  Electrodynamical coupled cluster (CC) methodologies have been formulated
employing standard QED Hamiltonian that is written in Coulomb gauge while using
the DF and the MCDF pictures of the matter field for closed-shell and
open-shell cases respectively. The general methodology employs a radiative
cluster, pure matter clusters and their pair modifications, and a number state
distribution of photons in thermal equilibrium. The closed-shell treatment
relies on the customary CC approach. For open shells, QED and electron
correlation through CC are treated on the same footing. An averaging over the
radiation state is done to generate Lamb, Breit and hyperfine interactions from
the radiative cluster. Because of the thermal distribution, it leaves a
residual transverse interaction that may modify the static correlation in open
shells. Dynamical correlation effects are determined next by using the
exponential matter cluster in traditional ways of single- and multi-reference
CC. When the matter cluster is extended to include de-excitations to
negative-energy levels, vacuum polarization effects are generated from the pair
part of Coulomb interaction. The dynamical correlation energy includes
relativistic corrections as well as QED contributions, namely, Lamb, Breit,
hyperfine and pair energy. This work has three novelties: (i) QED interactions
(Lamb, Breit and hyperfine) are obtained from a single procedure based on the
radiative cluster; (ii) pair energy is determined from an extended matter
cluster formalism; and (iii) additional correlation energy can be had from
radiative effects and pair terms, while the option for higher order pair energy
in high-Z atoms is kept open. The open-shell formalism has one more novelty in
finding an additional static correlation in certain cases when the radiation is
not isotropic.
