Ethane Example and Template

Introductory Level Exercises: Table of Contents

Advanced Level and Crystallographic CourseWare

Point Group Symmetry

Symmetry elements are points, lines or planes.

Symmetry operations move the species (molecule or ion) about the symmetry element.

Point Group Symmetry elements are those which coincide at the center (a point) of the species.

The point group symmetry elements are:


Proper Axes of Rotation

The Identity

Inversion Center (Center of Symmetry)

Improper Axes of Rotation



As you read down this page, pay attention to the colors as well as the words, pictures and animations.

The color blue will highlight discussion of symmetry elements.

The color red will highlight discussion of symmetry operations.





Your right and left hands are identical by reflection through a mirror plane.

Imagine σ as a plane pointing into the page. This mirror plane is the symmetry element.

The motion of taking one hand through the plane to give its reflection is the symmetry operation.


Rotation Axes: Proper

Consider a line perpendicular to this page through the oval.

Rotating each hand about the axis by 180 degrees gives the other hand.

Note that both are right hands.

360/180 = 2 = n

So this is a two-fold axis of rotation.


A Three-fold axis of Rotation.

The axis is perpendicular to the page through the triangle.

360/120 = 3 = n.


A four-fold axis of rotation.

360/90 = 4 = n.

Notice that every four-fold rotation axis contains a two-fold axis.


A six-fold axis of rotation.

360/60 = 6 = n.

Notice that every six-fold rotation axis contains a two-fold axis and a three-fold axis of rotation.


?-Fold axis of rotation.

Proper axes of rotation of order 2, 3, 4, and 6 are most common. However, those of order 5, 7, and 8 are also observed.

The Identity

The special case of 360/360 = 1 = n is called the identity, E.

If a molecule contains multiple axes of rotation, the axis with the highest value of n, the order, is called the principal axis.

Inversion Center


The center of symmetry, or inversion center, is a point through which the operation moves an atom at (x, y, z) to (-x, -y, -z).

Rotation Axes: Improper

The improper axis of rotation is a rotation followed by a reflection.

Shown here is a four-fold improper axis of rotation. The hands are rotated by 90 degrees, and then reflected through the mirror plane.



Every four-fold improper axis contains a two-fold proper axis of rotation.

Always look down a proper axis of roation to see if an improper axis with order 2n is presesnt.


Shown to the left is a six-fold improper axis of rotation. The hands are rotated by 60 degrees, then reflected through the mirror plane.

Every six-fold improper axis contains a three-fold proper axis of rotation.



Shown to the left is a three-fold improper axis of rotation. If a proper axis of rotation has a perpendicular mirror plane there will always be an improper axis of rotation of the same order.