A molecule has the following symmetries only: 𝐸, 𝐶₂, and 2𝜎 v. What is the point group of the molecule?
In this question, symmetry refers to symmetry elements, points, lines, or planes, that define the symmetry of the molecule. 𝐸 signifies the identity symmetry element, meaning we apply no change whatsoever to the molecule. All geometries have this symmetry element. 𝐶₂ indicates a two-fold proper rotation axis indicating that a rotation of 360 degrees divided by two, 180 degrees, produces the same molecule. 2𝜎 v signifies that the molecule has two planes of symmetry parallel to the principal rotational axis, our 𝐶₂ axis. These are said to be vertically aligned with the principal axis, thus the letter v.
A molecule of water displays all these symmetry elements and only these symmetry elements. The point group of a molecule is the summary of the symmetry elements that molecule has. In order to workout the point group, we need our point group flow chart. The first point on our flowchart is whether the molecule is linear or not. All linear molecules have a 𝐶 ∞ axis. Meaning, along their interatomic axis, they can be rotated by an infinitely fine angle and still produce the same molecule.
In the question, we’ve been told that this molecule only has 𝐸, 𝐶₂, and 2𝜎 v symmetry elements, no 𝐶 ∞. After answering no to the question linear, we arrive at the question whether our molecule has two or more 𝐶 n axes where n is greater than two . So we’d be looking for something like to 2𝐶₃ axes or 4𝐶₄ axes. Again, the answer is no. We only have the 𝐶₂ axis. So our n isn’t high enough, and we don’t have enough of them.
The next point on our flowchart is the question, do we have any proper rotational axes at all? We do; we have a 𝐶₂ axis. An answer yes leads us to the next question. Whether we have the same number of 𝐶₂ axes perpendicular to our principal axis. In this case, we’d be looking for 2𝐶₂ axes. We’d be looking for configuration, something like this. However, in this case we just have the one 𝐶₂ axis. So, we have to answer no to this question.
The next question is whether we have a plane of symmetry that is horizontal with respect to the principal axis. We only have 𝜎 vs and no 𝜎 hs. So, the answer to this question is no.
The last question on this trail is whether we have as many 𝜎 vs as the n in our principal axis. n is equal to two. So, we’re looking to have 2𝜎 vs. So our final answer is 𝐶 ₂v. Here, we take the general term 𝐶 nv and plug in our value for n for our principal axis. This question contain all the information necessary to guide you through the flowchart. You didn’t need to know necessarily that water was an example of a 𝐶 ₂v molecule. We’ve demonstrated that any molecule with the symmetry elements 𝐸, 𝐶₂, and 2𝜎 v only will have a point group of 𝐶 ₂v.