## 2 PDFT Simulations of Porphyrins in Nanostructures
### Porphyrins as Nano Wires, Sheets, Tubes, and Ladders  
As mentioned earlier, porphyrins can form many structural shapes due to their mechanical flexibility and rich chemistry. Hence, PDFT simulations were used to study the unique electronic structures and their applications that are possible because of the multidimensionality of porphyrins. Figure 5 shows typical shapes--various porphyrin nanowires, nanotubes, and nanosheets. [PERSON] et al. [61] studied the band structures of porphyrin nano sheets and tubes formed through covalent linkers. They used screened hybrid density functional theory simulations and Wannier function interpolation to obtain accurate band structures. The structural optimizations were performed with PBE-D2 [45; 53] functional and single point calculations for band structure were obtained using the HSE06 [46; 47] functional. It was reported that the electronic properties exhibit strong variations with the number of linking carbon atoms (C0 = no carbon atoms, C2 = two carbon atoms, C4 = four carbon atoms). For example, all C0 nanostructures exhibit gapless or metallic band structures, whereas band gaps open for the C2 or C4 structures. PDFT simulations showed that it is possible to design porphyrin nanostructures with tailored electronic properties such as specific band gap values and band structures by varying the type of the linkage used between each porphyrin units and the type of self-assembled formations (linear chains, nanosheets, nanotubes, and nanorings). A previous study on porphyrin nanotubes formed with acetyl linkers was done by [PERSON] et al. [62] who reported large oscillations in bandgaps of porphyrin nanotubes with increase in their size. The simulations were performed with a variety of periodic DFT functionals which show similar oscillation trend in the band gaps irrespective of the functional. Additionally, the authors report that the bandgap is a direct-bandgap which can be observed with photoelectron spectroscopic experiments. Various porphyrin nanosheet structures with variable porphyrin core metals and substituents were also studied extensively with PDFT calculations. Using the GGA-PBE functional [45], [PERSON] et al. [63] predicted the stable formation of a 2D ferromagnetic half-metal based on vanadium polymorphyrin (PP). The stability of the 2D metal was determined by comparing the Curie temperature (Tc) and phonon dispersion to other known 2D structures like manganese phthalocyanine (MnPc) and CrPP. The authors also note that the predicted 2D VPP is quite suitable for use in flexible spintronic devices. A similar study with Fe, Co, Li, Zn, and H\({}_{2}\)PPs was done by [PERSON] et al. [64] who reported that H\({}_{2}\), Li, and Zn PPs behave as direct bandgap semiconductors while Fe, and Co  
Figure 5: Typical structures of porphyrin (**a**) nanowire, (**b**) nanosheets, and (**c**) nanotubes. Reprinted with permission from reference [61] Copyright 2018 American Chemical Society. They also report that 2D PP systems behave like n-type semiconducting materials with strong electron mobilities, which were obtained using PDFT combined with Boltzmann transport method with relaxation time approximation. An interesting PDFT study of porphyrin nanosheets and nanowires fused at meso-meso-, \(\beta\)-\(\beta\)-, and \(\beta\)-\(\beta\)-linked array (referred to as SA) and a directly \(\beta\)-fused array (referred to as SB), was performed by [PERSON] [65]. In this study it was reported that the bandgaps of nanosheets is slightly lower than the nanowires and more importantly it was found that SA type linked arrays have significantly lower bandgaps than SB type. Another example of fused polyporphyrins (PP) with pyrazine linkage was performed by [PERSON] et al. [66] with transition metals Cr, Mn, Fe, Co, Ni, Cu, and Zn at the porphyrin cores of the nanosheet. Using PDFT calculations with GGA-PBE [45] and GGA+U [67; 68] approaches, it was shown that metal PPs have excellent thermal stability with the MnPP system having a ferromagnetic character and half-metallic behavior.