7 the anatase surface, shown in Fig. 8(b) and Fig. 9. This sug- gests boron and nitrogen doped 2D TiO$_{2}$ layers or surfaces may potentially be used for TiO$_{2}$ based electronics. On the other hand, for bulk anatase boron and nitrogen dopants both yield n -type semiconductors, as is seen exper- imentally [13]. We attribute this difference to the nitrogen- oxygen bond shown in Fig. 8(c), which may arise due to oxy- gen’s greater structural flexibility in the bulk. As a result, boron and nitrogen doping both yield mid-gap occupied Ti 2 d$_{xy}$ conduction band states with little weight on the doping sites. Here, the dopants donate their valence electrons fully to the conduction band, as seen in Fig. 8(c). Nitrogen thus acts as a donor for bulk-like clusters and anatase, in agreement with recent findings for nitrogen doping of both bulk anatase and thick bulk-like anatase nanopores [13]. This suggests that n -type or p -type semi- conductors may be produced by nitrogen doping depending on whether nearby oxygen atoms occupy lower-coordinated surface or higher-coordinated bulk sites. Further, we also find that the highest occupied states for boron and nitrogen doped systems become increasingly local- ized as the structure is dimensionally constrained from bulk → surface → layer → nanotube → nanorod. This may have im- portant consequences for the photocatalytic activity of these more localized states in lower dimensional structures. On the other hand, this may be partially alleviated by having these states located on the structure’s surface. In this way, electron- hole pairs need not travel significantly, as is the case for bulk materials. To analyze the qualitative trends in photocatalytic activity of boron and nitrogen doped TiO$_{2}$ nanostructures, we shall now compare the energy gaps obtained using different approx- imations within DFT, GW calculations, and experiment. As previous studies have shown, standard DFT tends to underestimate band gaps for bulk TiO$_{2}$ by approximately 1 eV, due in part to self-interaction errors [31, 32]. This may The DFT calculated energy gaps ε$_{gap}$ , for (TiO$_{2}$)$_{n}$ clusters ( n ≤ 9) and band gaps for TiO$_{2}$ (2,2) nanorods, (3,3) nan- otubes, and (4,4) nanotubes, in the undoped, boron doped, and nitrogen doped forms, are shown in Fig. 10. For the clusters, the gap is estimated by both the difference in energy between the HOMO and LUMO, and the difference between the ion- ization potential and electron affinity energies I$_{p}$ − E$_{a}$ . The band gap for TiO$_{2}$ nanorods, nanotubes, layers, surfaces and bulk phases is approximated by the indirect band gap between highest occupied and lowest unoccupied states. Although not as relevant for photoabsorption as the direct gap, the large size required for the doped super cells, shown in Fig. 6 and Fig. 9, leaves indirect and direct gaps indistinguishable. For the undoped nanorods and nanotubes and bulk rutile we find the band gap is direct, while for the undoped HexABC layer and anatase structures we find the band gap is indirect. For the same structures we also provide B3LYP band gaps, GW calculations, and experimental results for comparison. be partially addressed by the use of hybrid functionals such as B3LYP, which generally seem to improve band gaps for bulk systems [33–35]. However, B3LYP calculations for TiO$_{2}$ clusters largely underestimate the gap relative to the more re- liable I$_{p}$ − E$_{a}$ , while for the bulks the gap is overestimated by approximately 0.4 eV relative to experiment. Also, B3LYP and RPBE calculations provide the same qualitative descrip- tion of the trends in the energy gaps for TiO$_{2}$. GW is probably the most successful and generally appli- cable method to calculate quasi-particle gaps. For clusters it agrees well with I$_{p}$ − E$_{a}$ , but for bulk systems GW seems to D. Energy Gaps FIG. 10: Energy gap ε$_{gap}$ in eV versus TiO$_{2}$ structure for 0D (TiO$_{2}$)$_{n}$ clusters ( n ≤ 9), 1D TiO$_{2}$ (2,2) nanorods, (3,3) nanotubes, (4,4) nanotubes, 2D HexABC and anatase layers, and 3D anatase surface, anatase bulk, and rutile bulk phases. DFT calculations using RPBE of the highest occupied and lowest unoccupied state gaps / I$_{p}$ − E$_{a}$ for the (a) undoped ( # / l , red), (b) boron doped ( △ / ▲ , magenta), and $^{(c) nitrogen doped (}$GLYPH<31>/ ■ , blue) systems are compared with undoped B3LYP ( ▼ ), GW ( ▶ ), and experimental ( u ) results [6, 12, 18, 30]. Small open symbols denote transitions between highest fully occu- pied states and the conduction band. 0 2 4 6 8 10 RPBE I$_{p}$ - E$_{a}$ GW Exp B3LYP 0 2 4 Energy Gap  $_{ε}$gap  [eV] RPBE I$_{p}$ - E$_{a}$ GW (TiO 2 ) 2  Cluster (TiO 2 ) 3  Cluster (TiO 2 ) 4  Cluster (TiO 2 ) 5  Cluster (TiO 2 ) 6  Cluster (TiO 2 ) 7  Cluster (TiO 2 ) 8  Cluster (TiO 2 ) 9  Cluster (2,2) Nanorod (3,3) Nanotube (4,4) Nanotube HexABC Layer Anatase Layer Anatase Surface Anatase Bulk Rutile Bulk 0 2 4 RPBE I$_{p}$ - E$_{a}$ GW Exp (a) Undoped (b) B-doped (c) N-doped 0D 1D 2D 3D