### 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.