Originally posted by: simms
I need to know what's the difference between gerade and ungerade orbitals, and why we have sigma (gerade), pi (ungerade), etc, etc in molecular bonding and orbital mixing.
I know that gerade is orbitals symmetric to inversion, and ungerade is antisymmetric to inversion.. but what does it mean, and what is it's purpose?
Hmmm... Not sure I understand your question entirely. It seems that you've answered your first question in the second statement, namely, that gerade signifies an orbital that is symmetric with respect to inversion while ungerade signifies an orbital that is antisymmetric. To understand why it's necessary, you have to understand character tables. You know that a molecule of given symmetry has several possible representations (a la A1, A2, B1, E, etc.). Each of these representations transforms in a particular way with respect to the component transformations of the symmetry group (rotations, reflections, inversions, etc.) For example, the table for a molecule of C2 symmetry looks as follows:
C2 | E C2 |
-----------------
A | 1 1 |
B | 1 -1 |
-----------------
So the A representation transforms as (1, 1). It's called the A representation because it's symmetric with respect to a C2 transformation. Note that the representation that is antisymmetric with respect to the C2 transformation is denoted as something else - "B". This holds true for any symmetry - look it up... As you go to molecules of higher symmetry, there may be multiple representations that are symmetric with respect to the C2 transformation, but differ in their response to other transformations. Hence, to differentiate these you must add to the description, so you'll see things like A1, or A2. And finally, you may end up with two representations that transform just like A1, except that one is symmetric with respect to inversion, and the other is antisymmetric. To differentiate these, you add the gerade (g) and ungerade (u) notation and end up with A1g and A1u.
Hopefully that convinces you of why the notation is necessary. Now, the next question is how is it useful - what do we learn when we see it? Well, truthfully I've only come across one application, although there may be others. That application is the so-called "selection rules" for an electronic transition. You know that you can stimulate an electron from a molecular orbital to another of higher energy using light, and that the energy required to stimulate a certain transition can be calculated using the energies of the orbitals. When plotted, these energies should constitute the absorbance spectrum of the molecule (generally in the uv-vis region). But when you compare your calculations of all the possible transitions with an actual spectrum, you'll find that certain transitions are -missing- in the spectrum. This is because those transitions are forbidden as a result of quantum mechanics (e.g. they would violate the Heisenberg uncertainty principle). You can predict which transitions are allowed and which are forbidden using certain selection rules, one of which is that, "g -> g and u -> u transitions are forbidden, while g->u transitions are allowed", or something to that effect. So in this case, the g and u subscripts on an orbital help you to quickly determine whether a transition will be allowed or not, without having to look up the character table.
I know it's a long story to explain a single concept, but that's the easiest way I know how. Hey, this stuff isn't easy, as I'm sure you can appreciate. Hopefully it makes sense, I can elaborate a little if something doesn't click. Also,
Heisenberg - if you know of any other applications where the g/u notation gives you useful information, please share. I'd be interested to know.