SME Light Winter Quarter
Lecture 9: How Can We Use Symmetry?
Tuesday, February 1st, 2000


I. More on improper rotations
II. Polarity and chirality
III. Handedness defined
IV. Summary of symmetry and its chemical applications
V. Symmetry and physical laws

In this lecture we discuss how we can use symmetry. The practical applications of symmetry will set the stage for chirality, which we will start talking about in the next lecture.

I. More on improper rotations

The first thing we need to re-visit is improper rotations. Recall that you do an improper rotation Sn by rotating a molecule 360/n degrees about a Cn axis and then reflecting it through a plane perpendicular to that axis. The most common kind of improper rotation axis is an S1 axis, which is the same thing as a mirror plane. (show examples) The second most common kind of improper rotation axis is an S2 axis, which is the same thing as an inversion center. (show examples) If an object or molecule does not have a mirror plane or an inversion center, then there is a 99% chance that it does not possess any improper rotation axes.

II. Polarity and chirality

A. Polarity

A molecule is polar if one end of the molecule has a slight negative charge and the other end has a slight positive charge. Some atoms, like oxygen and nitrogen, are electronegative, meaning that they have a tendency to hog the electron density and pull negative charge toward themselves. Other atoms, like hydrogen and carbon, and not very electronegative, meaning that they get the short end of the stick where electron density is concerned. If carbons and hydrogens are in the same molecule with oxygens and nitrogens, then the oxygens and nitrogens will get a slight negative charge and the carbons and hydrogens will get a slight positive charge.

Any two different atoms have different degrees of attraction for electron density. This means that any molecule that has two or more different atoms in it will exhibit some degree of charge separation, where the more electronegative atom gets more electron density. But does this mean that the molecule will be polar? What if the vectors cancel out? It turns out that all molecules in the Cn, Cnv, and Cs point groups are polar. This is one of the ways in which symmetry can help us understand a chemical concept.

B. Chirality

Chiral molecules will be the subjects of our next lectures. "Chiral" comes from the Greek word meaning "hand." Spread out your hands in front of you with your fingernails on top. Notice that they are mirror images of each other. Notice also that you can't superimpose them one on top of the other because if you tried, your thumbs would stick out in opposite directions. In the same way as the molecules in the introduction, your two hands look the same, but they are geometrically different. You can appreciate this if you have ever tried to put a left-handed glove onto your right hand. It doesn't fit. For another example, turn to your neighbor and shake hands. Now try shaking left hands. Does it feel different? Now try shaking hands right to left. Does it feel different? Does it even work? The point of this exercise is to convince you that despite their apparent similarity, your hands are different from each other. It takes a hand to recognize a hand. Your hands are chiral. Chiral objects are handed.

The definition of a chiral object is "an object that is not superimposable on its mirror image." You can see that your hands are not superimposable, but sometimes it's difficult to tell for more complicated objects. That's where symmetry can help us out. If an object does not possess an improper rotation axis, it is chiral. It will further simplify matters to know that the most common improper rotation axes are S1 and S2 improper rotation axes. Remember that S1 is a mirror plane and S2 is an inversion center. Thus, if an object or molecule has a mirror plane or inversion center, you know right away that it is not chiral. If it does not possess a mirror plane or inversion center, then there's a good chance that it is chiral, unless it has one of the extremely rare higher-order improper rotation axes.

BREAK OUT SESSION: What are some chiral (handed) objects in everyday life? Are human beings chiral?

At first glance, you might imagine a mirror plane splitting you down the middle. But is it really there? Your heart is on the left, and you don't have a corresponding heart on the right. Moreover, as we will soon learn, all of the amino acids in your body are chiral.

Common chiral objects in everyday life include all things spiral: staircases, barbarshop signs, screws, propellers. Hurricanes and tornados are also chiral. Plants often grow in a spiral, as noted by Shakespeare in a Midsummer Night's Dream when he writes of the "tumbled intertwining of the right handed helical bindweed with the left-handed honeysuckle." One important chiral object is circularly polarized light.

III. The deeper meaning of handedness

Why all this fuss about whether a molecule is "chiral" or not? Chiral molecules have the property that they are "handed." Just as you only shake hands right-hand to left-hand, chiral molecules in your body can only "shake hands" (i.e. interact) with certain other chiral molecules. This has far-reaching biological implications that we will explore. Your enzymes, for example, are chiral proteins that are evolutionarily primed to recognize very specific molecules in very specific chiral configurations. We will talk about this extensively later; the point of this section is simply to make sure that you will be able to recognize chiral molecules when they come up.

Also, chiral molecules have the physically interesting property that they rotate plane polarized light. In fact, this is how they were first identified. Louis Pasteur, a French chemist, was puzzled by the fact that crystals that formed on some wine bottles rotated plane polarized light in opposite directions. By coincidence, the crystals themselves were handed too, and Pasteur was able physically to separate the ones that rotated light clockwise from the ones that rotated it counterclockwise. Jean Baptiste Biot, the deal of French optical rotation studies, was skeptical and made Pasteur do it in front of him before he believed the result. It was a great moment in the history of chemistry.

Why might chiral molecules rotate plane polarized light? It takes a hand to recognize a hand, and plane polarized light is composed of two circularly polarized (handed) components. A chiral molecule, itself being handed, recognizes one of the circularly polarized components of the plane polarized light and interacts with that component in a different way than it interacts with the other circularly polarized component. Remember, the wavelength of light is too small to interact with something as big as your hand, but it's just about the right size to interact with something as big as a molecule. It turns out that chiral molecules absorb one of the circularly polarized components of plane polarized light preferentially.

What happens to the plane polarized light when one of its circular components is preferentially absorbed? It's not easy to see (and you won't be responsible for showing how this works on a test), but it turns out that when a chiral molecule absorbs part of one component of circularly polarized light, that circular component gets out of phase with the other one, and the plane of polarization of the light gets rotated. So light that was straight up and down before you put it through a sample of chiral molecules will be tilted when it comes out the other side.

As a result, the plane of polarization of the original plane-polarized light rotates!

Chiral molecules rotate plane polarized light because the light itself is chiral.

IV. Summary of symmetry and its chemical applications

Following is a summary of some important chemical applications of group theory. Note that these are only a few of the possibilities; group theory can be extremely powerful when it is applied to chemical systems. In fact, there is a whole book titled Chemical Applications of Group Theory by Cotton that has been perplexing and frustrating inorganic chemistry students for decades. As you delve deeper into science and math, you will gain an increasingly deep appreciation for the power and beauty of symmetry in the universe. But before I start waxing philosophical, here's that list:

  • Only molecules belonging to the Cn, Cnv, and Cs point groups are polar, which means that they have a dipole moment. You can usually tell just by looking at a molecule whether it has a dipole moment, but it's always nice to be able to back it up mathematically. And sometimes it's not so obvious from inspection. For example, tetrahedral molecules such as carbon tetrachloride (CCl4) are non-polar despite the four very electronegative chlorines pointing off in different directions. If you were to calculate the dipole moment vectors and use vector addition to add them up, they would cancel each other out. But if you don't want to do the calculation, it's helpful to just know from group theory that tetrahedral molecules are non-polar and leave it at that.

  • Probably the most important chemical application of group theory is its application to bonding. We won't go into symmetry and bonding in this course. As a side note that you won't be held responsible for, the important thing about symmetry and bonding is that two orbitals must have the same symmetry about the internuclear axis in order to bond. A sigma bond, for example, is defined to be made up of two orbitals with C-infinity symmetry about the internuclear axis. A pi bond is made up of two orbitals with C2 antisymmetry about the internuclear axis. C2 antisymmetry means that a C2 rotation would invert the sign of the wavefunction (i.e. flip the shading) in the orbitals making up the pi bond.

V. Symmetry and Physical Laws

Physical laws don't change with time (as far as we know today). Because of this, we might say that they are "symmetric with respect to time." Physical laws also don't change depending on where you do an experiment (providing, of course, that all the particular conditions are the same in both places). Physical laws don't change if you rotate your experimental apparatus around either, so long as you rotate everything else that is relevant along with it. Because of this, we might say that physical laws are "symmetric with respect to space." Physical laws are even symmetric under uniform velocity in a straight line (but not if the line is curved). As Feynman puts it, "If we have a piece of apparatus working a certain way and then take the same apparatus and put it in a car, and move the whole car, plus all the relevant surroundings, at a uniform velocity in a straight line, then so far as the phenomena inside the car are concerned there is no difference: all the laws of physics appear the same."

This makes physical laws seem very symmetric. But they're not symmetric under any change. Feynman again: "After a long list of things that can be done without changing the phenomena, one might think that we could do practically anything; so let us give some examples to the contrary, just to see the difference. Suppose we ask: 'Are the physical laws symmetrical under a change of scale?' Suppose we build a certain piece of apparatus, and then build another apparatus five times bigger in every part, will it work exactly the same way? The answer is, in this case, no! The wavelength of light emitted, for example, by the atoms inside one box of sodium atoms and the wavelenght of light emmitted by a gas of sodium atoms five times in volume is not five times longer, but is infact exactly the same as the other. So the ratio of the wavelength to the size of the emitter will change."