**5. Concluding remarks**

**Ti atom position**

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

μ*<sup>x</sup>* 25.71 34.67 μ*<sup>y</sup>* 2.806 40.92 μ*<sup>z</sup>* 0.716 25.24 μ 25.87 59.29

+ +

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

Dipole moment (Debye)

76 Piezoelectric Materials

**Atom Ti is fixed in space Bond lengths Ti─O1 and Ti─O6 are shortened (mechanical stress)**

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

HOMO 0.10 2p O 3 0.212p O 4 0.42 2p O 4 0.14 2p O 4 0.112p O 5

LUMO 0.14 2p O 2 0.38 2p O 2 0.13 2p O 3 0.20 2p O 4 0.18 2p O 4

( ) ( ) ( )

HOMO 0.78 6s La 0.35 5p La 0.12 5p La 0.25 (5d ) La 0.18 5d La 0.41 5d La – 0.40 5d La

= ++ - -

( ) ( )

La 0.83 5d La 0.49 5d La

yz xz

LUMO 0.37 2p O 1 0.27 6s La 0.17 5p La 0.12 5p

= + -- +

( ) ( ) ( )

xz yz xy

( ) ( ) ( ) ( ) ( )

y x y

= + - -+ - + ++-

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


+ +

= + + + + +

( ) ( ) ( )

yyx

0.60 2p O 5 0.39 2p O 6 – 0.34 2p O 6

( ) ( ) ( ) ( ) ( )

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

x yz

2

zx yzx

xyzxy zxyzy

0.62 2p O 4 – 0.15 2p O 5 0.15 2p O 5 0.53 2p O 5 0.14 2p O9 6

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 repre‐ sentative 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.
