How does quantum electrodynamics explain the movement of nuclei in chemical reactions? In Feynman's third Douglas Robb Memorial Lecture (which took place in 1976 at University of Auckland), he states that:

*

*It's fine to approximate nuclei as a point, although they really aren't.

*Quantum electrodynamics explains everything except for radioactivity and gravity.

*Specifically, it explains the movement of electrons through space-time (a term that Feynman says he doesn't like, even though he uses it.)

In I. David Brown's book The Chemical Bond in Inorganic Chemistry: The Bond Valence Model, 2nd edition (Oxford University Press, 2016) he states:

Somewhat perversely, chemists have largely rejected this simple wave
picture of the atom in favor of a hybrid view in which the charge is
composed of a collection of electrons that are not waves but small
particles. The density of the charge wave merely represents the
probability that an electron will be found at a given location....
This model has many problems.... An electron is the smallest quantum
of charge that can have an independent existence, but the free
electrons that are attracted to a nucleus in order to form a neutral
atom cease to exist the moment they are captured by the nucleus. They
are absorbed into the charge wave and, like Lewis Carroll's (1865)
Cheshire Cat that disappears leaving only its smile behind, the
electron disappears bequeathing only its conserved properties: charge,
mass and spin, to the charge wave surrounding the nucleus.

Chemical reactions happen when there is a re-arrangement of electrons followed by a movement of nuclei. Clearly there is movement of nuclei involved: if there was not, there would be chemical reactions at a distance. Two molecules come together, react, and then two different molecules leave.
So how does quantum electrodynamics account for the movement of nuclei, or is that outside the scope of the theory?
 A: Quantum electrodynamics (QED) accounts for the movement of nuclei the same way it accounts for the movement of electrons. QED treats a nucleus as an elementary particle (that's the approximation that Feynman apparently meant), but that doesn't prevent it from moving.
Although they do move, the nuclei move slowly enough (in chemistry) that we can use non-relativistic QED for them. This is easier, and it allows us to freely choose properties like their spin and magnetic moments. Using the non-relativistic approximation for the electrons may also be good enough for many purposes in chemistry, or we can do better by using a hybrid model in which a non-relativistic approximation is used only for the nuclei.
Some of the literature might leave the impression that QED cannot account for nuclei. That's true in the sense that relativistic QED would predict the wrong magnetic moments (for example), and it's true in the sense that QED does not account for the Strong interaction between nucleons or between nuclei. However, as long as we don't mind treating each species of nucleus as an elementary particle, non-relativistic QED has no trouble accounting for the motion of nuclei and their electromagnetic interactions with the electrons, and this is good enough for chemistry.
