# Quantum Mechanical Study of YTiO3 to the Investigation of Piezoelectricity

Authors: Raimundo Dirceu de Paula Ferreira; Marcos Antonio Barros dos Santos; Maycon da Silva Lobato; Jardel Pinto Barbosa; Marcio de Souza Farias; Antonio Florêncio de Figueiredo; José Ciríaco Pinheiro; Oswaldo Treu-Filho; Rogério Toshiaki Kondo
Journal: Physics Research International
 (2011)

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## Abstract

In previous articles we reported through theoretical studies the piezoelectric effect in BaTiO3, SmTiO3, and YFeO3. In this paper, we used the Douglas-Kroll-Hess (DKH) second-order scalar relativistic method to investigate the piezoelectricity in YTiO3. In the calculations we used the [6s4p] and [10s5p4d] Gaussian basis sets for the O (3P) and Ti (5S) atoms, respectively, from the literature in combination with the (30s21p16d)/[15s9p6d] basis set for the Y (3D) atom, obtained by generator coordinate Hartree-Fock (GCHF) method, and they had their quality evaluated using calculations of total energy and orbital energies (HOMO and HOMO-1) of the 2TiO+1 and 1YO+1 fragments. The dipole moment, the total energy, and the total atomic charges in YTiO3 in Cs space group were calculated. When we analyze those properties we verify that it is reasonable to believe that YTiO3 does not present piezoelectric properties.


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## Body

## 1. Introduction

Even today barium titanate (BaTiO3), the first perovskite structure developed, is widely used in the industry. The polymorphic forms of BaTiO3 have been likened by Kay and Vousden [1] to a displacement of the central Ti+4 ion within its oxygen octahedron towards 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 understanding the structure. The piezoelectric properties in perovskite structure result from uncentersymmetric characteristics [2]. The structure is a network of corner-linked oxygen octahedral holes and large cation filling the dodecahedral holes [3]. For a review about the role of the perovskite structure in ceramic science and technology see Bhalla et al. [4].In previous reports [5–7], we performed Hartree-Fock (HF) studies to investigate the piezoelectric effect in BaTiO3, SmTiO3 and YFeO3. The obtained results offered evidences that the piezoelectricity can be caused by electrostatic interactions. The purpose of this paper is to present a quantum mechanical study of the YTiO3 and offer some insight into the investigation of this property in perovskites not experimentally investigated yet.In our study of YTiO3 we used the 6s4p and [10s5p4d] Gaussian basis sets for O (3P) and Ti (5S) atoms, respectively, from the literature [5].  For Y (3D) atom, we utilized the Generator Coordinate HF method [8] to obtain the (30s21p16d) Gaussian basis set which was contracted to [15s9p6d]. The quality of the contracted basis sets [6s4p]/[10s5p4d]/[15s9p6d] was evaluated in the study of the total energy and orbital energies for the 2TiO+1 and 1YO+1 fragments. Then it was done the addition of one d polarization function in the contracted basis set for the O atom and one diffuse function by symmetry for Ti and Y atoms. The [6s4p1d]/[11s6p5d]/[16s10p7d] molecular basis set was used to model the dipole moments, total energy, and total atomic charges in YTiO3 in Cs space group. In these work, we intend to investigate the possible piezoelectricity property of YTiO3 using theoretical calculations.


## 2. Computational Procedure

### 2.1. The Development of the (30s21p16d) Basis Set to Y (3D) Atom

By the GCHF approach, we choose the one-electron as the continuous superposition(1)Ψi=∫φi(1,α)fidα,i=1,2,…,n,
where φi are the generator functions (STFs and GTFs or other type of functions—in our case GTFs), the fi are unknown weight functions, and α is the generator coordinate.  The application of (1) to calculate expectation values using a Slater determinant leads to the Griffin-Wheeler-HF (GWHF) equations, which underlie the CGHF method
(2)∫[F(α,β)-εiS(α,β)]fi(β)dβ=0,i=1,2,…,m,
where the εi are the HF eigenvalues, and the Fock, F(α,β), and overlap S(α,β), kernels are defined in [8].In practical application the GWHF equations are solved through numerical integration. This is accomplished by discretization preserving the character of the CGHF method. This integration is implemented with a relabeling of theα space
(3)Ω=ln⁡αA,A>1,
where A is a scaling factor determined numerically.TheΩ space is discretized in an equally spaced mesh formed by Ω values so that
(4)Ωm=Ωmin⁡+(m-1)ΔΩ,m=1,2,…,N.
In (4), N corresponds to the number of discretization points defining the basis set size, Ωmin is the initial point, and ΔΩ the increment.In the solution of the discretized form of (2), the (30s21p16d) basis set was used as defined by the mesh of (3). The discretization parameters (which define the exponents) for the basis set built to Y (3D) are Ωmin(s)=-0.670, ΔΩ(s)=0.123, and N(s)=30; Ωmin(p)=-0.377, ΔΩ(p)=0.109, and N(p)=21; Ωmin(d)=-0.498, ΔΩ(d)=0.115, and N(d)=16.


### 2.2. Building the [15s9p6d] Contracted Basis Set Y (3D) Atom and Evaluating the Quality of the [6s4p]/[10s5p4d]/[15s9p6d] Contracted Basis Set in Polyatomic Calculations

The contraction of the (30s21p16d) basis sets was implemented by employing the segmented contraction scheme proposed by Dunning and Hey [9]. The (30s21p16d) basis set was contracted to 15s9p6d through the following scheme: (14,2,1,1,1,1,1,1,1,1,1,1,2,1,1/9,3,1,1,1,1,3,1,1/8,1,1,1,4,1).The (30s21p16d) and [15s9p6d] basis sets were obtained through a slightly modified version of the ATOMSCF program of Pavani and Clementi [10].In order to evaluate the quality of the [6s4p], [10s5p4d], and [15s9p6d] basis sets in polyatomic study, we performed calculations of total energies, highest occupied molecular orbital (HOMO) energy, and the energy of one level below the highest occupied molecular orbital (HOMO-1) for the2TiO+1 and 1YO+1 fragments. The results are compared with those obtained from the (24s14p), (30s19p14d), and (30s21p16d) basis sets.Table1 shows the HF total and orbital energies for 2TiO+1 and 1YO+1 fragments obtained with the uncontracted and contracted basis sets. From this table, one can see that the total energy obtained with [10s5p4d]/[6s4p] and [15s9p6d]/[6s4p] basis sets are close to the corresponding values of (30s19p14d)/(24s14p) and (30s21p16d)/(24s14p) basis sets. The total energies differ by 0.885 and 0.172 hartree for 2TiO+1 and 1YO+1 fragments, respectively. The HOMO energies obtained from contracted basis sets are closer to those of the uncontracted basis sets and differ by 0.00841 and 0.0138 hartree for the 2TiO+1 and 1YO+1 fragments, respectively. The HOMO-1 energies obtained from contracted basis sets are also close to the values of uncontracted basis sets and present deviations of 0.0128 and 0.00577 hartree.Table 1
Total and HOMO and HOMO-1 energies (hartree).
FragmentBasisTotal energyεHomoεHomo-12TiO+1[10s5p4d]a/[6s4p]b−922.0036930−0.2860−0.4573(30s19p14d)c/(20s14p)d−992.8885441−0.2944−0.58571YO+1[15s9p6d]e/[6s4p]b−3406.227040−0.4952−0.5191(30s21p16d)f/(20s14p)d−3406.3985754−0.5090−0.5249Ca, contracted and uncontracted basis for the Ti (3F) atom from [4].Cb, dontracted and uncontracted basis for  the O (3P) atom from [4].Ce, fontracted and uncontracted basis for  the Y (2D) atom from this paper.To better describe the properties of the YTiO3 in the implementation of the calculations it was necessary to include polarization function for the O (3P) atom. The polarization function value is αd=0.622507 [5].Inab  initio calculations of the transition metal compounds, the role of the basis set is a crucial point, since the description of the configuration of the metal in the compound differs from the neutral state. The adequate diffuse functions were obtained through the total energy optimization of the ground state anion Ti−1 (4F) and Y−1 (3F) by HF method.  The following diffuse functions for the [10s5p4d] and [15s9p6d] basis sets were determined for the Ti (5S) and Y (3D), respectively, using this strategy: αs=0.00177827; αp=0.0537032; αd=0.0263027 and αs=0.0081283; αp=0.0489779; αd=0.0323594.The [6s5p1d]/[11s6p5d]/[16s10p7d] basis set was used in the modelling of dipole moment, total energy, and total atomic charges from [YTiO3]2 fragment.


### 2.3. The Representation of the Crystalline 3D Periodic YTiO3 System

In previous papers [5–7], the fragment model shown in Figure 1 was used to simulate the necessary conditions to the existence of piezoelectricity in BaTiO3 as full solid. In this paper, we used the same model to represent the crystalline 3D periodic YTiO3 system. The Ti atom is located in the center of the octahedron, being wrapped up by six O atoms, disposed in the reticular plane (200), and two Y atoms arranged in the reticular plane (100).Figure 1
The octahedral [YTiO3]2 fragment. (Ia): (a) represents the [YTiO3]2 fragment with the Ti atoms fixed in the space; (b) represents the [YTiO3]2 being moved +0.005 Å in the symmetry x axis, while the Y and O atoms are maintained fixed; (c) it presents the [YTiO3]2 fragment in which the Ti atoms are moved −0.005 Å symmetry x axis, and Y and the O atoms are maintained fixed; (Ib), represents the [YTiO3]2 fragment with the bond lengths Ti1–O3, Ti1–O4, and Ti2–O4 shortened 0.005 Å.To study the crystalline 3D periodic YTiO3 system it is necessary to choose a fragment (or a molecular model), which represents adequately a physical property of the crystalline system as a whole. The fragment [YTiO3]2 was chosen because, after its optimization, the obtained structural parameters (interatomic distances) were closer to experimental values with good precision. The study was developed according to the following strategy: (i) first, it was carried out the geometry optimization of the [YTiO3]2 fragment in the Cs symmetry and 1A’ electronic state. (ii) Finally, calculations of single points were performed with the geometry optimized according to the descriptions shown in Figure 1.In Figure1 (I), (a) represents the [YTiO3]2 fragment with the Ti atoms fixed in the space. (b) represents the [YTiO3]2 fragment with the Ti atoms being moved +0.005 Å in the x axis, while the Y and O atoms are maintained fixed. (c) represents the [YTiO3]2 fragment in which the  Ti  atoms  are moved  −0.005 Å in the x axis, and Y and the O atoms are maintained fixed.Figure1 (II) represents the [YTiO3]2 fragment with the bond lengths Ti1–O3, Ti1–O4, and  Ti2–O4 shortened 0.005 Å.All molecular calculations in this work were carried out by Hartree-Fock method with inclusion of the DKH second-order scalar relativistic calculations [11–13] as implemented in the Gaussian 03 program [14].


## 2.1. The Development of the (30s21p16d) Basis Set to Y (3D) Atom

By the GCHF approach, we choose the one-electron as the continuous superposition(1)Ψi=∫φi(1,α)fidα,i=1,2,…,n,
where φi are the generator functions (STFs and GTFs or other type of functions—in our case GTFs), the fi are unknown weight functions, and α is the generator coordinate.  The application of (1) to calculate expectation values using a Slater determinant leads to the Griffin-Wheeler-HF (GWHF) equations, which underlie the CGHF method
(2)∫[F(α,β)-εiS(α,β)]fi(β)dβ=0,i=1,2,…,m,
where the εi are the HF eigenvalues, and the Fock, F(α,β), and overlap S(α,β), kernels are defined in [8].In practical application the GWHF equations are solved through numerical integration. This is accomplished by discretization preserving the character of the CGHF method. This integration is implemented with a relabeling of theα space
(3)Ω=ln⁡αA,A>1,
where A is a scaling factor determined numerically.TheΩ space is discretized in an equally spaced mesh formed by Ω values so that
(4)Ωm=Ωmin⁡+(m-1)ΔΩ,m=1,2,…,N.
In (4), N corresponds to the number of discretization points defining the basis set size, Ωmin is the initial point, and ΔΩ the increment.In the solution of the discretized form of (2), the (30s21p16d) basis set was used as defined by the mesh of (3). The discretization parameters (which define the exponents) for the basis set built to Y (3D) are Ωmin(s)=-0.670, ΔΩ(s)=0.123, and N(s)=30; Ωmin(p)=-0.377, ΔΩ(p)=0.109, and N(p)=21; Ωmin(d)=-0.498, ΔΩ(d)=0.115, and N(d)=16.


## 2.2. Building the [15s9p6d] Contracted Basis Set Y (3D) Atom and Evaluating the Quality of the [6s4p]/[10s5p4d]/[15s9p6d] Contracted Basis Set in Polyatomic Calculations

The contraction of the (30s21p16d) basis sets was implemented by employing the segmented contraction scheme proposed by Dunning and Hey [9]. The (30s21p16d) basis set was contracted to 15s9p6d through the following scheme: (14,2,1,1,1,1,1,1,1,1,1,1,2,1,1/9,3,1,1,1,1,3,1,1/8,1,1,1,4,1).The (30s21p16d) and [15s9p6d] basis sets were obtained through a slightly modified version of the ATOMSCF program of Pavani and Clementi [10].In order to evaluate the quality of the [6s4p], [10s5p4d], and [15s9p6d] basis sets in polyatomic study, we performed calculations of total energies, highest occupied molecular orbital (HOMO) energy, and the energy of one level below the highest occupied molecular orbital (HOMO-1) for the2TiO+1 and 1YO+1 fragments. The results are compared with those obtained from the (24s14p), (30s19p14d), and (30s21p16d) basis sets.Table1 shows the HF total and orbital energies for 2TiO+1 and 1YO+1 fragments obtained with the uncontracted and contracted basis sets. From this table, one can see that the total energy obtained with [10s5p4d]/[6s4p] and [15s9p6d]/[6s4p] basis sets are close to the corresponding values of (30s19p14d)/(24s14p) and (30s21p16d)/(24s14p) basis sets. The total energies differ by 0.885 and 0.172 hartree for 2TiO+1 and 1YO+1 fragments, respectively. The HOMO energies obtained from contracted basis sets are closer to those of the uncontracted basis sets and differ by 0.00841 and 0.0138 hartree for the 2TiO+1 and 1YO+1 fragments, respectively. The HOMO-1 energies obtained from contracted basis sets are also close to the values of uncontracted basis sets and present deviations of 0.0128 and 0.00577 hartree.Table 1
Total and HOMO and HOMO-1 energies (hartree).
FragmentBasisTotal energyεHomoεHomo-12TiO+1[10s5p4d]a/[6s4p]b−922.0036930−0.2860−0.4573(30s19p14d)c/(20s14p)d−992.8885441−0.2944−0.58571YO+1[15s9p6d]e/[6s4p]b−3406.227040−0.4952−0.5191(30s21p16d)f/(20s14p)d−3406.3985754−0.5090−0.5249Ca, contracted and uncontracted basis for the Ti (3F) atom from [4].Cb, dontracted and uncontracted basis for  the O (3P) atom from [4].Ce, fontracted and uncontracted basis for  the Y (2D) atom from this paper.To better describe the properties of the YTiO3 in the implementation of the calculations it was necessary to include polarization function for the O (3P) atom. The polarization function value is αd=0.622507 [5].Inab  initio calculations of the transition metal compounds, the role of the basis set is a crucial point, since the description of the configuration of the metal in the compound differs from the neutral state. The adequate diffuse functions were obtained through the total energy optimization of the ground state anion Ti−1 (4F) and Y−1 (3F) by HF method.  The following diffuse functions for the [10s5p4d] and [15s9p6d] basis sets were determined for the Ti (5S) and Y (3D), respectively, using this strategy: αs=0.00177827; αp=0.0537032; αd=0.0263027 and αs=0.0081283; αp=0.0489779; αd=0.0323594.The [6s5p1d]/[11s6p5d]/[16s10p7d] basis set was used in the modelling of dipole moment, total energy, and total atomic charges from [YTiO3]2 fragment.


## 2.3. The Representation of the Crystalline 3D Periodic YTiO3 System

In previous papers [5–7], the fragment model shown in Figure 1 was used to simulate the necessary conditions to the existence of piezoelectricity in BaTiO3 as full solid. In this paper, we used the same model to represent the crystalline 3D periodic YTiO3 system. The Ti atom is located in the center of the octahedron, being wrapped up by six O atoms, disposed in the reticular plane (200), and two Y atoms arranged in the reticular plane (100).Figure 1
The octahedral [YTiO3]2 fragment. (Ia): (a) represents the [YTiO3]2 fragment with the Ti atoms fixed in the space; (b) represents the [YTiO3]2 being moved +0.005 Å in the symmetry x axis, while the Y and O atoms are maintained fixed; (c) it presents the [YTiO3]2 fragment in which the Ti atoms are moved −0.005 Å symmetry x axis, and Y and the O atoms are maintained fixed; (Ib), represents the [YTiO3]2 fragment with the bond lengths Ti1–O3, Ti1–O4, and Ti2–O4 shortened 0.005 Å.To study the crystalline 3D periodic YTiO3 system it is necessary to choose a fragment (or a molecular model), which represents adequately a physical property of the crystalline system as a whole. The fragment [YTiO3]2 was chosen because, after its optimization, the obtained structural parameters (interatomic distances) were closer to experimental values with good precision. The study was developed according to the following strategy: (i) first, it was carried out the geometry optimization of the [YTiO3]2 fragment in the Cs symmetry and 1A’ electronic state. (ii) Finally, calculations of single points were performed with the geometry optimized according to the descriptions shown in Figure 1.In Figure1 (I), (a) represents the [YTiO3]2 fragment with the Ti atoms fixed in the space. (b) represents the [YTiO3]2 fragment with the Ti atoms being moved +0.005 Å in the x axis, while the Y and O atoms are maintained fixed. (c) represents the [YTiO3]2 fragment in which the  Ti  atoms  are moved  −0.005 Å in the x axis, and Y and the O atoms are maintained fixed.Figure1 (II) represents the [YTiO3]2 fragment with the bond lengths Ti1–O3, Ti1–O4, and  Ti2–O4 shortened 0.005 Å.All molecular calculations in this work were carried out by Hartree-Fock method with inclusion of the DKH second-order scalar relativistic calculations [11–13] as implemented in the Gaussian 03 program [14].


## 3. Results and Discussion

Aiming at the verification of possible piezoelectric properties of YTiO3, the Ti+3 central ion, should be centersymmetric and the [YTiO3]2 fragment polarises when submitted to mechanical stress. Firstly, we would like to call the attention for the comparison among the calculated bond lengths and the experimental values [15]. For Y–O and Ti–O bond lengths, our results are 2.009 and 2.421 Å, while the experimental values are 2.077 and 2.508 Å, respectively. The differences between the theoretical and experimental values are 0.0682 and 0.087 Å. This shows a very good performance of our 6s5p1d/11s6p5d/16s10p7d basis set to model the geometry of the fragment studied.In Table2 are shown the dipole moment (in Debye) and the total energy (in hartree) from [YTiO3]2 fragment. In this table, one can see that the fragment with Ti+3 at (b) is 0.0111 hartree more stable when compared at (a); this evidence is confirmed by the decrease of the dipole moment. Also, when the central ion is moved to the position c, it can be verified that the fragment is 0.00813 hartree more stable than at (a). This indicates that the Ti+3central ion is not centersymmetric. Still in Table 2 one can see that reducing the Ti1–O3, Ti2–O4, and Ti2–O4 bond lengths, the calculations in the atomic position d did not show either a deviation of atomic charges among the atoms of the [YTiO3]2 fragment, besides the decreasing of the dipole moment of the fragment.  These results suggest that there is not a probable polarization of YTiO3 crystal when submitted to mechanical stress. Configurations b, c, and d show lower total energy compared to configuration a, as well as, lower values of total dipole moment μ and its individual components μx, μy, and μz. We suggest in this work that symmetric distribution of atomic charges provides a more stability of studied molecular fragment.Table 2
Dipole moment (Debye) and total energy (hartree) of the [YTiO3]2 fragment.
Ti atom positionμxμyμzμEnergy totala−7.8176.5460.46310.21−8807.601595b−1.2005.8612.8786.639−8807.612686c−3.4852.4193.5155.510−8807.609726d−3.620−2.3193.7375.697−8807.610393Table3 shows the values of the total charges of the atoms in the [YTiO3]2 fragment with Ti+3 at a, b, and c positions and the Ti–O bond length reduced 0.005 Å. According to the table, when the Ti+3 central ion is deviated from the position a to b and c positions, it occurs the rearrangement of the electronic density, without being clearly established, however, a tendency in the migration of the atomic charges. On the other hand, in Table 3 one can see that when the Ti1–O3, Ti1–O4, and Ti2–O4 bond lengths are shortened, at atomic position (d), by a mechanical stress, it also occurs the rearrangement of the electronic density, without again verifying the clear tendency in the migration of the atomic charges in the material.Table 3
Total atomic charges from the [YTiO3]2 fragment.
AtomAtom positionabcdTi(1)2.12971.94741.90601.9082Ti(2)0.17860.27091.57780.1695Y(1)1.73911.68821.73621.7350Y(2)1.45341.46501.53511.5241O(1)−1.0417−1.1458−1.1290−1.1296O(2)−0.992−0.8898−0.8994−0.8967O(3)−0.5809−0.3688−0.3055−0.2873O(4)−0.6715−1.0558−1.0715−1.0788O(5)−1.1059−0.7628−0.8790−0.8976O(6)−1.1015−1.1484−1.0504−1.0467For the fragment a, which represents the probable configuration of this work, the HOMO and LUMO orbitals are HOMO = − 0.14 4s Ti1 − 0.27 2pz O1 − 0.48 2pz  O4 + 0.82 2px O4 + 0.11 2s O6 – 0.27 2pz O6 + 0.12 4dxz Y1; LUMO = + 0.17 4s Ti1 + 0.16 3px Ti1 + 0.14 2s O2 − 0.21 2py O2 + 0.16 2px O2 − 0.24 2px O3 + 0.24 4s Ti2+ 0.38 3px Ti2 + 0.43 3py Ti2  + 0.33 3dx2-y2 Ti2 + 0.10 3dxy Ti2 + 0.10 3dz2 Ti2 − 0.20 5s Y2 + 0.16 4px Y2 + 0.274py Y2.Thus, we suggest that the metal-O distance in the [YTiO3]2 fragment is formed by π-symmetry covalent bonds between 3d and 4d orbitals of the Ti (5S) and Y (3D) atoms, respectively, to 2p orbital of the O (3P) atom in opposition to BaTiO3 where it was verified that those bonds were ionic. The ionic characteristic between metal-O bonds is a fundamental condition to occurrence of piezoelectricity in our opinion, which is not observed in YTiO3. Therefore, previous theoretical studies of BaTiO3 [5] have suggested that known piezoelectric properties are caused by electrostatic effects, and for SmTiO3[6] they have suggested the possible piezoelectric effect as well as electrostatic effects in its chemical bonds. On the other hand, for YFeO3 [7] and YTiO3 (this work), the theoretical calculations suggest absence of piezoelectric effect with covalent bonds between their atoms. For us, it seems that ionic bonds contribute to the occurrence of piezoelectric effect.


## 4. Conclusions

In the theoretical modeling of YTiO3 in the Cs symmetry and 1A’ electronic state, the calculated results of the bond lengths showed a very good concordance with the experimental values from the literature. Obtained values of the moment of dipole, the total energy, and the total atomic charges showed that it is reasonable to believe that YTiO3 does not present behavior of piezoelectric material.



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*Source: 123492-2011-04-11.xml*