Gaussian basis sets to the theoretical study of the electronic structure of perovskite (LaMnO 3 )

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1 Journal of Molecular Structure (Theochem) 631 (2003) Gaussian basis sets to the theoretical study of the electronic structure of perovskite (LaMnO 3 ) O. Treu Filho a, J.C. Pinheiro b,c, *, R.T. Kondo d, R.F.C. Marques a, C.O. Paiva-Santos a, M.R. Davolos a, M. Jafelicci Jr. a a Instituto de Química, UNESP, CP 335, Araraquara, SP, Brazil b Laboratório de Química Teórica e Computacional, Departamento de Química, Centro de Ciências Exatas e Naturais, Universidade Federal do Pará, CP , Belém, PA, Amazônia, Brazil c Cooperativa Centro de Educação Científica e Empreendedora da Amazônia, Rua 15 de Novembro, 226, Sala 805, Belém, PA, Amazônia, Brazil d Secção detécnica de Suporte, Centro de Informática de São Carlos, Universidade de São Paulo, São Carlos, SP, Brazil Received 15 January 2003; accepted 27 February 2003 Abstract The Generator Coordinate Hartree Fock (GCHF) method is applied to generate extended (20s14p), (30s19p13d), and (31s23p18d) Gaussian basis sets for the O, Mn, and La atoms, respectively. The role of the weight functions (WFs) in the assessment of the numerical integration range of the GCHF equations is shown. These basis sets are then contracted to [5s3p] and [11s6p6d] for O and Mn atoms, respectively, and [17s11p7d] for La atom by a standard procedure. For quality evaluation of contracted basis sets in molecular calculations, we have accomplished calculations of total and orbital energies in the Hartree Fock Roothaan (HFR) method for 5 MnO 1þ and 1 LaO 1þ fragments. The results obtained with the contracted basis sets are compared with values obtained with the extended basis sets. The addition of one d polarization function in the contracted basis set for O atom and its utilization with the contracted basis sets for Mn and La atoms leads to the calculations of dipole moment and total atomic charges of perovskite (LaMnO 3 ). The calculations were performed at the HFR level with the crystal [LaMnO 3 ] 2 fragment in space group C 2v : The values of dipole moment, total energy, and total atomic charges showed that it is reasonable to believe that LaMnO 3 presents behaviour of piezoelectric material. q 2003 Elsevier B.V. All rights reserved. Keywords: Gaussian basis sets; Generator Coordinate Hartree Fock method; Perovskite; LaMnO 3 ; Piezoelectric 1. Introduction There is a considerable degree of freedom in choosing the basis functions for atomic and molecular * Corresponding author. Address: Laboratório de Química Teórica e Computacional, Departamento de Química, Centro de Ciências Exatas e Naturais, Universidade Federal do Pará, CP , Belém, PA, Amazônia, Brazil. address: ciriaco@ufpa.br (J.C. Pinheiro). calculations since any complete set of functions can be employed. The choice of the basis set in nonrelativistic atomic and molecular calculations is crucial as it determines the computational cost and the accuracy of the calculations. The conventional technique employed for calculations of the electronic structure of atoms and molecules involves the expansion of wave functions in terms of a finite number of basis functions containing /03/$ - see front matter q 2003 Elsevier B.V. All rights reserved. doi: /s (03)

2 94 O. Treu Filho et al. / Journal of Molecular Structure (Theochem) 631 (2003) a set of parameters that are specifically optimized for the particular system under investigation. Since the introduction in literature of the GCHF method made by Mohallem et al.[1], in the late eighties, it has posed as a powerful technique in the design of basis set for atomic and molecular calculations. After some years, some theoretical approaches derived from the GCHF method have been reported in the literature [2 5], meanwhile the quality of the basis sets and low computational cost involved in the choice of this basis sets still authorise the GCHF method as an excellent strategy for the generation of basis sets. Some applications of the basis sets designed with the GCHF method in ab initio studies of atoms and molecules have been reported in the literature [6 9]. However, the application of this basis in calculations of properties of atoms and molecules has been still accomplished in a timid way. Piezoelectricity is the ability of certain crystalline materials to develop an electric charge proportional to a mechanical stress. Soon it was realized that materials showing this phenomenon must also show the converse, a geometric strain (deformation) proportional to an applied voltage [10]. The most important piezoelectric ceramics crystalline is the perovskite structure. The structure is a network of corner linked oxygen octahedral holes and the large cation filling the dodecahedral holes [11]. The piezoelectric properties in perovskite structure result from uncentrosymmetrics characteristics [12]. Barium titanate (BaTiO 3 ) was the first perovskite structure developed and is still widely used. The polimorphic forms of BaTiO 3 have been likened by Kay et al. [13] to a displacement of the central Ti 4þ 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. In this article we explore the strategy of the GCHF method in the construction of basis sets to O ( 3 P), Mn ( 6 S), and La ( 2 D) atoms to the study of the electronic structure of the perovskite (LaMnO 3 ) and to verify possible piezoelectric properties of this material. 2. Computational methodology 2.1. The construction of the extended basis sets to O ( 3 P), Mn ( 6 S) and La ( 2 D) atoms In the GCHF method, the 1-electron functions are chosen as ð F i ¼ c i ð1; aþf i ðaþda i ¼ 1; 2; ; n ð1þ where c i ; a; and f i ðaþ in Eq. (1) are the generator functions, the generator coordinate (GC), and the WFs, respectively. Using Eq. (1) to build the Slater determinant for a multi-electronic wave function and minimizing the total energy in relation to the f i ðaþ; one arrives at the Griffin Wheeler HF (GWHF) equations: ð ½Fða; bþ 2 1 i Sða; bþšf i ðbþdb ¼ 0 ð2þ i ¼ 1; 2; ; m where the 1 i are the HF eigenvalues, and the Fock, Fða; bþ; and overlap Sða; bþ; kernels are defined in Ref. [1]. The GWHF equations are integrated numerically through discretization with a technique which 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 [14], i.e. V ¼ ln a=a A. 1 ð3þ with A being a scaling parameter determined numerically. The new GC V space is discretized, by symmetry, in an equally space mesh formed by V values so that V j ¼ V min þðj 2 1ÞDV j ¼ 1; 2; ; N ð4þ In Eq. (4), N corresponds to the number of discretization points defining the basis set size, V min is the initial point and DV the increment. The values of V min (lowest value) and the highest value V max ¼ V min þðn 2 1ÞDV are chosen in order to adequately encompass the integration range of f ðvþ: This is visualized by drawing the WFs from a preliminary calculation with arbitrary discretization parameters. Fig. 1 illustrates this situation and shows

3 O. Treu Filho et al. / Journal of Molecular Structure (Theochem) 631 (2003) Fig. 1. The 1s, 4s, and 5d WFs for O ( 3 P), Mn ( 6 S), and La ( 2 D) atoms obtained with extended (20s14p), (30s19p13d), and (31s23p18d0 Gaussian basis sets, respectively. therespective1s,4s,and5dwfsforo( 3 P),Mn( 6 S),and La ( 2 D) atoms. In the solution of the discretized form of Eq. (2) [1], the (20s14p), (30s19p13d), and (31s23p18d) basis sets were used as defined by the mesh of Eq. (3). The values of V min and V max were selected in order to satisfy the relevant integration range of each atom WF. In Table 1, the discretization parameters (which define the exponents) for the basis sets built are shown Contraction and evaluation of the quality of the basis sets in molecular calculations For molecular calculations, a basis set is usually developed in three stages: (i) construction of the extended basis set in atomic calculations, (ii) construction of the contracted basis set using a segmented or general scheme, and (iii) addition of supplementary functions of diffuse and polarization character. The first stage is usually a straightforward task. This part of construction of the Gaussian basis sets was discussed in Section 2.1. The second stage was implemented by employing the segmented contraction scheme proposed by Dunning et al. [15]. For O ( 3 P) atom, the (20s14p) Gaussian basis sets were contracted to [5s3p], through the contraction schemes 16, 1, 1, 1, 1/12, 1, 1. For Mn ( 6 S) atom, the (30s19p13d) Gaussian basis set was contracted to [11s6p6d] through the contraction scheme 14, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1/14, 1, 1, 1, 1, 1/8, 1, 1, 1, 1, 1. For La ( 2 D) Table 1 Discretization parameters (which define the exponents) for O ( 3 P), Mn ( 6 S) and La ( 2 D) atoms Symmetry O Mn La V min DV N V min DV N V min DV N S p d The scaling parameter used for s, p, and d symmetries for all atoms studied is equal to A ¼ 6:0:

4 96 O. Treu Filho et al. / Journal of Molecular Structure (Theochem) 631 (2003) atom, the (31s23p18d) Gaussian basis set was contracted to [17s11p7d] through the contraction scheme 10, 4, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1/13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1/10, 3, 1, 1, 1, 1, 1. The extended and contracted Gaussian basis sets were obtained through a slightly modified version of the ATOMSCF program of Pavani et al. [16]. In order to evaluate the quality of our contracted [5s3p], [11s6p6d], and [17s11p7d] basis sets in molecular studies, we accomplished calculations of total energy, highest occupied molecular orbital (HOMO) energy, and one level below to highest occupied molecular orbital (HOMO-1) energy for the 5 MnO 1þ and 1 LaO 1þ fragments at the HFR level [17]. The results are compared with those obtained from the extended (20s14p), (30s19p13d), and (31s23p18d) basis sets generated by GCHF method in this work. In Table 2, the total and orbital energies for 5 MnO 1þ and 1 LaO 1þ fragments are shown. From this table, we can see, for the fragments studied, that the total energies obtained with our [11s6p6d]/[5s3p] and [17s11p7d]/[5s3p] basis sets are close the corresponding values of extended (30s19p13d)/(20s14p) and (31s23p18d)/(20s14p) basis sets. The total energies differ by and hartree for 5 MnO 1þ and 1 LaO 1þ fragments, respectively. The HOMO energies obtained from contracted basis sets are closer to those of the extended basis sets and differ by and hartree for the 5 MnO 1þ and 1 LaO 1þ fragments, respectively. The HOMO-1 energies obtained from contracted basis sets are also closer to the values of extended basis sets and present deviations of and hartree. In order to better describe the molecular properties in the implementation of ab initio calculations it is necessary to include polarization functions in basis set. For the O atom of the [LaMnO 3 ] 2 fragment, it was added one polarization function of d ða ¼ 0: Þ symmetry in the contracted [5s3p] basis set. The polarization function is extracted from the Guassian primitive basis set. The adequate polarization function is chosen through successive calculations for the [LaMnO 3 ] 2 fragment by using different primitive functions, taking into account the minimum energy criterion. The molecular [17s11p7d]/[11s6p6d]/[5s3p1d] basis set was used to calculate dipole moment, total energy, and atomic charges (obtained with Mulliken s population analysis [18]). The octahedral [LaMnO 3 ] 2 fragment studied is shown in Fig. 2. The same represents the crystalline 3D periodic LaMnO 3 system. The Mn is located in the center of the octahedron being wrapped up for six O atoms, disposed in the reticular plane (200), and two La atoms arranged in the reticular plane (100). The study was developed according to the following strategy: (i) first, it was made the geometry optimization of the [LaMnO 3 ] 2 fragment in the symmetry C 2v and electronic state 5 B 1, (ii) finally, calculations single points were developed with the geometry optimized according to the descriptions shown in Fig. 2. Table 2 Total and orbital energies of the highest occupied HF molecular orbitals (in hartree) Fragment Basis set TE 1 HOMO 1 HOMO-1 5 MnO 1þ [11s6p6d] a /[5s3p] b (30s19p13d) c /(20s14p) d LaO 1þ [17s11p7d] e /[5s3p] b (31s23p18d) f /(20s14p) d a Contracted Gaussian basis sets, respectively, for Mn ( 6 S). b Contracted Gaussian basis sets, respectively, for O ( 3 P). c Extended Gaussian basis sets, respectively, for Mn ( 6 S). d Extended Gaussian basis sets, respectively, for O ( 3 P). e Contracted Gaussian basis sets, respectively, for La ( 2 D). f Extended Gaussian basis sets, respectively, for La ( 2 D).

5 O. Treu Filho et al. / Journal of Molecular Structure (Theochem) 631 (2003) Fig. 2. The octahedral [LaMnO 3 ] 2 fragment studied. (I): (a) represents the [LaMnO 3 ] 2 fragment with the Mn atoms fixed in the space; (b) represents the [LaMnO 3 ] 2 fragment with the Mn atoms being moved þ0.005 Å in the symmetry C 2 axis, while the La and O atoms being maintained fixed; (c) represents the [LaMnO 3 ] 2 fragment in that the Mn atoms are moved from Å in the symmetry C 2 axis and La and O atoms being maintained fixed; (II): represents the [LaMnO 3 ] 2 fragment with the bond lengths Mn 1 O 1,Mn 1 O 6, and Mn 2 O 6 shortened from Å. In Fig. 2 (I): (a) represents the [LaMnO 3 ] 2 fragment with the Mn atoms fixed in the space. (b) represents the [LaMnO 3 ] 2 fragment with the Mn atoms being moved þ0.005 Å in the symmetry C 2 axis, while the La and O atoms being maintained fixed. (c) it represent the [LaMnO 3 ] 2 fragment in which the Mn atoms are moved from Å in the symmetry C 2 axis and La and O atoms being maintained fixed. Fig. 2 (II) represents the [LaMnO 3 ] 2 fragment with the bond lengths Mn 1 O 1,Mn 1 O 6, and Mn 2 O 6 shortened from Å. The geometry optimization was performed by using the Berny s algorithm [19]. All molecular calculations in this work were carried out by HFR method as implemented in the Gaussian 94 program [20]. 3. Results and discussion For our purpose of verify possible piezoelectric properties of Perovskite [LaMnO 3 ]. First, we would like to bring attention for the comparison among the calculated bond lengths and the experimental values [21]. For Mn O and La O our results are and 2.67 Å, while the experimental values are and Å, respectively. The differences between the theoretical and experimental values are and Å. This show a good performance of molecular [17s11p7d]/[11s6p6d]/[5s3p1d] basis set to describe the geometry of the fragment studied. Table 3 presents the dipole moment and the total energy from [LaMnO 3 ] 2 fragment. As mentioned previously, the calculations at atomic positions (b) and (c) are concerned to Mn 3þ central ion deviation Table 3 Dipole moment (Debye) and Total energy (in hartree) from [LaMnO 3 ] 2 fragment Mn atom position m x m y m z m TE (a) (b) (c) (d)

6 98 O. Treu Filho et al. / Journal of Molecular Structure (Theochem) 631 (2003) from main axis of C 2 symmetry, meanwhile the calculation at atomic position (a) presents the fragment without this deviation. The results in Table 3 show that the fragment at (b) is hartree more stable than at (a), indicating the Mn 3þ central ion is not centersymmetric. The calculations also show that the atomic position (c) is hatree less stable than (a). Clearly, at (b), the presence of La atoms interacting with O and Mn 3þ atoms provides a best representation of the real chemical interactions present in the crystalline 3D periodic LaMnO 3 system. We can also observe at (b) a strong increase of the dipole moment over (a), indicating a polarization of [LaMnO 3 ] 2 fragment. Reducing the Mn 1 O 1, Mn 1 O 6 and Mn 2 O 6 bond lengths, the calculations in the atomic position (d) also show an increase of the dipole moment over (a). This fact suggests that there is a probable polarization of LaMnO 3 crystal when submitted under mechanical stress. Table 4 shows the values of the partial charges of the atoms. At atomic position (d) the decrease of Mn 1 O 1,Mn 1 O 6 and Mn 2 O 6 bond lengths, which could be caused by a mechanical stress, provokes a rearrangement of the partial charges in all atoms comparing with the atomic position (a). The rearrangement of the charges at (d) as well as the increase of the dipole moment can lead us to suppose that the decrease of Mn 1 O 1,Mn 1 O 6 and Mn 2 O 6 chemical bond lengths provokes a polarization of the [LaMnO 3 ] 2 fragment. We can also observe that the atomic charges do not change significantly at the atomic positions (b) and (c) the Mn 3þ central ion is Table 4 Total atomic charges from [LaMnO 3 ] 2 fragment Atom position Atom (a) (b) (c) (d) Mn Mn La La O O O O O O deviated from main axis of C 2 symmetry when compared with the atomic position (a). This suggests that the nature of La O and Mn O chemical bonds does not alter when the central ion Mn 3þ deviates from the C 2 symmetry of [LaMnO 3 ] 2 fragment. At (d), the decrease of M 1 O 1,Mn 1 O 6 and Mn 2 O 6 bond lengths provokes the deviation of atomic charges in the [LaMnO 3 ] 2 fragment, indicating that the nature of La O and Mn O chemical bonds was changed. Thus, we suggest that electrostatic interactions are very important in the electronic structure constitution of [LaMnO 3 ] 2 fragment. This is consistent because of two reasons: (1) due to the repulsive effect of d electrons in both high-spin and low-spin octahedral species of ML complexes (M ¼ metal and L ¼ ligant), all d electron density will repel the bonding electron density [22]; (2) the chemical bonds in lanthanide complexes is predominantly ionic [23]. To conclude, we presented in this work evidences that LaMnO 3 can present piezoelectric properties caused by electrostatic interactions. 4. Concluding remarks In the present work, highly accurate atomic basis sets were constructed using the GCHF method for O ( 3 P), Mn ( 6 S), and La ( 2 D) atoms. Theses basis sets were used to calculate the electronic structure of [LaMnO 3 ] 2 fragment. We show evidences that the [LaMnO 3 ] 2 fragment presents piezoelectric properties caused by electrostatic interaction of its atoms. The atomic basis developed by the GCHF method is an effective alternative to investigate electronic theoretical properties of LaMnO 3. Acknowledgements We gratefully acknowledge the financial support from the Brazilian agencies CNPq and FAPESP. We employed computing facilities at the IQ-UNESP, IBILCE-UNESP, CENAPAD-UNICAMP, and IF- UNESP.

7 O. Treu Filho et al. / Journal of Molecular Structure (Theochem) 631 (2003) References [1] J.R. Mohallem, R.M. Dreizler, M. Trsic, Int. J. Quant. Chem. Symp. 20 (1986) 45. [2] R. Custódio, M. Giordan, N.H. Morgon, J.D. Goddard, Int. J. Quant. Chem. 42 (1992) 411. [3] E.V.R. de Castro, F.E. Jorge, J.C. Pinheiro, Chem. Phys. 243 (1999) 1. [4] J.C. Pinheiro, F.F. Jorge, E.V.R. de Castro, J. Mol. Struct. (THEOCHEM) 491 (1999) 81. [5] J.C. Pinheiro, F.E. Jorge, E.V.R. de Castro, Int. J. Quant. Chem. 78 (2000) 15. [6] J.C. Pinheiro, A.B.F. da Silva, M. Trsic, J. Mol. Struct. (THEOCHEM) 394 (1997) 107. [7] J.C. Pinheiro, A.B.F. da Silva, M. Trsic, Int. J. Quant. Chem. 63 (1997) 927. [8] I.N. Jardim, O. Treu Filho, M.A.U. Martines, M.R. Davolos, M. Jafelicci Jr., J.C. Pinheiro, J. Mol. Struct. (THEOCHEM) 464 (1999) 15. [9] R.M.L. Savedra, J.C. Pinheiro, O. Treu Filho, R.T. Kondo, J. Mol. Struct. (THEOCHEM) 587 (2002) 9. [10] W.G. Cady, Piezoelectricity, McGraw-Hill, New York, [11] V.M. Goldschmidt, Mat.-Naturv. Kl. 2 (1926) 8. [12] B. Jaffe, W.R. Cook Jr., H. Jaffe, Piezoelectric Ceramics, Academic Press, New York, [13] H.F. Kay, P. Vousden, Philos. Mag. 40 (1949) [14] J.R. Mohallem, Z. Phys. D 3 (1986) 339. [15] T.H. Dunning Jr., P.J. Hey, in: H.F. Schaefer III (Ed.), Methods of Electronic Structure Theory, Plenum, New York, [16] R. Pavani, E. Clementi, Technical Report RJ 518, IBM Research, New York, [17] C.C.J. Roothaan, Rev. Phys. Mod. 23 (1951) 69. [18] R.S. Mulliken, J. Chem. Phys. 23 (1955) [19] H.B. Schlegel, J. Comput. Chem. 3 (1982) 214. [20] M.J. Frisch, G.W. Trucks, H.B. Schlegel, P.M.W. Gill, B.G. Johnson, M.A. Robb, J.R. Cheeseman, T.A. Keith, G.A. Petersson, J.A. Montgomery, K. Raghavachari, A. Al-Laham, V.G. Zakrzewski, J.V. Ortiz, J.B. Foresman, J. Cioslowki, B.B. Stefanov, A. Nanayakkara, M. Challacombe, C.Y. Peng, P.Y. Ayala, W. Chen, M.W. Wong, J.L. Andres, E.S. Replogle, R. Gomperts, R.L. Martin, D.J. Fox, J.S. Binkley, D.J. Defrees, J. Baker, P.J. Stewart, M. Head-Gordon, C. Gonzalez, J.A. Pople, Gaussian 94 (Revision D.1), Gaussian, Inc, Pittsburgh, PA, [21] R. Millini, M.F. Gagliardi, G. Piro, J. Mater. Sci. 29 (1994) [22] M. Gerloch, E.C. Contable, Transition Metal Chemistry, VCH, Weinheim, [23] N. Kaltsoyannis, P. Scott, The f Elements, Oxford University press, Oxford, 1999.