## 2 PDFT Simulations of Porphyrins in Nanostructures
### Porphyrin Nanostructures Using Crystal Geometries  
There are many known single crystal structures of porphyrins and metalloporphyrins in the literature but PDFT simulations on lattice structures of porphyrins are seldom found. Single crystal structures from x-ray crystallography of porphyrin nanostructures are even rare. [PERSON] et al. [42] determined the first single crystal structure of ionic porphyrin nanorods involving meso-tetra(N-methyl-4-pyridyl)porphyrin (TMPPyP) and meso-tetra(4-sulfonatophenyl)porphyrin (TSPP). Using various spectroscopic, diffraction, and imaging techniques, they reported the structure-function relationship of TMPPyP:TSPP nanorods. The crystal structure was correlated to the morphology and photoconductive behavior of the nanorods, and PDFT calculations on the crystal structure showed that the \(\pi\)-\(\pi\) stacking of TMPPyP:TSPP is responsible for their conductivity. Figure 2 shows the crystal structure and the corresponding band structure of TMPPyP:TSPP nanorods. The band structure was determined using optB88-van der Waals (vdW) GGA [43] functional with projector augmented wave (PAW) [35; 44] Perdew-Burke-Ernzerhof (PBE) [45] pseudopotentials. The authors note that the band structure obtained with PDFT underestimates the band gap (0.90 eV) and extended Huckel tight binding (EHTB) was used to determine the band gap (1.3 eV) which matches closer to experiment. Hybrid DFT functionals such as Heyd-Scuseria-Ernzerhof (HSE) [46; 47] have shown improved band gap prediction over GGA and LDA functionals. Although HSE can improve band gap prediction, geometric optimizations showed little change in contrast to GGA and LDA functionals. On the other hand, vdW-DF [43; 48] functionals were more reliable for geometric optimizations in systems with considerable dispersion interactions. In the case of TMPyP:TSPP, DFT calculations have been useful to predict the appropriate partial density of states (pDOS) that showed that the top of the valence band is populated by the contributions from TSPP and the bottom of the conduction band is populated by the TMPyP with no orbital hybridization in the vicinity of the bandgap. This prediction was consistent with the experimental data from UV-visible, diffuse reflectance and photoconductivity action spectra. This shows that PDFT calculations played a critical role in elucidating the photoconductive mechanism in TMPyP:TSPP nanorods. In the same study the authors reported that decreasing the porphyrin stacking distance would not necessarily change the band gap but would increase the dispersion in the band structure which would improve charge mobility in the nanorods. The work on TMPyP:TSPP nanorods has been expanded by [PERSON] et al. [49], by selective metalation of TMPyP and TSPP porphyrin cores with Ni and Cu transition metals. A single crystal structure of H\({}_{2}\)TMPyP:NiTSPP nanorods was determined and it was shown that metalation of ionic porphyrins led to exhibition of dark conductivity at moderately high temperatures and that conductivity increases upon photoexcitation. Additionally, the photoresponse of the H\({}_{2}\)TMPyP:CuTSPP substituted crystals is significantly higher than that of the CuTMPyP:H\({}_{2}\)TSPP and the Ni-substituted crystals. To understand the reasons behind this discrepancy in the conductive behavior with different metalation, PDFT calculations were performed on H\({}_{2}\)TMPyP:(M)TSPP and (M)TMPyP:H\({}_{2}\)TSPP systems, where M = Ni, Cu. The crystal structure of H\({}_{2}\)TMPyP:NiTSPP was used to create the lattice structure of other metalated binary ionic porphyrins by changing the core substitution of the H\({}_{2}\)TMPyP:NiTSPP crystals and then optimizing the structure. The band structure of each H\({}_{2}\)TMPyP:(M)TSPP and (M)TMPyP:H\({}_{2}\)TSPP systems are shown in Figure 3. It was reported that adding a metal to the freebase porphyrins reduces the band gap of the corresponding nanorods. Additionally, it was shown that Ni and Cu metalation causes distinct changes in the frontier bands of porphyrin nanostructures.