The hand-waving explanation that books*/Wikipedia** give for the splitting of the energy levels is the Pauli Exclusion Principle: when atoms are brought close together the energy levels must split in order to accommodate all the electrons. (Hand waving because they never give a mathematical derivation.)

My first encounter with energy splitting was through perturbation theory: the Hamiltonian is altered and so the eigenstates are correspondingly altered; sometimes in a way that removes degeneracy i.e. energy-splitting.

Are these two explanations one and the same? Is there a way to tie it all together?

*Semiconductor Physics and Devices by Donald A. Neamen - page 77 of pdf file; first paragraph

**Wikipedia article (link to specific section)

  • $\begingroup$ splitting of which energy levels, exactly? Could you quote which book or wiki page you are thinking of? $\endgroup$ – Rococo May 25 '17 at 17:14
  • $\begingroup$ I added the sources. I apologize that they are links. It would have been too long of a post to quote both sources. $\endgroup$ – YoA May 25 '17 at 17:24
  • 1
    $\begingroup$ Such explanations are a little hand waving because, well, they are. The point is that one has to start with Bloch's theorem and then think about how to try and connect Bloch states (representative of single electron states in a periodic potential) with single atom states. However, one gets the Bloch theorem without having to consider anything concrete about the periodic potential except that it is periodic. $\endgroup$ – Jon Custer May 25 '17 at 23:13

No, the two explanations are not the same. The Wikipedia article is alarmingly incorrect. To be clear: the splitting of energy levels as atoms are brought together to form a molecule, and the formation of energy bands in a solid, are single-particle effects and they are independent of both spin and the Pauli exclusion principle. The exclusion principle can dictate how those bands get filled (and, if the particles interact, even through a mean-field description, the filling can then influence the details of the bands), but the bands/splittings are already present even if you have a single electron in the system.

Why do I say that energy bands are single-particle effects? Because they are: if you take a situation where you only have a single electron to place on your system (be it a diatomic or an $N$-atomic molecule, or a crystal lattice), where your lone electron feels the influence of all the nuclei in the system, but no other electrons are present, then you will immediately get the relevant splittings / energy bands. These arise because the nuclear hamiltonian itself couples the different localized states together: if you have a localized orbital $|\varphi_1⟩$ at atom $1$ and a localized orbital $|\varphi_2⟩$ at atom $2$, then the matrix element $$ \left<\varphi_2\middle| \left(\hat V_{\mathrm{nuc},1}+\hat V_{\mathrm{nuc},2}\right) \middle|\varphi_1\right> \neq 0 $$ of the nuclear potential, due to either or both of the nuclei, between the two localized orbitals, is nonzero, whenever there is significant overlap between the two orbitals. This means that the localized orbitals are not eigenstates of the single-electron multi-nucleus hamiltonian, and to get the eigenstates we need to look at delocalized linear combinations with varying phases. For a molecule, these are the molecular orbitals (MOs); for a crystal lattice, they are the Bloch wavefunctions.

Now, what happens if we start adding electrons? Well, the first thing that will happen is that we need to choose where we put them, and the Pauli exclusion principle is going to be very stringent about trying to put more than two electrons on each orbital. (Note, however, that we could equally well do this with bosons, as implemented experimentally with BEC experiments on optical lattices, in which case the bosons will all happily sit on the same state if the temperature is right - but the energy bands still exist, as excited states.) However, the filling-up process only determines how the bands/MOs get populated, not their existence.

The next thing that will happen, of course, is that the electrons interact, and there are explicit Coulomb-repulsion operators in the hamiltonian that implement this. As a first stab at what will happen, we can just pretend that each electron sees a single-particle problem, which is only modified by the shielding provided by the other electrons. This changes the nuclear potential but it doesn't alter its fundamental structure, so the procedure will modify the bands/MOs, but it won't change the fact that they exist or their fundamental origin. (Moreover, this is where it starts getting a bit messy, because the orbitals depend on the shielding, which depends on the orbitals, so the single-particle problem becomes nonlinear and it needs to be solved through iterative processes known as mean-field, self-consistent theories, or more broadly as Hartree-Fock.)

In addition to this, if you're really unlucky, you can have interactions that are not describable at the level of a mean-field shielding, in which case the interactions are truly entangling, and you will require multiple Slater determinants to accurately describe each multi-electron eigenstate. For molecules, this is what quantum chemistry is all about, and it's mostly old news which can be solved so long as the system isn't too big. For solids, the presence of strong correlations is an unsolved problem whose applications include high-$T_C$ superconductivity and other hot topics. Depending on the situation, strongly-correlating interactions can really wreck some havoc on MO pictures and energy-band diagrams, though this can normally be ignored with the kinds of materials and molecules treated by introductory textbooks.


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