  First principles approaches have been successful in solving many-body
Hamiltonians for real materials to an extent when correlations are weak or
moderate. As the electronic correlations become stronger often embedding
methods based on first principles approaches are used to better treat the
correlations by solving a suitably chosen many-body Hamiltonian with a higher
level theory. Such combined methods are often referred to as second principles
approaches. At such level of the theory the self energy, i.e. the functional
that embodies the stronger electronic correlations, is either a function of
energy or momentum or both. The success of such theories is commonly measured
by the quality of the self energy functional. However, self-consistency in the
self-energy should, in principle, also change the real space charge
distribution in a correlated material and be able to modify the electronic
eigenfunctions, which is often undermined in second principles approaches. Here
we study the impact of charge self-consistency within two example cases:
TiSe$_{2}$, a three-dimensional charge-density-wave candidate material, and
CrBr$_{3}$, a two-dimensional ferromagnet, and show how real space charge
re-distribution due to correlation effects taken into account within a first
principles Green's function based many-body perturbative approach is key in
driving qualitative changes to the final electronic structure of these
materials.
