Symmetry Notes

  • Everything has an identity so technically everything has symmetry. The identity leaves everything unchanged.

  • Symmetry operation is an action that leaves an object looking the same after it has been carried out.

  • Symmetry element is a set of points through which the symmetry operation is performed. It could be an axis if it's a rotation - rotation axis or a plane if it's a reflection - plane of symmetry. This means that any points in the symmetry element doesn't move and stays in the same place.

  • Each symmetry operation has a corresponding symmetry element.

  • Cn represents the rotation index and is called the order. It corresponds to rotation of (360/n) degrees. So if the rotation is about 180 degrees, then n is equal to 2.

  • Reflection is represent by sigma.

  • Inversion is represented by i and it's when everything passes through a single point and has to going back to how it was before.

  • Sn stands for improper rotation and it's when you have a rotation and then have a cross on the plane of symmetry. Its order works the same way as Cn.

  • A symmetry operation must generate an unique symmetry operation to be counted.

  • There's a hierarchy- identity, rotation, reflection, inversion and then improper rotation. If an improper rotation produces the same orientation as an inversion then it's the improper rotation that doesn't count as it is after inversion in the hierarchy.

  • Ex 1: Water only have 1 rotation through the middle of the oxygen atom and it needs to be rotated by 180 degrees. Therefore, the order is n = 2 and it's C2.

  • Symmetry is about if the atoms was labelled, each atom would be in different positions but if the atoms wasn't labelled, the molecule would look the exact same as it started.

  • Ex 2: Although ammonia has one rotational axis through the middle of the nitrogen atom, it has 2 rotations. One is a rotation by 120 degrees and the other is by 240 degrees. The order this time is C3 and C32.

  • Rotations are always anticlockwise.

  • If you have rotations of 90 and 270 degrees, the order isn't C4 and C43, it's actually C4 and C4-1.

  • When asked about the number of rotations, you are counting the number of rotational axes. However, if one rotational axis have multiple axes then each rotation is counted as well.

  • Axis with the highest order so highest number of n is named the principal axis. Everything is based around this. When you are rotating, you need your principal axis to be through the z-axis.

  • Axis perpendicular to the principal axis are next in the hierarchy. Now if the axis is going through atoms then it is higher in the hierarchy than axes that don't go through the atoms.

  • Depending on the rotational axis is what the order of the rotation is called if they all have the same value of n. So if it's on the principal axis, the order is C2 e.g. If it's on the axes perpendicular to the principal axis but bisecting through atoms, it's called C2' . If it's on the axes perpendicular to the principal axis but not bisecting the atoms, it's called C2''.

  • Reflections also have a hierarchy in the same way as rotations. If the plane bisects the atoms it has the highest hierarchy. Next, it's planes that goes through the principal axis. Lastly it's dihedral reflection plane- this is where the plane bisects the angle between 2 C2 axes.

  • If a plane is vertical the order is written as sigma v.

  • If the plane is horizontal, it's written as sigma h.

  • If the plane is a dihedral reflection, then it's written as sigma d.

  • If the reflections are identical, for example, ammonia has 3 reflections but they are identical so they are all called sigma v. Only when the reflection is different is where you get your primes and remember the labeling of primes is due to the hierarchy.

  • In inversions, the symmetry element is a point, which is called the centre of inversion. There can only ever be 1 inversion. Water, if it was inverted, the molecule would look upside down. Therefore water doesn't have inversion as although the molecule looks upside down, it doesn't exactly look like how it started.

  • For improper rotations, the plane of symmetry must be perpendicular to the rotation axis. A rotation plane must be present to have improper rotation but a plane of symmetry isn't required.

  • The point group symbol represent in short hand all the symmetry operations of a molecule.