  For molecules and solids containing heavy elements, accurate electronic
structure calculations require accounting not only for electronic correlations
but also for relativistic effects. In molecules, relativity can lead to severe
changes in the ground-state description. In solids, the interplay between both
correlation and relativity can change the stability of phases or it can lead to
an emergence of completely new phases. Traditionally, the simplest illustration
of relativistic effects can be done either by including pseudopotentials in
non-relativistic calculations or alternatively by employing large all electron
basis sets in relativistic methods. By analyzing different electronic
properties (band structure, equilibrium lattice constant and bulk modulus) in
semiconductors and insulators, we show that capturing the interplay of
relativity and electron correlation can be rather challenging in Green's
function methods. For molecular problems with heavy elements, we also observe
that similar problems persist. We trace these challenges to three major
problems: deficiencies in pseudopotential treatment as applied to Green's
function methods, the scarcity of accurate and compact all-electron basis-sets
that can be converged with respect to the basis-set size, and linear
dependencies arising in all-electron basis-sets particularly when employing
Gaussian orbitals. Our analysis provides detailed insight into these problems
and opens a discussion about potential approaches to mitigate them.
