How can I do quantum chemistry with QED? So I am a theoretical and computational chemist by trade and my usual zone of operation in the domain of quantum mechanics is Hartree-Fock and Density Functional Theories. 
I was thinking if there was a way to apply QED on quantum chemical problems. In particular, I am interested in the computation of the electron density around bound to a molecule. 
In essence, I would like to know if this is at all feasible and, more specifically, what sequence of steps I would have to employ to get this done. I have no familiarity with the actual practise of using QED to solve actual problems, but I can follow appropriate references to understand the answer, if they are provided. 
Is there a sequence of steps for doing this? Any software I could use? Any algorithms I could implement? Perhaps only for $\operatorname{H}_2$ and its two electrons.
 A: From a theoretical chemists point of view, the inclusion of Quantum Electrodynamics (QED) can be very interesting if you are interested in radiaton-matter interactions, where the radiation field is quantized (commonly in the Coulomb gauge leading to a non-covariant quantization). This would include all types of absorption and emission processes but additionally also intermolecular dipole-dipole coupling, for example. If you are interested in this topic, I can highly recommend the textbook on Molecular Quantum Electrodynamics by Craig and Thirunamachandran.
Additionally, there is a very fascinating field studying atoms and molecules in cavities, i.e. in the framework of cavity-QED. In this context, a whole bunch of interesting new phenomena emerge, e.g. the Born-Oppenheimer-approximation needs to be reformulated, due to the full quantum description of the radiation field and the molecular degrees of freedom. You can consider https://arxiv.org/abs/1609.03901, if you are interested in getting an idea of this subject.
A: This is an interesting question. The clue is the in the acronym, Quantum Electro Dynamics. Usually in chemistry we average over the electrons to obtain an effective force on the nuclei. Situations where you might want to consider the dynamics of the electrons in a molecule generally involve strong radiation fields, such as higher harmonic generation or the cavity experiments ewf mentioned above.
A: There's certainly a lot that QED can bring to quantum chemistry. Coulomb's law is an approximation to a much more complicated law described using QED that does not follow an instantaneous, spin-independent interaction. For instance, there are other terms like the Breit- and Gaunt-coupling terms that are certainly relevant when doing calculations on heavy atoms or NMR spin-coupling constants. Getting these and other terms is the subject of effective field theory and renormalization techniques. This stuff is well understood in nuclear physics where they routinely derive effective 2-body interactions using the complex 3-body (and higher) interactions in QCD. In fact, most method we have in quantum chemistry originated in nuclear physics because people found effective interactions and needed good ways to solve 2-body problems.
As others have said, you can consider situations in which you quantize the EM field e.g. in a cavity, but that to me isn't really field theory. It's more of a Jaynes-Cummings type model where you couple a matter Hamiltonian to some bosonic Hamiltonian. It would be no different than dealing with quantized electronic and nuclear degrees of freedom like many vibronic excitation models do. The main problem there is the sheer size of the combined Hilbert space of fermionic and bosonic coordinates, which is why DFT is probably the only technique that makes use of these models beyond a testing or benchmarking capacity. There should also be some way to integrate out bosonic modes to create a spectral density like in Redfield (or other dynamics) theories. In that limit, the effect of the quantized field would just be some modification to your original 1- and 2-body integrals.
