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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.

Buttonwood
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Mäßige
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    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. – Mithoron Oct 18 '23 at 14:24
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    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. – Jon Custer Oct 18 '23 at 15:45
  • So the two theories can’t be combined or? – Mäßige Oct 19 '23 at 11:15
  • 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. – Ian Bush Oct 23 '23 at 06:48
  • 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. – permeakra Oct 23 '23 at 12:59
  • @permeakra Basis sets and the effective one-electron Hamiltonian are different things. You can perfectly well model localized states with plane-waves, you might just need a lot of them. The problem is that a LDA Hamiltonian has a spurious self-interaction from the lack of exact exchange (or similar) to cancel the coulomb term; the system tends to minimize the cost of this by spreading out the electron - hence the problem with localization. But this is not directly a problem of a plane-wave basis set. DFT+U is a way to try and introduce the required exchange terms. – Ian Bush Oct 23 '23 at 14:14
  • Secondly plane-waves can perfectly well do defects. You need supercells because otherwise the defect concentration would be unphysically high, not because you are using plane-waves. Consider an F centre in an NaCl cell. If you just used one cell the system would consist of the Na+ centre and an F centre - no Cl- at all. Hence the need for a supercell to bring the concentration of defects down towards something sensible - remember the maximum doping of Si used is around 0.01% – Ian Bush Oct 23 '23 at 14:20
  • @IanBush >You can perfectly well model localized states with plane-waves, you might just need a lot of them. || I would say you need infinitely many of them. Like continuum many. Plane waves are not square-integrable. Hence, you cannot use them to expand a square-integrable function in a countable series. At best, you can use them to expand a periodic function and hope it is a good enough model for your aperiodic system. This is a fundamental flaw in a plane-wave approximation. It's math, arguing with it is pointless. – permeakra Oct 23 '23 at 14:32
  • @IanBush Also, if you point at 'gaussian basis set' in programs like Elk code, they still use plane waves implicitly. – permeakra Oct 23 '23 at 14:37
  • I am an author of CRYSTAL, a Gaussian basis set based program for periodic systems. Please point me to the plane-waves in CRYSTAL. – Ian Bush Oct 23 '23 at 14:38
  • @IanBush "The Bloch function" . It is by definition a periodic plane wave. – permeakra Oct 23 '23 at 14:41
  • OK, last thing I will say, The Bloch function is purely a reflection of the translational symmetry of the crytsalline material being modelled. The eigenstates of the Hamiltonian of a periodic system are necessarily Bloch States. They are not being used by choice, they are a fundamental result of the mathematics underlying the model of the system being employed. F. Bloch, Zeitschrift für Physik 52, 555 (1928) – Ian Bush Oct 23 '23 at 14:51
  • I don’t know about those softwares. – Mäßige Oct 23 '23 at 15:59
  • Not my field but given the lack of translational coherence, I would answer "Are those bands vastly different than when those molecules are in solution" with a resounding "of course", and would argue that being in solution (highly mobile) is not conducive to band structures. – Buck Thorn Oct 24 '23 at 10:52

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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

M06-2x
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