Chemistry - What are the applications of quantum field theory to chemistry?
Have a look at this article: "The Physics behind Chemistry and the Periodic Table" Pekka Pyykkö Chem. Rev. 2012 112 (1), 371-384 (http://pubs.acs.org/doi/abs/10.1021/cr200042e) I think it answers your question and has most of the references to work done in that particular field. For most questions in chemistry the non-relativistic Schrödinger Equation is good enough. Even for systems with an appreciable amount of relativistic effects, the Dirac Equation is often not solved directly. So, the number of applications where Quantum Electrodynamics is used is really limited.
Quantum field theory is sometimes used in chemistry to derive relativistic corrections to non-relativistic quantum mechanical energies. And this is useful when computing energies of molecular matter with heavy atoms, due to electrons moving at high speeds near a massive nucleus.
In reality one does not use quantum field theory. One starts with a Schrodinger-like wavefunction expression
$$H \Psi = E \Psi$$
where the Hamiltonian operator is the sum of a kinetic part $K$ and a potential part $V$, and then one seeks for a potential, sometimes named an "effective potential", so that $V$ gives the same scattering amplitudes than quantum field theory. I.e. quantum field theory is used to verify that the chosen potential $V$ is physically admissible. Note that scattering equivalence does not imply dynamical equivalence. And sometimes incorrect potentials have been proposed in the belief that compatibility with quantum field theory was enough .
Basically there is no known expression for $V$ at arbitrary orders in powers of $(v/c)$. The problem is not only computational, it is technical, because quantum field theory cannot really describe bound states; the only rigorous states in quantum field theory are those of free particles. This limitation is fine for particle physics and the study of scattering events, but it is not for molecular sciences as chemistry.
An attempt to unify quantum field theory (specially QED) with the traditional many body methods used in chemistry is that from Lindgren and coworkers. Their research is summarized in book . The point is that QED does not consider all electron-electron correlation effects, which limits its applicability to systems where electron correlation effects are small. Lindgren and coworkers have developed a so named "covariant-evolution-operator method" which they claim
forms a suitable basis for a combined QED–MBPT procedure. This leads to a perturbative procedure that is ultimately equivalent to an extension of the relativistically covariant Bethe–Salpeter equation, valid also in the multireference case and referred to as as the Bethe– Salpeter–Bloch equation.
Their approach, however, doesn't solve fundamental questions associated to Bethe–Salpeter equations; they simply mention the difficulties and move away! Also, from a numerical point of view, renormalization of the radiative effects is only partial in their procedure, with hope to improve in future:
Schemes have been developed for this process but so far not been implemented in a QED-MBPT procedure. When the procedure is more developed, critical tests can be performed to find out to what extent the new effects will improve the agreement between theory and accurate experimental data.
The article from Pekka Pyykkö mentioned in another answer is freely available here. There are several objections to this article. For instance, in page 4 he claims that electron-electron interactions can be taken as instantaneous in the Coulomb gauge. This is not true. He seems to believe that electron-electron interactions are only given by the Coulomb term, when the Breit term also describes electron-electron interactions. Pyykkö then claims that for certain uses the CB potential has to be surrounded by projection operators. He omits to say that without those projection operators the standard CB potential gives invalid answers. Pyykkö only surrounds the interaction part of the DCB Hamiltonian, but in reality the Dirac Hamiltonian $h_i$ (equation 2 in his paper) also has to be surrounded like $Ph_iP$. That is why in other texts you will find projection operators around the full DCB Hamiltonian (check equation 2.113 in ) and not only about the CB term. There are similar objections to the rest of his paper, but the biggest objection, in my opinion, is that he never mentions that quantum field theory is not needed to explain the phenomena he covers.