# Symmetry in molecules

## Main Question or Discussion Point

So I'm trying to teach myself MO Theory and Spectroscopy. I was just wondering why symmetry was so important in understanding these and what characters are used to do. I know this is a broad question, but I've been reading a lot (Cotton, etc..) so I know they have to do with integrals and operators and stuff. But I'm having trouble explaining it when someone asks me -- I just want like a 30000 ft overview of why characters are useful in MO Theory and Spectroscopy. Can anyone help me out?

alxm
Well, you can always console yourself with the fact that lots of people disliked the introduction of symmetry and group theory into physics/chemistry. Schrödinger coined the term "Gruppenpest", and a lot of notable chemical physicists like Slater were skeptical of it. (and you know the math is abstract when the physicists start complaining about how abstract it is) But eventually its usefulness showed itself.

The basic starting point is this: The Schrödinger equation tells us everything we need to know about the system. But we can't solve it. But what can we say about the solutions without directly solving the equation?

So you take one step up in abstraction; from describing the wave function to describing its properties. Just to illustrate how that can be useful, you can go back to one-variable calculus and consider the simple case of even-vs-odd functions. Recall that if a function is even, and you're integrating over an even interval (e.g. $$\int_{-\infty}^\infty$$) then you can just take twice the integral over half the interval (2$$\int_0^\infty$$). And if it's odd and the interval even, you know the integral will be zero. You also know that if a function is even, its derivative is odd, and vice-versa. So obviously, there is quite a bit you may be able to say about a function even without knowing exactly what that function is.

So you can use this in quantum mechanics. Just to take the simplest example of MO theory, if you have two hydrogen atoms at +x and -x, which come together to form a molecule, what's its wave function like? Well, if we assume non-interacting (or mean-field interacting) electrons, then the Schrödinger equation is separable, and the total solution will be a linear sum of the wave functions of the two electrons, per the superposition principle.

So you have four possible combinations:
$$\psi = 1s_1 + 1s_2$$
$$\psi = -1s_1 + 1s_2$$
$$\psi = 1s_1 - 1s_2$$
$$\psi = -1s_1 - 1s_2$$

Where the subscript denotes which electron it is. But since the total sign of the wave function doesn't mean anything physically, two of these are equivalent to the other two.
$$\sigma = 1s_1 + 1s_2$$ and $$\sigma^* = 1s_1 - 1s_2$$
The first of these is an even function, the second is odd. We know that the second function must have a node in the plane x=0, so there is an area of zero electron density there. It's an antibonding orbital, and the other is bonding. We also know that 1s orbitals have no node themselves, so the bonding orbital has no nodes, and the antibonding has one node. The wave function is continuous (except at the nuclei) so the antibonding orbital has higher curvature, since from one nucleus to the other it has to go from +|value at nucleus| to -|value at nucleus| and pass through zero, whereas the bonding orbital goes from +|value at nucleus| to +|value at nucleus| and doesn't pass zero. The curvature of the wave function is related to momentum ($$\hat{p} = -i\hbar\nabla \psi$$), so the anti-bonding orbital has higher energy. So right there, without any explicit solving, I've predicted the properties of the ground-state wave function of H2, as well as that of the first excited state.

This kind of rationale is of course not limited to electronic states; it's equally applicable to vibrational and other states as well. The question is then, how can I systematize this? That's where group theory comes into play.

To begin with, you've got the simplest symmetries, e.g. a mirror plane and rotational symmetries. Consider a molecule like the methyl radical, which is planar, and let's just consider the two-dimensional symmetry of it. First you have a mirror plane, perpendicular to the plane of the molecule, going through one C-H bond and in between the other two C-H bonds. You also have symmetry under a rotation of 120 degrees in either direction. Now, every geometrical object that has that rotational symmetry will also have that mirror plane (but not vice versa). So this tells us that the rotational symmetry is a higher degree of symmetry than just the mirror plane alone, and also that the point group for this molecule (C3v, as it were) contains both this rotational symmetry C3 as well as the group for the mirror symmetry $$\sigma_v$$. (Being planar, it also contains CS, which in turn contains the $$\sigma_h$$ mirror symmetry) So this is how the character tables that define the various point groups are constructed.

Now what is this all good for? Well, to begin with, just for molecular identification, especially if you're trying to distinguish between isomers/rotamers. In crystallography, you can look for symmetries in the X-ray diffraction patterns and deduce the corresponding molecular symmetries. In spectroscopy you can (as per above) calculate the various energy levels and modes of vibration, and possibly say something about their internal order as well. You also use the symmetry to identify the various levels. Finally you also use this in quantum chemistry where, besides also using it for identification, you can apply these symmetry transforms to your calculations (much as in the initial example of even/odd integrals), and save quite a lot of computing. A calculation on a benzene molecule's ground state effectively becomes a calculation of a single carbon and hydrogen atom - the rest are determined by symmetry. A C60 molecule effectively becomes a single carbon atom!

Just to piggyback here, what's the most "general" way to describe the consequences of the symmetry of the Hamiltonian on other characteristics of the system (ie, the wave functions)? Also, is there a clear way to apply Noether's theorem and understand what conservation laws correspond to, say, molecular symmetries?

@alxm: Thank you so much for your post. When I took Inorganic Chem in school and we were learning group theory, I simply could not understand what contexts it fit under. I've been trying to relearn it on my own, but I'm faced with the same problem.