Each molecule has a set of symmetry operations that define the molecule’s overall symmetry. These sets of symmetry operations are known as point groups.

To understand point groups, it is important to have a general idea of symmetry elements and the operations that can be applied through them.

A symmetry element is a geometrical entity such as a point, a line, or a plane about which a symmetry operation is carried out to generate an equivalent orientation.

A symmetry operation produces a change in the orientation of a molecule, but the new orientation is indistinguishable from the original. Hence the two orientations are known to be equivalent to each other.

Repeating the same symmetry operation a specified number of times will yield the original orientation, which will now be called identical to the original.

Outline

## Symmetry Elements and Operations

The first of 5 symmetry elements is identity, represented by E. Its symmetry operation can be described as “doing nothing”. All objects possess this symmetry element, regardless of whether they have other symmetry elements or not.

In case a molecule possesses only identity, it will be classified into the groups of low symmetry and its point group would be C1.

Another symmetry element is proper rotation, and its symmetry operation is rotation about an n-fold axis at 360°/n. Proper rotation is represented by the symbol Cn. If a molecule has multiple rotation axes, the one with a higher n value will be the principal axis.

An example is boron trifluoride, BF3, which has a threefold principal axis, C3. This will allow rotation at 360°/3 = 120° to yield an equivalent orientation. Performing this operation of rotation 3 times will result in an orientation identical to the original one.

Also that this C3 axis in BF3 is associated with two rotation operations, one in the clockwise, and the other in the anti-clockwise direction, so it is written as 2C3.

BF3, being planar, also possesses 3C2 axes that allow rotation at 180° each. The point group of BF3 is therefore D3h.

The “h” in D3h represents σh, which is another symmetry operation: reflection through a plane. This “h” refers to a horizontal plane (perpendicular to the principal axis). In fact, the symmetry element plane of reflection also includes σv and σd (parallel to the principal axis), referring to vertical and dihedral planes, respectively.

The fourth symmetry element is improper rotation, represented by Sn. The operation involves rotation about an n-fold Cn axis followed by reflection through a σh plane.

The center of inversion is the final symmetry element and is denoted by i. The corresponding symmetry operation is an inversion in 3 dimensions through a point. Coordinates x, y, z will be transformed to -x, -y, -z.

## Point Groups

Symmetry elements are collectively used to classify molecules according to their symmetries. The name of a point group is determined by the symmetry elements it possesses.

### Groups C1, Ci, Cs

If a molecule has an identity (E) as the only symmetry element, it belongs to the C1 point group.

If it possesses a center of inversion (i) in addition to identity, it will belong to the Ci point group.

Moreover, it belongs to the Cs point group if it has an identity and a plane of reflection (σh).

### Groups Cn, Cnh, Cnv, C∞v

A molecule will belong to the Cn group if it has an n-fold principal axis as well as identity.

On the other hand, a molecule with a Cn principal axis and σh planes of reflection in addition to identity, belongs to the Cnh group.

Furthermore, molecules having a Cn principal axis, n σv planes of reflection, and identity belong to the Cnv point group. Ammonia, NH3, possesses a C3 principal axis and 3 σv planes of reflection, so it belongs to the C3v point group.

C∞v point contains the symmetry elements E, 2C (clockwise and anti-clockwise), and ∞σv.

### Groups Dn, Dnd, Dnh, D∞h

A molecule will belong to the Dn group if it has a Cn principal axis, nC2 axes perpendicular to the principal axis, and identity.

If it additionally contains σd planes of reflection, then Dnd group instead.

On the contrary, having n horizontal mirror planes σh along with a Cn axis, nC2 perpendicular axes, and identity, will classify the molecule into Dnh group. An example is benzene, C6H6. It has the symmetry elements E, C6, 3C2, 3C’2, and σh, so it belongs to the point group D6h.

Moreover, the D∞h group includes the symmetry elements E,  2C, ∞σv, i, 2S, ∞C’2.

### Sn Group

Molecules that are not classified into any of the groups given above but possess an Sn axis belong to the Sn group. Group S2 is the same as Ci, so it will already have been classified. Also that molecules with Sn for n > 4 are rare.

### The cubic groups: Tetrahedral (T, Td, Th), Octahedral (O, Oh), Icosahedral (I, Ih)

Molecules of high symmetry, possessing more than one principal axis, belong to one of the cubic groups.

Methane, CH4, belongs to the tetrahedral point group Td while sulfur hexafluoride, SF6, belongs to the octahedral point group Oh. Some boranes and buckminsterfullerene, C60 belong to the icosahedral (20-faced) group.

In fact, a flowchart can help assign the point groups to various molecules depending on the symmetry elements they possess.

### Drawing a Point Group

A point group is drawn by first identifying all of the symmetry elements and operations that a molecule possesses.

Then these operations are applied to the mathematical functions i.e. orbitals and rotations, which results in either a +1 (symmetric) or a -1 (antisymmetric) value for each of the symmetry operations.

This allows us to write down the all the distinct irreducible representations, followed by allotting them Mulliken’s symbols:

1-dimensional irreducible representation is denoted by A or B.

2-dimensional irreducible representation is denoted by E.

3-dimensional irreducible representation is represented by T.

“A” refers to symmetric rotation about the principal axis, while “B” refers to antisymmetric rotation about the principal axis.

Symmetric or antisymmetric with respect to the secondary axis is denoted by subscripts “1” and “2” respectively. In case of the absence of secondary axes, σv planes are used instead.

Single and double prime (‘ and “) are used to denote the symmetric and antisymmetric representations of the horizontal plane, σh.

Finally, in groups with a center of symmetry, symmetric or antisymmetric with respect to the inversion center is indicated with the subscript “g” or “u”.

After assigning Mulliken’s symbols, reducible representations are determined.

## Concepts Berg

What are point groups and examples?

There are five types of symmetry elements and their corresponding operations that result in similar orientations. Classifying objects using their set of symmetry elements and operations results in point groups.

There are various point groups: C1, Cs, Ci are groups of low symmetry whereas O, T, and I are examples of high symmetry point groups. Some examples of intermediate symmetry groups include Cn, Cnv, Cnh, Dn, Dnd, Dnh.

How do you identify point groups?

Point groups are identified using the set of symmetry elements they each possess.

Let us take an example of the point group D4h. The subscript “4” implies that there is a C4 principal axis of rotation. Moreover, the “D” refers to the presence of a further 4 perpendicular C2 axes of rotation (to the C4 principal axis). Finally, the subscript “h” indicates a σplane of reflection.

What are the two uses of point groups?

From the point groups of substances,  we can predict some of their physical properties in polarity and chirality.

How do you define a point group?

A point group is defined as the set of symmetry elements and their corresponding operations that define the molecule’s overall symmetry.

What is the point group of S8?

The point group of octasulfur, S8 is D4d. This is because it has the symmetry elements and operations E, a C4 principal axis (associated with 2 rotations: clockwise and counter-clockwise), a C2 axis (parallel to principal C4 axis), 4C2‘ axes (perpendicular to principal C4 axis), 4 σd planes of reflection, and finally one S8 rotary-reflection axis.

What is the point group for CIS PT?

Cisplatin belongs to the point group C2v. It contains the symmetry elements E, C2, σv, and σv‘.

What is the CN point group?

Cyanide anion possesses the symmetry elements/operations E, 2 C, and ∞ σv, so it belongs to the point group C∞v.

What point groups are chiral / optically active?

A chiral molecule is one that cannot be superimposed upon its mirror image. As chiral molecules rotate plane-polarized light, they are said to be optically active.

The theory of optical activity states that a molecule may be chiral if it does not possess a rotary-reflection axis, Sn. However, it may not be directly identified.

For example, Cnh point groups contain Cn and σh, which are the components of Sn. Any molecule with a center of inversion also possesses S2 (as i is equivalent to the combination of C2 and σh). Hence all molecules with centers of inversion are also achiral.

So the point groups that do not contain the symmetry elements mentioned above are bound to be chiral.

What is the point group of BF3?

The point group of BF3 is D3h. This is because BF3 possesses a C3 principal axis, 3 C2 axes perpendicular to the principal axis, and a σplane of reflection, in addition to identity.

What is the point group of H2O2?

H2O2 belongs to the point group C2, as it possesses only a C2 axis other than identity.

What is the point group of PdCl4-2?

PdCl4-2 has the symmetry elements E, 2C4, C2, 2C2‘, 2C2“, i, 2S4, σh, 2σv, 2σd, so its point group is D4h.

Reference Books

• Physical Chemistry | Fifth Edition, by P. W. Atkins (University of Oxford, Oxford, UK)