adsorbed by approaching the surface through the top site and its vicinity. This is in close agreement with previous theoretical studies performed on a PW91 based-PES, regardless of electronic nonadiabatic effects. A significant difference is that the redirection towards the top site is more pronounced because of the repulsive contribution of vdW functional when molecules get close to the surface. This leads to a lower sticking probability,in better agreement with experiments especially at energies higher than 150 meV. We have also found that including energy dissipation channels does not sensibly alter the dynamics with respect to rigid surface, neither improves the agreement between theory and experiments.

Then, the dynamics of the non-reactive scattering of N 2 on W(100) making use of the recently developed vdW-DF2 functional-based PES has been investigated. The simulations properly capture the scattering dynamics, even at low collision energy (below E n ∼0.1 eV normal energy), where dynamics relying on the PW91 functional is shown to significantly disagree with experiments. A great agreement with experimental data regarding the in-plane scattering angle distributions in the low energy regime is here achieved.

The analysis also allows to conclude that van der Waals forces render the interaction more attractive at distances above 3 Å from the surface, which leads to a significant fraction of trapping-desorption scattering at odd with a previously developed potential energy surface relying on the PW91 functional, which reveals repulsive structures at such altitudes. For this latter functional, a narrow distribution of scattered N 2 centered about the specular angle is observed because molecules bounce far away from the surface, where the corrugation is low. When using vdW-DF2 a significant broadening toward small angles is observed, in accordance with experiments.

Trapping is responsible for the broadening of the in-plane scattering angle distribution at low collision energies. On the other hand, we have observed that trapping-desorption is favored by surface temperature at the expense of sticking. In contrary to what is found at low collision energy regime, at higher energy (E n > 0.2 eV), both vdW-DF2 and PW91 PESs exhibit similar results. A qualitative agreement of the angle-resolved average translation energies as a function of the scattering angles is achieved once dissipation of energy to the phonons and electrons of the surface is accounted for. In the high energy regime (E n > 0.75 eV), dissipation through the lattice vibrations is found to govern energy transfers between the N 2 molecules and the W(001) surface, whereas electronic friction is responsible for ∼10% of the energy transfer.

Finally, to provide more insights into the elementary processes occurring between H and N atoms and the W(100) surface, we have analyzed their scattering dynamics. A special focus on the sticking mechanism, which is decomposed in adsorption and absorption processes, has been made. The H-atoms sticking is mediated mainly through energy dissipation to electrons, whereas coupling to lattice vibrations plays the major role for the case of N-atoms. We have observed that H-atoms can enter into the W slab through the bridge sites and either absorb on three well characterized sub-surface sites or diffuse as hot atoms deeper into the bulk. In agreement with experimental observations, the motion of the H-Atoms between the top layer and the third layer of the slab takes place following a pathway connecting Tetrahedral Interstitial Sites (TIS).

Part of the H-atoms adsorption mechanism occurs following the so-called penetration/resurfacing scheme. In this mechanism, H-atoms visit regions into the slabs, bounce and then get thermalized at the bridge surface site. Our simulations, performed within the energy range of initial collision energies below 1.0 eV, reveals that the sticking of N-atoms only proceeds through adsorption on the hollow site. No absorption species is observed as the N/W(100) interaction prevents any penetration below -1.5 Å.

Our results suggest that reflection is a very direct process for both H and N-atoms in contrast to sticking, which is characterized by indirect mechanisms involving many rebounds before thermalization after loss of the major part of the initial kinetic energy to lattice vibrations and/or electrons of the metal. The energy loss spectra of the reflected back H-atoms is not found to depend on the incidence angle. The simulated mean energy compares well with previous experimental and theoretical works on H scattering on several fcc metal(111) surfaces (Ni, Cu, Pd, Ag, Au, Pt). Our results for W(100), a bcc metal, reinforce the claim of an universal behavior of the electronically nonadiabatic adsorption process of atomic H on metal. For H and N-atoms atom scattering on the W(100) surface, the vdW interactions do not seem to play a significant role in the scattering .

To conclude, we have shown in this work that, if vdW forces are shown to influence significantly the scattering observables (both sticking and reflection) for H 2 and N 2 diatoms on W(100), the scattering of the corresponding atoms is much less affected.

Many open questions need to be solved and developments to be performed to get to a quantitative description of gas-surface scattering dynamics. On the long term, a method that would account for the motions of all the atoms of the system as well as the coupling

Figure 1 . 1 :

 11 Figure 1.1: Some elementary atom-diatom/surface. a) Reflection (in black); Adsorption; (in blue) c)Absorption (green)

  Chapter 3 focuses on the analysis of dissociative adsorption and molecular adsorption of H 2 on W(100). The effects of the long range van der Waals (vdW) forces, included in the representation of the potential energy surface (PES) by making use of density functional (DFT) calculations based on the vdW-DF2 functional, is studied. A detailed description of the construction of the PES is provided. The results are compared with the ones of previous simulations based on a PW91-PES and with available experimental data. • Chapter 4 presents the results of the non-reactive scattering of N 2 off W(100) including both energy dissipation channels (phonons and electronic excitations). In-plane scattering angles, the effect of including van der Waals interaction and that of the temperature on the trapping-desorption fraction are discussed, in view of MB experiments. The chapter especially adresses the low collision energy regime and the role played by van der Waals interactions and energy dissipation. • Chapter 5 deals with the atomic scattering of H and N on the W(100) surface. The effect of lattice vibration and electronic excitation on the dynamics is studied by means of classical trajectories on the electronically adiabatic ground state PES. A detailed analysis of the H,N/W(100) mechanisms of absorption and adsorption is 1.4. THEORETICAL SIMULATIONS carried out. The results of the dynamics are discussed in relation with observables

Figure 2 . 2 :

 22 Figure 2.2: Schematic representation of the coordinate system used in the construction of the PES. Surface atoms are depicted as large light brown spheres. The molecule as small dark blue spheres with its respective bond. The origin of the coordinate system is placed on top of a surface atom. X A,B , A,B , Z A,B corresponds to the coordinates of the center of mass of the atoms A and B. θ and φ are the polar and azimuthal angle, respectively.

38 ) 3 . 1

 3831 By combining both model in one calculation one can determine (by comparison) a possible coupling between these energy dissipation channel or if any of them dominates in the dynamic. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Potential Energy Surface for H 2 on W(100) . . . . . . . . . . . . . . . 3.2.1 Analysis of the PES . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Analysis of the Dynamics . . . . . . . . . . . . . . . . . . . . 3.3.2 Comparison with Experiments and previous theoretical results 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3. 1 .

 1 INTRODUCTIONstatic surface approximation, and no molecular energy loss was taken into account by lattice vibration or electron-hole pairs (e-h) excitations. Only qualitative agreement was reached with a large overestimation of S 0 at collision energies higher than 100 meV. The presence of non-activated reaction pathways have been evidenced and pure indirect dissociative adsorption mechanism (mediated through dynamic trapping) has been invoked for explaining the low energy decrease of the sticking. The authors have stressed on the strong influence of the PES representation in a region that corresponds to the molecule far from the surface (entrance channel) on the dynamics of both reflection and adsorption of H 2 on W(100) and W[START_REF] Stoer | bulirsch introduction to numerical analysis springer-verlag[END_REF].

for example N 2

 2 /W[START_REF] Kr Geethalakshmi | Non-reactive scattering of N 2 from the W(110) surface studied with different exchange-correlation functionals[END_REF][START_REF] Stoer | bulirsch introduction to numerical analysis springer-verlag[END_REF][START_REF] Alducin | Why N 2 molecules with thermal energy abundantly dissociate on W(100) and not on W(110)[END_REF][START_REF] Alducin | Dissociative dynamics of spin-triplet and spin-singlet O 2 on Ag(100)[END_REF][START_REF] Alducin | Dissociative adsorption of N2 on W(110): Theoretical study of the dependence on the incidence angle[END_REF][START_REF] Bocan | The role of exchange-correlation functionals in the potential energy surface and dynamics of N 2 dissociation on W surfaces[END_REF]. However Recent theoretical works[START_REF] Migliorini | Application of van der waals functionals to the calculation of dissociative adsorption of N 2 on W(110) for static and dynamic systems[END_REF][START_REF] Peña-Torres | Dynamics of N 2 sticking on W(100): the decisive role of van der Waals interactions[END_REF][START_REF] Peña-Torres | Energy dissipation effects on the adsorption dynamics of N 2 on W(100)[END_REF][START_REF] Martin-Gondre | Influence of the van der Waals interaction in the dissociation dynamics of N 2 on W(110) from first principles[END_REF] have shown the great interest of including vdW forces by means of new functionals such as vdW-DF[START_REF] Dion | Van der Waals density functional for general geometries[END_REF] or vdW-DF2[START_REF] Lee | Higher-accuracy van der waals density functional[END_REF]. As a matter of fact, significant improvement in the description of the interaction between the N 2 molecule and the W(100) surface has been achieved by using vdW-DF2 functional[START_REF] Peña-Torres | Dynamics of N 2 sticking on W(100): the decisive role of van der Waals interactions[END_REF][START_REF] Peña-Torres | Energy dissipation effects on the adsorption dynamics of N 2 on W(100)[END_REF][START_REF] Ibargüen | Inelastic scattering of N 2 off W(001): Reconciling experiment and theory at low collision energies[END_REF]. Dynamics simulations were in particular able to reproduce experiments of N 2 non-reactive scattering and dissociative adsorption on W(100), especially at low collision energies.

Fig. 3 .

 3 Fig. 3.1 and Table3.1 displays the 15 sites for which a total set of about 915

Figure 3 . 1 :

 31 Figure 3.1: Potential energy as a function of H altitude over the surface (Z) for three selected sites not included in the interpolation. The position of the sites over the unit cell is shown in colors on the right hand side and is specified in the legends. Solid circles corresponds to the DFT energies, whereas solid lines represent the energies obtained from the interpolation.

Fig. 3 . 2 ,

 32 shows potential energy as a function of the center of mass altitude for different configurations. Angular interpolation is illustrated in Fig. 3.3. A thorough analysis of the 6D-PES shows that interpolation errors are lower than 40 meV, with highest errors in strongly repulsive regions of the PES not relevant for the dynamics.

Figure 3 . 2 :

 32 Figure 3.2: Potential energy as a function of the H 2 center of mass altitude over the surface (Z) for 5 different molecular configurations. The position over the unit cell are shown on the right hand side in color code. Continuous lines and solid circles correspond to the interpolation and the DFT energies values respectively. The molecular orientation and internuclear distance for each configuration is depicted in the legend.

Figure 3 . 3 :

 33 Figure 3.3: Potential energy as a function of the H 2 polar angle θ. Comparison between interpolated potential energies (solid lines) and DFT energy values not used in the interpolation procedure (solid symbols). The three configuration used for the comparison are describes in the legend and the scheme of the unit cell shown on the right hand side.

Figure 3 . 4 :

 34 Figure 3.4: 2D-(Z,r) cuts of the H 2 /W(100) PESs from vdW-DF2 and PW91 functionals. Several molecular configurations depicted on insets are presented. Those configurations are from left to right: top vertical, T2B, bridge, and hollow in horizontal orientation. Thick red line indicate the zero potential energy isovalue i.e. the energy of the molecule far from the surface at its equilibrium interatomic distance. Thin full lines correspond to positive values of the potential whereas thin dashed lines correspond to negative values. Contour spacing is 0.1 eV.

Figure 3 . 5 :

 35 Figure 3.5: Sticking probability as a function of collision energy for two initial (J i ) rotational state of H 2 .

Figure 3 . 6 :

 36 Figure 3.6: Decomposition of sticking probability into dissociative and molecular adsorption.

Figure 3 . 7 :

 37 Figure 3.7: Probabilities of direct or indirect component to dissociative adsorption as a function of collision energy.

Figure 3 . 8 :

 38 Figure 3.8: P Zstop probabilities as a function of collision energy (see text).

Figure 3 . 9 :

 39 Figure 3.9: 2D-(X, Y )-cuts of the vdW-PES for a configuration on Top site with a parallel orientation con respect to the surface and r = r eq . The molecule is located at a distance of Z = 2 Å from the surface. Red thick line corresponds to the zero of the potential energy. Solid (dashed) black lines represent positive (negative) values separated by a spacing line of 50 meV.

Figure 4 . 1 :

 41 Figure 4.1: Molecular beam apparatus for determining scattering distributions and dissociative chemisorption probabilities. Key: R1-R5, pumping stages and vacuum regions; BS, beam source; C, high-speed chopper; F2 and F3, beam flags; QMS, doubly differential pumped quadrupole mass spectrometer; LBBD, low energy electron diffraction screen; AES, Auger electron spectrometer.The crystal is held on a manipulator which provides for heating and cooling and rotates about the same axis as the QMS. Taken from Rettner 1986[170].

Figure 4 . 2 :

 42 Figure 4.2: Schematic representation of the physical situation. Coordinates system and the W(100) unit cell. W atoms in blue. N 2 in red diatom. The polar angles Θ i and Θ f are defined with respect Z axis. The azimuthal angle Φ is defined with respect to the [001] surface axis. Θ and φ for internal coordinates of the diatoms are shown in figure within the blue circle.

45 • , 60 •

 60 , and the azimuthal angle (with respect to the[START_REF] Kr Geethalakshmi | Non-reactive scattering of N 2 from the W(110) surface studied with different exchange-correlation functionals[END_REF] surface axis) is set to Φ i = 13 • (see Fig. 4.2). Initial values of the molecular center-of-mass positions are chosen to sample uniformly the squared W(100) unit cell area. The maximum time for reflection is set to 10 ps: if N 2 molecules do not scatter back to vacuum within this time they are considered to be sticked on the surface. It was checked that this integration time is large enough to ensure convergence of the computed observables. The coordinate system, polar and azimuthal angles, and internal angles of the diatoms are depicted in Fig. 4.2. The molecule is supposed to dissociate whenever the N-N distance reaches the value of 2.24 Å (twice the N 2 gas-phase equilibrium distance) with positive radial momentum. As in the experiments, where only in-plane diffract is measured, here only molecules scattered within an angular deviation of ±1 • about in-plane angle are considered. Standard binning is used to semiclassically quantize the final rovibrational states. More than 3x10 6 trajectories have been launched to provide at least 50 000 in-plane scattering events, which ensure convergence of in-plane observables at each experimental condition. The full widths at half-maxima (FWHMs) have been extracted from the in-plane angular distributions with an estimated error of ±2 • . experimental one. Similar quality for the in-plane angular distribution were obtained for several initial collision energies in the 0.2-5.3 eV range.

  down) polar incidence angles in Fig. 4.4. QCT simulations within the BOSS, LDFA, GLO, and LDFA-GLO models, are also shown for both PESs based on PW91 (left) and vdW-DF2 (right). For both incidence angles, the experimental FWHMs evolve nonmonotonously, showing the largest width at the lowest collision energy (E i = 0.088 eV). As apparent from the Fig 4.4, the QCT simulations relying on the PW91 functional are in reasonable agreement with the experiment into the entire range of energy (rather more for Θ i = 60 • than Θ i = 45 • ), except for the lowest collision energy, for which the FWHM is the smallest. This deficiency is overcome by the use of the vdW-DF2 functional (right). The FWHMs are nicely described by all theoretical models over the whole collision energy range. By inspection of the Fig. 4.4 it is observed that all models reasonably describe experiments and including the energy dissipation to surface electrons and phonons does not seem to influence much this observable.

Figure 4 . 4 :

 44 Figure 4.4: Evolution of the width of the in-plane scattering angle distributions (FWHM) as a function of normal energy E n for Θ i =45 • (upper panels), Θ i =60 • (lower panels), and Φ i =13 • :Experiments (black dots)[START_REF] Ct Rettner | Dynamics of the chemisorption of N 2 on W(100): Precursor-mediated and activated dissociation[END_REF] and QCT simulations within the BOSS (red squares), GLO (blue up triangles), LDFA (cyan circles), and LDFA-GLO (green down triangles) models based on PW91 (left) and vdW-DF2 (right) functionals. Lines are guides to the eyes. The estimated uncertainty on the simulated FWHMs is about the size of the symbol used in the plot.

Figure 4 . 5 :

 45 Figure 4.5: Upper panel: Schematic representation of the minimum approach to the surface. Rebounds at Low and High Z are defined depending on whether the molecules bounce at altitudes above or below 5a 0 . Middle panels: In-plane scattering angle distributions from experiments (black dots) and LDFA-GLO simulations (green line) for E i =0.088 eV, Θ i =60 • , and Φ i =13 • . The blue and purple dashed lines are the contributions of scattered molecules bouncing at low Z and high Z, respectively. Lower panels: Distributions of the lowest rebound altitudes. Right (Left) panels correspond to the PES based on vdW-DF2 (PW91) functionals.

Figure 4 . 6 :

 46 Figure 4.6: One-dimensional cuts of PW91 (dashed line) and vdW-DF2 (solid line) for the parallel orientation of the N 2 molecule, at its gas-phase equilibrium geometry, along the [100] direction over the top (left), bridge (middle), and hollow (right) high-symmetry sites. The horizontal dotted line corresponds to the collision energy of Fig. 4.5 (E i = 0.088 eV)

  Fig. 4.7 for T s = 1200 K.)

Figure 4 . 7 :

 47 Figure 4.7: Trapping-desorption fraction as a function of surface temperature T s from experiments[180] (black dots) and simulations using PESs based on PW91 (magenta diamonds) and vdW-DF2 (blue squares) for E i =0.088 eV, Θ i =60 • , and Φ i =13 • . The inset illustrates the trapping-desorption cosine contribution for surface temperature T s =1200 K for the vdW-DF2based PES. Angles are in degrees.

4. 3 . 4

 34 Effect of the Energy Dissipation on the Final Translation EnergyAlthough dynamics calculations within the static surface approximation has been successfully used in many cases of gas-surface reactions, it is quite clear that including energy dissipation channels in the dynamics allows to obtain results with more physical meaning. With that in mind, to analyze the energy dissipation to the W(001) surface upon N 2 scattering, the average translation energies of the scattered molecules as a function of the scattering angles are displayed in Fig.4.8 for 1.5 and 2.5 eV collision energies and Θ i =45 for which experimental results are available. The BOSS model reveals a strong mixing of the N 2 translation, rotation, and vibration degrees of freedom but is not able to account for experiments as no energy can be transferred to the surface. When only dissipation to electrons (LDFA) is included, a minor effect is found, suggesting the weak influence of this energy transfer at such collision energies.

Figure 4 . 8 :

 48 Figure 4.8: Angle-resolved average translation energies as a function of the scattering angle.Experiments[START_REF] Ct Rettner | Dynamics of the chemisorption of N 2 on W(100): Precursor-mediated and activated dissociation[END_REF] (dots) and QCT simulations within the BOSS (red solid line), GLO (blue dashed line, T=800 K), LDFA-GLO (green dashed line, T=800 K), and LDFA (cyan dashed line) models are displayed. The dotted horizontal line in the lower panel represents the N 2 initial total energy. Energies are in electronvolts and angles in degrees.

Figure 5 . 1 :

 51 Figure 5.1: Top-view of the unit cell and the sites (green) for which DFT calculations have been carried out. W atoms in the first layer are in yellow. In orange, the W atom belongs to the second layer. Top, Bridge, Hollow, T2B, T2H, and B2H high symmetry sites are shown. The lattice constant is a = 3.239 Å

= 1 -

 1 P ref with P ref being the atomic reflection probability, is plotted in Fig. 5.3 as a function of collision energy. Results for both H and N + W(100) are included. A comparison is made between the sticking probabilities calculated using three models previously described: (i) the static surface approximation (Born-Oppenheimer static surface (BOSS) for which no exchange of energy is accounted for, (ii) the GLO model to allow energy dissipation to lattice vibrations and account for temperature (here 300 K) (iii) the GLO-LDFA model in which both surface motion and e-h pairs excitations are accounted for in the simulations

Figure 5 . 2 :

 52 Figure 5.2: 1D cuts of the potential energy surface as a function of atom/surface distance Z for several high symmetry sites. The upper panel corresponds to H/W(100). 1, 2, 3 refers to absorption sites. The Lower panel corresponds to N/W(100). Energies and distances are given in eV and Å, respectively.

Figure 5 . 5 :

 55 Figure 5.5: Distribution of final altitudes for H-atoms absorbed into the bulk at 0.3 and 0.8 eV of collision energy. Simula-tions are done within the GLO-LDFA model with a surface temperature of 300K.

5 _Figure 5 . 6 :

 556 Figure 5.6: Energetic scheme of the H-atoms in motion throughout the W(100) slab following the TIS-TIS-TIS pathway. Dashed line corresponds to the mean total energy of atoms reaching the -3.0 Å.

Figure 5 . 7 :Figure 5 . 8 :

 5758 Figure 5.7: Distribution of number of rebounds for H-atoms (top panel) and N-atoms (bottom panel) either reflected, adsorbed on the surface, or absorbed into the bulk for two different collision energies: 0.3, and 0.8 eV. Those distributions are computed after 2.0 ps of integration time. GLO-LDFA model

Figure 5 . 9 :

 59 Figure5.9: Total energy loss spectra (in eV) of reflected atoms for an initial collision energy of 1.92 eV. Two values of incidence angles are reported (normal incidence and off normal incidence with a polar angle of 45 degrees averaged over all azimuthal angles). The distributions are normalized to the area. GLO-LDFA model

  2 and H/N sticking have been investigated. To that end, global DFT-PESs have been built, using the vdW-DF2 functional, to account for long range van der Waals (vdW) interactions. Electronic structure energies have been interpolated via the Corrugation Reducing Procedure (CRP). The quasi-classical nuclei dynamics has been carried out, including non-adiabatic effects resulting from energy dissipation to lattice vibrations and electronic excitations by means of effective models, namely the Generalized Langevin Oscillator (GLO) and the Local Density Friction Approximation (LDFA) respectively.

Table 3 . 1 :

 31 Unit cell coordinates for all the sites on which the DFT were performed. Coordinates in blue corresponds to sites used to build the PES. All position are in unit of cell parameter a.

Table 4 .

 4 1 summarizes the average partition of the energy associated with in-plane scattering for both collision energies within the framework of the GLO and LDFA-GLO models for the vdW-DF2 PES. E tot , E rot , E vib and E tr denote the average total, rotational, vibrational, and translational energies, respectively. ∆E is the average energy transferred to the surface, and Θ is the average scattering angle. About 20% of the initial energy is transferred to the surface for both collision energies and slightly affected by accounting for dissipation to electrons. Energy transfers to phonons also mainly drive energy partition among the N 2 degrees of freedom. This partition is found

			4.3. RESULTS AND DISCUSSION
		Simulated Average Final Energies
			E i = 1.5 eV	E i = 2.5 eV
	vdW-DF2 GLO	LDFA-GLO	GLO	LDFA-GLO
	∆E	0.32	0.36	0.52	0.6
	E tot	1.31	1.28	2.12	2.04
	E rot	0.28(21) 0.27(21)	0.46(22) 0.42(21)
	E vib	0.17(13) 0.16(12)	0.28(13) 0.27(13)
	E tr	0.87(66) 0.85(67)	1.38(65) 1.34(66)
	Θ	45.3	45	46.6	45.5

Table 4 . 1 :

 41 E tot , E rot , E vib , and E tr respectively, the average total, rotational, vibrational, and translational energies. ∆E is the average energy transferred to the surface. Energies have been rounded to the second decimal place. The numbers in parentheses indicate (in %) the ratio of the respective energy to the average total energy E tot . Energies are in electronvolts. Θ is the average scattering angle in degrees.

Table 5 . 1 :

 51 Analysis of energy for absorption sites. The initial collision energy is 0.3 eV can reach in the simulations. In Table5.1, mean energy losses are compared for trajectories ending into absorption sites, with an initial total energy of 0.3 eV. The mean energy loss which corresponds to those trajectories still in diffusion is rather small (0.73 eV) leading to "hot atoms" characterized by mean total energy of about -0.43 eV, that is enough to overcome all the barriers to travel through the three first layers of the slab as can be seen on Fig.5.6, where diffusion path as a function of potential energy is depicted.

Slab Slab

put at a distance Z far from the surface. A representative scheme of this approach is depicted in Fig 2 .1. The size of the supercell is determined depending on the physical problem under scrutiny. The number of layers must be such that the properties of the

Summary

In this chapter an overview of the theoretical methods implemented in this work is presented.

Within the framework of the Born-Oppenheimer approximatino, the electronic problem is tackled by solving the Schrodinger equation, for a set of relevant positions for the nuclei, by plane-wave DFT calculations, adapted to periodi systems.

Potential energy surfaces are then built by interpolation using the corrugation reducing procedure. The theoretical description of the nuclei dynamics within the classical framework is then adressed, including GLO and LDFA effective models, to account respectively for energy dissipation into the vibrations and electron excitations of the solid.

Chapter 4

Inelastic Scattering of N 2 on W(100):

Reconciling Experiment and Theory at Low Collision Energies

Results and Discussion

In view of the significant improvement in the description on the dissociative adsorption of N 2 on W(100) with the vdW-DF2-PES presented in the previous studies [START_REF] Peña-Torres | Dynamics of N 2 sticking on W(100): the decisive role of van der Waals interactions[END_REF], in this section the results of dynamics simulations for the scattering of N 2 on W(100) are reported. The discussion is focused on the results obtained for dynamics at low collision energy regime and the role of the van der Waals interactions in this dynamics. Although the scattering angle tends to be slightly overestimated, especially at angles greater than 50 • when allowing energy transfers to the surface, (regardless if it is via phonons or e-h), a good agreement between QC simulation and experimental data is found. The shape and width of in-plane angular distributions match quite well with the

In-plane Scattering Angle Distribution (FWHM)

Sticking Probability

Collision Energy (meV) From the information contained in Fig. 5.3 arises that, within the BOSS model, the sticking probability of H decreases sharply when initial collision energy increases to approximately 200 meV and then remain almost constant up to 1 eV. In the case of N/W, S 0 decreases monotonically throughout the range when energy is increased, as has been observed in earlier theoretical simulations [START_REF] Volpilhac | Dissociative adsorption of N 2 on the W(100) surface[END_REF], where it has been shown that dynamic trapping governs the sticking of atoms at low energy. Within the BOSS level, only transfer of energy between the perpendicular and parallel motion of the atom can lead to trapped atoms. By increasing collision energy this transfer becomes rapidly inefficient and the sticking probability gets small. At this level of approximation, one may compare the present BOSS results with the ones obtained by Volpilhac et al [START_REF] Volpilhac | Scattering of atomic nitrogen on W(100)[END_REF] on a PW91 based PES. In their study, absorption was supposed to occur whenever trajectories reached the top surface layer located at Z=0. Then, calculation was stopped and those trajectories were accounted for in the sticking probability. For comparison purposes, we performed a calculation of the sticking probability using the same criteria.

An almost identical sticking curve is obtained with a sharp decrease and a plateau at about 0.5. If atoms are allowed to travel between Z=-3.0 and 0 Å in the simulations, the sticking probability is largely reduced (with a value of 0.1 for 1 eV of collision energy (see Fig. 5.3) and composed uniquely of long-lived species bouncing on the surface with high kinetic energy. This result shows that resurfacing [START_REF] Oliver Bünermann | Electron-hole pair excitation determines the mechanism of hydrogen atom adsorption[END_REF] and sub-surface motion plays an important role in the dynamics and should be accounted for in the simulations.

Going beyond the BOSS model by including energy dissipation channels increases significantly the trapping of atoms in a larger range of collision energies. A comparison of the GLO and GLO-LDFA curves reveals that electronic effects are clearly predominant for the H/W(100) case, whereas coupling to lattice vibrations is the main channel of energy loss for N/W(100). The high sticking probability (close to 1) obtained for the whole energy range has been observed in other theoretical simulations making use of different PES and different methodologies. Recent theoretical work of Y.

Dorenkamp et al. [START_REF] Dorenkamp | Hydrogen collisions with transition metal surfaces: Universal electronically nonadiabatic adsorption[END_REF] exhibits the same trend with a uniformly high sticking probability close to unity for H on Ni [START_REF] Ac Luntz | CH 4 dissociation on metals: A quantum dynamics model[END_REF]. A slight decrease of the probability from 1 to 0.6 is only observed at very high collision energy (more than 3 eV).

In Fig. 5.4, a decomposition of the sticking probability in two mechanisms is proposed. The first mechanism corresponds to adsorption (P ADS ) of the atom on the surface. The second mechanism is associated to an absorption (P ABS ) of the atom between the first and the third layer of the slab. Atoms that reach the third layer with velocity vector pointing towards the bulk are also counted as absorbed. For both H,N/W(100) the sticking probability is mainly due to atoms adsorbed on the surface. As expected from the analysis of the PES, H atoms are able to penetrate into the bulk and being absorbed. The absorption process is rather independent of the H collision energy and represents about 10% of probability. The N/W(100) dynamics are much simpler since only adsorption onto the surface is observed.

Collision Energy (meV)

Insights into the Adsorption and Absorption Processes

An analysis of the final Z distribution of the absorbed atoms reveals three sites of absorption below the top layer of the slab (see Fig. 5.5 where Z-distributions are plotted for 0.3 eV and 0.8 eV of initial collision energies). A detailed analysis, allow us to locate those absorption sites on bridge site for the Z=-0.40, and -2.37 Å distances, and on top-to-brigde (T2B) site for Z = -1.55 Å.

The 3 sites of absorption are represented for rather low collision energy (0.3 eV) and higher energy (0.8 eV) (Fig. 5.5). The most populated site is the T2B, especially at high energy, even if it is not the site characterized by the highest binding energy (see Table 5.1 for a comparison of the binding energy over the 3 adsorption sites). A non negligible proportion of the adsorbed atoms diffuse into the bulk, as revealed by the histograms at Z=-3.0 Å, which altitude corresponds to the lowest altitude trajectories

Summary and Remarks

Scattering dynamics of H and N atoms on W(100) has been investigated by means of theoretical simulations.

A special focus has been made on sticking mechanism discriminated into adsorption and absorption processes.

As also demonstrated elsewhere, the H-atoms sticking is mediated through energy dissipation to electrons whereas coupling to lattice vibrations plays the major role for the case of N-atoms.

H-atoms can penetrate the W slab through the bridge sites and either end absorbed on various sub-surface sites (bridge and T2B) between topmost and third layer, or diffuse as hot atoms deeper into the bulk. The H-Atoms motion between the top layer and the third layer of the slab occurs following a TIS-TIS-TIS pathway. Part of the H-atoms adsorption mechanism follows the so-called penetration/resurfacing scheme already proposed in earlier works. Sub-surface adsorbed H-atoms have been characterized and could potentially lead to recombination events with atoms coming from the gas phase, producing highly vibrationally excited molecules. Further theoretical works would be required to confirm this possible scenario.

Within the energy range considered in the simulations (i.e. collision energy below 1 eV), the sticking of N-atoms is only described through the adsorption. No absorption is observed as the N/W(100) interaction prevents any penetration below -1.5 Å. For both H and N atoms the reflection is a direct process whereas sticking is characterized by indirect mechanisms involving many rebounds before the atoms remain thermalized on the surface.

The energy loss spectra of the reflected back H-atoms has been studied, exhibiting total energy scaling behavior. The mean energy loss determined in our simulations is in close agreement with previous experimental and theoretical works on H scattering on several metal surfaces (Ni, Cu, Pd, Ag, Au, Pt). Further studies would be required to fully understand the features of the energy loss spectra in relation with surface structure and symmetry.

with all the electrons would be the ultimate goal. But, such a method is today very far and we still need to work with approximate approaches to rationalize gas-surface processes.

A first perspective from my work is to understand the origin of the discrepencies between theory and experiments regarding the H 2 scattering from W(100). The H 2 scattering from Pd [START_REF] Ac Luntz | CH 4 dissociation on metals: A quantum dynamics model[END_REF] has been studied recently in the group by using quasi-classical trajectories in the quantum spirit [START_REF] Crespos | Classical molecule-surface scattering in a quantum spirit: Application to H 2 /Pd(111) nonactivated sticking[END_REF]. To that end, the final actions of reflected molecules are pseudo-quantized in Bohr's spirit (using the Gaussian Binning procedure)

and the diffraction-mediated-trapping of the incoming wave is adequately treated (submitted J. Phys. Chem. Lett.). Such classical simulations get to a spectacular agreement with state-of-the-art quantum dynamics calculations. Given the discrepencies of our quasi-classical dynamics results with experiments, it could be thus very useful to performed quantum dynamics calculations for H 2 on W(100), using for example Multi-Configuration Time Dependent Hartree wave-packet propagation, and compare with our quasi-classical trajectory simulations analyzed in a quantum spirit.

A second perspective is to analyse diatom recombination of H 2 from Tungsten. This surface process has been intensively investigated in the past ten years both theoretically [START_REF] Galparsoro | Energy dissipation to tungsten surfaces upon Eley-Rideal recombination of N 2 and H 2[END_REF][START_REF] Galparsoro | Hydrogen abstraction from metal surfaces: When electron-hole pair excitations strongly affect hot-atom recombination[END_REF][START_REF] Oihana Galparsoro | Energy dissipation to tungsten surfaces upon hot-atom and Eley-Rideal recombination of H 2[END_REF] and experimentally [START_REF] Markelj | Production of vibrationally excited hydrogen molecules by atom recombination on Cu and W materials[END_REF]. In particular, experiments by Markejl et al [START_REF] Markelj | Production of vibrationally excited hydrogen molecules by atom recombination on Cu and W materials[END_REF] have suggested that such recombination reactions could be characterized by a large exothermicity (of about 4 eV), producing vibrationally very hot molecules. Such energetics is not consistent with recombination of gas-phase atoms with atoms adsorbed on the surface but would be possible through recombination with an hydrogen atom sitting in a TIS sub-surface site. It would be then very nice to investigate recombination dynamics with subsurface atoms. Both perspectives require a sizeable amount of work but would help to tackle unsolved questions of the field.