3. Materials modeling based on DFT

The employment of DFT on material modeling offers an excellent relation time � efficiency. The efficiency is denoted by the results precision to prevision of material properties. In case of photocatalysis applications of solid-state materials, a theoretical methodology based on DFT provides important information about band gap, band-gap nature, photoinduced behavior, density and mobility for charge carriers; thus, it is possible to determine the applicability of each material in photoinduced devices and processes.

In order to evaluate these properties, it is necessary to investigate the material electronic structure. In solid-state materials, the electronic structure sorts materials as insulators, conductors and semiconductors. The formation of electronic structure of solid-state materials is originated from interactions between all constituent atoms since the atomic energy levels are perturbed by the neighbor atoms. Thereby, in these materials, the overlap of a large number of valence atomic orbitals results in the formation of molecular orbitals with very close energy, forming a quasi-continuous band; that is, an energy band is a continuum of closely spaced electron states [39–41]. Each energy band is occupied by two electrons (α and β) as predicted by Pauli exclusion principle. In solid-state materials, the electrons are observed in the lowest energy states; the last occupied energy level refers to the top or VBM, whereas the first unoccupied energy level is the bottom or CBM. There is no electronic state in the region between the VBM and CBM and it is called the band-gap which has direct influences on conduction properties of a solid [42]. According to band-gap energy, a material is classified as conductor (band gap > 1.0 eV), semiconductor (band gap from 1.0 to 4.0 eV) and insulator (band gap 4.0 eV or higher) [40, 43].

#### 3.1. Density of states (DOS) and band structure

(a ¼ 0.20, b ¼ 0.72 and c ¼ 0.81) to adequate the theoretical results to experimental results combined to Perdew and Wang correction gradient (PW91). This new functional has excellent results compared to half-half functional due to its more precise description of EXC energy and

In 1994, Frisch and co-authors adapted the LYP corrections rather than PW91 creating the B3LYP that uses the same empirical parameters of B3PW91. Such functional employs the Slater exchange plus Vosko, Wilk, Nusair (SVWN) to improve the correction proposed by LYP. In the last few years, B3LYP has become one of the most used for computational calculations because of the excellent results obtained. Another factor responsible for the wide use of hybrid functional is its versatility, and once such functional was used to investigate semiconductors, pro-

The hybrid functionals are differentiated according to the percentage of parameters of the Hartree-Fock method employed in the determination of EXC. For example, B3LYP uses 20% of HF parametrization in its formalism. In 1996, Perdew and co-workers increased the HF percentage to 25% in PBE functional and created the PBE0 functionals. The increase in HF aims to minimize the over-estimation of factors arising from the electrons interaction; the PBE0 uses the GGA to evaluate the EXC energy and it does not present adjustable parameters in its formulation (empirical parameters are defined as a ¼ 0.25, b ¼ 0.75 and c ¼ 1). Other differences regarding B3LYP is less sophisticated formulation to PBE0 [27, 36, 37]. Moreover, PBE0

Another functional that is largely used is the HSE which was developed by Heyd, Scuseria and Ernzerhof. This functional investigates the EXC energy by slitting it in two parts: a short-range part and a long-range part. Particularly, the HSE hybrid functional employs the parameter ω in function of Bohr radius. Thus, the short-range part is treated in the same form as that in the PBEh global hybrid, that is, it uses 25% short-range exact exchange and 75% short-range PBE exchange; whereas, the long-range part is investigated by PBE. If ω goes to 0, the short-range part dominates and HSE reduces to PBEh; if ω goes to infinity, the short-range part disappears and HSE reduces to PBE. The use of HSE functional reduces the computational cost to obtain

The employment of DFT on material modeling offers an excellent relation time � efficiency. The efficiency is denoted by the results precision to prevision of material properties. In case of photocatalysis applications of solid-state materials, a theoretical methodology based on DFT provides important information about band gap, band-gap nature, photoinduced behavior, density and mobility for charge carriers; thus, it is possible to determine the applicability of

In order to evaluate these properties, it is necessary to investigate the material electronic structure. In solid-state materials, the electronic structure sorts materials as insulators, conductors and semiconductors. The formation of electronic structure of solid-state materials is

parameterization [21, 27, 28, 31, 32].

218 Titanium Dioxide

teins, organic compounds and others [20, 21, 24, 26–28, 31–35].

has great efficiency in material properties determination.

the exact exchange-correlation energy [38].

3. Materials modeling based on DFT

each material in photoinduced devices and processes.

The DOS analysis (Figure 1) consists of a graphical representation of packing level of energy states in a quantum system, that is, the number of states in each region of energy. A high amount of energetic states results in a high density of states on projected DOS. In general, the DOS analysis is employed to investigate the energy levels nearest to band gap. As observed in Figure 1, the band of lowest energy refers to valence band and the higher energy band is the conduction band. Besides, the DOS analysis can also provide the contribution of each atom to compose the valence and conduction bands; by contribution of this analysis, it is also possible

Figure 1. Projected DOS on Ti and O atoms of anatase TiO2. The region before Fermi energy (EF) is the valence band and the region after 1 eV is the conduction band.

to clarify the chemical bond composition and predict the electronic configuration of atoms [40– 43]. Furthermore, applying the one-third of Simpson rule [44] on projected density of states for pure and doped models, it is possible to evaluate the number of available states in VB and CB. Such methods consist of numeric integration of the area under the DOS curve. Then, the number of states obtained from numeric integration is divided by unit cell volume. The values for CB and VB represent the number of available states for the formation of holes and electrons in electronic structure, respectively. According to the number of available states, a semiconductor material is classified as p-type (greater number of holes) or n-type (greater number of electrons) [44].

However, the energy bands for solid materials are not regular in all crystalline structures, and the same band can show a different energy level at each high symmetry point of Brillouin zone. Such symmetry points vary according to the spatial group of the crystalline structure and are labeled to their coordinates in space. The band structure analysis provides the energy of bands and the position of VBM and CBM. Thus, the band structure evaluation offers the band-gap nature to a material, that is, if the electron excitation occurs directly or indirectly [40]. Figure 2 presents the band structure for anatase TiO2; it is observed as an indirect band-gap and that the bands are not regularly distributed at high symmetry points.

The band structure study can also provide the effective mass for a solid-state material. In general, the transfer rate of electron-hole pair is inversely proportional to their effective mass. Thereby, a great effective mass denotes a low transfer rate of carriers, whereas, a small effective mass indicates that the charge carriers are extremely stable. The stability for these carriers also promotes the migration of electron and holes, as well as inhibits their recombination. The

Figure 2. The band structure analysis for anatase TiO2 shows an indirect band-gap.

effective mass of electrons and holes allows to assess indirectly the rate of charge carriers as presented in Eq. (8); in this formula, m\* is the effective mass of the charge carrier, k is the wave vector, ℏ is the reduced Planck constant and ν is transfer rate of photo-generated electrons and holes. Besides, the effective mass for electrons (me� ) and holes (mh� ) should be investigated regarding the CBM and VBM; such points are very important to determine the E<sup>g</sup> for a material. The values for effective mass were calculated by a fitting parabolic function around these points (Eq. (9)); in such an equation, m\* is the effective mass of the charge carrier, k is the wave vector, ℏ is the reduced Planck constant and E refers to the energy of an electron at wave vector k in that band. In order to guarantee the validation of the parabolic approximation within the CBM and VBM, the parabolic fitting is performed considering a difference of 1 meV around the CBM and VBM regions [45–48].

$$
\boldsymbol{\nu} = \hbar \mathbf{k}/\mathfrak{m}^\* \tag{8}
$$

$$
\hbar \mathbf{m}^\* = \hbar^2 \left(\frac{d^2 E}{d\mathbf{k}^2}\right)^{-1} \tag{9}
$$

#### 3.2. Photoinduced properties

to clarify the chemical bond composition and predict the electronic configuration of atoms [40– 43]. Furthermore, applying the one-third of Simpson rule [44] on projected density of states for pure and doped models, it is possible to evaluate the number of available states in VB and CB. Such methods consist of numeric integration of the area under the DOS curve. Then, the number of states obtained from numeric integration is divided by unit cell volume. The values for CB and VB represent the number of available states for the formation of holes and electrons in electronic structure, respectively. According to the number of available states, a semiconductor material is classified as p-type (greater number of holes) or n-type (greater number of

However, the energy bands for solid materials are not regular in all crystalline structures, and the same band can show a different energy level at each high symmetry point of Brillouin zone. Such symmetry points vary according to the spatial group of the crystalline structure and are labeled to their coordinates in space. The band structure analysis provides the energy of bands and the position of VBM and CBM. Thus, the band structure evaluation offers the band-gap nature to a material, that is, if the electron excitation occurs directly or indirectly [40]. Figure 2 presents the band structure for anatase TiO2; it is observed as an indirect band-gap and that

The band structure study can also provide the effective mass for a solid-state material. In general, the transfer rate of electron-hole pair is inversely proportional to their effective mass. Thereby, a great effective mass denotes a low transfer rate of carriers, whereas, a small effective mass indicates that the charge carriers are extremely stable. The stability for these carriers also promotes the migration of electron and holes, as well as inhibits their recombination. The

the bands are not regularly distributed at high symmetry points.

Figure 2. The band structure analysis for anatase TiO2 shows an indirect band-gap.

electrons) [44].

220 Titanium Dioxide

The photoinduced properties of a solid compound determine how it interacts with electromagnetic radiation. In case of solid-state materials, a material interacts only with radiation which has energy equal to or higher than the material band gap [40, 41]. Thus, through evaluation of Eg values, it is possible to determine the characteristic type of radiation that interacts with each investigated material. Allied to the deep electronic structure investigation, it is possible to determine the bulk and surfaces of photoinduced behavior of materials and its applicability on photocatalytic process. Such a condition is essential to form the electron-hole pair (e– -h• ), which is the precursor to start the photocatalysis process. Another point is focused on energetic stability of e– -h• pair, and this factor can be estimated by reduced mass; however, the half-life time is not provided.

A practical example and much discussed in literature is comparing the electronic structures of rutile and anatase TiO2 to clarify these concepts. Analyzing the band structures of both materials, we found band gaps of 3.50 and 3.80 eV (Figure 3), respectively. The minor band gap for anatase structure shows a photoinduction of e– -h• pair from wavelength close to 326 nm, whereas, for rutile polymorph the e– -h• pair is photogenerated around to 354 nm. Such difference is possible because of the major disorder associated with the TiO6 cluster of anatase phase in relation to rutile structure; the displacement of Ti atom from central position of the unit cell in anatase causes modification in electronic structure through overlap between oxygen 2p orbitals and titanium 3d orbitals. This overlap orients to TiO6 cluster to a TiO5 cluster configuration, while, in rutile phase this effect is smaller. However, the anatase TiO2 has an indirect band gap and rutile TiO2 shows a direct band gap indicating that for anatase the e– -h• pair recombination process is less possible because it requires a photon-phonon coupling and in rutile structure such phenomenon is direct, that is, localized in the same symmetric region. Thus, for anatase structure as photoinduction energy required to make the e– –h• pair as possibility of its recombination are more favorable than rutile phase. Now, let us evaluate the

Figure 3. The crystalline structure and band structure for titania materials. (a) anatase crystalline phase and (b) rutile crystalline phase.

energetic stability of the e– -h• pair photoinduced through effective mass. The effective mass for electron and hole charge carriers calculated for anatase phase was 0.46 and 0.14, respectively; then, the electron/hole ratio is 3.29, indicating that e– -h• pair is favorable in anatase phase as carriers. Nevertheless, for rutile structure, this effective mass for electron and hole carriers is 0.65 and 0.32, respectively; likewise, the electron/hole ratio is 2.03 showing that the e– -h• pair is less favorable than anatase structure. Therefore, every electronic structure of anatase TiO2 material from its crystalline structure is more inclined to photocatalytic process in comparison to rutile TiO2 polymorph because anatase structure has small and indirect band gap and the e– -h• pair is more favorable.

#### 3.3. Surface properties

Surface chemistry is an interesting and challenging topic in photocatalytic applications. In this case, the molecular level can be described by the combination between the surface structure and the interface region, where the main reactions involving photoinduced charge carriers occur. Therefore, the understanding of surface chemical reactions plays a fundamental role in the description of chemical bond between surface and adsorbed molecules, which are the fundamental basis to clarify the material photocatalytic activity [14, 15].

Recently, a lot of theoretical studies show the benchmark of DFT methods to reproduce the main effects behind the catalytic behavior of transition metal surfaces [49, 50]. For example, Cheng and co-workers systematically investigate the water splitting along rutile (110) surfaces combining DFT and molecular dynamics [51]. In addition, Migani and Blancafort studying the photocatalytic oxidation of methanol on TiO2 (110) surfaces trough DFT/HSE06 calculations highlighted the role of excitons to produce formaldehyde [52]. Several related reports have appeared recently in the literature, as described by Nolan and co-authors that summarize the main advances in ab initio-based design of photocalytic surfaces [53].
