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the systems, showing the DOS and energy gaps, followed by a concluding section. 

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 

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## II. METHODOLOGY 

## III. ENERGETIC STABILITY 

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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 

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