Interactions in atomic and molecular systems are dominated
by electromagnetic
forces and the theoretical framework must be in the quantum regime.
The physical theory for the combination of quantum mechanics and electromagnetism,
quantum electrodynamics has been “established” by the
mid-twentieth century, primarily as a scattering theory. To describe
atoms and molecules, it is important to consider bound states. In
the nonrelativistic quantum mechanics framework, bound states can
be efficiently computed using robust and general methodologies with
systematic approximations developed for solvingwave equations. With the sight of the development of a computational quantum electrodynamics
framework for atomic and molecular matter, the field theoretic Bethe–Salpeter
wave equation expressed in space–time coordinates, its exact
equal-time variant, and emergence of a relativistic wave equation,
is reviewed. A computational framework, with initial applications
and future challenges in relation with precision spectroscopy, is
also highlighted.