Symmetry holds a power to fascinate us; this fascination is especially strong among chemists.  The symmetry of a molecule can have profound effects on its spectroscopic signature and even its chemical reactivity.  An important aspect of symmetry is chirality among chemical compounds.  Chirality refers to a sort of handedness.  Your left and right hands display chirality, they are mirror images of each other but they cannot be superimposed on each other.

A molecule like methane is superimposable on its mirror image.  However, if we substitute other atoms or groups for the hydrogen atoms we will arrive at a molecule that cannot be superimposed on its mirror image.  Chirality is illustrated in the image below – the mirror image is referred to as the enantiomer o the original and vice versa.

There is much written about the importance of one chiral form over another.  There is yet another feature (or should I say apparent lack of a feature) of chiral systems that I would like to address.   This feature is known as Hund’s paradox.  The problem is rather abstract and is entirely quantum mechanical.  It can be understood without mathematics, but we do need to be comfortable with a quantum mechanical peculiarity.  This peculiarity is known as superposition.  Superposition is an entirely foreign concept to beings that live in a predominantly classical world.

Before I get to Hund’s paradox, it helps consider superposition in a simpler system.  The simplest system that displays this oddity is the spin on an electron.  Spin is just a type of angular momentum, for our purposes think of it a simply a vector (an arrow) pointing up or down.  There can be spin in each coordinate direction: spin up or down in the x direction, spin up or down in the y direction, and spin up or down in the z direction.   Classically, there is nothing special; an electron will have a well defined spin in each direction.  However, quantum mechanics tells us we can’t do that.  Quantum mechanics tells us we can only know one component with a certainty at any given time.  Let’s say we know the electron has spin up in the x direction.  What about the spin in the y direction?

Here is where we need to think differently.  Quantum mechanics tells us the spin in the y direction is up + down.  If this bothers you, then good.  We are not used to things being in superpositions of definite states (i.e. we never see a cat that is alive + dead and we never see a pendulum swinging left + right).

Suppose now we measure the spin in the y direction.  We will get a definite value of up or down – this odd quality that making a measurement in quantum systems entails.  We now know that the y spin with certainty but now the x direction spin is in a superposition of up + down.  We can go back and forth all day.

Getting back to our chiral system, it turns out that under most circumstances one enantiomer is as good as the other (by “as good” I mean that one is not more energetically favorable than the other).  Hund’s paradox is this, why do we not find the chiral molecule in a superposition of left handed and right handed forms?  If the spin thought experiment above is valid, we would expect to find chiral molecules in superpositions of their enantiomeric forms all the time.  A jar of right handed sugars would soon decay into a racemic (1:1 left handed: right handed) mixture.  Yet we don’t see this in nature.

A hint towards resolving this paradox lies in the act of measurement I mentioned above.  In the spin example we were able to force the electron to have a definite spin in the y direction by measuring the said quantity.  In a similar way we could force the chiral molecule to stay in one enantiomer state by constantly measuring it.

But measuring need not be done by a human observer – in fact this is one terrible fallacy that leads to some ridiculous interpretations of quantum mechanics.  The environment provides plenty of opportunities for measurements to take place.  Molecules are always coming into contact with other bits of matter or radiation, each time they do a measurement of the molecule’s handedness occurs.  In this manner it is not hard to see that once a molecule is in a particular enantiomeric form it stays there (for the most part).  This is an example of what physicists call decoherence.

Still, this apparent resolution to the paradox is less satisfying than one might suspect.  There is little addressed about the mechanism that causes decoherence – “interacts with environment” is too vague – and nothing that really tells us under what conditions we might see the molecule go back into its superposition state.

The decoherence solution to the paradox has recently received some computational attention.  A group in Munich has performed some calculations with a simple chiral molecule (D₂S₂).  The calculation focuses on London dispersion forces; these intermolecular interactions are caused by molecular polarizability and are the dominant interactions at large distances.  “London dispersion forces” is a bit of an umbrella term and there are many interactions (all electrical) that contribute to what we would call a London dispersion force.  The paper turns its attention to a specific molecular interaction which gets weaker as the 7^th power of the distance between the molecules.  This interaction is often neglected as it mainly affected by the molecule’s chirality – because of the two enantiomers it is often averaged away.

The details are very technical and a little bit boring.  The result, however, is quite cool.  We can think of the molecule as having a decoherence cross section (i.e. the chiral molecule has some area around it, and if another molecule enters this area then chiral molecule will decohere).  Again, this is not different from the original solution to Hund’s paradox but it does pin down a particular cause for the decoherence effect.  Best of all, this paper uses these calculations to predict under what circumstances we might observe a lack of decoherence for a molecule.

The paper even proposes an experiment that could be performed to verify their results.  Conceptually the experiment is simple.  First we would take a molecular beam of D₂S₂ and sort the beam according to handedness.  One of the pure enantiomer beams would pass through a gas chamber and the handedness of the beam would be checked again.  We could control the pressure of the gas and thus we could control the mean intermolecular distance – the distance between the gas molecule and the chiral molecule.  At some pressure and temperature the beam will remain pure (stays in a decoherent state) or will turn into a racemic mixture (tunnel back to a superposition state).  I can see a lot of technical issues with trying to do this, but I hope someone with fondness for molecular beams will attempt it.

As an aside for more technical readers, I realize I have neglected the fact that thermal energy can cause a flip from one enantiomeric state to another.  The paper addresses this and the energy barrier is high enough that it should not prevent such an experiment from being performed.

References:

Hund’s paradox and the collisional stabilization of chiral molecule; Johannes Trost and Klaus Hornberger; Arnold Sommerfeld Center for Theoretical physics theory Ludwig-Maximilians-Universitat Munchen; Phys. Rev. Lett. 103; 023202 (2009)