2 the systems, showing the DOS and energy gaps, followed by a concluding section. II. METHODOLOGY All DFT calculations have employed the RPBE exchange correlation (xc)-functional [20]. The plane-wave code DA- CAPO[21, 22] was used for structural minimization, the real-space code OCTOPUS[23] for charged calculations, and YAMBO[24] with ABINIT[25] or PWSCF[26] for GW calcu- lations. A plane-wave cutoff of 340 eV was used, with a Monkhorst-Pack k -point sampling of 1 × 1 × 12 for TiO$_{2}$ nan- otubes, where the nanotube axis is parallel to the z -axis, 12 × 12 × 1 for TiO$_{2}$ layers and surfaces, where the normal direction is parallel to the z -axis, and 12 × 12 × 12 for TiO$_{2}$ bulk phases. All structures have been relaxed until a maxi- mum force below 0.04 eV/ ˚ A was obtained. The occupation of the one electron states was calculated at a temperature of k$_{B}$T ≈ 0.1 eV for the periodic systems and k$_{B}$T ≈ 0.01 eV for the clusters, with all energies extrapolated to T = 0 K. Spin unpolarized calculations have been performed for the un- doped TiO$_{2}$ systems, while spin polarized calculations have been performed for all doped TiO$_{2}$ systems, since the unit cells for the doped systems contain an odd number of elec- trons. Doping of (TiO$_{2}$)$_{n}$ clusters has been modeled by substitut- ing a single boron or nitrogen atom in each geometrically in- equivalent oxygen site of the most stable isomer to determine the most stable doping site. Only clusters of sufficient size to obtain experimentally realizable doping fractions ≲ 10 % ( 5 ≤ n ≤ 9 ) have been considered [11–15]. To model TiO$_{2}$ nanotube doping, we have repeated the min- imal unit cell four times along the tube axis, and substituted a single boron or nitrogen atom in each geometrically inequiv- alent oxygen site to obtain the most stable doped structure. In this way, dopant-dopant interactions are minimized by main- taining a dopant separation of approximately 12 ˚ A. This corre- sponds to experimentally realizable doping fractions of 3.1%, 2.1%, and 1.6% for TiO$_{2}$ (2,2) nanorods, (3,3) nanotubes and (4,4) nanotubes, respectively. Doping of TiO$_{2}$ layers, surfaces, and bulks has been simi- larly modeled by repeating the minimal unit cell twice in each periodic direction, and substituting a single boron or nitrogen atom in each geometrically inequivalent oxygen site to obtain the most stable structure. Experimentally realizable doping fractions of 5.6%, 3.1%, and 3.1% were thus obtained for the layers, surfaces, and bulks respectively. We define the formation energy E$_{form}$ for a given structure consisting of n TiO$_{2}$ functional units as III. ENERGETIC STABILITY E$_{form}$ = 1 n E [ Ti$_{n}$O$_{2}$$_{n}$$_{−}$$_{p}$$_{−}$$_{q}$B$_{p}$N$_{q}$ ] − E [ TiO$_{2}$ ] − 1 n ( pE [ B ] + qE [ N ] − ( p + q ) E [ O ]) , (1) E [ O ] = E [ H$_{2}$O ] − E [ H$_{2}$ ] − Δ H [ H$_{2}$O ] , (2) E [ B ] = 1 2 ( E [ B$_{2}$H$_{6}$ ] − 3 E [ H$_{2}$ ] − Δ H [ B$_{2}$H$_{3}$ ]) , (3) where E [ Ti$_{n}$O$_{2}$$_{n}$$_{−}$$_{p}$$_{−}$$_{q}$B$_{p}$N$_{q}$ ] is the total energy for the system, E [ TiO$_{2}$ ] is the energy of an isolated TiO$_{2}$ molecule, and E [ O ] , E [ B ] , and E [ N ] are the respective energies with reference to gas phase species for O, B, and N. These have been obtained using the experimental doping reactions FIG. 1: Formation energy E$_{form}$ in eV per TiO$_{2}$ functional unit versus TiO$_{2}$ structure for 0D (TiO$_{2}$)$_{n}$ clusters ( n ≤ 9), 1D TiO$_{2}$ (2,2) nanorods, (3,3) nanotubes, and (4,4) nanotubes, 2D HexABC and anatase layers, and 3D anatase surface, anatase bulk, and rutile bulk phases. DFT calculations using RPBE for the undoped ( # , red), $^{nitrogen doped (}$GLYPH<31>, blue), and boron doped ( △ , magenta) systems are shown. Note the sp 2 character of oxygen in the more stable rutile and (2,2) nanorod structure, and the sp 3 character of oxygen in the less stable layer and larger nanotube structures. (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 -5 -4 -3 -2 Formation Energy  E form  [eV/ TiO 2 ] Undoped B-doped N-doped 0D 1D 2D 3D  3  2 sp  character sp  character (2,2) Nanorod Rutile Bulk (3,3) Nanotube (4,4) Nanotube HexABC Layer