5 0 1 2 3 0 1 2 Total Density of States [eV -1 /TiO 2 ] -1 0 1 2 3 $_{VB}$ [eV] 0 1 2 (a) (2,2) Nanorod (b) (3,3) Nanotube (c) (4,4) Nanotube B−doped N−doped (3,3) Nanotube (4,4) Nanotube (2,2) Nanorod FIG. 5: RPBE calculation of the total density of states in eV -1 per TiO$_{2}$ functional unit versus energy in eV for undoped (thin black solid line), boron doped (magenta dashed line) and nitrogen doped (blue thick solid line) TiO$_{2}$ (a) (2,2) nanorods, (b) (3,3) nanotubes and (c) (4,4) nanotubes, with 3.1%, 2.1%, and 1.6% doping respec- tively. Energies are measured from the top of the valence bands of anatase TiO$_{2}$ ε$_{VB}$ , and the DOSs are shifted to align the lowest eigen- state with that for anatase TiO$_{2}$. Occupancy is denoted by curve fill- ing for states in the band gap. FIG. 6: Schematics of boron doped (left) and nitrogen doped (right) TiO$_{2}$ (2,2) nanorods, (3,3) nanotubes, and (4,4) nanotubes, with 3.1%, 2.1%, and 1.6% doping respectively. The highest occupied states for boron and nitrogen doped TiO$_{2}$ 1D structures are depicted by isosurfaces of ± 0.05 e/ ˚ A$^{3}$. B. DOS of 1D Nanorods and Nanotubes FIG. 7: GW calculation of the total density of states in eV -1 per TiO$_{2}$ functional unit versus energy in eV for undoped (thin black solid line), boron doped (magenta dashed line) and nitrogen doped (blue thick solid line) TiO$_{2}$ (2,2) nanorods with 3.1% doping. Energies are measured from the top of the valence bands of the undoped TiO$_{2}$ nanorod ε$_{VB}$ , and the DOSs are shifted to align the lowest eigenstate with that for the undoped TiO$_{2}$ nanorod. Occupancy is denoted by curve filling for states in the band gap. 0 2 4 6 8 $_{VB}$ [eV] 0 1 2 3 Total Density of States [eV -1 / TiO 2 ] (2,2) Nanorod We shall now discuss how boron and nitrogen doping in- fluences the electronic structure of stable 1D nanorods and nanotubes. Although the the most stable doping sites are the same as those obtained for doped TiO$_{2}$ rutile surface and bulk [5, 16, 17], the influence of dopants on the DOS is rather dif- ferent. Figure 5 shows the DFT calculated DOS and Fig. 6 the structures of the most stable boron doped and nitrogen doped TiO$_{2}$ (2,2) nanorods, (3,3) nanotubes, and (4,4) nanotubes. The highest occupied state is also shown as isosurfaces of ± 0.05 e/ ˚ A 3 in the side views of the doped structures. As with TiO$_{2}$ clusters, the influence of boron dopants on TiO$_{2}$ nanorods and nanotubes may be understood in terms of boron’s weak electronegativity, especially when compared with the strongly electronegative oxygen. We find that boron prefers to occupy oxygen sites which are 2-fold coordinated to neighboring titanium atoms. However, as with the 0D clus- ters, boron’s relatively electropositive character induces sig- nificant structural changes in the 1D structures, creating a stronger third bond to a neighboring three-fold coordinated oxygen via an oxygen dislocation, as shown in Fig. 5. This yields three occupied mid-gap states localized on the boron dopant, which overlap both the valence band O 2 p$_{π}$ and con- duction band Ti 2 d$_{xy}$ states, as shown in Fig. 5. Boron dopants