Question

Sir i need help for understanding symmetry elements which is the basis of group theory.i have read many books containing this subject , however i can't imagine these elements.please help me.thanking you?

Answer 1

Many good group theory texts should have images... but this website is great for additional visualization practice. Bookmark it! You can perform the operations by clicking the button next to the symmetry element.

Also, you will find this website useful later; it'll help you check your reduced reducible representations, so keep this website in mind as well. For instance, `D_(4h)` can be a pain to work with, and this page helps immensely.

Symmetry operations can be categorized in general as:

• Identity, `hatE`, symmetry element = `E` (nothing)
• Rotation, `hatC_n`, symmetry element = `C_n` (an axis)
• Reflection, `hatsigma`, symmetry element = `sigma` (a plane)
• Inversion, `hati`, symmetry element = `i` (a dot)

We can use `"NH"_3` (`C_(3v)` point group) and cyclobutane (`D_(4h)` point group) as examples (because they're on Otterbein).

Note: `hatR` is the symmetry operation, and `R` is its symmetry element.

IDENTITY

The identity operation `hatE` is rather simple. It is otherwise known as the "do nothing" operation.

There really is no point in identifying what the symmetry element `E` is for this (because you don't have to use a symmetry element to perform a "do nothing" operation).

ROTATION

The rotation operation, `hatC_n`, rotates the molecule `(360^@)/n` degrees so that the new orientation is identical to the previous orientation, and its symmetry element is the `C_n` axis.

For example, `"NH"_3` has a `C_3` axis through the nitrogen's lone pair:

![)

When you rotate `"NH"_3` so that you see a top-view, the definition of `C_3` will become more apparent:

![)

From this angle, it is more noticeable that you can rotate `(360^@)/3 = 120^@` to return the same molecular orientation. In other words, it has a three-fold rotation axis `C_3`, demonstrable through the `hatC_3` rotation operation.

REFLECTION

The reflection operation, `hatsigma`, has three variations: `hatsigma_v` (vertical), `hatsigma_h` (horizontal), and `hatsigma_d` (dihedral/diagonal). Obviously, the symmetry element is the plane itself.

• `sigma_v` is colinear with the principal `C_n` axis (of the highest `n`), and lines up with an outer atom.
• `sigma_h` is perpendicular to the principal `C_n` axis.
• `sigma_d` bisects two outer atoms, crossing through the center of the molecule, and is in between two `sigma_v` planes. It must not directly line up with an outer atom (otherwise it is `sigma_v`).

Cyclobutane (`"C"_4"H"_8`) is a nice example that has all three of these elements (its `C_4` axis is through the plane formed by the four carbons):

![)

ROTATION-REFLECTION (IMPROPER ROTATION)

This is its own operation, `hatS_n`, the improper rotation (rotation-reflection), but it really is just a combination of rotation and reflection in either order.

For example, `hatS_4` is really the compound operation `hatsigma_h hatC_4`, i.e. we rotate `360^@/4 = 90^@` around the principal rotation axis (`C_4`), and then reflect through the horizontal plane (`sigma_h`).

You should convince yourself though that `hatS_2` is really the same as `hati`, which we'll talk about next.

INVERSION

The inversion operation `hati` may be the hardest to visualize, with a symmetry element `i` that is a dot at the center of the molecule.

The easiest way I can think of to describe it is that it takes the coordinates `(x,y,z)` and transforms them into `(-x,-y,-z)`. In other words, take each coordinate and change its sign.

Here's an example of inversion with a molecule that doesn't have inversion symmetry, like `"NH"_3`:

This is hard to visualize for molecules with inversion symmetry, because it looks like it does nothing. Practice with this one.