From molecular orbitals to band diagram Let’s say we have a periodic crystal structure. We could, in theory, treat this system as if it were a large molecule. Therefore, we could use Hartree Fock theory or other methods to get the molecular orbitals of this system and also the energy levels.
My question is:
If we a have all of these molecular orbital energies, how can we get to the band diagram in k space? How are these two concepts related qualitatively and mathematically?
Comment: I know how to calculate band structures or molecular orbital energies. My question is not about how to calculate them but about how to switch between these two ‚representations’. Maybe one could list a step by step solution, this would be really helpful for the understanding of the topic.
 A: It is basically means that you should write a Hamiltonian of the system with periodic potential, and solve the Schrodinger equation for the Hamiltonian. Then you get the wave functions corresponding to the system. However, if the interaction between molecules is not very strong you can say that you are going to use the wave functions of lone molecules.
Another thing about a band diagram, the point is that a limited crystal will have a certain number of energy levels, however if number of atoms is high they will form bands with very small distances between every level.
As for bands, you can solve the Schrodinger equation for an large number of delta-atoms (with a single level) separated by a  distance, and you will see that a  this level will split.
A: There aren't two different representations. The translational symmetry in the crystal gives you another good "quantum number" to label your orbitals. This quantum number is $k$. You can label each orbital in a crystal by its energy and its k value. A molecule does not have translational symmetry and the states are not eigenvalues of a translation operator, so you can't label the states with k in addition to energy. But of course there is a regime where this is approximately possible, when your molecular system is very large and quasi-periodic. Then you may find orbitals that are very similar to the orbitals in a crystal and the orbitals should approach the crystal orbitals with increasing size of the system. You could then look at two specific values of a orbital, evaluated at $r$ and $r+a$, the ratio of the values should be $\exp(ika)$, allowing you to determine $k$.
There is also the point of interaction between atoms/molecules. The molecular orbitals of a single molecule and a crystal do not need to be similar if there is a strong interaction possible once you have more than one atom/molecule. Just like the orbitals of a monomer can be very different to a dimer or trimer, in a molecular system.
If the lattice sites are only weakly interacting, you may retain the original shape of the orbitals on each site and the crystal orbitals can be approximated by linear combinations of the localized orbitals, where $k$ determines how the phase of the LCAO coefficients varies.
$$
\psi_k(r) = \sum _{n=-\infty}^\infty\exp(ikan)\chi_n(r)
$$
where $\psi_k(r)$ is a crystal orbital, $a$ is the lattice spacing and $\chi_n(r)$ is a orbital centered on lattice site n. Think for example of a chain of hydrogen atoms with a spacing of $a$ and with a s-orbitals on each atom. When $k=0$ you would get a linear combination where all orbitals on all hydrogen atoms are in phase. When $k=\pi/a$ you would get the anti-bonding orbital where the sign between neighboring s-orbitals changes between $\pm 1$. $k$ can take on values between $-\pi/a$ to $\pi/a$.
This would lead to a band that rises in energy when $k$ increases from $0$ to $\pi/a$. The steepness of the band should depend on the spacing $a$. For small $a$ you should get a steep band since the interaction between sites is strong. For a large spacing $a$ you should get a flat band since the interaction is reduced.
You can make similar arguments for different types of orbitals. For example a band formed by p-Orbitals that point in the direction of the spacing. There you would have the highest energy at $k=0$ since neighboring orbitals are then antibonding (... -+  -+  -+ ...) and $k=\pi/a$ would be the lowest energy of the band since the sign-flip between neighbors would then "align" the phases of the p-orbitals(... -+  +-  -+ ...). In this way you can make predictions about bands and their shape based on the orbitals of a single atom/molecule.
I cannot quantify all of this with equations since solid state physics/chemistry is not my field of expertise but I think those are some of the basic ideas to connect the crystal picture with the molecular picture.
