**4.1. Computational**

The calculations were done using the basis set of Lan2DZ (Los Alamos National Laboratory dupla zeta) [45–47] at the density functional theory (DFT) level. In the DFT calculations, we have employed the Becke's 1988 functional [48] using the LYP (Lee-Yang-Parr) correlation functional [49] as implemented in the Gaussian 98 program [50].

**3. Revision of basis sets and theoretical methods used to study the**

The literature reports different methods and basis sets built by GCHF method for investigation of perovskites' piezoelectric effects. The RHF (Restricted Hartree-Fock) method together with the 17s11p7d/11s6p6d/5s3p1d GTOs basis was used for investigation of perovskites' piezo‐ electricity of lanthanum manganite [38]. Besides, the ROHF (Restricted Open-shell Hartree-Fock) method and the 14s7p7d/11s7p7d/9s7p1d GTOs basis set allowed the calculations of electronic structure of yttrium manganite [39]. After, studies of piezoelectric effects were done in praseodymium manganite using ROHF method and 18s12p5d3f/9s6p4d/9s5p1d GTOs basis set [40], in BaTiO3 using HF/16s9p5d/10s5p4d/6s4p1d GTOs method/basis set [41], as well as in lanthanum ferrite using HF/19s14p8d/14s9p7d/7s6p1d GTOs approximation/basis set [33]. Calculations for theoretical investigation of piezoelectric effect in samarium titanate and in yttrium ferrite were developed using DKH level (Douglas-Kroll-Hess second-order scalar relativistic) with 17s12p8d4f/10s6p3d/5s4p1d GTOs [42] and 14s9p8d/14s8p6d/6s4p1d [37] basis sets, respectively. On the other hand, quantum chemical studies for piezoelectricity of yttrium titanate and gadolinium niquelate using DKH approximation were been reported in literature. For the yttrium titanate, the 16s10p7d/11s6p5d/6s5p1d GTOs basis set was used [43], while, for gadolinium niquelate, the used basis set was 20s14p10d6f/13s8p7d/6s4p1d GTOs

In the developed studies for piezoelectricity investigation, standard basis sets from literature mostly showed inefficient for theoretical description of studied perovskites, and, therefore, the electronic structure description of these materials is inadequate for a satisfactory interpretation of this effect. For this reason, we recommend the use of GCHF method strategy; even standard basis sets are inappropriate to describe the geometric parameters of studied perovskites under the piezoelectric effect perspective. Furthermore, using GCHF method basis sets allows

**4. Investigation of piezoelectricity in perovskites using standard basis sets**

In this section, we investigate the piezoelectricity in BaTiO3 and LaTiO3 perovskites. Initially, we apply the methodology for the BaTiO3 to verify if the results show the piezoelectricity property, once it is known this perovskite has it. After, we apply the same methodology for LaTiO3 and the results are analyzed to check the property. **Figure 2** shows crystallographic

The calculations were done using the basis set of Lan2DZ (Los Alamos National Laboratory dupla zeta) [45–47] at the density functional theory (DFT) level. In the DFT calculations, we

choosing the ideal basis set to better describe the studied polyatomic environment.

**in barium titanate (BaTiO3) and lanthanum titanate (LaTiO3)**

**piezoelectric effect in perovskites**

70 Piezoelectric Materials

[44].

units of studied perovskites.

**4.1. Computational**

For the study of the crystalline 3D periodic BaTiO3 [51] and LaTiO3 [52] systems (**Figure 2**), it is necessary to choose a fragment (or a molecular model), which represents adequately a physical property of the crystalline system as a whole.

**Figure 2.** Crystallographic units BaTiO3 (a), features seen, O atoms (red), Ti atoms (white), and Ba atoms (brown) and LaTiO3 (b), features seen, O atoms (red), Ti atoms (white), and La atoms (blue), from which the fragments were extract‐ ed for study.

**Figure 3** shows the molecular models used to simulate the necessary conditions of the existence of piezoelectricity in BaTiO3 [51] and LaTiO3 [52] as full solids. The [BaTiO3]2 and [LaTiO3] fragments were chosen because, after its optimization, the obtained structural parameters (interatomic distances) were close to experimental values with very good precision. For the two systems, the Ti is located in the center of the octahedron being wrapped up for six O atoms, disposed in the reticular plane (2 0 0), and two Ba or La atoms arranged in the reticular plane (1 0 0).

In this study, the following strategy was used: (i) initially, it was made the geometry optimi‐ zation of the [BaTiO3]2 and [LaTiO3]2 fragments in the Cs symmetry and 1A' electronic state; (ii) at last, with the geometry optimized according to the descriptions presented in **Figure 3**, single-point calculations were developed.

In **Figure 3**, (a) represents the [BaTiO3]2 fragment with the Ti atom fixed in the space and Ti atom being moved 0.003 Å in the direction to O1, O2, O3, O4, O5, and O6 atoms and Ba atom and all O atoms are fixed; (b) represents the [BaTiO3]2 fragment with the bond lengths Ti─O1, Ti─O6 shortened from 0.003 Å; (c) represents the [LaTiO3]2 fragment with the Ti atom fixed in the space and Ti atom being moved 0.005 Å in the direction to O1, O2, O3, O4, O5, and O6 atoms and La atom and all O atoms are fixed; and (d) represents the [LaTiO3]2 fragment with the bond lengths Ti─O1, Ti─O6 shortened from 0.005 Å.

**Figure 3.** (a) represents the [BaTiO3]2 fragment with the Ti atom fixed in the space and Ti atom being moved 0.003 Å in the direction to O1, O2, O3, O4, O5, and O6 atoms and Ba atom and all O atoms are fixed; (b) represents the [BaTiO3]2 fragment with the bond lengths Ti─O1 and Ti─O6 shortened from 0.003 Å; (c) represents the [LaTiO3]2 fragment with the Ti atom fixed in the space and Ti atom being moved 0.003 Å in the direction to O1, O2, O3, O4, O5, and O6 atoms and La atom and all O atoms are fixed; and (d) represents the [LaTiO3]2 fragment with the bond lengths Ti─O1 and Ti─O<sup>6</sup> shortened from 0.003 Å.

#### **4.2. Results and discussion**

**Table 2** shows the theoretical (calculated) bond lengths and the experimental values from literature [51] for BaTiO3. The theoretical values are closer to the experimental data. The deviations between the theoretical and literature values are 6.32 × 10−2 and 6.39 × 10−2 Å for Ti─O1 and Ba─O1, respectively.


**Table 2.** Experimental and theoretical geometric parameters obtained for BaTiO3 by [BaTiO3]2 fragment optimization.

**Table 3** presents the total energy of [BaTiO3]2 fragment. As it mentioned previously, the cal‐ culations are at atomic positions: Ti is fixed in space, Ti is moved toward the O1 atom and Ba atom and the O other atoms are fixed, Ti is moved toward the O2 atom and Ba atom and the O other atoms are fixed, and so on. The results in **Table 3** show that when the Ti atom is displaced relative to the fixed position, the fragment is 4.32 × 105 , 8.20 × 10−5, 1.08 × 10−4, 8.38 × 10−4 , 8.54 × 10−4, 7.39 × 10−4 hartree more stable, indicating the Ti+4 central ion is not centersymmetric. Also we can see that decreasing the Ti─O1 and Ti─O<sup>6</sup> bond lengths (me‐ chanical stress), the energy calculation shows less stable fragment compared to the system without mechanical stress (Ti fixed in space).


**Table 3.** Total energy of [BaTiO3]2 fragment.

**Figure 3.** (a) represents the [BaTiO3]2 fragment with the Ti atom fixed in the space and Ti atom being moved 0.003 Å in the direction to O1, O2, O3, O4, O5, and O6 atoms and Ba atom and all O atoms are fixed; (b) represents the [BaTiO3]2 fragment with the bond lengths Ti─O1 and Ti─O6 shortened from 0.003 Å; (c) represents the [LaTiO3]2 fragment with the Ti atom fixed in the space and Ti atom being moved 0.003 Å in the direction to O1, O2, O3, O4, O5, and O6 atoms and La atom and all O atoms are fixed; and (d) represents the [LaTiO3]2 fragment with the bond lengths Ti─O1 and Ti─O<sup>6</sup>

**Table 2** shows the theoretical (calculated) bond lengths and the experimental values from literature [51] for BaTiO3. The theoretical values are closer to the experimental data. The deviations between the theoretical and literature values are 6.32 × 10−2 and 6.39 × 10−2 Å for

**Bond length (Å) Theoretical (this work) Experimental [51] Δ**

─O<sup>1</sup> 1.93431 1.99750 6.32 × 10−2 Ba─O<sup>1</sup> 2.77480 2.83871 6.39 × 10−2

**Table 2.** Experimental and theoretical geometric parameters obtained for BaTiO3 by [BaTiO3]2 fragment optimization.

shortened from 0.003 Å.

72 Piezoelectric Materials

Ti

**4.2. Results and discussion**

Ti─O1 and Ba─O1, respectively.

Δ = |Theoretical − experimental|


**Table 4.** Total atomic charges and dipole moment of the [BaTiO3]2 fragment.

**Table 4** shows the values of the total charges of the atoms for Ti atom fixed in space and for the fragment under the influence of mechanical stress and the dipole moments, respectively. Also, it shows the rearrangement of the charges in all atoms caused by a mechanical stress comparing with the Ti position fixed in space. As well as we can notice the change in the dipole moment resulting from this mechanical stress. The rearrangement of the charges and the variation of the dipole moment can lead us to suppose that the decrease of Ti─O<sup>1</sup> and Ti─O6 chemical bond lengths provokes a polarization of the [BaTiO3]2 fragment, indicating the nature of Ti─O and Ba─O chemical bonds was changed.

For the [BaTiO3]2 fragment with Ti atom fixed in space, the HOMO (Highest Occupied Molecular Orbital) and the LUMO (Lowest Unoccupied Molecular Orbital)are represented, respectively, by

$$\begin{aligned} \text{HOMO} &= 0.63 \, \text{(2p}\_x\text{)} \, \text{O(1)} - 0.46 \, \text{(2p}\_x\text{)} \, \text{O(2)} + 0.15 \, \text{(2p}\_x\text{)} \, \text{O(3)} + 0.16 \, \text{(2p}\_x\text{)}\\ \text{O(4)} + 0.57 \, \text{(2p}\_x\text{)} \, \text{O(5)} - 0.21 \, \text{(2p}\_x\text{)} \, \text{O(6)} \end{aligned}$$

$$\begin{aligned} \text{LUMO} &= +0.17 \left( 2\text{p}\_{\text{x}} \right) \text{O} \text{(1)} + 0.33 \left( 2\text{p}\_{\text{y}} \right) \text{O} \text{(1)} + 0.12 \left( 2\text{p}\_{\text{z}} \right) \text{O} \text{(1)} + 0.42 \left( 2\text{p}\_{\text{x}} \right) \text{O} \text{(2)} + 0.41 \left( 2\text{p}\_{\text{x}} \right) \text{O} \text{(2)} \\ \text{O} \text{(3)} + 0.34 \left( 2\text{p} \right) \text{O} \text{(4)} + 0.36 \left( 2\text{p}\_{\text{x}} \right) \text{O} \text{(5)} + 0.56 \left( 2\text{p}\_{\text{z}} \right) \text{O} \text{(6)} \end{aligned}$$

For the fragment [BaTiO3]2 under mechanical stress, the HOMO and the LUMO are written as

$$\text{HOMO} = +0.64\left(2\text{p}\_{\text{y}}\right)\text{O}\left(1\right) - 0.44\left(2\text{p}\_{z}\right)\text{O}\left(2\right) + 0.35\left(2\text{p}\_{z}\text{O}\left(3\right) - 0.12\left(2\text{p}\_{z}\right)\text{O}\left(3\right) + 0.18\left(2\text{p}\_{\text{y}}\right)\text{O}\left(4\right)\right)$$

$$\text{O}\left(4\right) - 0.35\left(2\text{p}\_{z}\right)\text{O}\left(4\right) + 0.16\left(2\text{p}\_{\text{y}}\right)\text{O}\left(5\right) + 0.44\left(2\text{p}\_{z}\right)\text{O}\left(5\right) - 0.34\left(2\text{p}\_{\text{y}}\right)\text{O}\left(6\right)$$

$$\begin{aligned} \text{LUMO} &= +0.18 \, \text{(2p}\_{\text{x}}\text{)} \, \text{O} \, \text{(1)} - 0.31 \, \text{(2p}\_{\text{x}}\text{)} \, \text{O} \, \text{(1)} + 0.41 \, \text{(2p}\_{\text{x}}\text{)} \, \text{O} \, \text{(2)} + 0.42 \, \text{(2p}\_{\text{x}}\text{)} \\ \text{O} \, \text{(3)} + 0.38 \, \text{(2p}\_{\text{x}}\text{)} \, \text{O} \, \text{(5)} + 0.56 \, \text{(2p}\_{\text{x}}\text{)} \, \text{O} \, \text{(6)} \end{aligned}$$

The analysis of the HOMO and the LUMO of the [BaTiO3]2 fragment shows the mechanical stress does not cause appearance of chemical bonds with contributions of barium atom's 4d orbitals. This fact together with the changing of the nature of Ti─O and Ba─O chemical bonds leads us to suggest that electrostatic interactions are very important in electronic structure of [BaTiO3]2 fragment. This is consistent because the repulsive effect of d electrons in both highspin and low-spin octahedral species of ML complexes (M = Metal and L = Ligand), all d electron density will repel the bonding electron density [53]. This shows that the piezoelec‐ tricity in BaTiO3 can be caused by electrostatic interactions.

The experimental [52] and theoretical geometric parameters for LaTiO<sup>3</sup> are shown in **Table 5**. According to **Table 5**, the theoretical results are 2.02413 and 2.58928 Å for Ti─O1 and La─O1, respectively, while the experimental values are 2.01556 and 2.59918 Å. The differences between the theoretical and experimental values are 8.57 × 10−3 and 9.99 × 10−3 Å.


**Table 5.** Experimental and theoretical geometric parameters obtained for LaTiO3 by [LaTiO3]2 fragment optimization.

**Table 6** presents the total energy from [LaTiO3]2 fragment. As mentioned previously, the calculations are at atomic positions: Ti is fixed in space, Ti is moved toward the O1 atom and La atom and the O other atoms are fixed, Ti is moved toward the O2 atom and La atom and the O other atoms are fixed, and so on. According to **Table 6**, the fragment is stable 8.16 × 10−2, 3.36 × 10−2, 5.70 × 10−2, 7.04 × 10−2, 1.72 × 10−2, and 2.87 × 10−2 hartree to the titanium atom moving to the positions O1, O2, O3, O4, O5, and O6, respectively, when fixed in space. Also, for the [LaTiO3]2 fragment, we can note that, with the shortening of Ti─O1 and Ti─O<sup>6</sup> chemical bonding (mechanical stress), the system becomes less stable.


**Table 6.** Total energy of [LaTiO3]2 fragment.

**Table 4** shows the values of the total charges of the atoms for Ti atom fixed in space and for the fragment under the influence of mechanical stress and the dipole moments, respectively. Also, it shows the rearrangement of the charges in all atoms caused by a mechanical stress comparing with the Ti position fixed in space. As well as we can notice the change in the dipole moment resulting from this mechanical stress. The rearrangement of the charges and the variation of the dipole moment can lead us to suppose that the decrease of Ti─O<sup>1</sup> and Ti─O6 chemical bond lengths provokes a polarization of the [BaTiO3]2 fragment, indicating

For the [BaTiO3]2 fragment with Ti atom fixed in space, the HOMO (Highest Occupied Molecular Orbital) and the LUMO (Lowest Unoccupied Molecular Orbital)are represented,

HOMO 0.63 2p O 1 – 0.46 2p O 2 0.15 2p O 3 0.16 2p

LUMO 0.17 2p O 1 0.33 2p O 1 0.12 2p O 1 0.42 2p O 2 0.41 2p

For the fragment [BaTiO3]2 under mechanical stress, the HOMO and the LUMO are written as

= + + + + +

= + +

x x

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

O 4 – 0.35 2p O 4 0.16 2p O 5 0.44 2p O 5 – 0.34 2p O 6

+ +

x x

( ) ( ) ( ) ( ) ( )

O 3 0.38 2p O 5 0.56 2p O 6

tricity in BaTiO3 can be caused by electrostatic interactions.

+ +

zyzy

HOMO 0.64 2p O 1 – 0.44 2p O 2 0.35 2p O 3 – 0.12 2p O 3 0.18 2p

LUMO 0.18 2p O 1 0.31 2p O 1 0.41 2p O 2 0.42 2p

The analysis of the HOMO and the LUMO of the [BaTiO3]2 fragment shows the mechanical stress does not cause appearance of chemical bonds with contributions of barium atom's 4d orbitals. This fact together with the changing of the nature of Ti─O and Ba─O chemical bonds leads us to suggest that electrostatic interactions are very important in electronic structure of [BaTiO3]2 fragment. This is consistent because the repulsive effect of d electrons in both highspin and low-spin octahedral species of ML complexes (M = Metal and L = Ligand), all d electron density will repel the bonding electron density [53]. This shows that the piezoelec‐

The experimental [52] and theoretical geometric parameters for LaTiO<sup>3</sup> are shown in **Table 5**. According to **Table 5**, the theoretical results are 2.02413 and 2.58928 Å for Ti─O1 and La─O1,

= + -+ +

= + + +

( ) ( ) ( ) ( ) ( ) ( ) ( )

zz zz

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

xzx x

y z zz y

xyzx x

the nature of Ti─O and Ba─O chemical bonds was changed.

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

O 3 0.34 2px O 4 0.36 2p O 5 0.56 2p O 6

O 4 0.57 2p O 5 0.21 2p O 6

+ -

+ ++

z z

respectively, by

74 Piezoelectric Materials

**Table 7** shows the total atomic charges and the dipole moment values for the [LaTiO3]2 when the Ti atom is fixed in space and under mechanical stress. In **Table 7**, we can see, when the Ti atom is fixed in space, the La atom presents positive charge as expected. However, with mechanical compression, the La atom receives electrons, presenting negative charge. It can be characterized by appearance of a pair of free electrons in its 5d orbitals. We can also notice a wide variation of charges in oxygen atoms. Therefore, as BaTiO3, LaTiO3 also presents piezoelectric property due to electrostatic effects with strong variation of the dipole moment as shown in **Table 7**.


**Table 7.** Total atomic charges and dipole moment of the [LaTiO3]2 fragment.

For the LaTiO3, the HOMO and the LUMO, when the Ti atom is fixed in space, are, respectively,

$$\begin{aligned} \text{HOMO} &= +0.10 \, 2\text{p}\_{\text{x}}\text{O} \, \text{(3)} + 0.21 \, 2\text{p}\_{\text{x}}\text{O} \, \text{(4)} + 0.42 \, 2\text{p}\_{\text{y}}\text{O} \, \text{(4)} + 0.14 \, 2\text{p}\_{\text{x}}\text{O} \, \text{(4)} + 0.11 \, 2\text{p}\_{\text{x}}\text{O} \, \text{(5)} \\ &+ 0.60 \, 2\text{p}\_{\text{y}}\text{O} \, \text{(5)} + 0.39 \, 2\text{p}\_{\text{y}}\text{O} \, \text{(6)} - 0.34 \, 2\text{p}\_{\text{x}}\text{O} \, \text{(6)} \end{aligned}$$

$$\begin{aligned} \text{LUMO} &= +0.14 \, 2\text{p}\_{\text{x}}\text{O} \text{(2)} - 0.38 \, 2\text{p}\_{\text{y}}\text{O} \text{(2)} - 0.13 \, 2\text{p}\_{\text{x}}\text{O} \text{(3)} + 0.20 \, 2\text{p}\_{\text{x}}\text{O} \text{(4)} - 0.18 \, 2\text{p}\_{\text{y}}\text{O} \text{(4)} \\ &+ 0.62 \, 2\text{p}\_{\text{z}}\text{O} \text{(4)} - 0.15 \, 2\text{p}\_{\text{x}}\text{O} \text{(5)} + 0.15 \, 2\text{p}\_{\text{y}}\text{O} \text{(5)} + 0.53 \, 2\text{p}\_{\text{z}}\text{O} \text{(5)} - 0.14 \, 2\text{p}\_{\text{y}}\text{O} \text{(6)} \end{aligned}$$

For the LaTiO3 fragment under mechanical stress, the HOMO and the LUMO are

$$\text{HOMO} = +0.78 \left(\text{6s}\right) \text{La} + 0.35 \left(\text{5p}\_{\text{x}}\right) \text{La} - 0.12 \left(\text{5p}\_{\text{y}}\right) \text{La} - 0.25 \left(\text{5d}\_{\text{x}}\right)^{2}$$

$$\text{La} - 0.18 \left(\text{5d}\_{\text{x}}\right) \text{La} + 0.41 \left(\text{5d}\_{\text{y}}\right) \text{La} - 0.40 \left(\text{5d}\_{\text{xy}}\right) \text{La}$$

$$\text{LUMO} = +0.37 \left( 2\text{p}\_{\text{y}} \right) \text{O} \left( 1 \right) - 0.27 \left( 6 \text{s} \right) \text{La} - 0.17 \left( 5 \text{p}\_{\text{x}} \right) \text{La} + 0.12 \left( 5 \text{p}\_{\text{y}} \right)$$

$$\text{La} + 0.83 \left( 5 \text{d}\_{\text{yz}} \right) \text{La} + 0.49 \left( 5 \text{d}\_{\text{xz}} \right) \text{La}$$

Analyzing the HOMO and the LUMO of LaTiO3 fragment, with Ti atom fixed in space and under compression, we can notice that, initially, the system does not present contributions of 5d orbital of La atom for the HOMO and the LUMO. Nevertheless, the mechanical stress has led to the appearance of contributions of this orbital (5d) in the HOMO and the LUMO of the fragment. This confirms that these orbitals work as a pair of free electrons causing the appearance of negative charge in lanthanum atom as shown in **Table 7**.
