These systems will be studied in the next chapter.
approach is nonetheless accomplished at a considerably smaller computational effort. The total elapsed time required by our formalism for the computation of the low-lying transition energies of Thiel’s compounds was roughly 60 times shorter than that needed using TD-PBE. For large systems this factor is increased to circa 250. We finally discussed the implementation of or-bital constraints within TD-DFTB to further reduce the computational time. The calculation of low-lying excitation energies with marked collective character using orbital constraints was roughly three times faster than for the unrestricted scheme, at no additional loss of accuracy.
For systems displaying single-particle excitations, restricted calculations can be further sped up by a factor of 7.
Chapter 5
PHOTOCATALYTIC ACTIVITY OF TITANIUM DIOXIDE
In this chapter we investigate the photocatalytic degradation of nitric oxide and acetaldehyde on TiO2 under UV and visible illumination. We consider two titania surfaces, namely, rutile (110) and anatase (001). The rutile (110) surface consists of alternating rows of fivefold coordinated (5c) Ti and twofold coordinated (2c) O atoms (see Fig. 5.1). This surface is not only the most investigated single-crystal TiO2surface but has also become the prototypical metal oxide surface for fundamental studies [252]. However, for the purpose of photocatalysis the anatase phase of titania enjoys more relevance due to its generally higher photocatalytic activity [253–256]. The (001) facet of anatase is well known by its high reactivity [257,258]. This surface is structurally simpler than rutile (110), with every Ti atom on the surface being pentacoordinated (Fig. 5.1).
Although (101) is the naturally dominant facet of anatase, different techniques have emerged for the synthesis of TiO2 structures with exposed{001}crystal facets in order to enhance their photocatalytic properties [256, 259–261].
O2c Ti5c
O3c
Ti6c
Ti5c O2c
O3c [110]
[001]
[110]
[001]
[100]
[010]
Figure 5.1: DFTB optimized structures of rutile (110) (left) and anatase (001) (right) surfaces of titania.
83
The TD-DFTB approach is employed for the computation of the excitation energies and oscil-lator strengths of the pollutant-TiO2 complexes. PBE and HSE calculations are performed for the validation of DFTB for the description of the ground-state properties of the systems under investigation. We finally compare our predictions to experimental results.
5.1 Ground State Properties of the Pollutant-TiO
2Com-plexes
The study of the excited-state properties of ligand adsorption on TiO2 surfaces is currently not feasible within a first-principle methodology without compromising the accuracy of the results with the use of oversimplified models. The utilization of small clusters to describe a periodic material introduces a significant error amounting to sampling the Brillouin zone with a single k-point. This error is alleviated by increasing the model size, thus decreasing the dimensions in the reciprocal space. To afford the use of reliable models for the ambitioned computational assignment we employ TD-DFTB. For the accurate description of the systems under investiga-tion, a proper parametrization set has to be defined. In this section we test the suitability of DFTB parameters for the description of the ground-state properties of TiO2 and its interaction with NO. This represents an important step towards the appropriate characterization of the photocatalytic properties of titania.
5.1.1 Validation of DFTB parameters
For the study of TiO2 surfaces and its interaction with NO and acetaldehyde, we have modified the set of DFTB parameters tiorg [262]. Pure TiO2 surfaces interact weakly with the investi-gated ligands, which physisorb at rather large distances (over 2 ˚A) from the surface. We have improved the Ti-O and Ti-N repulsive potentials, laying special emphasis on the function tail in order to reproduce interatomic distances and adsorption energies of the adsorbates.
As reference systems we employed NO attached to the Ti5c atom of rutile (110) and anatase (001) surfaces. We considered both orientation of the molecule, that is, either O or N pointing towards the metal site. Additionally, the oxygen-reduced rutile (110) surface was employed to explore a different interatomic distance regime. The defective surfaces were modeled by removing a bridging O2catom per each (2×1) surface unit cell. Furthermore, we compare bulk properties of both considered morphologies of titania with DFT and experimental findings.
Reference calculations were performed at the PBE level of theory as implemented in the Vienna ab initio Simulation Package (VASP) [263–266]. Plane wave basis set with an energy cutoff of 420 eV as well as the projector augmented-wave (PAW) method [267, 268] have been used.
For the surface systems periodic calculations were performed according to the slab approach, where a fairly large vacuum region is created along the normal to the surface in the supercell, thus avoiding spurious interaction between the periodically repeated slabs. For Brillouin zone integrations, k-point meshes were sampled using a (8×8×4) and (8×8×8) Monkhorst-Pack (MP) [269, 270] grids for bulk anatase and rutile, respectively. For rutile (110) and anatase (001) surfaces, (8×8×1) MP grids were employed. During the geometry optimization, all atoms were allowed to move till the interatomic forces were smaller than 10−3 eV/˚A for bulk
calculations and 10−2 eV/˚A for surface calculations. Spin polarized calculations were performed for those systems involving NO.
Similar supercells were employed for the DFTB calculations where the supercell dimensions were re-optimized. The Brillouin zone was sampled with a (4×4×8) and (4×4×2) MP grids for bulk rutile and anatase, respectively, whereas for surface calculations, (3×1×3) and (4×4×1) MP grids were respectively used for rutile (110) and anatase (001). For the optimization of the geometries, every atomic position was relaxed till the interatomic forces were smaller than 5×10−4 eV/˚A for bulk and 5×10−3 eV/˚A for the pristine and modified TiO2 surfaces. Single-point calculations were conducted using VASP optimized geometries and then varying the Ti5c-molecule distance in a vertical configuration (that is, N-O bond oriented along the surface normal) with a step of 0.1 ˚A. Analogue VASP single-point calculations were also performed with the aim of construction of the repulsive potentials.
The cohesive energies and lattice constants of anatase and rutile as calculated with the mod-ified tiorg set are given in Table 5.1. We further provide PBE and experimental values for comparison. DFTB results are overall in line with the ab initio and experimental findings with some tendency of underestimation of the cohesive energies and overestimation of the lat-tice constants. The calculated band gaps are 3.1 eV and 3.2 eV for bulk rutile and anatase, respectively, which agree well with the experimental values of 3.0 eV for rutile [271] and 3.2 eV for anatase [272]. More remarkably, they are in better agreement with experiment than the typically underestimated values obtained within DFT using local or semi-local XC func-tionals [273–275] or the overestimated values obtained with hybrid funcfunc-tionals [262, 273]. This originates from a fortunate error compensation within the method.
Property DFTB PBE Exp.
anatase
a (˚A) 3.90 3.79 3.78
c (˚A) 9.49 9.74 9.52
c/a 2.43 2.57 2.51
Ecoh (eV) 18.88 21.54 19.73 rutile
a (˚A) 4.72 4.63 4.59
c (˚A) 3.00 2.96 2.96
c/a 0.64 0.64 0.64
Ecoh (eV) 19.16 21.44 19.79
Table 5.1: Comparison between DFTB, PBE and experimental results concerning the properties of bulk rutile and anatase.
The adsorption properties of NO on rutile and anatase surfaces are summarized in Table 5.2.
The adsorption energies were calculated as Eads = 1
n(ET−Ebare−n ENO), (5.1)
whereET is the total energy of the complex, Ebare is the total energy of the bare TiO2 surface, ENO is the energy of an isolated NO molecule in the gas phase and n is the number of NO molecules absorbed on the surface. To determine the most stable ground-state structure of the ligand-TiO2 complexes several molecular orientations and binding sites were considered.
According to our findings, NO adsorbs favorably on the perfect (110) surface in a tilted con-figuration with the N atom oriented towards the surface and bonded to the Ti5c atom. The distance from the N atom to the metal site is 2.56 ˚A. These results are in line with previous theoretical predictions [18, 21, 22] and our PBE results. For the adsorption through the oxygen atom the O-Ti5c distance (2.80 ˚A) also compares well to our PBE findings, which returned an optimized distance of 2.75 ˚A. For the oxygen-reduced surface the N-Ti5c and O-Ti5c distances are shortened to 1.95 and 1.83 ˚A, respectively. In the case of adsorption on the anatase (001) surface, ligand-substrate distances are also in agreement with PBE. In terms of adsorption en-ergies, the correspondence of DFTB with higher-level theory is also remarkable with absolute errors equal or smaller than 0.08 eV.
Configuration Property DFTB PBE
rutile (110)
on perfect surface via N-Ti5c N-Ti distance 2.56 2.55
Eads -0.25 -0.27
on perfect surface via O-Ti5c O-Ti distance 2.80 2.75
Eads -0.15 -0.14
on O2c-reduced surface via N-Ti5c N-Ti distance 1.95 1.87
Eads -0.89 -0.86
on O2c-reduced surface via O-Ti5c O-Ti distance 1.83 1.99
Eads -0.29 -0.34
anatase (001)
on perfect surface via N-Ti5c N-Ti distance 2.25 2.26
Eads -0.47 -0.35
on perfect surface via O-Ti5c Eads -0.11 -0.16
Table 5.2: Comparison between DFTB and PBE in terms of optimized structures and energetics of the NO adsorption on TiO2. Distances are given in ˚A and energies in eV.