A rectangular graphyne sheet is composed of units similar to phenyl rings that
are linked by acetylenic chains, as in hexagonal γ-graphyne. This system is organized
over a rectangular lattice similar to that of the recently synthesized biphenylene network.
We investigate the stability of this sheet from different perspectives and study its
electronic structure. Rectangular graphyne is a semiconducting system in its pristine
form and features a pair of highly localized states. These characteristics are correlated
with the structural anisotropy of the system, since its frontier states behave like quasi-1D
states embedded in the 2D lattice. We further consider modified systems in which longer
acetylenic links are introduced. We discuss how a strategic choice of the position of
these longer bridges can lead to specific changes of the electronic structure of the
rectangular graphyne sheet.