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Astron. Astrophys. 334, 363-375 (1998)

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2. Hamiltonian and potential energy surface

2.1. The model

The graphite surface has been modeled by a rigid cluster. Consequently, only 6 degrees of freedom are necessary to describe the reactive collision in the vicinity of the surface. The hamiltonian of the whole system can be written as:


where [FORMULA] and [FORMULA] are respectively the position and the conjugated momentum of the [FORMULA] H atom; m is the mass of one hydrogen atom.

From this Hamiltonian, the equations of motions of the H atoms near the surface can be easily derived from Hamilton's equations:


The choice of a rigid surface for this system is justified by the small mass of the hydrogen atom with respect to the carbon one, so that a small energy transfer to the surface (small coupling with the phonons) is expected. A comparison between rigid and non-rigid surface is in progress and will published later.

The crucial point for predicting the reactivity of one H atom with another H atom adsorbed on the graphite surface is the interaction potential. For the purpose of calculation we need a realistic description of the interaction through an empirical potential model. The model potential energy surface (PES) has been built from atom-atom potentials with parameters which naturally depend on the chemical nature of the binding of the involved atoms with the surface. The interaction potential [FORMULA] can be decomposed as a sum of two terms that are detailed heredown.

Surface-H interaction
The first term ([FORMULA]) corresponds to the interaction between all the H atoms and the surface. The binding energy of one isolated chemisorbed H atom on graphite has been calculated by different authors using ab-initio procedure (Bennet et al. (1971); Aronowitz & Chang (1985); Klose (1989) and Fromherz et al (1993)). These numerical values are strongly different depending on the methods used by these authors. In this work, we have chosen to reproduce qualitatively the ab-initio results obtained by Fromherz et al. (1993). The interaction potential has been written as:


The first contribution [FORMULA] is calculated from a sum of pairwise atom-atom terms between each of the H atoms and the [FORMULA] (= 67) effective centers of forces located at the centers of the benzenic rings. We have:


As the interaction between the surface and an H2 molecule or a H atom is strongly different, the progressive evolution from the net chemisorption of H to the much weaker adsorption of H2 is accounted for by the addition of a purely repulsive ([FORMULA]) term whose magnitude depends on the H -H distance by:


where [FORMULA] is the coordinate of the [FORMULA] H atom along the axis perpendicular to the surface, and [FORMULA] is depending on the H -H distance ([FORMULA]) by the following relation:


The values of all the parameters used to describe the surface-H interaction are reported in Table 1.


Table 1. Potential parameters for the H -graphite interaction.

H-H interaction
The second contribution ([FORMULA]) to the total potential energy is the lateral interaction between all the H atoms. A Morse potential has been used to describe this interaction:


[FORMULA] is the distance between the [FORMULA] and the [FORMULA] H atoms. [FORMULA] is the equilibrium distance of the diatomic molecule in the gas phase. [FORMULA] is the binding energy for the Morse potential. In our model, we assume that the magnitude of [FORMULA] depends on the localization of the diatomic with respect to the surface as:


[FORMULA] is the z coordinate of the [FORMULA] H atom and [FORMULA] the binding energy for the isolated H2 molecule in the gas phase in its ground electronic state [FORMULA].

2.2. Characteristic features of the reactive surface

Values of all the [FORMULA] potential parameters are reported in Table 2. As noted just before, the potential model has been built in order to reproduce the main characteristics of the H -graphite potential calculated by Fromherz et al. (1993). These authors have found that the binding energy of the most stable configuration was equal to -1.31 eV which corresponds to a weak chemisorption. In this equilibrium structure, the H atom is just located above a carbon atom and the z coordinate is equal to 1.15 Å . In this minimum energy configuration, they have found that graphite surface was locally reconstructed: the carbon atom is attracted to the H atom. This tendency to form a tetrahedron results from the evolution from a sp2 (graphite) to a sp3 character.


Table 2. Potential parameters for the H -H interaction.

Concerning the chemisorption of the atomic hydrogen, one only site has been found with the two models described in the previous section. The H atom is on a top position just above a carbon atom. In the potential model (I), the binding energy is equal to [FORMULA] = -1.32 eV and the distance between the H atom and the carbon atom is equal to 1.19 Å . In Fig. 1 is plotted the potential energy curve along the z coordinate for a single H atom. A small activation barrier ([FORMULA] = 0.081 eV) appears at z = 3.20 Å . A saddle point between two equivalent top sites appears in the middle of the C-C bond. The height of this barrier for the H diffusion (or migration) is equal to 0.25 eV. The middle of the C-C bond and the center of the ring correspond to saddle points in the PES.

[FIGURE] Fig. 1. Plot of the graphite-H interaction potential as a function of the z desorption coordinate. The solid curve and the dashed curve correspond respectively to model (I) and model (II).

In the case of model (II), the topology of the PES is not strongly modified. In the minimum energy configuration, the H atom is always located just above a carbon atom at a slightly larger value of z ([FORMULA] = 1.22 Å). The corresponding energy is equal to [FORMULA] = -1.35 eV. In this second model, the activation barrier along the z coordinate is slightly smaller ([FORMULA] =0.028 eV) and at a larger z value ([FORMULA] = 4.20 Å). Comparison between the two proposed models for the dependence of the potential along the z coordinate is displayed in Fig. 1. The consequence of the small barrier at large z for the H -graphite potential is that a small activation barrier is present in the entrance channel of the reactive PES. As pointed out by Fromherz et al. (1993) the reality of this small activation barrier has to be confirmed by more refined ab-initio calculations. Indeed very recent ab-initio calculations based on the density functional theory (Jeloaica & Sidis, private communication) have just confirmed its existence.

About the adsorption site for the [FORMULA] molecule, the two models give a minimum energy configuration with the H2 molecule parallel to the surface. With model (I) the binding energy is equal to [FORMULA] = - 0.112 eV, and slightly smaller [FORMULA] = - 0.088 eV in the case of model (II). This is in good agreement with the value proposed by Ross et al. (1964)

(- 0.084 eV), and of the same order of magnitude as the values derived by Mattera et al. (1980) from scattering experiments (-0.052 eV). Unfortunately, to our knowledge, no experimental information is available about the localization of this site with respect to the surface.

Since we are interested in the recombination process it is useful to look at a representation of the total reactive H -Surface-H PES. Fig. 2 is a contour plot of the total PES of model (I) in the collinear approach above a top site, using as reaction coordinates the [FORMULA] distance r and the C-H distance R from the chemisorbed H atom to the surface. In this graph an ER process is represented by a trajectory coming from the top at large r and with R [FORMULA] 1.2 Å and ending at the lower right corner at large R with r [FORMULA] 0.7 Å . The activation barrier in the entrance channel is too small to be seen in this figure. All the data about the minimum energy configurations for H on graphite with the two potential models have been summarized in Table 3.

[FIGURE] Fig. 2. Contour plot of the reactive PES in a collinear collision for potential model (I). The spacing between contour lines is equal to 0.45 eV.


Table 3. Characteristics of the H -graphite potential energy surface. [FORMULA]: equilibrium distance of the chemisorbed H from the surface; [FORMULA]: binding energy in the chemisorption site; [FORMULA]: distance of the barrier in the entrance channel from the surface; [FORMULA]: energy at the top of the barrier.

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© European Southern Observatory (ESO) 1998

Online publication: May 12, 1998