[farlsbarkley]

This statement/question is related to a post from earlier this year.  Since it was so old, I decided to start a new topic in case anyone else had similar confusion.

Basically, the original poster asked a question that I have been struggling with for some time.  Namely, is the phenomenon of resonance a result of the quantum superposition of the contributing resonance structures?

The reply, posted by Yggdrasil (great name, btw) went as follows:
"No, the resonance structures do not represent a superposition of states.  They are eigenstates of the hamiltonian (so they have a defined energy).  They basically represent chemical bonds (or more precisely, molecular orbitals) that span more than two atoms, something which conventional organic chemists' drawings have trouble representing."

At first, I thought he was wrong, but after thinking about it, I see that it is the correct answer.  My reasoning, however, seems to be fundamentally different from Yggdrasil's explanation, unless I am misinterpreting him.  It is possible that what he means by using the phrase "resonance structures" is actually what I am calling the "resonance hybrid" in what follows.  If so, then I think our explanations are the same, but I would still like to know if my clarification is accurate.

My reasoning follows, and I welcome any and all corrections or criticisms:

Basically, the resonance hybrid is a complete, exact description of (or wavefunction for) the full molecule.  It is a solution to the Schrodinger Equation for the molecule, an eigenstate of the Hamiltonian, and therefore a pure quantum state.  This resonance hybrid (wavefunction) is a linear combination (or superposition) of the resonance structures (i.e., the individual Lewis diagrams).  These structures are only approximations to the true state, and no one of them exactly solves the Schrodinger Equation by itself.  Therefore, they are not pure quantum states for the molecule in question.  The full resonance hybrid is thus a linear superposition of these in the mathematical sense (a weighted average), but not a quantum superposition in the physical sense.

Does that make sense?

Another way that I thought of visualizing this is by considering the way the wavefunction collapses during a measurement.  For a system that is in a quantum state which is a quantum superposition of pure states (think Schrodinger's cat), a measurement will result in a collapse of the system to one of those pure states with a certain probability.  This is not what happens in resonance, as we are emphatically reminded by our Organic Chem teachers.  The molecule is never observed to be behaving as one resonance structure or another, but always to be the weighted average or hybrid state, which is thus a pure state of the Hamiltonian.

[Yggdrasil]

That sounds like an excellent explanation to me.  And yes, in my original post I should have said that the resonance hybrid is not a superposition of states, and that the resonance hybrid is an eigenstate of the hamiltonian.  Thank you for clarifying.

[josh]

Sorry if I revive this dead thread, but the question urges me.
The question whether the phenomenon of resonance has something to do with QM or not, has been presented to me in the following terms:

If someone wanted to give a sketchy review of QM he could say that QM is, in a nutshell, the theory of superpositions, and he could introduce this notion by taking in consideration the classical account of benzene ring. So the benzene ring is a hexagon of carbon atoms, each with a hydrogen atom attached. Replace two adjacent hydrogens with 2 different substituents, so that we can unambiguously number the sides of the hexagon. Two structures conform to the rules of valence: one with double bonds on the odd-numbered sides, another with double bonds on the even-numbered sides.
The molecule reacts sometimes as if it has one structure, sometimes as if it had the other. Yet we do not think that a (statistical) population of molecules is a mixture of the two structures. Neither do we think that each molecule oscillates rapidly between one structure and the other. Neither do we think that the molecule has a betwixt-and-between structure - there is no such thing as a bond midway between double and single. Rather, we think that each molecule is in a superposition: a state objectively indeterminate between the two structures.

What's wrong in here?

[mjc123]

If someone wanted to give a sketchy review of QM he could say that QM is, in a nutshell, the theory of superpositions

He could say whatever he liked, but would he be right?

he could introduce this notion by taking in consideration the classical account of benzene ring.

Are you suggesting that for molecules without resonance, e.g. ethane, the bonding is not described by QM? How would you describe it classically?

The molecule reacts sometimes as if it has one structure, sometimes as if it had the other.

No it doesn't; its chemistry is quite different from that of a molecule with localised double bonds, e.g. it reacts with Br₂ to give bromobenzene, not dibromocyclohexadiene. (The statement would be more appropriate to e.g. an enolate ion.)

there is no such thing as a bond midway between double and single.

Yes there is. Atomic orbitals can overlap to give molecular orbitals spread over more than two atoms; filling a bonding orbital with two electrons gives a bond with an average order of less than 1 per linked atom pair. That's what happens in benzene - six 2p atomic orbitals on six C atoms overlap to give six MOs, three bonding, three antibonding. Putting the 6 electrons into the bonding orbitals gives the equivalent of 3 pi bonds, or half a bond per C-C, or 1.5 including the sigma bonds.
It is the conventional Lewis structures that cannot easily show bonds of intermediate order.

Rather, we think that each molecule is in a superposition: a state objectively indeterminate between the two structures.

See the statement of farlsbarkley above: "The full resonance hybrid is thus a linear superposition of these in the mathematical sense (a weighted average), but not a quantum superposition in the physical sense."
To summarise: the actual state is not a statistical mixture of the resonance structures, nor a rapid oscillation between them, nor a quantum superposition of them (they are not true quantum states), but (if you like) a betwixt-and-between; something intermediate between the resonance structures, which is harder to describe in conventional valence-bond terms, but closer to the truth.