In order to identify the molecular point-group the system must be oriented in a
specific way. Four families of point-groups are checked for: (1) the infinite
groups, (2) the cubic groups, (3) groups with one high-symmetry axis, and (4)
the Abelian groups. Each family is treated differently. First, the moments of
inertia are calculated. If all are zero, the system is a single atom, and the
associated group is R_{3}. If two moments are zero, the system is either
C
or D
;
the presence of a horizontal plane of
symmetry distinguishes between them.

Having eliminated the infinite groups, the three moments of inertia are checked
to see if they are all the same. If they are, then the system is cubic. Cubic
systems are oriented by identifying atoms of the set nearest to the center of
symmetry. If there are 4, 6, 8, 12, or 20 of these, and the number of
equidistant nearest neighbors is 3, 4, 3, 5, or 3, respectively, then the
atoms are
at the vertices of one of the Platonic solids (tetrahedron,
octahedron, cube,
icosahedron,
dodecahedron),
and therefore all atoms of the set lie on high-symmetry axes. The first atom
is selected and used to define the *z* axis.

If the number of atoms in
the set does not correspond to any of the Platonic solids, then the set is
checked for the existence of a equilateral triangle, a square, or a regular
pentagon. When one of these is found, the center of the polygon is used to
define the *z* axis. An example of this type of system is C_{60},
Buckminsterfullerene, which has a five-fold axis going through the center of a
pentagonal face.

Once the *z* axis is identified, the system is checked for C_{n} axes, *n*=3 to
*n*=8. To complete the orientation, the system is rotated about the *z* axis
so that two atoms, having equal *z* coordinates, have equal *y* coordinates.
The existence of rotation axes which are not coincidental with the *z* axis
and the presence or absence of a center of inversion are then used to identify
which cubic group the system belongs to.

If the system has still not been identified, then the two equal moments of
inertia indicate a degenerate point group. As with the cubic groups, the *y* and
*z* axes (and, by implication, the *x* axis) are identified. The system is
oriented, and the C_{n} and S_{n} axes identified.

The degenerate groups, C_{n}, C_{nv}, C_{nh}, D_{n}, D_{nd}, D_{nh},
S_{n}, are distinguished by the existence or absence of C_{2} axes
perpendicular to the *z* axis, and by planes of symmetry.

All that remains are the Abelian groups, C_{1}, C_{2}, C_{i}, C_{s}, C_{2v},
C_{2h}, D_{2}, and D_{2h}. After orienting the molecule, the axes are
swapped around so that the normal convention for orienting Abelian systems is
obeyed. For groups C_{1}, C_{2}, C_{i}, and C_{s}, there is no possibility for
ambiguity. For C_{2v} and D_{2}, however, the orientation of the system
affects the labels of the irreducible representations. To prevent ambiguity,
the convention for orienting Abelian molecules is:

- The axis with the largest number of atoms is the
*z*axis. - The plane with the largest number of atoms that includes the
*z*axis is the*yz*plane.

Thus for ethylene, the * π*
orbitals point along the *x* axis.