Discretization parameters (which define the exponents) for O (^{3}P), Fe(^{5}D), and La(^{2}D) atoms^{a}.
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 HartreeFock (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.
Keywords
 basis sets
 theoretical methods
 quantum mechanical approaches
 piezoelectricity
 perovskites
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 ABO_{3}, where A is a large cationlike alkaline metals, earth metals, and rare earth metals and B is a small cationlike 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 (BaTiO_{3}). The polymorphous forms of BaTiO_{3} 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 HartreeFock (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 HartreeFock 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.,
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
The GCHF method was introduced in 1986 [3], and one of the first applications was in the generation of Gaussian and Slatertype orbitals (GTO and STO) universal basis sets [15–17]. The GCHF method was used to build contracted GTF (Gaussian TypeFunction) basis sets for the first and secondrow atoms which were applied in calculations of various properties at the HF, CISD (configuration interaction with single and double excitations), and MP2 (MöllerPlesset 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 H_{2}, N_{2}, and Li_{2} [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]. Firstprinciples (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 oneelectron functions as the continuous superposition:
where ψ_{i} are the generator functions (GTOs for the case of perovskites) and
The φ_{i} are then employed to build a Slater determinant for the multielectronic wave function and minimizing the total energy with respect to
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.,
with
The new GC Ω space is discretized, by symmetry, in an equally space mesh formed by ü values so that
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
In the solution of the discretization of Eq. (2), the (22s14p), (30s19p13d), and (32s24p117d) GTOs basis sets were used to O (^{3}P), Fe (^{5}D), and La (^{2}D) 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 

Ωmin  ΔΩ 

Ωmin  ΔΩ 


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 
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 (^{3}F), Fe (^{5}D), and La (^{2}D) atoms in LaFeO_{3} [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 (HOMO1) 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 LaFeO_{3}, the ^{2}FeO^{+1} and ^{1}LaO^{+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 LaFeO_{3} [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 [ABO_{3}]_{2} fragment for different primitive functions, taking into account the minimum energy criterion. For the [LaFeO_{3}]_{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 groundstate anions of the metals present in perovskite structure [37].
For LaFeO_{3}, 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 LaFeO_{3} 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 HartreeFock) method together with the 17s11p7d/11s6p6d/5s3p1d GTOs basis was used for investigation of perovskites’ piezoelectricity of lanthanum manganite [38]. Besides, the ROHF (Restricted Openshell HartreeFock) 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 BaTiO_{3} 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 (DouglasKrollHess secondorder 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 (BaTiO_{3}) and lanthanum titanate (LaTiO_{3})
In this section, we investigate the piezoelectricity in BaTiO_{3} and LaTiO_{3} perovskites. Initially, we apply the methodology for the BaTiO_{3} to verify if the results show the piezoelectricity property, once it is known this perovskite has it. After, we apply the same methodology for LaTiO_{3} 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 (LeeYangParr) correlation functional [49] as implemented in the Gaussian 98 program [50].
For the study of the crystalline 3D periodic BaTiO_{3} [51] and LaTiO_{3} [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 3 shows the molecular models used to simulate the necessary conditions of the existence of piezoelectricity in BaTiO_{3} [51] and LaTiO_{3} [52] as full solids. The [BaTiO_{3}]_{2} and [LaTiO_{3}] 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 [BaTiO_{3}]_{2} and [LaTiO_{3}]_{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, singlepoint calculations were developed.
In Figure 3, (a) represents the [BaTiO_{3}]_{2} fragment with the Ti atom fixed in the space and Ti atom being moved 0.003 Å in the direction to O_{1}, O_{2}, O_{3}, O_{4}, O_{5}, and O_{6} atoms and Ba atom and all O atoms are fixed; (b) represents the [BaTiO_{3}]_{2} fragment with the bond lengths Ti─O_{1}, Ti─O_{6} shortened from 0.003 Å; (c) represents the [LaTiO_{3}]_{2} fragment with the Ti atom fixed in the space and Ti atom being moved 0.005 Å in the direction to O_{1}, O_{2}, O_{3}, O_{4}, O_{5}, and O_{6} atoms and La atom and all O atoms are fixed; and (d) represents the [LaTiO_{3}]_{2} fragment with the bond lengths Ti─O_{1}, Ti─O_{6} shortened from 0.005 Å.
4.2. Results and discussion
Table 2 shows the theoretical (calculated) bond lengths and the experimental values from literature [51] for BaTiO_{3}. 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─O_{1} and Ba─O_{1}, respectively.
Bond length (Å)  Theoretical (this work)  Experimental [51]  Δ 

T_{i}─O_{1}  1.93431  1.99750  6.32 × 10^{−2} 
Ba─O_{1}  2.77480  2.83871  6.39 × 10^{−2} 
Table 3 presents the total energy of [BaTiO_{3}]_{2} fragment. As it mentioned previously, the calculations are at atomic positions: Ti is fixed in space, Ti is moved toward the O_{1} atom and Ba atom and the O other atoms are fixed, Ti is moved toward the O_{2} 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 × 10^{5}, 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─O_{1} and Ti─O_{6} 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 O_{1} atom; Ba and O atoms are fixed  −534.259066114 
Ti is moved toward the O_{2} atom; Ba and O atoms are fixed  −534.259104826 
Ti is moved toward the O_{3} atom; Ba and O atoms are fixed  −534.259130429 
Ti is moved toward the O_{4} atom; Ba and O atoms are fixed  −534.259860527 
Ti is moved toward the O_{5} atom; Ba and O atoms are fixed  −534.259877355 
Ti is moved toward the O_{6} atom; Ba and O atoms are fixed  −534.259762234 
Bond lengths Ti─O_{1} and Ti─O_{6} are shortened (mechanical stress)  −534.257326207 
Ti atom position  

Atom  Ti is fixed in space  Bond lengths Ti─O_{1} and Ti─O_{6} are shortened (mechanical stress) 
Ti  +0.885  +0.887 
Ba  +1.837  +1.841 
O_{1}  −0.636  −0.633 
O_{2}  −0.564  −0.563 
O_{3}  −0.664  −0.663 
O_{4}  −0.291  −0.295 
O_{5}  −0.296  −0.300 
O_{6}  −0.271  −0.274 
Dipole moment (Debye)  
μ_{x}  23.28  −23.51 
μ_{y}  −0.3622  0.1374 
μ_{z}  −2.0703  0.0585 
μ  23.38  23.51 
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_{1} and Ti─O_{6} chemical bond lengths provokes a polarization of the [BaTiO_{3}]_{2} fragment, indicating the nature of Ti─O and Ba─O chemical bonds was changed.
For the [BaTiO_{3}]_{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
For the fragment [BaTiO_{3}]_{2} under mechanical stress, the HOMO and the LUMO are written as
The analysis of the HOMO and the LUMO of the [BaTiO_{3}]_{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 [BaTiO_{3}]_{2} fragment. This is consistent because the repulsive effect of d electrons in both highspin and lowspin 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 BaTiO_{3} can be caused by electrostatic interactions.
The experimental [52] and theoretical geometric parameters for LaTiO_{3} are shown in Table 5. According to Table 5, the theoretical results are 2.02413 and 2.58928 Å for Ti─O_{1} and La─O_{1}, 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 

T_{i}─O_{1}  2.02413  2.01556  8.57 × 10^{−3} 
La─O_{1}  2.58928  2.59918  9.99 × 10^{−3} 
Table 6 presents the total energy from [LaTiO_{3}]_{2} fragment. As mentioned previously, the calculations are at atomic positions: Ti is fixed in space, Ti is moved toward the O_{1} atom and La atom and the O other atoms are fixed, Ti is moved toward the O_{2} 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 O_{1}, O_{2}, O_{3}, O_{4}, O_{5}, and O_{6}, respectively, when fixed in space. Also, for the [LaTiO_{3}]_{2} fragment, we can note that, with the shortening of Ti─O_{1} and Ti─O_{6} 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 O_{1} atom; La and O atoms are fixed  −540.329458501 
Ti is moved toward the O_{2} atom; La and O atoms are fixed  −540.281387651 
Ti is moved toward the O_{3} atom; La and O atoms are fixed  −540.304861376 
Ti is moved toward the O_{4} atom; La and O atoms are fixed  −540.318198767 
Ti is moved toward the O_{5} atom; La and O atoms are fixed  −540.265005660 
Ti is moved toward the O_{6} atom; La and O atoms are fixed  −540.276475167 
Bond lengths Ti─O_{1} and Ti─O_{6} are shortened (mechanical stress)  −532.224697936 
Table 7 shows the total atomic charges and the dipole moment values for the [LaTiO_{3}]_{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 BaTiO_{3}, LaTiO_{3} 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─O_{1} and Ti─O_{6} are shortened (mechanical stress) 
Ti  +0.967  +2.229 
La  +1.451  −0.212 
O_{1}  −0.492  −0.691 
O_{2}  −0.010  +4.16 
O_{3}  −0.553  −0.527 
O_{4}  −0.267  −1.803 
O_{5}  −0.297  −1.766 
O_{6}  −0.799  −1.390 
Dipole moment (Debye)  
μ_{x}  25.71  34.67 
μ_{y}  2.806  40.92 
μ_{z}  0.716  25.24 
μ  25.87  59.29 
For the LaTiO_{3}, the HOMO and the LUMO, when the Ti atom is fixed in space, are, respectively,
For the LaTiO_{3} fragment under mechanical stress, the HOMO and the LUMO are
Analyzing the HOMO and the LUMO of LaTiO_{3} 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 wellsupplemented 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.
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