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where the formation reaction enthalpies are Δ H [H$_{2}$O] ≈ − 2.506 eV and Δ H [B$_{2}$H$_{6}$] ≈ 0.377 eV, as taken from Ref. [27]. This avoids difficulties associated with modeling isolated atoms and the triplet state of molecular oxygen. 

Fig. 1 shows DFT calculated formation energies E$_{form}$ for undoped, boron doped, and nitrogen doped forms of 0D (TiO$_{2}$)$_{n}$ clusters ( n ≤ 9), 1D ( n , n ) nanotubes ( n ≤ 4 ), 2D HexABC and anatase layers, and 3D anatase surface, anatase bulk, and rutile bulk phases. 

From the formation energies, we see that (TiO$_{2}$)$_{n}$ clusters become generally more stable with increasing size, as ex- pected. On the other hand, the formation energies of the ( n , n ) TiO$_{2}$ nanotubes ( n ≤ 4) suggest that they are all surprisingly stable, being more stable than the 2D structures and within ap- proximately 0.2 eV of the least stable bulk phases. Even more surprising, the TiO$_{2}$ nanotubes seem to become more stable with decreasing size. This is clearly seen in Fig. 2, which shows the radial dependence of the formation energies for the TiO$_{2}$ (2,2) nanorod and TiO$_{2}$ ( n,n ) nanotubes ( 3 ≤ n ≤ 8 ). 

In a simple classical picture, the strain energy to be over- come when bending a layer into a tube should be inversely proportional to the square of the tube radius ( ∼ R − $^{2}$). Such models have been shown to accurately describe TiO$_{2}$ anatase and lepidocrocite layer nanotubes [28]. However, the Hex- ABC TiO$_{2}$ nanotube system is complicated by the fact that TiO$_{2}$ is not found naturally in a HexABC layered structure. As a result, the strain energy introduced by bending the layer may be overcome by changes in bonding which better resem- ble the bulk TiO$_{2}$ structures. This is seen in Fig. 1, where we find oxygen atoms have sp 3 character in both the infinite layer structure and the (4,4) TiO$_{2}$ nanotube, but sp 2 character in the more stable bulk rutile and (2,2) TiO$_{2}$ nanorod. 

As shown in Fig. 2, the radial dependence of the nanotube formation energies may be well approximated by 

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Here E$_{layer}$ ≈ 4.63 eV/TiO$_{2}$ is the formation energy of the nanotube layer calculated using the optimized unit cell pa- rameters for the nanotubes, E$_{rutile}$ ≈ 4.89 eV/TiO$_{2}$ is the formation energy of the bulk rutile phase, R$_{(}$$_{2,2}$$_{)}$ is the radius of a TiO$_{2}$ (2,2) nanorod, and R is the nanotube radius. The first term in (5) ensures the nanotube formation energy tends to that for the infinite layer at large nanotube radii. The sec- ond term models the increase in stability as the nanotube is ‘bent’ into a more rutile-like structure at smaller radii. This reaches a maximum for a (2,2) nanorod ( R = R$_{(}$$_{2,2}$$_{)}$ ), where the structure becomes that of the rutile phase. 

It should be noted that DFT formation energies have been shown to not accurately describe the relative stability of anatase and rutile phases, independent of xc-functional used [29]. This should be borne in mind when comparing forma- tion energies. 

Comparison of the doped formation energies with those of the undoped TiO$_{2}$ structures shows that although less stable, doped forms of both TiO$_{2}$ clusters and nanotubes should be experimentally realizable. This is further justified by recent 

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nitrogen doping experiments of large TiO$_{2}$ nanotubes which yielded doping fractions between 1% and 10% [12, 14, 15]. 

The formation energies of the doped systems also appear to correlate well with both those of the undoped structures, and the relative doping fraction. Further, structures are typ- ically less stable when doping induces oxygen dislocations, which occur for boron and nitrogen doped (TiO$_{2}$)$_{7}$ clusters, (TiO$_{2}$)$_{8}$ clusters, and bulk phases. On the other hand, systems are more stable when the TiO$_{2}$ structure is unchanged by the introduction of a dopant, as is the case for nitrogen doping of (TiO$_{2}$)$_{n}$ clusters with surface sites ( n = 5 , 6 , 9 ), nanotubes, layers, and surfaces. Such a distinction will also prove useful when we analyze the electronic structure of the doped systems in Sec. IV. 

We shall begin our analysis of the electronic structure of boron and nitrogen doped TiO$_{2}$ nanostructures by first con- sidering (TiO$_{2}$)$_{n}$ clusters (5 ≤ n ≤ 9). As well as being of interest in its own right, the study of such systems provides insight into how local changes in nanostructure may change whether a dopant behaves as a donor or acceptor site. Such in- formation will prove useful when trying to form p -type TiO$_{2}$ semiconducting materials. 

The DFT calculated DOS and structures for the most sta- ble boron and nitrogen doped (TiO$_{2}$)$_{n}$ clusters are shown in Figs. 3 and 4, for 5 ≤ n ≤ 9. In general, boron prefers to replace the most highly coordinated oxygen in TiO$_{2}$ clusters, 

## IV. ELECTRONIC STRUCTURE 

## A. DOS of 0D Clusters 

