# Ozone Activation on TiO\({}_{2}\) Studied by IR Spectroscopy and Quantum Chemistry
## 3 Results
### Results of Calculations
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & **Isotopomer** & **v\({}_{1}\), cm\({}^{-1}\)** & **v\({}_{3}\), cm\({}^{-1}\)** & **v\({}_{1+3}\), cm\({}^{-1}\)** & **v\({}_{1}\) + v\({}_{3}\), cm\({}^{-1}\)** & **v\({}_{1}\) + v\({}_{3}\) – v\({}_{1+3}\), cm\({}^{-1}\)** & **v\({}_{1}\) – v\({}_{3}\), cm\({}^{-1}\)** \\ \hline \multirow{11}{*}{**V\({}_{3}\)**} & & 16 16 & 1298.9 & 1068.7 & 2377.1 & 2367.6 & \(-9.5\) & 230.2 \\  & & 16 16 18 & 1263.4 & 1066.2 & 2335.5 & 2329.6 & \(-6.0\) & 197.2 \\  & & 16 18 16 & 1262.9 & 1034.3 & 2305.7 & 2297.2 & \(-8.5\) & 228.6 \\  & & 18 16 16 & 1295.2 & 1042.5 & 2346.5 & 2337.7 & \(-8.8\) & 252.7 \\  & & 16 18 18 & 1226.9 & 1032.4 & 2264.8 & 2259.2 & \(-5.5\) & 194.5 \\  & & 18 16 18 & 1263.5 & 1041.6 & 2314.6 & 2305.2 & \(-9.4\) & 221.9 \\  & & 18 18 16 & 1259.3 & 1008.3 & 2275.9 & 2267.6 & \(-8.3\) & 251.1 \\  & & 18 18 18 & 1226.3 & 1007.0 & 2242.3 & 2233.3 & \(-9.0\) & 219.3 \\ \hline \multirow{11}{*}{**V\({}_{3}\)**} & & 16 16 & 1255.0 & 1086.8 & 2346.3 & 2341.8 & \(-4.6\) & 168.2 \\  & & 16 16 18 & 1222.6 & 1084.8 & 2309.9 & 2307.4 & \(-2.5\) & 137.9 \\  & & 16 18 16 & 1221.0 & 1052.4 & 2277.1 & 2273.4 & \(-3.7\) & 168.6 \\  & & 18 16 16 & 1252.6 & 1063.3 & 2321.4 & 2315.9 & \(-5.5\) & 189.4 \\  & & 16 18 18 & 1187.5 & 1050.1 & 2239.0 & 2237.5 & \(-1.5\) & 137.4 \\  & & 18 16 18 & 1220.5 & 1061.7 & 2287.1 & 2282.2 & \(-4.9\) & 158.7 \\  & & 18 18 16 & 1219.3 & 1027.7 & 2252.8 & 2246.9 & \(-5.9\) & 191.6 \\  & & 18 18 18 & 1184.8 & 1026.1 & 2215.2 & 2210.9 & \(-4.3\) & 158.7 \\ \hline \multirow{11}{*}{**V\({}_{3}\)**} & & 16 16 & 1310.3 & 1051.5 & 2371.6 & 2361.8 & \(-9.8\) & 258.9 \\  & & 16 16 18 & 1275.7 & 1050.0 & 2333.6 & 2325.7 & \(-7.9\) & 225.8 \\  & & 16 18 16 & 1273.9 & 1018.7 & 2301.5 & 2292.5 & \(-9.0\) & 255.2 \\  & & 18 16 16 & 1308.7 & 1028.4 & 2347.9 & 2337.1 & \(-10.9\) & 280.2 \\  & & 16 18 18 & 1237.8 & 1017.2 & 2261.8 & 2255.0 & \(-6.8\) & 220.6 \\  & & 18 16 18 & 1275.1 & 1026.5 & 2311.6 & 2301.6 & \(-10.0\) & 248.5 \\  & & 18 16 1273.0 & 994.5 & 2278.3 & 2267.5 & \(-10.7\) & 278.5 \\  & & 18 18 18 & 1237.2 & 994.2 & 2240.7 & 2231.4 & \(-9.3\) & 243.0 \\ \hline \multirow{11}{*}{**V\({}_{3}\)**} & & 16 16 & 1263.8 & 1070.6 & 2339.6 & 2334.4 & \(-5.2\) & 193.3 \\  & & 16 16 18 & 1230.4 & 1067.5 & 2301.5 & 2297.9 & \(-3.6\) & 162.9 \\ \cline{1-1}  & & 16 18 16 & 1228.0 & 1035.7 & 2267.2 & 2263.7 & \(-3.5\) & 192.4 \\ \cline{1-1}  & & 18 16 16 & 1262.2 & 1047.2 & 2316.8 & 2309.3 & \(-7.5\) & 215.0 \\ \cline{1-1}  & & 16 18 18 & 1194.9 & 1033.0 & 2230.5 & 2227.9 & \(-2.6\) & 161.9 \\ \cline{1-1}  & & 18 16 18 & 1228.0 & 1045.8 & 2278.9 & 2273.7 & \(-5.1\) & 182.2 \\ \cline{1-1}  & & 18 18 16 & 1228.1 & 1012.4 & 2247.2 & 2240.5 & \(-6.8\) & 215.8 \\ \cline{1-1}  & & 18 18 18 & 1190.6 & 1010.6 & 2205.3 & 2201.2 & \(-4.1\) & 180.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Calculated frequencies of ozone isotopomers forming complexes with the relaxed anatase \(\text{Ti}_{20}\text{O}_{40}\). Figure 7: Positions of the bands of \(\text{v}_{1}\) (**a**) and \(\text{v}_{3}\) (**b**) vibrations of ozone isotopomers forming complexes with the \(\text{Ti}_{20}\text{O}_{40}\) cluster. Monodentate with 4- (**1**) and 5-coordinated (**2**) \(\text{Ti}^{4+}\) ions, and bidentate complexes with the same sites of the latter cluster (**3**,**4**). Besides the frequency values of fundamental vibrations \(\
u_{1}\) and \(\
u_{3}\), as well as the combination mode \(\
u_{1+3}\), in Table 3 we present the difference between the sum \(\
u_{1}+\
u_{3}\) and \(\
u_{1+3}\) as a measure of anharmonicity. For gaseous ozone, this value was \(1042.1+1103.1-2110.8=34.4\). Our calculation for a free molecule provided a value of 21 cm\({}^{-1}\), but as seen from the table, for all the isotopomers of ozone complexes this value is always negative and varies between \(-1\) and \(-11\), thus diminished by 22-32 cm\({}^{-1}\) with respect to free molecules.