Symmetry: Point Groups
HOW TO FIND OUT POINT GROUP
Dr. Cyriac Mathew
Symmetry elements can combine in a definite number of ways. For example consider BF3 molecule. It has the C3 axis as the principal axis. Also, there are 3 C2 axes perpendicular to the C3 axis,
3 σv planes, and one σh plane. All these symmetry elements can combine. Take NH3. It also has C3 axis as principal axis. There are 3 σv planes, but no C2 perpendicular to C3 or a σh plane. For the N2F2 molecule there exist one C2 axis and a σh plane but no σv planes.
Thus in general we can say a Cn can combine with either nC2 or no C2 perpendicular to it; it can combine with either one σh or no σh; or it can combine with n vertical planes or no vertical planes.
These are relevant only in the case of systems where we can identify a principal axis. Such systems are called axial systems.
In the case of tetrahedral, octahedral, cubic, icosahedral and dodecahedral objects we cannot identify the principal axis. Solids of these structures are called platonic solids. Crystals and molecules of these shapes are highly symmetric and can be called multi higher order axial systems. They have several higher order axes than C2 axis. The symmetry elements combine in definite ways in these systems also.
Point Groups
Point groups are possible combinations of symmetry elements. Since symmetry elements can combine only in a definite pattern, there will be only a finite number of point groups possible. For crystals only
32 points groups exist. Crystals cannot have axes of symmetry order 5 or higher than 6. On the other hand molecules can have proper axes of symmetry of order 5, 7 and ∞ also. Hence for molecules some additional point groups are possible which are not possible for crystals.
Redundant symmetry elements: symmetry elements which combine to form a point group are known as essential symmetry elements. There exist some symmetry elements which are present as a consequence of the essential symmetry elements. Consider
Dr. Cyriac Mathew
Symmetry elements can combine in a definite number of ways. For example consider BF3 molecule. It has the C3 axis as the principal axis. Also, there are 3 C2 axes perpendicular to the C3 axis,
3 σv planes, and one σh plane. All these symmetry elements can combine. Take NH3. It also has C3 axis as principal axis. There are 3 σv planes, but no C2 perpendicular to C3 or a σh plane. For the N2F2 molecule there exist one C2 axis and a σh plane but no σv planes.
Thus in general we can say a Cn can combine with either nC2 or no C2 perpendicular to it; it can combine with either one σh or no σh; or it can combine with n vertical planes or no vertical planes.
These are relevant only in the case of systems where we can identify a principal axis. Such systems are called axial systems.
In the case of tetrahedral, octahedral, cubic, icosahedral and dodecahedral objects we cannot identify the principal axis. Solids of these structures are called platonic solids. Crystals and molecules of these shapes are highly symmetric and can be called multi higher order axial systems. They have several higher order axes than C2 axis. The symmetry elements combine in definite ways in these systems also.
Point Groups
Point groups are possible combinations of symmetry elements. Since symmetry elements can combine only in a definite pattern, there will be only a finite number of point groups possible. For crystals only
32 points groups exist. Crystals cannot have axes of symmetry order 5 or higher than 6. On the other hand molecules can have proper axes of symmetry of order 5, 7 and ∞ also. Hence for molecules some additional point groups are possible which are not possible for crystals.
Redundant symmetry elements: symmetry elements which combine to form a point group are known as essential symmetry elements. There exist some symmetry elements which are present as a consequence of the essential symmetry elements. Consider