# A Deeper Insight on the Stability of Water-Induced Reconstruction of Anatase (001) Surface
## 3 Results  
The formation energy of the unreconstructed surface \(\gamma\) is calculated according to the following equation:  
\[\gamma=\frac{E_{slab}-n_{TiO_{2}}E_{TiO_{2}}}{A}-\gamma_{f}=\frac{2E_{slab}- E_{slab}^{f}-n_{TiO_{2}}E_{TiO_{2}}}{2A} \tag{1}\]  
where \(\gamma_{f}\), \(E_{slab}\), \(E_{slab}^{f}\) are the surface formation energy of the unrelaxed slab and the total energy values respectively of the relaxed and the unrelaxed slab, \(n_{TiO_{2}}\) is the number of TiO\({}_{2}\) units in the slab, \(E_{TiO_{2}}\) is the energy of the TiO\({}_{2}\) unit calculated in the anatase bulk structure, and, finally, A is the surface area [27]. The unreconstructed surface energy is, of course, independent on the surface periodicity employed, i.e., on the \(n\) value (with \(n\) = 2, 3, 4, 6) of the (2\(xn\)) supercell. At the present level of the theory and using the above specified MP sampling scheme in the reciprocal space, we get \(\gamma\) = 1.105 J/m\({}^{2}\), which is quite a large value reflecting the reactivity of the ideal unreconstructed surface. Concerning the (2\(xn\)) water reconstructed surface, the surface energy can be calculated provided the average adsorption energy of dissociated waters is known: Indeed, if this is calculated as  
\[E_{(2 xn)}^{a}=\frac{E_{(2 xn)}^{wet}-E_{(2 xn)}^{dry}-n_{H_{2}O}E_{H_{2}O}}{n_ {H_{2}O}} \tag{2}\]  
where \(E_{(2 xn)}^{wet}\) and \(E_{(2 xn)}^{dry}\) are the total energy values of respectively the water reconstructed surface and the unreconstructed one, \(n_{H_{2}O}\) is the number of water molecules dissociatively adsorbed and \(E_{H_{2}O}\) is the energy of an isolated water molecule, the surface energy can be obtained through:  
\[\gamma_{(2 xn)}=\gamma+n_{H_{2}O}\frac{E_{(2 xn)}^{a}}{A} \tag{3}\]  
The resulting values of the average adsorption surface energy are reported in Table 1.  
The above values, which have been obtained in vacuum, can be further corrected by taking into account the dispersion energy contribution that should be considered in water. In bulk water, indeed, the adsorption process and energy, and the corresponding surface energy, must contain also the contribution from the dispersion energy change due to the water molecules that move from the bulk water to the surface where they are dissociatively adsorbed. This aspect has been detailed in a recent article [27]: According to this model, an estimate of such correction can be done with the following assumptions:  
* Provided that the total energy of the real system is obtained by summing the DFT and the dispersion energy terms, \(E=E^{DFT}+E^{d}\), the interaction between the water molecules in bulk water is accounted for by the dispersion energy only (\(E^{DFT}_{\mu H_{2}O}=nE^{DFT}_{H_{2}O}\));
* the dispersion energy change occurring in a real system when one water molecule is dissociatively adsorbed from the bulk water to the surface is obtained in a Born Oppenheimer Molecular Dynamics (BOMD) run as detailed recently [27]; this contribution is supposed to be due only to the change of the dispersion energy interaction between water molecules and between the TiO\({}_{2}\) slab and water molecules (the dispersion energy change due to the TiO\({}_{2}\) re-arrangement is neglected); BOMD simulations were performed on on the anatase unreconstructed and fully hydrated (001) surface with the mixed Gaussian and plane wave (GPW) method as implemented in the CP2K package [38].