By identifying all symmetry elements in ammonia NH₃, assign the point group of the molecule.
Symmetry elements are points, lines, or planes through which symmetry operations are performed. Before we can figure that out, we need to know the structure of the ammonia molecule. Using valence-shell electron-pair repulsion theory, we can predict a structure for the ammonia molecule of trigonal pyramidal. Since nitrogen has five electrons in its outer shell, after forming three single bonds to the three hydrogen atoms, there’s a lone pair left over. As with all molecules, ammonia has the identity symmetry element. Now, we can look at all the others and see which ones apply.
Let’s start with a symmetry axis, C. The obvious place to start is an axis going directly between the hydrogens and through the nitrogen atom. If we look from above, we can easily see the three equivalent hydrogens. If we rotate the molecule about this axis, the hydrogens move into equivalent positions. If we perform that twice more, we get back to our starting position. So, the angle between this rotation is 120 degrees. And, we’re dealing with our principle axis as a C₃ axis. And, this is how we can label each rotation operation moving from one configuration to the next. The little hat above the symbol indicates that it’s an operation and not a symmetry element.
There are no other locations to put other symmetry axes. And, the C₃ axis is not going to give rise to any others. So, we can move on to our next symmetry operation. Now that we know our principal C₃ axis, we can start looking for planes of symmetry. Again, our best vantage point is viewing from the top where we can see all the hydrogens in equivalent positions. We can draw one mirror plane through one nitrogen–hydrogen bond, and we can see on both sides the image is equivalent. This mirror plane is parallel with the principal axis. So, we can label this 𝜎v where the v means vertical.
If we draw this plane on our 3D diagram, we can see that it is indeed parallel with the principal axis. All the hydrogens are equivalent as demonstrated by the C₃ axis. So we do, in fact, have three 𝜎v planes of symmetry. Before we move on, we should see if there are any symmetry elements we’re missing. Ammonia doesn’t have an inversion center or improper rotational axes.
Now, that we’ve confidently identified all the symmetry elements in ammonia, let’s move on to assigning the point group. A point group is essentially a summary of all the symmetry elements. The point group allows us to understand relationships with similar molecular geometries. In order to find the point group, we’ll need our point group flow chart. The first question is whether the molecule is linear or not. The next question is whether our molecule contains two or more C 𝑛 axes where 𝑛 is greater than two. This is a no because ammonia has only one C₃ axis.
The next point in the flow chart is whether we have a C 𝑛 axis to which the answer is yes. Ammonia has a C₃ axis. The next question is whether there are 𝑛 C₂ axes perpendicular to our principal axis. So, we’re looking for three C₂ axes perpendicular to our C₃ axis. We only have the C₃, no C₂s. So, the answer to this is no, leading us to our next question.
Whether we have any planes of symmetry that are horizontal with respect to the principal axis. Ammonia only has 𝜎vs, no 𝜎hs. Which leads us to our last question, whether we have three 𝜎vs. That’s as many 𝜎vs as the 𝑛 in our principal axis. With the answer to this being yes, our final point group for ammonia is C₃v. So, we’ve identified all the symmetry elements that are present in the ammonia molecule. And assigned the point group systematically by going through the flow chart arriving at our answer, C₃v.