*Astron. Astrophys. 334, 363-375 (1998)*
## 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 2*L*
2*L*. 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 -
and + .
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 calculated from the
equipartition theorem:
is the Boltzmann constant and
corresponds to the time averaged kinetic energy
of the chemisorbed atom before the other *H* atom is in
interaction with the surface. 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 defined by:
, and
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:
is randomly chosen in the [O;2
] interval. =
) is the pulsation.
The same relations are to be considered for *y*,
, *z* and .
Consequently for a set of vibrational quantum numbers
, we can select trajectories in the phase space
corresponding to the hamiltonian. To transform
the harmonic hamiltonian into the "real" hamiltonian, we built a
time-dependent hamiltonian defined as:
is the harmonic potential defined previously
and is the "real" potential. g(t) is a slowly
varying time dependent function, defined for t
[0; ], by:
In this work, was taken equal to 1
*ps*. During the adiabatic switching, quantum numbers
, and
are conserved if the transformation is a slow
enough process ( has to be larger than
100 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 *H*_{2} 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 *H*_{2} molecules were evaluated. The
translational energy is given by =
in which is the linear
momentum of the *H*_{2} molecule center of mass and
is the mass of the diatomic molecule. The
vibrational part is calculated as =
, and the rotational contribution is calculated
by = .
is the angular momentum of the
*H*_{2} diatomic molecule and *µ* its reduced
mass.
A semi-classical vibrational number v was calculated from the
semi-classical quantization of the action integral:
is the internuclear potential of the
*H*_{2} isolated molecule in its ground electronic
state. is the rovibrational energy of the
molecule and h is the Planck's constant.
For a given set of parameters (,
, or
()), 2000-8000 trajectories have been run to
obtain statistically significant results.
© European Southern Observatory (ESO) 1998
Online publication: May 12, 1998
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