# Electronic band structure - Pauli Exclusion principle and perturbation theory

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)

• splitting of which energy levels, exactly? Could you quote which book or wiki page you are thinking of? – Rococo May 25 '17 at 17:14
• I added the sources. I apologize that they are links. It would have been too long of a post to quote both sources. – YoA May 25 '17 at 17:24
• 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. – Jon Custer May 25 '17 at 23:13

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.
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.