This report describes, for the selected structure, the point
group of symmetry, the symmetry operations and the symmetry
elements. When
Periodicity is set to
Yes and
various crystallographic point groups are possible, they are
listed in sequence of increasing symmetry.
Point Group
The chemical or crystallographic point group of symmetry
is reported here. When
Periodicity is set to
No
(the default), the chemical group is shown, using the Schoenflies
notation (or
Undefined, if no point group can be determined).
Chemical groups for linear systems, with a rotation axis of
infinite order, are named
C0v and
D0h. The spherical
group, corresponding to a single atom, is named
Kh.
When
Periodicity is set to
Yes, the group is named
using first the International and then the Schoenflies notation
(or
Undefined, if no point group can be determined. In
this case, rotation axes of order different from 6, 4, 3, 2
are discarded, all the other elements are shown).
Gamgi can find axes with any rotation order, so any chemical
(infinite) or crystallographic (32) group of symmetry can
be determined.
When users require the crystallographic point group, Gamgi
determines first the chemical group and then applies the periodic
restrictions to obtain the point group in a crystal with the
highest possible symmetry. When more than one option is available,
Gamgi shows the various solutions. For example, a C24 rotation
axis in a molecule can be restricted to 6 or 4 axes
in a periodic crystal.
Gamgi tries to find all the symmetry elements independently,
and in general each of these elements requires a different
tolerance. Thus, for a given tolerance, some elements
may be recognized and others may go missing, resulting in
an inconsistent set of symmetry elements. When this happens,
the group is reported
Undefined. The solution is to
increase the tolerance (so valid elements might be found)
or to decrease it (so fake elements might be discarded).
Symmetry Operations
The complete set of symmetry operations, generated from the symmetry
elements found (forming a group, in the mathematical sense), is reported
in abreviated format. For example, a
C4 axis generates operations
C41,
C42 (equal to
C21),
C43 and
C44
(equal to
E), so only two
C4 operations are new,
described as
2C4.
In groups with infinite rotation orders,
C0v and
D0h,
rotation operations are presented, as
2C0 and
2S0,
symbolizing the two directions of rotation. In these cases, a single
mirror plane
m is considered (although they are infinite).
For the
Kh spherical group, a single rotation axis and plane
are considered (although they are infinite).
Symmetry Elements
All symmetry elements that were found are individually reported
here. The inversion center is described by its coordinates.
Mirror planes are described indicating the corresponding
normal vectors. For rotation axes, normal and improper, the
rotation order is reported, plus the normal vector describing
the axis, starting from the center.
For all symmetry elements, the error produced when applying
the operations of symmetry, is reported (this error is smaller
than the tolerance, otherwise the element would have been
rejected).
Rotation axes are sorted according to increasing rotation
order, so more symmetric axes come first. When present,
infinite order axes are always the first in the list.
Mirror planes and axes with the same rotation order are
sorted according to decreasing element error, so better
defined elements come first.
When present, a horizontal mirror plane is always listed
before the other planes. When present, a C2 axis along the
main direction is always listed before the other C2 axes.