Computational catalysis in nanotubes - Catalysis - page 127

Fig. 11 (a) The formation energies of pristine and various reconstructed rutile (110) single
trilayer thin sheets. (b) Optimized geometries of rutile TiO 2 (110) sheet with different b
value and constant a (a = 2.97 Å) (1) b = 6.56 Å, (2) b = 5.46 Å, (3) b = 5.36 Å, (4) b = 5.16 Å, (5)
b = 5.06Å. 73

so much. Indeed, the coordination environments of oxygen and titanium
are the same in two kinds of structures (1 and 2). The only difference is
the distance between O 2f and Ti 4f changing from 3.50 to 3.02 Å and the
thickness from 2.45 to 2.56 Å. However, a stable structure 3 is obtained
when the length of b is 5.36 Å. The formation energy is 0.34 eV per TiO 2 ,
which is much smaller than that of the pristine sheet. It should be noted
that all oxygen and titanium atoms in 3 are O 3f and Ti 6f , which are the
same as those of bulk TiO 2 . In particular, we found that this structure is
very similar to that of the single layer of b-PtO 2 . 77 The structure with
saturated coordination oxygen and titanium is very stable within the
range of b value from 5.36 to 5.16 Å. Once the saturated coordination
environments of oxygen or titanium are changed, the structure of the
thin rutile sheet becomes unstable again. For example, the formation
energy of structure 5 is 1.63 eV per TiO 2 , which is much higher than that
of structure 3 and 4. Due to the very short b value (5.06 Å), half of the four
oxygens and all of the titanium oxygens are O 1f and Ti 5f . The stable single
trilayer sheets (structure 3 and 4) can be obtained by choosing the suit-
able length of b, which is highlighted in Fig. 21. We found the structure is
same as that of a single layer of b-PtO 2 . 85 The hexagonal ring of the most
stable sheet is very similar to that of graphene although they have three
or one layers, respectively. Therefore, we can define various TiO 2 NTs
using the same rules of the carbon NTs by rolling the graphene sheet. 86
Two primitive lattice vectors R 1 , R 2 are defined in Fig. 12. A pair of in-
tegers (n 1 ,n 2 ) can define a vector R:

R = n 1 R 1 þ n 2 R 2 (8)