  We explore the electronic band structure of free standing monolayers of
chromium trihalides, CrX\textsubscript{3}{, X= Cl, Br, I}, within an advanced
\emph{ab-initio} theoretical approach based in the use of Green's function
functionals. We compare the local density approximation with the quasi-particle
self-consistent \emph{GW} approximation (QS\emph{GW}) and its self-consistent
extension (QS$G\widehat{W}$) by solving the particle-hole ladder Bethe-Salpeter
equations to improve the effective interaction \emph{W}. We show that at all
levels of theory, the valence band consistently changes shape in the sequence
Cl{\textrightarrow}Br{\textrightarrow}I, and the valence band maximum shifts
from the M point to the $\Gamma$ point. However, the details of the transition,
the one-particle bandgap, and the eigenfunctions change considerably going up
the ladder to higher levels of theory. The eigenfunctions become more
directional, and at the M point there is a strong anisotropy in the effective
mass. Also the dynamic and momentum dependent self energy shows that
QS$G\widehat{W}$ adds to the localization of the systems in comparison to the
QS\emph{GW} thereby leading to a narrower band and reduced amount of halogens
in the valence band manifold.
