  In this thesis are shown developments in the random phase approximation (RPA)
in the context of range-separated theories. We present advances in the
formalism of the RPA in general, and particularly in the "dielectric matrix"
formulation of RPA, which is explored in details. We show a summary of a work
on the RPA equations with localised orbitals, especially developments of the
virtual localized orbitals that are the "projected oscillatory orbitals" (POO).
A program has been written to calculate functions such as the exchange hole,
the response function, etc on real space grid (parallelepipedic or of the "DFT"
type) ; some of those visualisations are shown here. In the real space, we
offer an adaptation of the effective energy denominator approximation (EED),
originally developped in the reciprocal space in solid physics. The analytical
gradients of the RPA correlation energies in the context of range separation
has been derived. The formalism developped here with a lagrangian allows an
all-in-one derivation of the short- and long-range terms that emerge in the
expressions of the gradient. These terms show interesting parallels. Geometry
optimisations at the RSH+dRPA-I and RSH+SOSEX levels on a set of 16 molecules
are shown, as well as calculations and visualisations of correlated densities
at the RSH+dRPA-I level.
