Non-relativistic chemical bond models This is based on this discussion in Stack Exchange Worldbuilding, specifically a discussion in the comments about whether in a nonrelativistic universe there could be chemical bonds without quantum electrodynamics. Specifically, someone asked for an approximate model of chemical bonds that worked in non-relativistic quantum mechanics. Does such a model exist?
Edit: If spin and the Pauli exclusion principle could be put in somehow it would be great, but if it's absolutely impossible I'll have to find out if there are ways around it.
Another edit: Here is a link to an article suggesting that spin can be found even to a nonrelativistic theory. I don't know, however, whether the explanation is completely free of errors, or whether it keeps the Pauli exclusion principle. http://quantumchymist.blogspot.co.uk/2014/04/is-spin-relativistic-effect-l-and-first.html
 A: The best answer I can give is that there are plenty of models of chemical bonds that don't include relativity.  In fact, many of the commonly used models ignore relativity or treat it as a 2nd or 3rd order correction.  
The aforementioned LCAO model a good example.  There are actually many types of LCAO models that make various kinds of approximations.  Many compare pretty well with experiment with no relativistic correction.  An important exception is in zinc blende materials, where Spin-Orbit Coupling is important.  You can derive a spin-orbit coupling term with just classical relativity, so your "relativity-free" world may still be safe for zinc blende materials.
An import caveat is that most of these models ignore or heavily approximate electron-electron interactions because they are so hard (or impossible) to calculate, and I'm not sure what impact neglecting relativity will have on phenomena that depends on those effects.
A: With the classical theory of electrodynamics, something like the hydrogen atom is not stable. Non-relativistic quantum mechanics is able to give a stable description of electrons in a hydrogen atom, and we can solve for the hydrogen energy levels. Furthermore, basically all the intro chemistry descriptions of atomic orbitals, such as S vs P vs D orbitals and the basics of bonding, come from non-relativistic quantum mechanics and extrapolating from those single electron hydrogen like wavefunctions.
There are many computational chemistry models that don't include relativity explicitly. One caveat that I'd mention is that they do include it implicitly as soon as there are multiple electrons.  For example the Hatree-Fock method describes molecular structures by describing molecular orbitals in some basis (a linear combination of atomic oribitals about each atom, or gaussian approximations of atomic orbitals, etc).  The wavefunction is then written as a Slater determinant of these molecular orbitals.  With only a single Slater determinant, the majority of the electron-electron correlation is ignored. Basically, the only correlation you get is that, due to the structure of a determinant, it provides that the probability that two electrons of the same spin are found at the same location is zero.  These somewhat crude models have been quite successful in the initial exploring of computational quantum chemistry.
In other words, the quantum chemistry models can be quite successful with just non-relativistic quantum mechanics and without including relativistic terms or corrections in the Hamiltonian (there are not even magnetic interactions in Hartree-Fock, the momentum operation is non-relativistic, and so on). However, relativity is implicitly snuck in at the fundamental level by using the Slater determinants. This is what enforces the Pauli exclusion principle. It is essentially put in by hand.
Without relativity, you would not be able to derive the spin statistics theorem. This is incredibly important to the structure of molecules (otherwise all the electrons would essentially just pile up in the same lowest energy orbital).
That being said, I see no reason you couldn't just postulate spin and the Pauli exclusion principle, and let non-relativistic quantum mechanics take you the rest of the way to quantum chemistry.
Alternatively, if you are worried about quantizing the electromagnetic field and getting into quantum field theory, I see no reason you could not (at least in principle) just obtain a well defined theory by taking the $c \to \infty$ limit of our current theories.  In the $c \to \infty$ limit, special relativity reduces to Galilean relativity of Newtonian mechanics.  This limit is a bit counter-intuitive in that some everyday things may change, so I cannot speak confidently on the results.  For instance I think the effect of the magnetic field drops to zero as the speed of light is taken to infinity, and I'm not sure if radio communication could work anymore.  If you are just fantasy "WorldBuilding" and going full Newtonian, you could try to use the old idea of a medium for electromagnetism and have the speed of light still be finite (the same way sound propagating in a material is much less than c), even if the relativity parameter "c" goes to infinity.
