Quantum Mechanical Approaches for Piezoelectricity Study in Perovskites

Abstract

In this chapter, we show the procedures we have been used to theoretically investigate the piezoelectric effects in perovskites. The construction of extended basis sets using the generator coordinate Hartree-Fock (GCHF) method is shown, as well as the strategies used to contract extended basis sets and to evaluate their quality in molecular calculations. Besides, we show adequate procedures to choice polarization and diffuse functions to best represent the studied crystal. In addition, we also discuss conditions under which GCHF basis sets and standard basis sets from literature can be used to theoretical investigation of piezoelectricity in perovskites. We finalize the chapter presenting and discussing the results for investigations of piezoelectricity with standard basis sets for barium and lanthanum titanates. To conclude, we present evidences that BaTiO3 and LaTiO3 may have piezoelectric properties caused by electrostatic interactions.

1. Introduction

The worldwide growing demand for energy has led to increasing dependence on fossil fuels from few and unstable regions of the globe. Oppositely, clamors for more restrictive environmental regulation have pointed out the development of cleaner power supplies that meet the increasing demand for electricity.

The use of natural gas or their liquid fuels, the first and second generation of biofuel, and, even, hydrogen has been pointed as “clean” power supplies for the growing demand. Other classic solutions cover hydropower, solar, wind, and nuclear supplies which have enforcement power and limitations described by literature.

Piezoelectricity is a good alternative for clean and environmental power supply. It does not generate waste or pollutants because it does not need fuels or additives. Some revolutionary applications of this type of energy can be enumerated: piezoelectric plates in shoe soles can generate energy for charging portable electronic devices; piezoelectric plates in floors can be an alternative for lighting squares or dance clubs; and, even in a futuristic vision, piezoelectric materials can be used in pavements of highways or streets to generate inexpensive energy. This type of power supply can bring benefits to society and to environment because it can be obtained sustainably and be capable to replace other power supplies.

The piezoelectric effects were discovered in 1880 by brothers Pierre and Jacques Currie in quartz crystals. Ever since, piezoelectricity has been leading numerous researches in development of electronic transducer systems. The effect consists basically in conversion of mechanical energy to electrical energy (from Greek term “piezo” for pressure). In 1881, Lippman, using the thermodynamic analysis, predicted the existence of inverse piezoelectric effect, which consists in appearance of material deformation submitted to an electric field.

Two conditions must be presented simultaneously in a crystal to have piezoelectric property. First, it is the presence of uncentersymmetric characteristic in the crystalline structure of the material. Second, it is the existence of a material polarization when submitted under mechanical stress.

The piezoelectric effect is a reversible process for materials which present direct (internal generation of electric charge resulting from applied mechanical stress) and reverse (internal generation of mechanical stress resulting from applied electric field) effects. For example, lead zirconate titanate crystals generate measurable piezoelectricity when static structure is deformed by 0.1% from initial dimension. Oppositely, these same crystals change 0.1% of their static dimensions submitted to an external electric field.

Oxides as perovskites have a general formula ABO₃, where A is a large cation-like alkaline metals, earth metals, and rare earth metals and B is a small cation-like transition metals. Most common perovskites are those where A is a cation of rare earth metal with oxidation state +3 and B is a transition metal with the same valence state [1].

The perovskite structure is the most important piezoelectric crystalline ceramic. This structure is a network of cornered linked oxygen octahedral holes with a large cation filling the dodecahedral holes. The piezoelectric properties in perovskite structure result from uncentersymmetric characteristic, since this physical property is originated from crystal anisotropy.

The first perovskite structure developed was barium titanate (BaTiO₃). The polymorphous forms of BaTiO₃ have been likened displacing the central Ti⁺⁴ ion within its oxygen octahedron toward one, two, and, then, three of the six adjacent oxygen ions as the temperature is lowered. This is a simplification of the actual atomic displacements, but it is a useful first approximation for structure understanding. For a revision about the role of the perovskite structure in ceramic science and technology, see literature [2].

The pragmatic application of theory in science follows two strategies: (i) there is readiness of experimental information of the interested system’s properties (the application’s results for the theory in study of those properties can be confronted to experimental data which will serve as guide to corroborate the applied concepts or to suggest changes) and (ii) experimental data are not available (the resulting forecasts of the application of theory can be used by experimentalists as guide to facilitate the rational money application and the time reduction for the system under investigation).

In last decade, we have been reported in literature a series of articles about theoretical studies of perovskites. Basically, our purpose is to investigate the possible existence of piezoelectric properties in those materials, using developed basis sets for the appropriate environment of their crystals. In our approaches, different theoretical methods have been used and the results suggest or not this property.

In this chapter, we show that the strategies have been used to study piezoelectricity in ceramic materials as perovskites using basis sets obtained by our research group. To obtain extended basis sets, we will show computer details using generator coordinate Hartree-Fock (GCHF) method [3] and the procedures used to contract extended basis sets as well as the strategy used to evaluate their quality in molecular environment. In addition, we will show the supplement of polarization and diffuse functions to best represent the studied crystal environment and the theoretical methods used in our articles in literature. We will also discuss conditions how our obtained basis sets and standard basis sets from packages in literature can be used to develop studies of piezoelectricity in perovskites.

Finally, we will present and discuss the obtained results for investigation of piezoelectricity with standard basis sets for barium and lanthanum titanates and last considerations related to this chapter.

2. The generator coordinate Hartree-Fock method and the construction of basis sets

In this section, we provide a brief history of the scientific scene that gave rise to the GCHF method and the atmosphere in which it has developed. We will also make a presentation of the GCHF method as strategy for building extended and contracted basis sets as well as the procedure used to evaluate their quality and the selection of polarization and diffuse functions used in calculations of perovskites.

2.1. Brief history of scientific scene and atmosphere for the development of GCHF method

In 1957, the generator coordinate (GC) method for the nuclear bound state [4] was introduced in literature. According to this method, the variational trial function is written as an integral transform over a nucleonic wave function and a weight function depending on a parameter (GC), i.e., f(α). In this way, the common variational principle, δE/δα=0 (where E is the total energy of system and α is the GC), a priori, was made more powerful with the requirement δE/δfα, leading to an integral equation. Most applications in nuclear physics relied on the Gaussian overlap approximation (GOA). Although there were some attempts in numerical solution [5], the discussion of the GC method against the background of the Fredholm theory of linear integral equations was reported in literature [6, 7]. Probably the first application of GC to electrons in a molecular system (hydrogen molecule) was reported in literature in the second half of the 1970s [8]. After, the GC method was applied to several model problems, including the He atom, with special emphasis on various aspects of the discretization technique [9]. Besides, further developments in discretization techniques were reported in literature [10, 11].

On the other hand, in the late 1960s, the integral method for atomic and molecular systems was introduced in the literature, closely related to GC [12, 13]. In these applications, explicit forms were chosen for f(α) (as the delta function) leading to a variational treatment for the integration limits. Extensive bibliography on this method was found in literature [14].

The GCHF method was introduced in 1986 [3], and one of the first applications was in the generation of Gaussian- and Slater-type orbitals (GTO and STO) universal basis sets [15–17]. The GCHF method was used to build contracted GTF (Gaussian Type-Function) basis sets for the first- and second-row atoms which were applied in calculations of various properties at the HF, CISD (configuration interaction with single and double excitations), and MP2 (Möller-Plesset perturbation theory to second order) levels for a group of neutral and charged diatomic species [18, 19]. Also in the 1990s, efforts were concentrated on the development of GCHF formalism for molecular systems and the first applications have been focused on building basis for H₂, N₂, and Li₂ [20] and LiH, CO, and BF [21]. Applying GCHF basis sets for calculation of properties of polyatomic systems with the first application being concerned with the study of electronic properties and IR spectrum of high tridymite began in the second half of the 1990s [22]. First-principles (ab initio) calculations of electron affinities of enolates were also performed [23]. Theoretical interpretation of IR spectrum of hexaaquachromium (III) ion, tetraoxochromium (IV) ion, and tetraoxochromium (VI) ion [24] and theoretical interpretation of the Raman spectrum [25] and the vibrational structure of hexaaquaaluminum (III) ion [26] were also conducted with GCHF basis. Process of adsorption of sulfur on platinum (2 0 0) surface [27], infrared spectrum of isonicotinamide [28], and transition metal complexes [29–32] were also studied with GCHF basis sets.

2.2. Construction of extended and contracted basis sets for calculations in perovskites

The GCHF approach is based in choosing the one-electron functions as the continuous superposition:

${\text{φ}}_{\text{i}}(\text{1})\text{=}\int {\text{ψ}}_{\text{i}}\text{(1,}\text{α)}{\text{f}}_{\text{i}}\text{(α)}\text{dα}\text{i=1,}\text{2,}\text{3,}\mathrm{....}\text{n}$E1

where ψ_i are the generator functions (GTOs for the case of perovskites) and f_i are the weight functions (WFs) and ü is the GC.

The φ_i are then employed to build a Slater determinant for the multi-electronic wave function and minimizing the total energy with respect to f_i(α), which arrives to the HF-Griffin-Wheeler (HFGW) equations:

$\int \left[\text{F}(\text{α,}\text{β}){\text{-ε}}_{\text{i}}\text{S}(\text{α,}\text{β})\right]f_{\text{i}}(\text{β})\text{dβ}=\text{0}{}_{\text{i}}=\text{1,}\text{2,}\text{3,}\mathrm{........}\text{m}$E2

where ü_i are the HF eigenvalues and the Fock kernels, F(α,β) and S(α,β) are defined in Refs. [3, 16].

The HFGW equations are integrated numerically through discretization with a technique that preserves the integral character of the GCHF method, i.e., integral discretization (ID). The ID technique is implemented with a relabeling of the GC space [15], i.e.,

$\Omega =\text{In α/A A>1}$E3

with A is a scaling parameter numerically determined. For perovskites A = 6.0.

The new GC Ω space is discretized, by symmetry, in an equally space mesh formed by ü values so that

$\Omega ={\Omega }_{\min }+(\text{k+1})\Delta \Omega \text{k = 1,2,3,}\mathrm{.......}\text{N}$E4

In Eq. (4), N corresponds to the number of discretization points defining the basis set size, Ω_min is the initial point, and ΔΩ is the increment.

The values of Ω_min (lowest value) and the highest value Ω_max=Ω_min+(N − 1)ΔΩ are chosen in order to adequately encompass the integration range of f(Ω). This is visualized by drawing the WFs from preliminary calculations with arbitrary discretization parameters. To illustrate the application of the GCHF method in the choice of basis sets for perovskites, we refer to our publication that investigates the piezoelectricity in LaFeO₃ [33]. Figure 1 shows the respective 2s, 3p, and 5d weight functions for O (³P), Fe (⁵F), and La (²D) atoms.

Figure 1.

2s, 3p, and 5d WFs for O (³P), Fe (⁵D), and La (²D) atoms obtained with (20s14p), (30s19p13d), and (31s23p18d0 Gaussian) basis sets, respectively. Reproduction authorized by authors [33].

In the solution of the discretization of Eq. (2), the (22s14p), (30s19p13d), and (32s24p117d) GTOs basis sets were used to O (³P), Fe (⁵D), and La (²D) atoms, respectively, as defined by the mesh of Eq. (3). The values of Ω_min and Ω_max were selected in order to satisfy the relevant integration range of each WF atom. In Table 1, the discretization parameters (which define the exponents) for the built basis sets are shown.

Symmetry	O			Fe			La
	Ωmin	ΔΩ	N	Ωmin	ΔΩ	N	Ωmin	ΔΩ	N
s	−0.3969	0.122	22	−0.5998	0.114	30	−0.6304	0.115	32
p	−0.4387	0.119	14	−0.2758	0.110	19	−0.3938	0.104	24
d	−	–	–	−0.3771	0.117	13	−0.4532	0.113	17

Table 1.

Discretization parameters (which define the exponents) for O (³P), Fe(⁵D), and La(²D) atoms^a.

^aThe scaling parameter used for s, p, and d symmetries for all atoms studied is equal to A = 6.0. Reproduction authorized by authors [33].

For perovskite calculations, a basis set is usually developed in three stages: (1) construction of the extended basis set in atomic calculations, (2) construction of the contracted basis set using a segmented or general scheme (at this stage, due to increased availability of software access, we use the segmented contraction scheme), and (3) addition of supplementary functions of diffuse and polarization character. The first stage is usually a straightforward task. To illustrate the second step, we consider the segmented contraction [31] of basis sets to O (³F), Fe (⁵D), and La (²D) atoms in LaFeO₃ [33]. The 20s14p basis set to O atom was contracted to 7s6p as follows: 16, 1, 1, 1, 1/9, 1, 1, 1. For Fe atom, the 30s19p13d GTO basis set was contracted to 13s8p6d according to 14, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1/12, 1, 1, 1, 1, 1, 1, 1/8, 1, 1, 1, 1, 1. For La atom, the 32s24p17d GTO basis set was contracted according to 13, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1/10, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1/10, 2, 1, 1, 1, 1, 1.

2.3. Quality evaluation of the basis sets in perovskite calculations

In order to evaluate the quality of the contracted GTO basis sets in perovskite studies, the calculations of total energy, the highest occupied molecular orbital (HOMO) energy and the one level below to highest occupied molecular orbital (HOMO-1) energies for perovskites fragments at the HF level [35], are performed and the results are compared with those obtained from the extended GTOs basis sets. For LaFeO₃, the ²FeO⁺¹ and ¹LaO⁺¹ fragments were studied. Comparison of the calculated values with the contracted and extended GTOs basis sets, respectively, shows differences of 0.2534 and 0.2827 hartree for the total energy and 8.8 × 10⁻⁴ and 1.26 × 10⁻³ hartree for the energy orbital. These values show a very good quality of the contracted GTOs basis sets for the study of properties of perovskite LaFeO₃ [33].

2.4. Supplementation of the basis sets with polarization and diffuse functions for calculations in perovskites

In order to better describe the properties of perovskite systems in the implementation of ab initio calculations, the inclusion of polarization functions in GTO basis sets is necessary. A methodology that has been a good strategy in the choice of polarization function for contracted GTOs bases sets is to extract the polarization function from the own Gaussian primitive basis set using successive calculations for the [ABO₃]₂ fragment for different primitive functions, taking into account the minimum energy criterion. For the [LaFeO₃]₂ fragment, the polarization function was included in the contracted GTO basis set for the O atom, i.e., α_d = 0.30029 [33].

The role of the basis set is a crucial point in ab initio calculations of systems containing transition metals, since the description of the metal atom’s configuration in complex is different from neutral state. In our studies with perovskites, the adequate diffuse functions for supplementation of contracted GTOs basis sets have been selected via one of the following methods: (1) the exponents of the basis sets are ranked according to magnitude and plotted on a logarithmic scale with equally spaced abscissas. Then the extrapolation of curve was done to smaller values of exponents, thereby obtaining exponents for a diffuse function [36] and (2) using the total energy optimization of the ground-state anions of the metals present in perovskite structure [37].

For LaFeO₃, the adequate diffuse functions were chosen using the first methodology described, i.e., for the contracted GTO basis set of Fe and La atoms, the diffuse functions are, respectively, α_s = 0.0138038, α_p = 0.1000000, and α_d = 0.054954; α_s = 0.0125892, α_p = 0.0446683, and α_d = 0.0112201.

The results obtained in the study of the perovskite LaFeO₃ with contracted GTO basis sets constructed with GCHF method strategy are well documented in literature [33].

3. Revision of basis sets and theoretical methods used to study the piezoelectric effect in perovskites

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’ piezoelectricity 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 [44].

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

4. Investigation of piezoelectricity in perovskites using standard basis sets in barium titanate (BaTiO₃) and lanthanum titanate (LaTiO₃)

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 units of studied perovskites.

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].

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 extracted 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 optimization of the [BaTiO₃]₂ and [LaTiO₃]₂ fragments in the C_s 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─O₆ 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.

Bond length (Å)	Theoretical (this work)	Experimental [51]	Δ
Ti─O1	1.93431	1.99750	6.32 × 10^−2
Ba─O1	2.77480	2.83871	6.39 × 10^−2

Table 2.

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

Δ = |Theoretical − experimental|

Table 3 presents the total energy of [BaTiO₃]₂ fragment. As it mentioned previously, the calculations 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─O₆ bond lengths (mechanical stress), the energy calculation shows less stable fragment compared to the system without mechanical stress (Ti fixed in space).

Ti atom position	TE (hartree)
Ti is fixed in space	−534.259022802
Ti is moved toward the O1 atom; Ba and O atoms are fixed	−534.259066114
Ti is moved toward the O2 atom; Ba and O atoms are fixed	−534.259104826
Ti is moved toward the O3 atom; Ba and O atoms are fixed	−534.259130429
Ti is moved toward the O4 atom; Ba and O atoms are fixed	−534.259860527
Ti is moved toward the O5 atom; Ba and O atoms are fixed	−534.259877355
Ti is moved toward the O6 atom; Ba and O atoms are fixed	−534.259762234
Bond lengths Ti─O1 and Ti─O6 are shortened (mechanical stress)	−534.257326207

Table 3.

Total energy of [BaTiO₃]₂ fragment.

	Ti atom position
Atom	Ti is fixed in space	Bond lengths Ti─O1 and Ti─O6 are shortened (mechanical stress)
Ti	+0.885	+0.887
Ba	+1.837	+1.841
O1	−0.636	−0.633
O2	−0.564	−0.563
O3	−0.664	−0.663
O4	−0.291	−0.295
O5	−0.296	−0.300
O6	−0.271	−0.274
Dipole moment (Debye)
μx	23.28	−23.51
μy	−0.3622	0.1374
μz	−2.0703	0.0585
μ	23.38	23.51

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─O₁ 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

$\mathrm{HOMO}=0.63\left(2{\mathrm{p}}_{\mathrm{z}}\right)\mathrm{O}\left(1\right)–0.46\left(2{\mathrm{p}}_{\mathrm{z}}\right)\mathrm{O}\left(2\right)+0.15\left(2{\mathrm{p}}_{\mathrm{z}}\right)\mathrm{O}\left(3\right)+0.16\left(2{\mathrm{p}}_{\mathrm{z}}\right)\mathrm{O}\left(4\right)+0.57\left(2{\mathrm{p}}_{\mathrm{z}}\right)\mathrm{O}\left(5\right)-0.21\left(2{\mathrm{p}}_{\mathrm{z}}\right)\mathrm{O}\left(6\right)$E5

$\mathrm{LUMO}=+0.17\left(2{\mathrm{p}}_{\mathrm{x}}\right)\mathrm{O}\left(1\right)+0.33\left(2{\mathrm{p}}_{\mathrm{y}}\right)\mathrm{O}\left(1\right)+0.12\left(2{\mathrm{p}}_{\mathrm{z}}\right)\mathrm{O}\left(1\right)+0.42\left(2{\mathrm{p}}_{\mathrm{x}}\right)\mathrm{O}\left(2\right)+0.41\left(2{\mathrm{p}}_{\mathrm{x}}\right)\mathrm{O}\left(3\right)+0.34\left(2\mathrm{p}\mathrm{x}\right)\mathrm{O}\left(4\right)+0.36\left(2{\mathrm{p}}_{\mathrm{x}}\right)\mathrm{O}\left(5\right)+0.56\left(2{\mathrm{p}}_{\mathrm{x}}\right)\mathrm{O}\left(6\right)$E6

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

$\mathrm{HOMO}=+0.64\left(2{\mathrm{p}}_{\mathrm{y}}\right)\mathrm{O}\left(1\right)–0.44\left(2{\mathrm{p}}_{\mathrm{z}}\right)\mathrm{O}\left(2\right)+0.352{\mathrm{p}}_{\mathrm{z}}\mathrm{O}\left(3\right)–0.12\left(2{\mathrm{p}}_{\mathrm{z}}\right)\mathrm{O}\left(3\right)+0.18\left(2{\mathrm{p}}_{\mathrm{y}}\right)\mathrm{O}\left(4\right)–0.35\left(2{\mathrm{p}}_{\mathrm{z}}\right)\mathrm{O}\left(4\right)+0.16\left(2{\mathrm{p}}_{\mathrm{y}}\right)\mathrm{O}\left(5\right)+0.44\left(2{\mathrm{p}}_{\mathrm{z}}\right)\mathrm{O}\left(5\right)–0.34\left(2{\mathrm{p}}_{\mathrm{y}}\right)\mathrm{O}\left(6\right)$E7

$\mathrm{LUMO}=+0.18\left(2{\mathrm{p}}_{\mathrm{x}}\right)\mathrm{O}\left(1\right)-0.31\left(2{\mathrm{p}}_{\mathrm{z}}\right)\mathrm{O}\left(1\right)+0.41\left(2{\mathrm{p}}_{\mathrm{x}}\right)\mathrm{O}\left(2\right)+0.42\left(2{\mathrm{p}}_{\mathrm{x}}\right)\mathrm{O}\left(3\right)+0.38\left(2{\mathrm{p}}_{\mathrm{x}}\right)\mathrm{O}\left(5\right)+0.56\left(2{\mathrm{p}}_{\mathrm{x}}\right)\mathrm{O}\left(6\right)$E8

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 high-spin 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 piezoelectricity in BaTiO₃ can be caused by electrostatic interactions.

The experimental [52] and theoretical geometric parameters for LaTiO₃ 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.

Bond length (Å)	Theoretical (this work)	Experimental [52]	Δ
Ti─O1	2.02413	2.01556	8.57 × 10^−3
La─O1	2.58928	2.59918	9.99 × 10^−3

Table 5.

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

Δ = |Theoretical − experimental|

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─O₆ chemical bonding (mechanical stress), the system becomes less stable.

Ti atom position	TE (hartree)
Ti is fixed in space	−540.247808392
Ti is moved toward the O1 atom; La and O atoms are fixed	−540.329458501
Ti is moved toward the O2 atom; La and O atoms are fixed	−540.281387651
Ti is moved toward the O3 atom; La and O atoms are fixed	−540.304861376
Ti is moved toward the O4 atom; La and O atoms are fixed	−540.318198767
Ti is moved toward the O5 atom; La and O atoms are fixed	−540.265005660
Ti is moved toward the O6 atom; La and O atoms are fixed	−540.276475167
Bond lengths Ti─O1 and Ti─O6 are shortened (mechanical stress)	−532.224697936

Table 6.

Total energy of [LaTiO₃]₂ fragment.

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.

	Ti atom position
Atom	Ti is fixed in space	Bond lengths Ti─O1 and Ti─O6 are shortened (mechanical stress)
Ti	+0.967	+2.229
La	+1.451	−0.212
O1	−0.492	−0.691
O2	−0.010	+4.16
O3	−0.553	−0.527
O4	−0.267	−1.803
O5	−0.297	−1.766
O6	−0.799	−1.390
Dipole moment (Debye)
μx	25.71	34.67
μy	2.806	40.92
μz	0.716	25.24
μ	25.87	59.29

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,

$\mathrm{HOMO}=+0.102{\mathrm{p}}_{\mathrm{z}}\mathrm{O}\left(3\right)+0.212{\mathrm{p}}_{\mathrm{x}}\mathrm{O}\left(4\right)+0.422{\mathrm{p}}_{\mathrm{y}}\mathrm{O}\left(4\right)+0.142{\mathrm{p}}_{\mathrm{z}}\mathrm{O}\left(4\right)+0.112{\mathrm{p}}_{\mathrm{x}}\mathrm{O}\left(5\right)+0.602{\mathrm{p}}_{\mathrm{y}}\mathrm{O}\left(5\right)+0.392{\mathrm{p}}_{\mathrm{y}}\mathrm{O}\left(6\right)–0.342{\mathrm{p}}_{\mathrm{x}}\mathrm{O}\left(6\right)$E9

$\mathrm{LUMO}=+0.142{\mathrm{p}}_{\mathrm{x}}\mathrm{O}\left(2\right)-0.382{\mathrm{p}}_{\mathrm{y}}\mathrm{O}\left(2\right)-0.132{\mathrm{p}}_{\mathrm{z}}\mathrm{O}\left(3\right)+0.202{\mathrm{p}}_{\mathrm{x}}\mathrm{O}\left(4\right)-0.182{\mathrm{p}}_{\mathrm{y}}\mathrm{O}\left(4\right)+0.622{\mathrm{p}}_{\mathrm{z}}\mathrm{O}\left(4\right)–0.152{\mathrm{p}}_{\mathrm{x}}\mathrm{O}\left(5\right)+0.152{\mathrm{p}}_{\mathrm{y}}\mathrm{O}\left(5\right)+0.532{\mathrm{p}}_{\mathrm{z}}\mathrm{O}\left(5\right)-0.142{\mathrm{p}}_{\mathrm{y}}\mathrm{O}9\left(6\right)$E10

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

$\mathrm{HOMO}=+0.78\left(6\mathrm{s}\right)\mathrm{L}\mathrm{a}+0.35\left(5{\mathrm{p}}_{\mathrm{x}}\right)\mathrm{L}\mathrm{a}-0.12\left(5{\mathrm{p}}_{\mathrm{y}}\right)\mathrm{L}\mathrm{a}-0.{{25(5\mathrm{d}}_{\mathrm{z}}}^2)\mathrm{L}\mathrm{a}-0.18\left(5{\mathrm{d}}_{\mathrm{x}\mathrm{z}}\right)\mathrm{L}\mathrm{a}+0.41\left(5{\mathrm{d}}_{\mathrm{y}\mathrm{z}}\right)\mathrm{L}\mathrm{a}–0.40\left(5{\mathrm{d}}_{\mathrm{x}\mathrm{y}}\right)\mathrm{L}\mathrm{a}$E11

$\mathrm{LUMO}=+0.37\left(2{\mathrm{p}}_{\mathrm{y}}\right)\mathrm{O}\left(1\right)-0.27\left(6\mathrm{s}\right)\mathrm{L}\mathrm{a}-0.17\left(5{\mathrm{p}}_{\mathrm{x}}\right)\mathrm{L}\mathrm{a}+0.12\left(5{\mathrm{p}}_{\mathrm{y}}\right)\mathrm{L}\mathrm{a}+0.83\left(5{\mathrm{d}}_{\mathrm{y}\mathrm{z}}\right)\mathrm{L}\mathrm{a}+0.49\left(5{\mathrm{d}}_{\mathrm{x}\mathrm{z}}\right)\mathrm{L}\mathrm{a}$E12

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.

5. Concluding remarks

We present a methodology we developed using the GCHF to build basis sets and to study piezoelectric effects in ceramic materials as perovskite. The GCHF method is a legitimate alternative for the standard basis sets available in packages for polyatomic system calculations. Even for cases where standard basis sets do not present computational problems, the GCHF method is still a good alternative due to the possibility of building basis sets from the own polyatomic system environment. Besides, the availability of basis sets built by GCHF method or other variant methodologies is rich documented in literature. Therefore, it is possible, having components of the atomic basis sets for perovskite systems, to apply portion of the presented methodology to obtain contracted basis sets with well-supplemented and representative polarization and diffuse functions to study piezoelectric properties of this type in ceramic material. We also demonstrated the use of different possibilities of theoretical approximations for calculation of these properties, which embrace approximations with more or less inclusion of electronic correlation energy as well as approximations with relativistic correction effect for better description of the studied property.

To conclude this chapter, we presented our strategy to investigate the piezoelectric effect of two perovskite (barium and lanthanum titanates) using standard basis set and the DFT. Thus, the presented methodology is a legitimate alternative to investigate theoretically piezoelectric properties in ceramic materials as perovskites.