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typically in a central location, and forms boron-oxygen bonds via oxygen dislocations, as shown in Fig. 4. Due to its elec- tropositive character, boron acts as a donor, introducing three occupied mid-gap states near the lowest unoccupied molecu- lar orbital (LUMO), as seen in Fig. 3. 

However, for (TiO$_{2}$)$_{n}$ clusters the influence of nitrogen dopants is not as straightforward. As with rutile TiO$_{2}$ surfaces[16, 17], nitrogen prefers to occupy sites which are 3-fold coordinated to titanium. For clusters consisting only of surface sites ( n = 5, 6, 9), oxygen is tightly constrained, and no significant changes in the cluster’s structure occur via oxygen dislocations, as shown in Fig. 4. As with rutile TiO$_{2}$ surfaces[16, 17], nitrogen acts as an acceptor, introducing an unoccupied mid-gap state near the highest occupied molecular orbital (HOMO). 

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For clusters where the nitrogen dopant’s oxygen neighbors occupy interior/bulk sites ( n = 7, 8), oxygen is more mo- bile, and may form new nitrogen-oxygen bonds, as shown in Fig. 4. Since oxygen is more electronegative than nitrogen, nitrogen may transfer charge to the bonding oxygen through these bonds. In this case, nitrogen instead acts as a donor, introducing occupied states near the LUMO. 

In general, we find when dopants form bonds with oxy- gen atoms via oxygen dislocations, they act as donor sites. On the other hand, when dopants such as nitrogen leave the TiO$_{2}$ structure unchanged, they act as acceptor sites, yield- ing a somewhat more stable structure. We shall find such a description useful in understanding why nitrogen dopants be- have differently in low dimensional systems and bulk systems. 

