Wavefunction of isomers

Question

In quantum chemistry, the wavefunction for a molecule can be viewed as the output of a function $\xi(m, n_1,..., n_k)$ with $m, n_i \in \mathbb{Z}^+$ that returns a $|\psi\rangle$ that satisfies a $H|\psi\rangle = E|\psi\rangle$. $H$ is the electrostatic Hamiltonian for $m$ electrons and $k$ nuclei with charges $\{+n_k\}$ (and the appropriate masses). I believe there is an injective mapping between $\xi$ and $|\psi\rangle$, right?

In which case, how are chemical isomers explained? All isomers for a given chemical formula are stable and have the same $\xi$, but they have different energy eigenvalues and different structures.

Answer 1

Well, the problem is easy from the standpoint of chemistry. Yes, there are multiple isomers.

So why are there different structures and different energy eigenvalues?

The function you describe is high-dimensional. For even something "small" like water, we're talking about 3 atoms and 10 electrons. If we restrict ourselves to a Born-Oppenheimer picture and assume the nuclei are fixed, we still have a 13-body problem.

It's clear that different chemical isomers are local minima on a highly multi-dimensional space.

So then the question from a physics standpoint would be whether it's possible to "hop" from different isomeric structures. Of course in the general case, this should be possible, and of course chemistry knows of many such things (chirality, stereochemistry, cis-trans isomerization, etc.)

And of course, given a specific structure/isomer, the eigenvalues and eigenvectors from quantum mechanics will differ simply because of the different electrostatic and electron correlation effects.

Answer 2

The paradox occurs because of quantum superposition. After some simplification (namely, factoring the E(3) global symmetry out) one may say that an eigenstate of H may be, although not necessarily is, a superposition of several isomers or conformations.

For example, a ground-state wave function of ammonia (NH₃) is a superposition of nitrogen-up and nitrogen-down conformations, and obviously neither of those is an energy eigenstate. For some other (than NH₃) compounds such distinct conformations are chiral. If something remained unclear, just ask.