# 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.

• For crystals and semiconductors, the pseudopotential method has been successful in calculating band structures. The key to this method is replacing the atom core and inner electrons with an effective screening potential that allows the calculation of the wavefunctions for the valence electrons only. Commented Jan 18, 2023 at 14:02
• The article How Chemistry and Physics Meet in the Solid State by Roald Hoffmann starts with orbitals and the typical orbital description used in chemistry and shows how bands emerge if the system size increases. Commented Jan 18, 2023 at 14:19
• Maybe some chemistry book. It might be somewhere on molecular orbitals or before organic chem in the following book. goodreads.com/book/show/25007809-chemistry Commented Jan 21, 2023 at 7:48

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.

• I am not sure what you mean by: ‚ 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 $k$.‘ What is r and r+a? I think you were using Blochs theorem? Commented Jan 26, 2023 at 11:20
• @Lockhart I am referring to an orbital of an extended but non-periodic system. For example 10^20 hydrogen atoms with a spacing of a. It is not immediately clear which $k$ value is associated with each of the 10^20 orbitals. To determine the $k$ value of one of these orbitals, you could do what I have described. The orbital with a specific value of $k$ should closely correspond to the same crystal orbital/bloch function of a periodic crystal with the same value of k. The point is to associate orbitals of systems that have no strict translation symmetry with values of k from periodic systems. Commented Jan 26, 2023 at 12:03

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.

• and if you have a periodic potential you can search for the solutions in the form of $\psi(r) = e^{ikr}u(r)$. To see the Bloch's theorem. en.wikipedia.org/wiki/Bloch%27s_theorem Commented Jan 18, 2023 at 12:39
• which simplifies differentiation $\frac{d}{dx}$, it is like to take Fourier transform which, basically gives $\frac{d}{dx}\rightarrow jk$ Commented Jan 18, 2023 at 12:42
• this is how $\vec{k}$ appears Commented Jan 18, 2023 at 12:44
• Thank you for your answer. It is clear to me how to calculate band structures or molecular orbital energies. But I do not understand how these are related. How can I get from one representation to the other? Commented Jan 18, 2023 at 18:59
• @ Lockhart, sorry I do not understand the question. There is a $\vec{k}$ and an energy E and and there is an orbital associated to those quantities. Commented Jan 21, 2023 at 7:50