The procedure used for arranging clusters of rigid
octahedral molecules and computing molecular trajectories is described
in
^{2}.
For highly symmetrical molecules, it is reasonable to
neglect the intramolecular force and to deal only with the intermolecular
potential that is a sum of an atom-atom Lennard-Jones term and
a Coulomb term
^{2}.
Initially, the cluster was arranged in a base-centered
cubic lattice at elevated temperatures (below the melting point).
The classical equations of motion for a cluster of N molecules
interacting via this potential are written in a Hamiltonian form.
Computing center-of-mass trajectories requires solving a set of
6N first-order differential equations. These equations
are integrated using the velocity Verlet algorithm
^{8}.
Cluster's phase diagrams are obtained by successive decrease and increase
of the temperature. The cluster is thermalized at a particular
temperature using the velocity scaling algorithm which has been
proved
^{9}
to give the correct canonical distribution in the coordinate
space with an accuracy of order
if the scaling is carried out in every time step during the thermalization
^{3}.
The time step= 10 fs is chosen to
keep the total energy constant during the run. After the thermalization
is switched off, the system's equilibration is monitored by recording
instantaneous values of the potential and kinetic energy during
this period. These are taken every 0.1 ps, in order to avoid collecting
correlated values of the quantities. Adequate equilibration is
especially important when the initial configuration is a lattice
with one type of symmetry and the state point of interest is in
another type of symmetry. Examples of a 89 molecule Tef_{6} cluster
at two different temperatures are shown in
Figure 2 and
Figure 3.

In our computations, the orientation of the molecules relatively
to the cluster center-of-mass (space system) is defined by the
four quaternions suggested by Evans
^{10}
to solve the problem of divergence in the orientation equations for the
three Euler angles when transforming between the space and the body system.
The orientation of the rigid molecule specifies the relation
between an axis system fixed in the space and one fixed with respect
to the body. We have chosen the body system of each molecule
to have its origin at the center of mass of that molecule so
the inertia tensor is diagonal. Any unit vector e can be
expressed in terms of components in the body
or space coordinate system.
Conversions from the body-fixed
to space-fixed systems is handled by the
Equation 1.

We have proposed
^{3}
a novel method for representation of
the molecular orientation in order to study its time dependence
and to compute the density of molecules oriented in a given direction.
The orientations of the molecules in the 3D space are projected
onto a 2D spherical surface. Each projection point is represented
with an oriented cross. In such a way we can preserve the complete
information.

In a rigid octahedral molecule TeF_{6},
Figure 1,
the six fluorine atoms are at the vertices of a octahedron and
the Te atom is at the center. In an instant configuration the
orientation of the molecule is defined by the three orthogonal
F-Te-F directions in the body frame. At a given temperature these
three directions differ from one configuration to the other.
If we take a snap shot of one configuration,
Table 1, Figure 1.1,
we have information about the instant orientation of the molecules
in the cluster. If we project many configurations, we get the
time evolution of the cluster,
Table 1, Animation 1.1.

The step-by-step procedure to project a configuration is given
in
^{3}. Briefly: a unit vector
from the molecule is converted in
the space system using , where
is the transpose
matrix. The vector located either
in the octant (x, y, z) or in the octant (-x,-y,-z)
is taken as the unit vector of the projection. When it is in
the first octant, it is not changed in the further calculations.
If the projected vector belongs to the complementary octant (-x,-y,-z),
then it is rotated. Two of the spherical coordinates
in the 2D spherical surface are
determined by the position of this
vector. In order to keep the information about the third angle,
we use the second unit vector of the molecule. Its relative orientation
with respect to the first one (in the space system) determines
the third angle . This last angle is
taken as modulus of due to the
molecular symmetry.
In such a way we obtain three coordinates
that are plotted onto 2D surface as
follows: the coordinate
is along the angular axis and is
along the radial axis. Thus we obtain a point with coordinates
.
In this point we draw a cross which is rotated at an angle
with respect to the radius-vector of the point.

This method makes possible to follow the dynamics of each molecule
during the phase transformations when cluster is cooling or heating
when the projections are used for animation.
Table 1 and Table 2
show two clusters containing different number of molecules.