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As my last question (Semiconductors and their electronic bands) was badly structured, I decided to elaborate my questions a bit.

As I now know, every solid/liquid forms a band structure, so all condensed matter. For simple elements this is easy enough to illustrate.

But what happens when there are complicated molecules like organic molecules with many different bonds? Are those bands vastly different than when those molecules are in solution?

For example, azo dyes come to mind whose color is the same in the solid state as in solution; also what about complexes for example Cu(II)-hexahydrate, whose color we usually interpret with CFT. Is there even crystal field splitting in a solid or does is form crystal field ‚bands‘? Somehow I am at a loss connecting the band concept to this.

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    $\begingroup$ This question isn't the greatest, too. In general, you always have some anti-bonding, or higher atomic orbitals, available for electron transiting between molecules or atoms. In context of band structure, they all are treated as bands. $\endgroup$
    – Mithoron
    Oct 18, 2023 at 14:24
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    $\begingroup$ I've found somewhat generally that trying to interpret the solid state physics of crystal energy bands with chemistry bonding principles can rapidly lead to incomplete or false analogies. Even for supposedly "simple" elements. $\endgroup$
    – Jon Custer
    Oct 18, 2023 at 15:45
  • $\begingroup$ So the two theories can’t be combined or? $\endgroup$
    – Mäßige
    Oct 19, 2023 at 11:15
  • $\begingroup$ Actually I disagree to an extent with Jon Custer. You can give a perfectly adequate description of the bands in materials using LCAO and symmetry adaption of the basis (aka k point sampling in the solid state world) - all bonding is ultimately the same. But you do have to be careful, a simple A bonds to B picture is often incomplete; delocalisation is rife, and a simple bond/anitbonding pair is not always a complete picture. $\endgroup$
    – Ian Bush
    Oct 23, 2023 at 6:48
  • $\begingroup$ Basic DFT in general is ill-suited for calculation of one-electron properties and excited states, it requires patches. Plane-wave expansion does not account for localized states and defects. The common approach to model them is supercells. However, even this approach fails to model the so-called topological insulators, which requires further patching such as DFT+U method. All this is to be expected for formalisms derived from description of isotropic electron gas. $\endgroup$
    – permeakra
    Oct 23, 2023 at 12:59

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I personally think and I could be wrong, that this question is a mixture of very inhomogeneous concepts. There is indeed a correlation between quantum chemistry and solid state physics. Quantum chemistry in the simple formulation evoked in the comments deals mainly with localised states that are not very too far from atomic orbitals, the expansion of the unknown WF is a set of localised orbitals (LCAO). Solid-state physics is concerned with delocalized states. Their difference is a matter if size and the symmetry of the system under study.

In solid-state physics, the most crucial symmetry is translation, and the single-particle translational operator $\hat{T}$ linked to the linear moment operator $\hat{p}=i\hbar \nabla \hat{T}$. The eigenfunctions here are often plane-waves (PW) or a combinaison of PW. $\hat{T}$ produces the band structure and Bloch states by assuming that it commutes with the Hamiltonian. Everything that affects translational symmetry leads directly to chemical localized states : pairwise $U$ Coulomb interactions, defects ...

In the chemical limit, the operator position $\hat{x}$ is important and the symmetries such as rotations or mirrors that commute with the Hamiltonian.

I don't know where this argument about band structure in liquids where there is no translational symmetry comes from. CFT in solids like transition metal oxides (cupper oxide) is in my opinion more of a chemistry problem, I could go into mathematical details using the hubbard model to show that in this configuration the metal cation has a quasi-atomic structure for the d-orbitals simply because the overlap of the d-orbitals is too small due to impurities compared to the coulombic interaction $U$.

The number of bonds is not the generator of a band structure but the symmetry of these bonds. Large molecules such as fullerenes $C_{200}$ do not have a band structure but energy levels because there is not enough translational symmetry. The band gap is just due to how this translational symmetry is affected by the potential, but this gap may differ from the splitting responsible for the colors of these oxides due to CFT.

Even if it's not the direct answer to the question I hope it helps

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