1. Introduction

Over the last few decades, the field of photocatalysis has assumed a remarkable importance in environmental topics, mainly on technologies based on the capability of use of the main supply of energy on Earth: solar irradiation. In this case, the solar light can be used to activate a chemical process of radicals species, destroy undesired compounds or be transformed in chemical energy; for example, hydrogen production from H2O [1, 2]. The aim of this field was initially motivated by the oil crisis, which promoted the research about alternative energy sources. In addition, the concern about pollution has attracted a recent economical, political and scientific interest because of the toxicological potential of recalcitrant compounds for environmental contamination, as well as for animal and human health. Thus, eco-friendly methods became a fundamental topic, being the advanced oxidation processes (AOPs) the most powerful tool to destroy recalcitrant synthetic species due to high reactive potential of photoinduced radicals, which allows the extinction of a wide range of compounds [1, 3, 4].

Nowadays, heterogeneous photocatalysis proved its efficiency to degrade chemical contaminants through a photoinduced reaction in the presence of a semiconductor photocatalyst. This method was originated in the decade of 1970 when Fujishima and Honda described the water splitting by photoelectrochemical cell containing TiO2 [5]. Posteriorly, the scientific and technological interest in such methods has exponentially increased, TiO2 being the widely used semiconductor material for charge carriers generation due to the photostability, chemical and biologically inert nature, availability and low cost. Such material can be found/synthetized in anatase, rutile and brookite phases, anatase being a polymorph, the most prominent candidate for photocatalytic applications. Anatase TiO2 is an n-type semiconductor with indirect band gap of around 3.2 eV. Several reasons were proposed to explain the higher photocatalytic efficiency of anatase polymorph, such as increased surface area, –OH group concentrations and electronic structure features. In particular, the electronic features have been shown as the key advantages of anatase TiO2 mainly related to the band edge positions, which allow the reduction or oxidation of protons from water creating –OH radicals as intermediate in the photo-oxidation reactions to break organic compounds because of high oxidation potential [1, 6, 7].

However, due to large band gap (3.2 eV), anatase TiO2 absorbs only UV light above 400 nm, which amounts for 4–5% of the solar photons. To enable the effective use of solar radiation in photocatalytic process with TiO2, much effort has been directed to the narrowing of band gap from UV to the visible spectra range. Such efforts include the crystal shape engineering and the doping or co-doping process. Recently, a lot of theoretical and experimental studies have been developed to investigate metals (Fe, Cr, V, Mo, Re, Ru, Mn, Co, Rh) and non-metals (N, S, B, C, F) doping effects on electronic structure of anatase TiO2 photocatalysts [7–9]. In the metal doping process, the theoretical and experimental reports suggest that a redshift of the band gap occurs due to the insertion of a new electronic band closer to the conduction band minimum (CBM) improving the photocatalytic properties. Despite narrowing of the band gap, foreign cations frequently act as recombination center, suggesting that significant improvements in the photocatalytic efficiency are possible only at low concentration of dopants [10–12]. Alternatively, the coupling of TiO2 with another semiconductor (SnO2, WO3, CdS and others) is commonly used as an approach to improve the photocatalytic efficiency through the band structure mismatch, which allows a physical separation between the photoinduced charge carriers, reducing the recombination rate. On the other hand, for non-metal doping, there are three different main opinions about the band-gap narrowing that can be related to the shift of valence band maximum (VBM), creation of impurity levels or oxygen vacancies [7–9].

Therefore, the main challenge in titania-assisting photocatalytic process remains on the tailoring of their electronic structure in order to improve and stabilize the photoinduced reactions. From a lot of experimental and theoretical researches, it is possible to investigate the understating of the main features of these complex reactions, such as the control and engineering of band gap, band edges, charge transport and recombination rate. In this aspect, theoretical analyses are helpful due to the precision of quantum mechanics-based methods to predict structural-electronic properties and its changes from chemical compounds. For example, recently it was published in manuscripts announcing the reliability of density functional theory (DFT) methods to research the band structure distribution, band-gap evaluation and engineering, electron-hole transport and dielectric constant, which are key properties of semiconductor photocatalytics. Besides the theoretical endeavor to investigate and predict bulk, surface chemistry, electronic structure and other properties have played a fundamental role in the understanding of morphological transformations and crystal growth, as well as for adsorption and reaction phenomena [13–15].

In this chapter, we propose a theoretical point of view about the photocatalytic properties of TiO2, focusing on the evaluation of electronic structure parameters for both bulk and surfaceoriented materials. The following sections are dedicated to explain (i) the mechanism of titaniaassisted photocatalytic properties, (ii) benchmark of DFT methods on the investigation of photoinduced properties and (iii) theoretical results and discussion for different TiO2 models.

## 1.1. Basic mechanism of TiO2 photocatalysis

1. Introduction

212 Titanium Dioxide

tial [1, 6, 7].

Over the last few decades, the field of photocatalysis has assumed a remarkable importance in environmental topics, mainly on technologies based on the capability of use of the main supply of energy on Earth: solar irradiation. In this case, the solar light can be used to activate a chemical process of radicals species, destroy undesired compounds or be transformed in chemical energy; for example, hydrogen production from H2O [1, 2]. The aim of this field was initially motivated by the oil crisis, which promoted the research about alternative energy sources. In addition, the concern about pollution has attracted a recent economical, political and scientific interest because of the toxicological potential of recalcitrant compounds for environmental contamination, as well as for animal and human health. Thus, eco-friendly methods became a fundamental topic, being the advanced oxidation processes (AOPs) the most powerful tool to destroy recalcitrant synthetic species due to high reactive potential of photoinduced radicals, which allows the extinction of a wide range of compounds [1, 3, 4]. Nowadays, heterogeneous photocatalysis proved its efficiency to degrade chemical contaminants through a photoinduced reaction in the presence of a semiconductor photocatalyst. This method was originated in the decade of 1970 when Fujishima and Honda described the water splitting by photoelectrochemical cell containing TiO2 [5]. Posteriorly, the scientific and technological interest in such methods has exponentially increased, TiO2 being the widely used semiconductor material for charge carriers generation due to the photostability, chemical and biologically inert nature, availability and low cost. Such material can be found/synthetized in anatase, rutile and brookite phases, anatase being a polymorph, the most prominent candidate for photocatalytic applications. Anatase TiO2 is an n-type semiconductor with indirect band gap of around 3.2 eV. Several reasons were proposed to explain the higher photocatalytic efficiency of anatase polymorph, such as increased surface area, –OH group concentrations and electronic structure features. In particular, the electronic features have been shown as the key advantages of anatase TiO2 mainly related to the band edge positions, which allow the reduction or oxidation of protons from water creating –OH radicals as intermediate in the photo-oxidation reactions to break organic compounds because of high oxidation poten-

However, due to large band gap (3.2 eV), anatase TiO2 absorbs only UV light above 400 nm, which amounts for 4–5% of the solar photons. To enable the effective use of solar radiation in photocatalytic process with TiO2, much effort has been directed to the narrowing of band gap from UV to the visible spectra range. Such efforts include the crystal shape engineering and the doping or co-doping process. Recently, a lot of theoretical and experimental studies have been developed to investigate metals (Fe, Cr, V, Mo, Re, Ru, Mn, Co, Rh) and non-metals (N, S, B, C, F) doping effects on electronic structure of anatase TiO2 photocatalysts [7–9]. In the metal doping process, the theoretical and experimental reports suggest that a redshift of the band gap occurs due to the insertion of a new electronic band closer to the conduction band minimum (CBM) improving the photocatalytic properties. Despite narrowing of the band gap, foreign cations frequently act as recombination center, suggesting that significant improvements in the photocatalytic efficiency are possible only at low concentration of dopants [10–12]. Alternatively, the coupling of TiO2 with another semiconductor (SnO2, WO3, CdS and others) is commonly Photo-driven processes, such as used in photocatalytic and photovoltaic devices, are based on the conversion of light energy into other forms of energy such as electricity (solar cells) or chemical compounds (photocatalysis, water splitting, CO2 reduction and others). These technologies, mainly photocatalytic processes, require the semiconductor electronic structure for the light absorption and conduction of the photoinduced charge carriers. In a molecular point of view, the general mechanism behind such devices can be divided in three steps: (i) light adsorption; (ii) electron-hole dissociation and (iii) charge carriers dynamic [1, 2, 13].

In the first step, commonly described as light adsorption, the light interacts with the electronic structure of the semiconductor. In this case, the light wavelength must be equal to or higher than the band gap, the energy difference between VB and CB, to promote an electron (e) from the VB to CB, inducing an electronic vacancy in the VB, denominated as hole (h• ). The electronhole pair interacts through a Coulomb attraction and plays a fundamental role in subsequent steps, controlling the photocatalytic efficiency of semiconductors. Especially for sunlight absorption, an optimum band gap is required due to the light wavelength commonly found in solar radiation. For the solar-driven photocatalysis, the optimum band gap belongs to the range between 1.6 and 2.5 eV. In addition, for photoinduced reactions, another compromise needs to be achieved: besides the band-gap value, the energy of the electron/hole should be high enough to perform the given reactions. These chemical potentials depend on the position of energy levels in the semiconductor, which is one of the key advantages of TiO2 among other semiconductors because both the reduction of protons (ENHE(H<sup>þ</sup>/H2) ¼ 0.0 ev) and the oxidation of water (ENHE(O2/H2O) ¼ 1.2 ev) can be activated simultaneously. Moreover, the superficial –OH groups can act as donor species to generate OH• radicals that have a very high oxidation potential, which enables the subsequent reactions used in chemical decontamination [1, 6, 9, 13].

Another important feature associated with the optical excitation corresponds to the band-gap nature, which plays an important role in the absorption coefficient of the semiconductor. For semiconductors the band gap can be direct or indirect, depending upon the localization of the VBM and CBM along the Brillouin zone. In photocatalysis, the band-gap nature is important in the recombination of photo-generated electrons and holes due to differences in the electron decay from CBM. For semiconductors with direct band gap, such as rutile TiO2, the recombination of the charge carriers emits a photon once the CBM and VBM are located in the same k vector. However, for indirect band-gap semiconductors such as anatase, the recombination is assisted by a phonon due to the difference between CBM and VBM, making the direct recombination difficult between excited electrons and holes, which results in an increased electronhole lifetime. As a key result, the diffusion rate and the reaction time of the excited electron hole in indirect semiconductor also increase, making them promising candidates with superior photocatalytic performance than direct semiconductors [16, 17].

In the next step, the electron and hole have to be dissociated to obtain free charge carriers, which will be used in the electron transport of the device. This dissociation depends on the electron-hole binding energy (Eb) that is inversely proportional to the semiconductor dielectric constant. If we assume that the electron-hole dissociation will be made by the thermal energy, E<sup>b</sup> should be lower than kBT (kB ¼ Boltzmann constant and T ¼ absolute temperature) around 25 meV at room temperature, and the semiconductor must have a dielectric constant around 10. For TiO2, both rutile and anatase polymorphs have a superior dielectric constant, which enables a lower binding energy between the electron holes, making the dissociation easier [13, 16].

The final step corresponds to the diffusion of free charge carriers. In this step, the electron and the hole are transported to their active sites where they will be used before the recombination [1, 6, 13, 16]. The diffusion (D) is strictly related to the mobility (μ—Eq. (1)) of the charge carrier which in turn is linked to the effective mass (m\* ) and the collision time (τ) of the charge carrier (Eq. (2)). Therefore, D is increased if the effective mass of the photogenerated carriers becomes lighter resulting in enhanced photocatalytic efficiency. Furthermore, the ratio between the effective mass of electrons (m<sup>e</sup> \* ) and holes (m<sup>h</sup> \* ) is a powerful tool to predict the electron/hole pair stability with respect to the recombination process. In this case, a larger effective mass difference induces distinct mobility, which reduces the electron-hole pair recombination, increasing the photocatalytic efficiency [13, 14, 17, 18].

$$D = \frac{k\_B T}{e} \tag{1}$$

Quantum Chemistry Applied to Photocatalysis with TiO2 http://dx.doi.org/10.5772/intechopen.69054 215

$$
\mu = e \frac{\pi}{m} \tag{2}
$$

However, the simulations involving scattering process are very expensive and the evaluation of mobility of charge carriers requires an alternative approach. The effective mass can be associated with the band curvature at the top of the VB or at the bottom of CB. For a single isotropic and parabolic band, the effective mass can be obtained through the expression:

$$\frac{1}{2m} = \frac{1}{\hbar} \frac{\partial^2 E}{\partial \mathbf{k}^2} \tag{3}$$

Therefore, m<sup>e</sup> \* can be obtained by fitting the bottom of the conduction band, whereas m<sup>h</sup> \* corresponds to the fitting along the top valence band. In order to acquire the validity of the parabolic approximation within the CBM and VBM regions, the parabolic fitting must be done within an energy difference around 26 meV near to the CBM and VBM, corresponding to the thermal dissociation energy of carriers at room temperature [18, 19].

#### 2. Theoretical methods and density functional theory

The theoretical methods based on quantum mechanical simulations are an important tool to evaluate material properties, mainly at the molecular level. Historically, the development of new materials to technological applications appears as a difficult task which requests a long time of studies. Meanwhile, the theoretical-computational method plays the traditional role of study of materials already discovered. However, the technological advancement is extremely dependent on development of new materials, once such materials are responsible for the improvement of available technologies. Front of such need, the theoretical methods helped chemists and physicist on the development materials at higher speed; it was possible once the theoretical investigation provides the materials characteristic and the limitations to its applications [20–22]. A large number of computational methods is available for investigation of material properties; however, in the last three decades, the use of DFT has changed the world because it offers an excellent relation between results precision and calculation time. The importance of this theory on material investigation is evidenced by the number of manuscripts based on its application since is very superior than the number of manuscripts based on Hartree-Fock (HF) and semi-empirical methods simulations as for inorganic chemistry as for organic compounds [20, 21, 23].

#### 2.1. Density functional theory

needs to be achieved: besides the band-gap value, the energy of the electron/hole should be high enough to perform the given reactions. These chemical potentials depend on the position of energy levels in the semiconductor, which is one of the key advantages of TiO2 among other semiconductors because both the reduction of protons (ENHE(H<sup>þ</sup>/H2) ¼ 0.0 ev) and the oxidation of water (ENHE(O2/H2O) ¼ 1.2 ev) can be activated simultaneously. Moreover, the superficial –OH groups can act as donor species to generate OH• radicals that have a very high oxidation potential, which enables the subsequent reactions used in chemical decontamina-

Another important feature associated with the optical excitation corresponds to the band-gap nature, which plays an important role in the absorption coefficient of the semiconductor. For semiconductors the band gap can be direct or indirect, depending upon the localization of the VBM and CBM along the Brillouin zone. In photocatalysis, the band-gap nature is important in the recombination of photo-generated electrons and holes due to differences in the electron decay from CBM. For semiconductors with direct band gap, such as rutile TiO2, the recombination of the charge carriers emits a photon once the CBM and VBM are located in the same k vector. However, for indirect band-gap semiconductors such as anatase, the recombination is assisted by a phonon due to the difference between CBM and VBM, making the direct recombination difficult between excited electrons and holes, which results in an increased electronhole lifetime. As a key result, the diffusion rate and the reaction time of the excited electron hole in indirect semiconductor also increase, making them promising candidates with superior

In the next step, the electron and hole have to be dissociated to obtain free charge carriers, which will be used in the electron transport of the device. This dissociation depends on the electron-hole binding energy (Eb) that is inversely proportional to the semiconductor dielectric constant. If we assume that the electron-hole dissociation will be made by the thermal energy, E<sup>b</sup> should be lower than kBT (kB ¼ Boltzmann constant and T ¼ absolute temperature) around 25 meV at room temperature, and the semiconductor must have a dielectric constant around 10. For TiO2, both rutile and anatase polymorphs have a superior dielectric constant, which enables a lower binding energy between the electron holes, making the dissociation eas-

The final step corresponds to the diffusion of free charge carriers. In this step, the electron and the hole are transported to their active sites where they will be used before the recombination [1, 6, 13, 16]. The diffusion (D) is strictly related to the mobility (μ—Eq. (1)) of the charge

carrier (Eq. (2)). Therefore, D is increased if the effective mass of the photogenerated carriers becomes lighter resulting in enhanced photocatalytic efficiency. Furthermore, the ratio

electron/hole pair stability with respect to the recombination process. In this case, a larger effective mass difference induces distinct mobility, which reduces the electron-hole pair recom-

<sup>D</sup> <sup>¼</sup> kBT

) and holes (m<sup>h</sup>

\*

\*

) and the collision time (τ) of the charge

<sup>e</sup> <sup>ð</sup>1<sup>Þ</sup>

) is a powerful tool to predict the

photocatalytic performance than direct semiconductors [16, 17].

carrier which in turn is linked to the effective mass (m\*

bination, increasing the photocatalytic efficiency [13, 14, 17, 18].

between the effective mass of electrons (m<sup>e</sup>

tion [1, 6, 9, 13].

214 Titanium Dioxide

ier [13, 16].

The material properties can be evaluated through several computational approaches, such as molecular dynamics, ab initio methods and semi-empirical methods. The calculations based on molecular dynamics evaluated the system properties based on the behavior of ball-and-springs model under application of a force external field to atoms representation. In turn, the ab initio and semi-empirical methods employ different approaches to solve the Schrödinger Equation and for the obtainment of system wave function (Ψ) [24]. Particularly, the DFT system interpretation is not based on wave-function (Ψ), once it assumes the system total energy as a single functional of electronic density (ρ). Poorly, this theory can be simplified in two basic postulates [24, 25]:

i. The density functional (ρ) determines exactly and completely all the ground state properties for a system. Thus, ρ is only dependent on three variables that determine the position (x, y and z);

$$
\rho\_{\text{(x, y, z)}} = E\_0 \tag{4}
$$

ii. Any function for electronic density will have energy greater than or equal to the ground state energy for a real system.

$$E\_{\left(\nu\right)}\left[\rho\_0\right] \succeq E\_{\left(\mathcal{O}\right)}\left[\rho\_0\right] \tag{5}$$

Nevertheless, the analytical function for electronic density is not yet known and the electronic density is obtained by HF equations for achievement of ρ by a self-consistent field (SCF) method. Hence, the HF method is very similar to DFT so that the difference lies in the equation's formalism. Although this similarity was observed, the DFT shows highest precision and low time (computational cost) regarding HF simulations due to number of variables in each methodology. The HF and semi-empirical methods employ a high number of variables for a system investigation; the number of variables is in the 4n order, where n refers to the number of electrons in the system. In turn, DFT is dependent on three variables [20, 22, 24, 25].

Actually, the quantum calculation based on DFT applied the theorem developed by Kohn and Shan (KS) in 1965; the KS equation describes all the functional theories (Eq. (6)) and their representation for molecular orbitals (Eq. (7)). In such equations, ∇<sup>2</sup> is the kinetic energy for non-interacting electrons; u(r) refers to classic Coulomb potential for a density of n electrons; d refers to the system space and ϕ corresponds to molecular orbital [26]. The KS equation was applied on two different systems; the first considers that there are no interactions between electrons, whereas the other assumes that such interactions are observed. The obtained results for both systems indicate a significant difference of energy between them. In order to correct this difference, the exchange-correlation term (EXC) was inserted in DFT formulism; the EXCconsists of the sum of kinetic and potential energy difference between interacting and non-interacting systems. In terms of system interpretation, this energy refers to 1% of system total energy, and its physical meaning is the interaction between electrons in the investigate compound. Thus, DFT describes 100% of system total energy.

$$E\_{KS} = -\frac{1}{2} \sum\_{i} \int d^3 r \phi\_i^\*(r) \nabla^2 \phi\_i(r) + \int d^3 r \left[\frac{1}{2} u(r) + V\_{ext}(r)\right] \rho(r) + E\_{XC} \left[\rho\right] \tag{6}$$

$$\left[-\frac{1}{2}\nabla^2 + V\_{\ell f}(r)\right]\phi\_i(r) = E\_i^{\text{KS}}(r)\phi\_i(r) \tag{7}$$

The main characteristic of EXCis the possibility of description by several forms dependent on what exchange-correlation functional was employed. Thereby, the choice of a functional to describe EXC has a giant effect on material properties evaluation, as offering better results and as offering a reduction in computational cost. The first functionals are called local functionals and are based on an electron cloud model to represent the system electronic density; among its class of functionals, stands out the local density approximation (LDA)/local spin density approximation (LSDA) and generalized gradient approximation (GGA) [20, 22, 25, 27, 28]. Further ahead, the use of hybrid functionals gains force due to high proximity to experimental results.

#### 2.1.1. Exchange-correlation functionals

pretation is not based on wave-function (Ψ), once it assumes the system total energy as a single functional of electronic density (ρ). Poorly, this theory can be simplified in two basic postu-

i. The density functional (ρ) determines exactly and completely all the ground state properties for a system. Thus, ρ is only dependent on three variables that determine the

ii. Any function for electronic density will have energy greater than or equal to the ground

� � <sup>≥</sup> <sup>E</sup>ð Þ<sup>0</sup> <sup>ρ</sup><sup>0</sup>

Nevertheless, the analytical function for electronic density is not yet known and the electronic density is obtained by HF equations for achievement of ρ by a self-consistent field (SCF) method. Hence, the HF method is very similar to DFT so that the difference lies in the equation's formalism. Although this similarity was observed, the DFT shows highest precision and low time (computational cost) regarding HF simulations due to number of variables in each methodology. The HF and semi-empirical methods employ a high number of variables for a system investigation; the number of variables is in the 4n order, where n refers to the number

<sup>E</sup>ð Þ<sup>ν</sup> <sup>ρ</sup><sup>0</sup>

of electrons in the system. In turn, DFT is dependent on three variables [20, 22, 24, 25].

compound. Thus, DFT describes 100% of system total energy.

� 1 2

<sup>∇</sup><sup>2</sup> <sup>þ</sup> Vef fð Þ<sup>r</sup> � �

φi

ð Þ¼ <sup>r</sup> EKS

ð d3 rφ� <sup>i</sup> ð Þ<sup>r</sup> <sup>∇</sup><sup>2</sup> φi ð Þþ r ð d3 r 1 2

EKS ¼ � <sup>1</sup> 2 X i

Actually, the quantum calculation based on DFT applied the theorem developed by Kohn and Shan (KS) in 1965; the KS equation describes all the functional theories (Eq. (6)) and their representation for molecular orbitals (Eq. (7)). In such equations, ∇<sup>2</sup> is the kinetic energy for non-interacting electrons; u(r) refers to classic Coulomb potential for a density of n electrons; d refers to the system space and ϕ corresponds to molecular orbital [26]. The KS equation was applied on two different systems; the first considers that there are no interactions between electrons, whereas the other assumes that such interactions are observed. The obtained results for both systems indicate a significant difference of energy between them. In order to correct this difference, the exchange-correlation term (EXC) was inserted in DFT formulism; the EXCconsists of the sum of kinetic and potential energy difference between interacting and non-interacting systems. In terms of system interpretation, this energy refers to 1% of system total energy, and its physical meaning is the interaction between electrons in the investigate

> u rð Þþ Vextð Þr � �

> > <sup>i</sup> ð Þr φ<sup>i</sup>

ρð Þþ r EXC ρ

ð Þr ð7Þ

� � <sup>ð</sup>6<sup>Þ</sup>

<sup>ρ</sup>ð Þ x, y, <sup>z</sup> <sup>¼</sup> <sup>E</sup><sup>0</sup> <sup>ð</sup>4<sup>Þ</sup>

� � <sup>ð</sup>5<sup>Þ</sup>

lates [24, 25]:

216 Titanium Dioxide

position (x, y and z);

state energy for a real system.

The EXC term can be described in several forms according to the employed functional. Recently, the functionals are classified as local, non-local and hybrid functionals. Among the local approximations, stands out the LDA that is a general approach to represent the electron gas model and offers a simple description of exchange correlation at local character. The employment of LDA functional results in exact values for system kinetic energy shows better results when applied to a system where ρ slowly changes similar to a uniform electron gas. It is observed because of the LDA evaluates a real system, in which electronic density is E[ρ](r), by a homogeneous electrons gas system with the same density. Although such representation is valid, the density ρ becomes independent of position since the electrons are evenly distributed in the system. Hence, the investigation of real system by LDA functional is not accurate and over-estimates the correlation energy at 100%. The non-local functionals are developed in order to correct this failure [24, 25]. Furthermore, the LSDA is also a local description of the EXC term and is very similar to LDA; both functionals are based on electron gas model but the LSDA includes the electron spins to system properties determination [20, 24].

The non-local functionals were developed aiming to correct the LDA failures in representation of real system. These approaches are based on a charge gradient and are also commonly known as corrected gradient functional. Among the non-local approximations, stands out the GGA which uses the electronic density gradient; this gradient consists of the first derivative of ρ as a function of its position. The GGA functional obtains the EXC terms by the sum of correlation energy (EC) and exchange energy (EX), both negative values. The GGA functionals offer better results than LDA and LSDA approximations since the relation between ρ and position is considered. Compared to LDA and LSDA approximations, the GGA shows a deviation of 1% for system EXC once the electronic density distribution is treated in real form and not as an electron gas model [25, 29, 30]. However, the results presented by GGA approximations can be improved from Correlations (Becke Functionals) and Exchange Functionals (LYP and P86) [24].

Ultimately, the other class of functionals is known as hybrid functionals. Such functional employs Hartree-Fock method parameters to determine the EXC energy. The first hybrid functional was the half-half functional which was developed by Becke, and it was developed by a linear description of electronic density. However, this functional shows several limitations. Thus, in 1993, Becke proposes the B3PW91 model that uses three empirical parameters (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 parameterization [21, 27, 28, 31, 32].

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

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 has great efficiency in material properties determination.

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 exact exchange-correlation energy [38].
