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

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3. Dynamical calculations

The equations of motion (Eqs. (2)) for the H atoms have been numerically integrated with an Adams-Moulton fifth order predictor-corrector algorithm. Consequently, a constant energy trajectory was run in the phase space. The time step for this propagation was equal to 0.10 fs. The energy conservation was satisfied to about 10-5 for one typical trajectory.

The initial coordinates of the colliding H atom was randomly chosen in a (x,y) square cell of size 2L [FORMULA] 2L. Most of the calculations have been made with L = 5 Å. The z coordinate of this atom was initially set at 15 Å, a distance where the interaction between the H atom and the surface was negligible. The initial momentum of this H atom was also randomly chosen to describe a given gas phase equilibrium temperature following a Maxwell-Boltzmann distribution.

In this work, most of the results have been obtained with the direction of the initial momentum normal to the surface. However, to analyse the influence of an initial angular distribution, we have generated some trajectories with the angle between the normal vector and the velocity vector randomly chosen between - [FORMULA] and + [FORMULA].

For the other hydrogen atom, two different procedures have been used. The first one, which we call the classical one, consists to prepare the initially chemisorbed H atom on carbon cluster at a given kinetic temperature [FORMULA] calculated from the equipartition theorem:


[FORMULA] is the Boltzmann constant and [FORMULA] corresponds to the time averaged kinetic energy of the chemisorbed atom before the other H atom is in interaction with the surface. [FORMULA] has been chosen around 70 K which means that the H adsorbed atom was prepared near its equilibrium configuration with only a very small amount of vibrational energy. In this approach the zero-point energy was not considered.

The second procedure corresponds to a quasi-classical trajectory (QCT) in which the zero-point energy (ZPE) of the H atom on the graphite surface is taken into account by a semi-classical method based on the Ehrenfest adiabatic theorem. We built a separable harmonic potential [FORMULA] defined by:


[FORMULA], [FORMULA] and [FORMULA] are the cartesian coordinates of the equilibrium configuration in the "real" H -graphite potential. In this harmonic separable hamiltonian, one can select trajectories which satisfy the semi-classical quantization by:


[FORMULA] is randomly chosen in the [O;2 [FORMULA] ] interval. [FORMULA] = [FORMULA]) is the pulsation.
The same relations are to be considered for y, [FORMULA], z and [FORMULA]. Consequently for a set of vibrational quantum numbers [FORMULA], we can select trajectories in the phase space corresponding to the [FORMULA] hamiltonian. To transform the harmonic hamiltonian into the "real" hamiltonian, we built a time-dependent hamiltonian [FORMULA] defined as:


[FORMULA] is the harmonic potential defined previously and [FORMULA] is the "real" potential. g(t) is a slowly varying time dependent function, defined for t [FORMULA] [0; [FORMULA] ], by:


In this work, [FORMULA] was taken equal to 1 ps. During the adiabatic switching, quantum numbers [FORMULA], [FORMULA] and [FORMULA] are conserved if the transformation is a slow enough process ([FORMULA] has to be larger than [FORMULA] 100 [FORMULA] characteristic vibrational period of the system). The final coordinates and associated momenta satisfy the semi-classical quantization and then are associated to a set of vibrational quantum numbers in the "real" hamiltonian. They are taken as initial coordinates and momenta for the collisional trajectories.

The maximum duration of one collisional trajectory was equal to 12 ps. During this microcanonical trajectory, if the z coordinate of one H atom was found greater than 15 Å , this atom was considered as desorbed. A distance criterion was then used to test if a H2 molecule had been formed: if the distance between the two H atoms was smaller than 2.0 Å , a molecule was considered as desorbed to the gas phase.

If this was the case, the tranlational, rotational and vibrational energies of the H2 molecules were evaluated. The translational energy is given by [FORMULA] = [FORMULA] in which [FORMULA] is the linear momentum of the H2 molecule center of mass and [FORMULA] is the mass of the diatomic molecule. The vibrational part is calculated as [FORMULA] = [FORMULA], and the rotational contribution is calculated by [FORMULA] = [FORMULA]. [FORMULA] is the angular momentum of the H2 diatomic molecule and µ its reduced mass.

A semi-classical vibrational number v was calculated from the semi-classical quantization of the action integral:


[FORMULA] is the internuclear potential of the H2 isolated molecule in its ground electronic state. [FORMULA] is the rovibrational energy of the molecule and h is the Planck's constant.

For a given set of parameters ([FORMULA], [FORMULA], [FORMULA] or ([FORMULA])), 2000-8000 trajectories have been run to obtain statistically significant results.

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© European Southern Observatory (ESO) 1998

Online publication: May 12, 1998