Question Video: Atoms, Structure, and Symmetry

What is the point group of water?

04:58

Video Transcript

What is the point group of water?

A point group is a summary of all the symmetry elements of a certain geometry. What we’re being asked to do is find the geometry and the symmetries of a water molecule H₂O. A molecule of water has an oxygen atom in the middle with two singly bonded hydrogens and two lone pairs. According to Valence Shell Electron Pair Repulsion theory, it has a bent geometry. So, what we’re going to do to find the point group is assess all the symmetry operations that apply to water, and then we’ll use the point group flow chart to assign its point group.

All geometries have the identity symmetry element labeled E, which leaves it unchanged. To start off, we can analyze the molecule of water looking for its proper rotational axes. The easiest way of finding the principle rotation axis is to look from the top. Here, we can see if we drive a line down through the oxygen, that we can rotate the hydrogens into equivalent positions. If we rotate about this axis, we can move one hydrogen into the position of the other, swapping them round. If we repeat this, we end up in the starting position.

Since it took two rotations to return to the starting position, we must have rotated each time through 180 degrees. So, each time we’re performing a C₂ symmetry operation. We indicate that it’s an operation with the hat symbol so it’s not confused with the symmetry element. So, our principle axis is a C₂ axis. There aren’t any positions where we can see such rotational equivalents between the hydrogen, so this is our only rotation axis. So, we can move on to looking for planes of symmetry.

If we look from the side on the oxygen molecule, we can see that we can reflect the hydrogens on the left to look like the hydrogens on the right. This gives us a plane of symmetry, which is dubbed 𝜎. This particular 𝜎 is called a 𝜎ᵥ because it’s parallel with our principle C₂ axis. If we change the side we’re looking at so that we see only one hydrogen atom, we see another 𝜎ᵥ. And we see no 𝜎_hs. No planes horizontal with respect to the principle axis produce the same molecule on reflection.

So, there we have our water molecule with its identity symmetry element, its C₂ principle axis, and two 𝜎ᵥ planes of symmetry. There’s no center of inversion where we can move all the particles to the opposite side and still have the same molecule. And we have no improper rotational axes where we can rotate and then reflect. So, now, it’s time to move on to a point group flow chart to assign the point group.

The first question on the way is whether our molecule is linear. It’s clearly not linear because it’s bent. We can’t draw a single line through all the atoms in the molecule. The next question is whether we have two or more C_𝑛 proper rotational axes where 𝑛 is greater than two. A molecule of water only has one C₂ axis. Therefore, the answer to this question is also no. The next stop is whether we have any C axes at all, which, of course we do; water has a C₂ axis.

The next question is whether we have as many C₂ axes perpendicular to our principle axis as the 𝑛. In this case, 𝑛 is two, so we’re looking for two C₂ axes perpendicular to our original C₂ axis. In this case, we’d be looking for something like this, which, of course, a molecule of water doesn’t have because it has only one C₂ axis. The next question is whether a molecule of water has at least one 𝜎_h. A 𝜎_h is a plane of symmetry horizontally aligned with respect to the principle rotation axis. We only have 𝜎ᵥs.

This leads us to our last question. Whether we have as many 𝜎ᵥs as we have 𝑛 in our principle rotational axis, which of course we do. A water molecule has two planes of symmetry that are vertically aligned with the principle rotation axis. Here, we’ve taken the general C_𝑛v from our flow chart and plucked in the value of two for 𝑛. By finding the symmetry elements of a water molecule, we’ve been able to assign the point group as C₂ᵥ.

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