Schwarz P-surface via isolated sp² carbon heptagons: Design and properties

Described are the first two molecular designs of the triply periodic Schwarz P surface using merely the Schläfli t{3,7} pattern of sp² -hybridized carbon atoms. Each atom is exactly part of one heptagon and two hexagon rings so that two heptagons do not share the same edge or vertex. Such pattern, called hyperbolic soccer ball obeys the isolated-heptagon rule with the minimum possible number of hexagons between heptagons similar to the isolation of pentagons from hexagons in the C60 fullerene. Both of the designed P surfaces are unbalanced, that is, they have two unequal sides, and belong to space groups P432 of the cubic system, and P4/ncc of the tetragonal system, respectively. Unit cells have a multiple of 24 heptagons similar to the only one previously known in literature Schwarzite with the hyperbolic soccer ball pattern-the D surface of Vanderbilt and Tersoff. The geometry of the periodic structures and unit cell parameters were fully optimized by DFT calculations using CASTEP software with PBE and PBESOL functionals under generalized gradient approximation. The effect of P and D surface dilution by hexagons on the calculated density, elastic and electronic properties is discussed.