  Imagine that there is a gapless plane tessellated by irregular, convex
pentagons with their side lengths at the sub-nanoscale, and tiny balls are
placed at the vertices of each pentagon. If there are no interactions among
these balls, one would expect that they stand still at the vertices of the
pentagons. But what if these balls symbolize atoms? The electrons and nuclei of
the atoms will start to interact following the laws of quantum mechanics. As a
result, the atoms will relocate leading to a system of energy minimum. While
performing an experiment of manipulating individual atoms and observing the
resulting configurations may be challenging, we apply ab initio density
functional theory (DFT) calculations to carry out a virtual experiment, where
we assign carbon atoms at the vertices of 14 types of convex pentagons and then
optimize the positions of these atoms. We obtain a variety of unexpected
structures; almost all of them except types 2 and 4 structures{these two
structures are equivalent{differ from the initial pentagonal geometries. We
compute the band structure of each resulting geometry representing the crystal
structure of a two-dimensional (2D) material. We find diverse electrical
properties among these 2D materials: metallic, semiconducting, and
semi-metallic with a Dirac cone. Our work shows that encoding quantum mechanics
into the geometries of irregular, convex pentagons and with the help of DFT
calculations open up a novel route for accelerating discovery of new 2D
materials.
