A Density Functional Theory based Analysis on the Electronic, Mechanical, and Optical Properties of Cubic TiO2

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ScienceDirect Materials Today: Proceedings 18 (2019) 596–605

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ICN3I-2017

A Density Functional Theory based Analysis on the Electronic, Mechanical, and Optical Properties of Cubic TiO2 Debashish Dasha*, Saurabh Chaudhuryb a

b

Research Scholar, Dept. of Electrical Engg., NIT Silchar, Assam, 788010, India Associate Professor, Dept. of Electrical Engg., NIT Silchar, Assam, 788010, India

Abstract This paper presents an analysis on electronic, mechanical and optical properties of cubic titanium dioxide using Orthogonalized Linear Combinations of Atomic Orbitals (OLCAO) basis set under the framework of Density Functional Theory. The structural property, namely lattice constant ‘a’, and the electronic properties such as, the band diagram, density of states (DOS) have been studied and analyzed. Whereas, the mechanical properties like, bulk moduli, Shear moduli, Young’s Moduli, poison’s ratio have also been investigated thoroughly. Moreover, optical properties such as refractive index, extinction co- efficient, reflectivity, absorption coefficient have been studied and analyzed thoroughly. The results are compared with previous theoretical and experimental results. It is found that, DFT based simulation produces results which are approximation to experimental results, whereas, the calculated values of elastic constants are better than the previous theoretical and experimental values. © 2019 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of International Conference on Nanotechnology: Ideas, Innovations & Initiatives-2017 (ICN:3i2017). Keywords: Density Functional Theory; Cubic TiO2; Electronic Properties; Mechanical Properties

1. Introduction Naturally, TiO2 is found in many different structures and crystalline forms such as, hexagonal, tetragonal, monoclinic, orthorhombic etc. [1]. It provides different utilization ways for different polymorphic forms. In nature titanium dioxide forms in three different ways such as rutile, anatase and brookite [2]. Since last decade, various advanced materials have been explored by applying high pressure on the polycrystalline structure for various electronic applications. Yugui et al. synthesized platinum nitride as a noble nitride material, under high pressure and high temperature in its crystalline form [3]. They found out the electronic properties using Raman scattering and mechanical properties like, bulk modulus, Shear modulus using Voight, Reuss and Hill (VRH) theory. Li and Liu et al. used laser light excitation to investigate the Raman and photoluminescence spectra of Y2O3/ Eu3+ and Y2O3/ 2214-7853 © 2019 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of International Conference on Nanotechnology: Ideas, Innovations & Initiatives-2017 (ICN:3i2017).

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Eu3+/Mg2+ nano rods under high pressure and observed that a structural transition from cubic to amorphous takes place, when pressure is more than 24 GPa [4]. They also found that, when pressure is equal to 8 GPa, distorted phases of Raman and photoluminescence spectra becomes clearly visible [4]. Thus, it is clear that high pressure plays a vital role of catalyst in converting a material from one form to other for different applications. However, in last decade, titanium dioxide has been extensively studied both theoretically and experimentally due to its numerous advantages like non-toxicity, low cost, long term stability and also due to its superb catalytic feature [5]. It is observed that, when high pressure is applied to anatase type polymorph, they get converted to isostructural type of fluorite (CaF2) and pyrite (FeS2) structure. Investigations are still in the early stages to explore different applications of titanium dioxide polymorphs. Some distinguished industrial and commercial applications of TiO2 such as, selfcleaning of tiles, self- cleaning paints, windows panes, transparent windows and glasses, high efficiency solar cells, dynamic random access modules and creating super hard materials are highly encouraged in present era [6]. However, considering future conservation of energy and cleanliness of environment, solar cell is the most important application for which cubic TiO2 is highly preferred [7]. The cubic TiO2 can be obtained from natural anatase TiO2 by synthesizing at a high temperature of 1900 to 2100 K in a diamond-anvil cell under a pressure of 48 GPa [8, 9]. Investigations on both phases of cubic TiO2 have been carried out by many researchers both experimentally and theoretically. Initially, Swamy and Muddle [10] concluded that, except bulk modulus, most of the properties of pyrite TiO2 is closer to experimental values. Later, Mattesini et al. [11] found that, there is little difference between both the structures of cubic TiO2. They have found out that, little difference in the place of oxygen makes them different from each other. In pyrite TiO2, oxygen atoms are located at 0.34, 0.34, 0.34 whereas, in fluorite structure, these are at 0.25, 0.25, 0.25 . Later on, Kim et. al. [12] performed number of experiments and concluded that pyrite TiO2 is unstable due to the imaginary frequencies in the phonon spectra during the entire pressure range, however fluorite TiO2 is stable compared to pyrite TiO2 because of absence of such imaginary frequencies. In the same way, Hu. et al. carried out a deep study on electronic and other properties using Density Functional Theory (DFT) and different ultra-soft pseudopotentials [13]. From the literature, it is observed that, various studies were carried out for natural TiO2 whereas very less investigations were carried out on the cubic phase of TiO2. As it is already observed that, ambiguity exists between experimental and theoretical results. Hence, detailed investigation on electronic, elastic and optical properties of cubic TiO2 under high pressure is highly motivated from various applications point of view. Here, in this paper, a systematic ab- initio investigation of electronic, elastic and optical properties of cubic TiO2 under high pressure are studied in detail using (LCAO) basis set. Rest of the paper is organized as follows. Section 2 describes computational method adopted to study TiO2. Results and discussions are illustrated in section 3 and finally conclusion are drawn in section 4.

Figure 1 (a) Total Energy Vs Total Volume for Fluorite structured TiO2 (b) Total Energy Vs Total Volume for Pyrite structure TiO2

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White balls represents Ti (Titanium) atoms and red balls represents O (oxide) atoms. Figure 2 (a) Optimized Fluorite TiO2 with distance between Ti’s, O’s and Ti-O atoms (b) Optimized Pyrite TiO2 with distance between Ti’s, O’s and Ti-O atoms. 2. Computational Details The lattice parameters considered here are = = = . Å for fluorite and = = = . Å for pyrite TiO2. Limited Memory Broyden- Fletcher- Goldfarb- Shanno (LBFGS) algorithm [14-18] is utilized as optimization method. The relationship between these quantities are shown Figure -1 (a) and (b) for fluorite and pyrite TiO2 respectively. The lowest energy lattice constants are considered for simulation because minimizing the total energy of the crystal determines an appropriate set of linear combination of coefficients. For simulation, Orthogonalzed Linear Combinations of Atomic Orbitals (OLCAO) [19] method is adopted here which is an all- electron technique adopted for calculating 3p64s23d2 and 2s22p4 states as valence electrons for Ti and O respectively. The optimized lattice structure of fluorite and pyrite TiO2 is shown in Figure 2 (a) and (b) respectively. The LCAO is implemented in the framework of Density Functional Theory (DFT) having exchange correlation type of Local Density Approximation (LDA). For the simulation of various parameters, Dirac Bloch exchange with Perdew-Zunger (PZ) [20] correlation is considered. The grid mesh cut-off was taken as 1633 eV (60 Hartree). The monkhorst- pack [21] scheme is used for the k-point sampling, which is for fluorite TiO2 and point sampling for pyrite TiO2 [22]. 3. Results and Discussions With the framework presented in the previous section, the simulation results and analysis on structural, electronic, and mechanical properties of fluorite and pyrite TiO2 are presented in this section in detail. This simulations are carried out in a HP workstation with 12 GB RAM (Z380) using Virtual Nano Lab software Atomistix Tool Kit (ATK) [23-25] from Synopsys Quantumwise. 3.1 Structural Properties First of all, a systematic structured optimization has been carried out for both phases of TiO2 over a wide range of lattice parameters. As all the physical properties are related with the total energy, so those lattice constants which have a minimum total energy are considered as the equilibrium lattice constants. If the total energy is obtained, then any of the physical properties related to the total energy can be easily calculated. The geometrical configuration for = 1.97 plus two O ions at a far pyrite phase is given with the distance between Titanium and oxygen atoms = 2.57 and the distance between two titanium atoms given by = 3.42 . distance given with = 2.07 plus two O ions at far with distance = 2.39 and the distance Similarly, for fluorite phase = 3.38 . All the values shown in Figure – 2 (a) and Figure -2 (b). between any titanium atoms is

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3.2 Electronic Properties To illustrate the electronic properties, it is focussed primarily on two major parameters, namely, the band structure and total density of state (TDOS). The band structure usually gives a detail idea about electronic and optical properties of both the TiO2 crystals. The energy band diagram as calculated using LDA for both fluorite TiO2 and pyrite TiO2 are shown in Figure 3 (a) and (b). It can be seen from Figure 3 (a) that the Fermi level lies at the lower edge of the conduction band in the fluorite structured TiO2 and has a direct band gap with a simulated value of band-gap energy of 0.89 eV which is the characteristics of a narrowband semiconductor material. The top of valence band i.e. Valence Band Maximum (VBM) and the bottom of conduction band i.e. Conduction Band Minimum (CBM) lies near to Γ point of optimized maximum peak of crystal i.e. Brillouin Zone (BZ). The estimated value of the band gap energy is found to be very close to [8], while compared with some other work [26] the result is contradictory too. The bandgap as calculated by [26] is indirect in nature and is found to be 1.061 eV and 1.16 eV respectively. From the simulation, the band gap values found for fluorite TiO2 vary from 0.89 eV to 1.79 eV. The energy band gap values as found in this study is ~ 20% lesser than DFT- GGA based simulation [24] and ~16% less than [8]. LDA provides mostly the direct band gap whereas GGA provides indirect band gap in most of the simulation studies [8, 24, and 26]. Thus it can be inferred that the simulated value of energy band gap so obtained here in this study is quite close to other established approaches. The simulated energy band diagram for pyrite type TiO2 shown in Figure 3 (b). It can be seen from the figure that the energy band gap has a quite similar behaviour as with fluorite type TiO2 and exhibits an indirect band gap transition with an energy of 1.18 eV. The top of valence band i.e. Valence Band Maximum (VBM) lies near to Γ and the bottom of conduction band i.e. Conduction Band Minimum (CBM) lies very close to R point of BZ. According to [8, 24, and 26], the indirect band gap transition value varies from 1.438 eV to 1.451 eV. Here, the simulated band gap values are 17.94%- 20.27% lower than other simulated results. This is an inherent drawback of DFT and associated difference in crystal structure. Another most important material property is the Density of State (DOS) which also describes the electronic property of a material. In general, it shows the energy representation for describing molecular dynamics and spectroscopy. In other words, the density of states (DOS) is a measure of number of electron (or hole) states per unit volume at a given energy. The total density of state of fluorite TiO2 and pyrite TiO2 are illustrated in Figure 4 (a) and (b). From the Figure 5, it can be seen that, the lowest valence band occur at -20 eV for fluorite TiO2 whereas for pyrite TiO2, it starts well before -20 eV. The valence band between -20 eV to -15 eV in case of fluorite TiO2 is mainly contributed by O-s state whereas, in the energy range between -7.5 eV and 0 eV, it is mainly contributed by O-p state. However, for Pyrite TiO2, the valence band range >-20 eV and -13eV which is mainly contributed from O-s state whereas, from -10 eV to 0 eV, it is contributed by O-p state. The highest occupied valence bands for fluorite are essentially dominated by O-1s and Ti-2p states. Ti-1s state is also contributing to the valence band but density of this state is so small compared to O-1s and Ti-2p states. In the case of pyrite, the highest occupied valence bands are dominated by O-1s and Ti-1s states. A little contribution is also seen from Ti-2p and Ti-3d orbitals towards valence band formation. The lowest occupied conduction band just above Fermi energy level is mainly contributed by Ti-3d and O-2p states for both the crystals. Other orbital atoms also contributes but the values of density of these states are negligibly small compared to Ti-3d and O-2p orbital atoms. 3.3 Elastic Properties Many information regarding material properties can be obtained from the elasticity matrix. Generally, it provides bonding character between adjacent atomic planes, the bonding anisotropy, stability of the structure and stiffness of the material. So their ab-initio calculation requires precise ways of computation. Basically, the forces and elastic constants are the functions of first-order and second order derivatives of the potentials [27-30]. , assumptions made for small lattice distortions in order to Here, to compute the independent elastic constants remain within the elastic domain of the crystal. As both the crystals are cubic in nature, it has only 3 independent

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elastic constants such as , , [26]. With the help of these three elastic independent constants, mechanical stability can be determined. Initially, positive definiteness of the stiffness matrix should be checked before proceeding to further calculations. The conditions are as follows [31, 32]: > | |, > 0, (1) > 0, >0 +2 If all the above conditions are satisfied, then the material is mechanically stable. The Bulk modulus and Shear modulus is calculated using two different theories, such as Reuss theory and Voigt theory. and are used here to represent Bulk and Shear modulus applying Reuss theory [33] whereas, and represents Bulk and Shear modulus applying Voigt theory [34]. From the Reuss and are expressed as: approximation, = 3 +6 (2) =5 4 −4 +3 (3) Where , and is the compliance matrix values of both phases of TiO2 materials. and are expressed as: Again, using the Voigt theory, = (4) =

(5)

Table 1: Structural, electronic, elastic and optical constants of cubic TiO2 using the OLCAO and other exchange correlation methods used with LDA. Column 1 indicates different phases of cubic TiO2, column 2 indicates specific method of simulation, column 3 indicates about optimized lattice constant ‘a’ in Å3, column 4 indicates optimized volume, column 5 indicates bulk modulus ‘B ’ in GPa, column 6 indicates bandgap ‘E ’ given in eV, and finally last column indicates the referred article.

Phases

Fluorite

 Fm3m 

Pyrite

 Pa 3 

Methods

a (Å)

Vol. (Å3)

B (GPa)

E (eV)

OLCAO-PZ

4.787

109.696

296.2

0.89 (D)

LCAO-LDA

4.75

107.04

308

LDA-CAPZ

4.739

106.429

285

1.065(D)

642

[8]

LDA-CASTEP

4.749

107.104

289, 266

1.08(D)

298

[9]

PP-PW-PBE

4.762

107.986

EXPERIMENT

4.870

115.5

202

LDA

4.7412

106.577

OLCAO-PZ

4.844

LCAO-LDA

Reference 431.03

This Study [6]

1.042 (I) 1.79(D), 1.04(I)

[11]

284.5

1.061(I)

[25]

113.661

270.83

1.18(ID)

4.8

110.66

273

LDA-CAPZ

4.807

111.1

276.43

CASTEP-LDA LCAO-LDA LDA LDA

4.805 4.800 4.813 4.809

110.95 110.66 111.49 111.215

304 273 279.7

[13]

313.47

This study [6]

1.451(ID)

1.44(ID) 1.438(ID)

333

[8]

328

[8] [9] [13] [25]

299.2

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(a)

(b)

Figure 3 (a) Band Diagram for Fluorite TiO2 (b) Band Diagram for Pyrite TiO2

Figure 4 (a) Density of States for Fluorite TiO2 (b) Density of States for Pyrite TiO2 Hill (1952) [35] in this context suggested that the actual effective elastic moduli of anisotropic crystalline material should be approximated by the arithmetic mean of Bulk and Shear moduli. As per the Hill approximation, the Bulk and Shear modulus are given as: modulus =

(6)

=

(7)

It is well known that, Bulk modulus and Shear modulus are measure of the hardness of any crystalline solid. The Bulk modulus is a measure of resistance to volume change by some applied pressure. However, Shear modulus is a measure of resistance to reversible deformations upon Shear stress. Thus, Shear modulus is better predictor of . As much large will be, hardness than the Bulk modulus. Generally, Shear modulus mainly depends on larger will be shear modulus . of pyrite TiO2 is larger because the deformity of Oxygen atoms from cubic to rhombohedral which largely enhances the rigidity against the shape deformations at some directions. If Shear modulus is larger, the material is harder. Thus, the Shear modulus is a measure of resistance to the shape change and is more pertinent to hardness. Here, for fluorite structured TiO2, the Bulk modulus is found out to be 296.2 whereas, Shear modulus is 68.126. However, for pyrite TiO2, Bulk modulus is 270.83 but Shear modulus is about 100 which

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is quite large compared to fluorite type TiO2. So pyrite is much harder than fluorite. Later, using Vickers hardness, it is also proved that, pyrite is harder than fluorite. All the calculated values are listed in Table 1. Young’s modulus can be defined as the ratio of stress and strain, which basically is used to calculate the measure of stiffness of the solid. If the Young’s modulus is high, then the material is stiffer. The Young’s modulus also has an impact on the ductility. When it increases, the covalent nature of the material increases. Direct calculation of Young’s modulus from Bulk and Shear modulus is avoided at the time of simulation due to existence of many different alternative way of calculation. Here the Young’s modulus is calculated from compliance matrix directly, as it is not influenced by any conventions. Thus it is given as: =

ℎ

=

, ,

(8)

Where is the inverse of the elastic constants matrix which is also known as elastic compliance matrix. The calculated values of Young’s modulus is presented in Table 1. 3.4 Optical Properties Here, main focus is on the unpolarised light wave, as such the direction of this electric field varies randomly with and complex dielectric time. The optical properties can be calculated considering real dielectric constant, constant, using the following equation: = + (9) As TiO2 is considered as a super - photo catalytic material, a detailed investigation is needed to know its future applications. This can be accomplished if study on the dielectric properties of it is carried out. As in Eq. (9), the imaginary part, that is the complex dielectric constant ε2, consists of two parts: one due to intraband transition of the incident electromagnetic (EM) wave and the other due to interband transition of the incident EM wave. For metals, intraband transition was considered whereas interband transition is for semiconducting materials. Again, the interband transition is of two types, direct band and indirect band transitions. Due to little contribution towards dielectric function, indirect interband transitions can be neglected, although it provides information regarding electron-phonon scattering. The direct interband transitions contributes mainly to the dielectric function (imaginary part ) which can be found out from the momentum matrix between occupied and unoccupied wave functions. For evaluating real and imaginary parts of the dielectric function, the Kubo- Greenwood formalism has been used [36]. Mathematically, it can be expressed as: =−

ℏ

∑

ℏ

ℏ

∏

∏

0

(10)

component of the dipole matrix element between state and , v is the volume, Γ is the where ∏ is the broadening factor, and the fermi function. All the optical properties shown in Fig. 5 (a – j) are obeying Drude theory [37] which directly related with the real and imaginary parts of optical properties for both the cubic structures. The real part of dielectric function (ε1) shows how much a material becomes polarized when an electric field is applied. This is due to creation of electric dipoles in the material. From Fig. 5 (a), a major peak occurring at ~2 eV can be observed for the fluorite TiO2. At this point, electron transition occurs at the Ti 3d states in the CB minimum to O 2p states in the VB maximum. The observed dielectric constant is 4.7 eV for fluorite type TiO2. Similarly, it can be seen that from Fig. 5 (b) for the pyrite TiO2 real dielectric constant, it can be seen that, there is two peak points situated at ~1.8 eV and 2 eV. The first peak is weak and corresponds to the Ti 3p and O 2s states in the lower valence band whereas the second peak is the highest occurring point which is caused due to transition of electron from conduction band minimum at Ti 3d states and valence band maximum at O 2p states. For pyrite, the observed dielectric constant is lower than fluorite structure having value of 3.5 eV. of the dielectric constant as in The absorption property of any material can be known from the imaginary part Eq. (9). If this value is 0, then the material is transparent. When absorption begins, this value becomes non-zero. The peak absorption occurs at 2.1 eV which is due to direct interband transitions as well as direct excitons of the fluorite structured TiO2 whereas for pyrite structured TiO2, the peak occurs at 2.45 eV. The imaginary part of dielectric constant for both the structure shown in Fig. 5 (c) and (d) respectively.

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Figure 5 (a) Real part of Fluorite TiO2 (b) Real part of Pyrite TiO2 (c) Imaginary part of Fluorite TiO2 (d) Imaginary part of Pyrite TiO2 (e) Refractive Index of Fluorite TiO2 (f) Refractive Index of Pyrite TiO2 (g) Extinction coefficient of Fluorite TiO2 (h) Extinction coefficient of Pyrite TiO2 (i) Reflectivity of Fluorite TiO2 (j) Reflectivity of Pyrite TiO2

The calculated values of dielectic constants as obtained from equation (10) also validate the equation for dielectric constant in terms of refraction [38] as given below: (11) = where ƞ is the index of refraction. Refractive Index is a dimensionless quantity, which describes, how light propagates through that medium. It is a function of photon energy and it can be given as: = + (12) where n = the complex refractive index η = the refractive index k = the extinction co- efficient When light passes through a material, it interacts with the constituent molecules or atoms. These interactions has a direct effect on the bending of light. Further, light causes a change in temperature of the material, due to which refractive index changes. As temperature increases, the interaction of molecules decreases. From Fig. 5 (e) and (f), it can be seen that the calculated value of refractive index is 2.16 for the fluorite TiO2 and 1.9 for pyrite TiO2 respectively. The refractive index for other type of exchange correlation methods given in Table 1. When light goes from one medium to another medium, it is called refracted light. As per Snell’s law states, if light moves from a

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medium with refractive index n1 to another medium having refractive index n2 with incident angle to the surface normal of , the refraction angle can be given as: = (13) As we know, when light enters a material with higher refractive index, the angle of refraction gets smaller compared to angle of incidence and the light will be refracted towards the normal of the surface. If any material has higher refractive index, closer to the normal direction of the light will travel and vice versa. With lower refractive index, more and more light absorption takes place. Moreover, due to this light also travels faster. Because of these reasons, efficiency of solar energy conversion increases. So, pyrite type TiO2 is much more efficient compared to fluorite type TiO2. In other words, refractive index is a measure of how much the light is refracted or how much the beam is diverted. In general, the refractive index of air is 1.00. So, larger the difference between the refractive index of air compared to refractive index of the material, more is the change in angle of refraction of light. In this case, the fluorite structure will have more bending effect as compared to pyrite TiO2, because it has more refractive index [39]. The extinction coefficient for both the structures is displayed in Fig.5 (g) and (h). Basically, indicates the amount of absorption loss when the electromagnetic wave propagates through the material. Normally, similar characteristics for and absorption coefficient can be observed. It is found from Fig. 5 (g) that, for fluorite type TiO2, molar absorptivity starts from 0 eV and continues up to 9 eV. Thus in the energy range,0 < ≤9 , fluorite type TiO2 behaves as a transparent material. At 3 eV, rises to its global maximum value, and then gradually falls to its minimum value at around 9 eV. Similarly, for pyrite structured TiO2, molar absorptivity starts from 0 eV and continues up to 7 eV. In the energy range,0 < ≤ 7 , pyrite type TiO2 behaves as a transparent material. At ~2.5 eV, k rises to its global maximum value, and then gradually falls to its minimum value at around 7 eV as seen from Fig. 5 (h). Fig. 5 (i) and (j) shows the reflectivity curve of fluorite and pyrite structures of TiO2. The term reflectivity means when a photon beam is bombarded with the material, some portion of the beam is reflected back or scattered at the interface between the two media even if both materials are transparent in nature. The reflectivity of any material with normal light incidence can be calculated using refractive indices (ƞ) and extinction coefficient (k) by the following Fresnel equation [39]: =

(14)

From Eq. (13), it can be observed that at lower energy, fluorite type TiO2 shows strong reflectivity. For fluorite structure TiO2, the global maximum occurs at 4.15 eV which arises due to inter-band transition. This global maxima occurs at an energy level within the Infrared and Ultraviolet region, from which it can be inferred that, fluorite type TiO2 can be best used as a coating material. This minimized reflectivity is due to a collective plasma resonance. This plasma resonance can be determined by the imaginary part of dielectric constant which also represents the degree of overlapping between the inter-band absorption regions. When total energy increases towards higher range, fluorite material shows no reflectivity. Similarly, for pyrite structure TiO2, a strong reflectivity can be observed in the energy range 0 – 5 eV. The maximum reflectivity also happens around 4.7 eV. In pyrite type TiO2 also reflectivity lies within IR-visible-UV region, due to which it is also suitable as coating material. 4. Conclusion A simulation based study is presented here on the structural, electronic, and mechanical as well as frequency dependent dielectric properties within the dipole approximation for fluorite and pyrite TiO2 material using OLCAO basis sets under the framework of DFT. Most results are compared with previously calculated LDA works and found competitive results. The calculated lattice parameters of this binary compound is in good agreement with experimental data with a deviation of less than 1%. Moreover, in this work, investigation on the dielectric properties with LDA found to be very much closer to other researchers calculated values. Some underestimated value for some of the dielectric properties can be observed because a single exchange – correlation potential is not continuous across the band gap. By following two individual ways, error can be made less. First one is to implement Green’s function

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instead of density functional theory. The Green’s function can help to study self-energy of quasi particles in a many particle system. The second way can be, applying of self-interaction correction (SIC). Here, self-interaction in the Hartree term is removed by an orbital-by-orbital correction to the exchange-correlation potential. From the results, it has been observed that the dielectric functions agree with experimental values in case of both the cubic -TiO2 with a minimal error. These underestimation can be avoided if better optimization algorithms are used. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]

[39]

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