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GCHF basis sets and their application in the electronic structure study of PrMnO3
Journal of Molecular Structure (Theochem) 668 (2004) 113–117 www.elsevier.com/locate/theochem
GCHF basis sets and their application in the electronic structure study of PrMnO3 Oswaldo Treu Filhoa, Jose´ Cirı´aco Pinheirob,*, Roge´rio Toshiaki Kondoc, Miguel Jafelicci Jr.a a
Instituto de Quı´mica, UNESP, CP 335, 14801-140 Araraquara, SP, Brazil Laborato´rio de Quı´mica Teo´rica e Computacional, Departamento de Quı´mica, Centro de Cieˆncias Exatas e Naturais, Universidade Federal do Para´, CP 101101, 66075-110 Bele´m, PA, Amazoˆnia, Brazil c Sec¸a˜o de Suporte, Centro de Informa´tica de Sa˜o Carlos, Universidade de Sa˜o Paulo, 13560-970 Sa˜o Carlos, SP, Brazil b
Received 21 July 2003; accepted 20 October 2003
Abstract The approach called generator coordinate Hartree –Fock (GCHF) method is used in the selection of Gaussian basis set [25s18p for O (3P), 31s21p14d for Mn (6S), and 33s22p16d9f for Pr (4J)] for atoms. The role of the weight functions in the assessment of the numerical integration range of the GCHF equations is shown. These basis sets are contracted to (25s18p/9s5p), (31s21p14d/9s6p4d), and (33s22p16d9f/18s12p5d3f) by segmented contraction scheme of Dunning and they are utilized in calculations of Restricted-Open-HF (ROHF) Total and Orbital energies of the 3MnOþ1 and 1PrOþ1 fragments, to evaluate their quality in molecular studies. The addition of one d polarization function in the contracted (9s5p) basis set for O(3P) atom and their application with the contracted (9s6p4d), (18s21p5d3f) basis sets for Mn (6S) and Pr– Pr (4J) atoms lead to the electronic structure study of PrMnO3. The dipole moment, the total energy, and total atomic charges properties were calculated and were carried out at ROHF level with the [PrMnO3]2 fragment. The calculated values show that PrMnO3 does not present piezoelectric properties. q 2004 Elsevier B.V. All rights reserved. Keywords: Generator coordinate Hartree –Fock basis sets; Contracted basis sets; Electronic structure of PrMnO3; Piezoelectric
1. Introduction The generator coordinate Hartree –Fock (GCHF) method was developed in 1986 [1] and it has been used as a powerful technique in design of basis sets in atomic and molecular environments [2,3]. One of the early applications was in the construction of universal basis sets [2,4]. Also, the GCHF method was tested as a tool for choosing polarization functions to be used in Gaussian basis sets for the first- [5] and the second-row [6] atoms, and applied in the calculations of electronic properties and IR-spectrum of high trydimite [7]. In addition, the GCHF method is used in the generation of GTF basis sets, for ab initio calculations of electronic affinities of some enolates [8], and in the generation of contracted GTF basis sets to the theoretical interpretation of the Raman spectrum of hexaaquachromium(III) ion [9]. * Corresponding author. E-mail address: [email protected] (J.-C. Pinheiro). 0166-1280/$ - see front matter q 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2003.10.031
In previous reports we used the GCHF method as a tool to develop basis sets for calculations of the electronic structure of YMnO3 and perovskite LaMnO3 and to verify possible piezoelectric properties of those materials [10,11]. Now, in this article, we explore the strategy of the GCHF method in the selection of basis sets and their application in the electronic structure study of PrMnO3. Our intent is also to verify possible piezoelectric properties in the crystal structure of PrMnO3.
2. Theoretical methodology 2.1. GCHF basis sets The GCHF approach [1] is based in choosing the oneelectron functions as the continuous superposition, i.e. ð Ci ¼ ci ð1; aÞfi ðaÞda i ¼ 1; …; n ð1Þ
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Fig. 1. The 2s, 3p, and 4f weight functions for O (3P), Mn (6S), and Pr (4J) obtained with 25s18p, 31s21p14d, and 33s22p16d9f GCHF basis sets, respectively.
where ci ð1; aÞ are the generator functions, the generator coordinate (Slater or Gaussian type orbitals or other functions), the fi ðaÞ are unknown weight functions and a is the generator coordinate, a parameter or set of parameters relevant to the description of the system. The application of the variational principle to the energy expectation value built with such one-electron functions leads to the Griffin – Wheeler HF (GWHF) equations ð ½Fða; bÞ 2 e i Sða; bÞfi ðbÞdb ¼ 0 i ¼ 1; …; m ð2Þ where the 1i 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 a technique called integral discretization (ID) [12]. This technique is implemented through a relabling of the generator coordinate space, i.e.
V ¼ ln a=A A . 1
ð3Þ
with A being a scaling parameter determined numerically. The new generator coordinate space, V; is discretized for each s, p, d, f, symmetry in an equally space mesh {Vi } so that
Vl ¼ Vmin þ ðq 2 1ÞDV q ¼ 1; 2; …; N
ð4Þ
In Eq. (4), N is an option that allows the definition of the size of the basis sets, and DV is the numerical integration interval. The values of Vmin (V lowest value) and Vmax (V highest value) ¼ Vmin þ ðN 2 1Þ DV are chosen so as to adequately encompass the integration range of f ðVÞ: This is clearly visualized by drawing the weight functions from a preliminary calculation with arbitrary discretization parameters in Fig. 1. It shows the 2s, 3p, and 4f weight functions for O (3P), Mn (6S), and Pr (4J), respectively. The lowest values at the left and the decay at the right indicate clearly the integration range relevant for the correct numerical integration of the GWHF equations. Plots of weight functions were thus used to build rather large GCHF basis sets for the referred atoms. The size (25 for s symmetry and 18 for p symmetry for O atom, 31 for s symmetry, 21 and 14 for p and d symmetries, respectively, for Mn atom, and 33 for s symmetry, 22, 16, and 9 for p, d, and f symmetries, respectively, for Pr atom), ensuring the error for the Total HF ground state energy, was 9.8 £ 1027, 1.7 £ 1025, and 1.35 £ 1023 hartree for O, Mn, and Pr atoms, respectively, compared to numerical HF values [13,14]. Table 1 shows the discretization parameters (which define the exponents) for the GCHF basis sets.
Table 1 Discretization parameters (basis set exponents) for O (3P), Mn (6S), and Pr (4J) atoms ðA ¼ 6:0Þ Sy‘mmetry
s p d f
O (3P)
Mn (6S)
Pr (4J)
Vmin
DV
N
Vmin
DV
N
Vmin
DV
N
20.47680 20.45420 – –
0.11977 0.10916 – –
25 18 – –
20.59427 20.28200 20.38600 –
0.11214 0.10327 0.11339 –
31 21 14 –
20.64383 20.40552 20.13618 20.26615
0.11021 0.10950 0.10430 0.12900
33 22 16 9
O. T. Filho et al. / Journal of Molecular Structure (Theochem) 668 (2004) 113–117 Table 2 Total and orbital energies of the highest occupied molecular orbitals (in hartree) 21HOMO 21HOMO21
Fragment GCHF basis sets
2ET
3
MnO1þ
(9s6p4d)/(9s5p)a,b 31s21p14d/25s18pa,b
1224.026402 0.20227 1224.05978 0.33251
0.20668 0.36292
1
PrO1þ
(18s12p5d3f)/(9s5p)b,c 8993.578112 0.40467 33s22p16d9f/25s18pb,c 8995.464772 0.40565
0.52845 0.57981
a b c
Contracted and uncontracted GCHF basis sets for Mn (6S) atom. Contracted and uncontracted GCHF basis sets for O (3P) atom. Contracted and uncontracted GCHF basis sets for Pr (4J) atom.
2.2. Contraction, evaluation of quality in molecular calculations, and polarization function of the GCHF basis sets For O atom, the 25s18p primitive GCHF basis set was contacted to (9s5p) through the contraction scheme 16,1,1,1,2,1,1,1,1/14,1,1,1,1 For the Mn atom the 31s21p14d primitive GCHF basis set was contracted to (9s6p4d) through the contraction scheme 14,3,3,1,1,1,3,4,1/15,2,1,1,1,1/11,1,1,1. For the Pr atom the 33s22p16d9f primitive GCHF basis set was contracted to (18s12p5d3f) through the contraction scheme 10,1,1,1,1,1,1,1,1,1,11,2,3,3,2,1,1/6,1,1,1,1,1,1,3,3,2,1,1/9,2,3,1,1/7,1,1. The total energy for O, Mn, and Pr atoms, obtained with contracted GCHF basis sets, were compared to the numerical HF values [13,14] and the differences were 2.9 £ 1024, 4.6 £ 1022, and 0.24 hartree, respectively. For the contraction of the GCHF basis sets, we used the contraction scheme proposed by Dunning and Hey [15]. The uncontracted and contracted GCHF basis sets were designed using a modified version of the ATOMSCF of MOTEC-89 [16]. The quality of the contracted basis sets in molecular studies was evaluated through of the application in calculations of the total energy, the highest occupied molecular orbital energy ð1HOMO Þ; and one level below to
115
the highest occupied molecular orbital energy ð1HOMO21 Þ for the 3MnOþ1 and 1PrOþ1 fragments in the ROHF level with the HF-Roothaan approach [17]. In this case the calculations were compared with that obtained with uncontracted GCHF basis sets. In Table 2 the total, 1HOMO ; and 1HOMO21 energies are shown for 3MnOþ1 and 1PrOþ1. The discrepancies are 0.0333 and 1.89 hartree for the fragments studied, respectively. The values of 1HOMO energy obtained with contracted GCHF basis sets presented discrepancies of 0.1302 and 0.1562 hartree, respectively, when compared with the uncontracted GCHF basis sets. For 1HOMO21 energy this discrepancies are 9.8 £ 1024 and 0.05134 hartree, respectively. In order to describe better the electronic properties of the studied [PrMnO3]2 fragment we included one polarization function of d ða ¼ 0:4675168Þ symmetry in contracted GCHF basis set for O atom. This basis set along with the other contracted GCHF basis sets (Mn and Pr atoms) were used in the calculations of the dipole moment, the total energy, and the total atomic charges (obtained with Mulliken’s population analysis [18]) of fragment of our interest (see Section 3).
3. Results and discussion The atomic basis sets obtained by the Generated Coordinate Hartree – Fock method were used to calculate the dipole moment, the total energy and the atomic charges of [PrMnO3]2 fragment (Fig. 2). The results for the dipole moment and the total energy are shown in Table 3 and the total atomic charges are shown in Table 4. The geometry optimization was calculate by the ROHF level [17] using the Benry optimization algorithm of Schlegel [19]. The molecular calculations performed in this work were done by using the GAUSSIAN 94 routine [20]. The Mn – O and Pr –O bond length obtained after the optimization are 1.934
˚ displaced, (c) the Fig. 2. The [PrMnO3]2 fragment: (a) the manganese atoms are in the geometric center of the fragment, (b) the manganese atoms are þ 0.005 A ˚ displaced to the x-axis of the spatial coordinators and Pr and O atoms being kept freeze in their positions, (d) represents the manganese atoms are 20.005 A ˚ smaller then normal. fragment having the same geometry but with bond lengths Mn1 –O2, Mn2 – O1, Mn2 –O2, and Pr2 –O3 0.005 A
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Table 3 Dipole moments (in Debye) and total energy (in hartree) of [PrMnO3]2 fragment Mn atom position mx
my
mz
m
23.744 20.067 0.976 3.870 20.57 16.44 20.037 26.33 20.91 11.55 22.170 23.99 24.317 5.858 2.207 7.605
(a) (b) (c) (d)
2TE 20584.4267756 20584.3430380 20584.3326061 20584.4121449
˚ , while the experimental bond length [21], are and 2.687 A ˚ , respectively. The respective discrepan1.991 and 2.773 A ˚. cies are 0.057 and 0.087 A With the purpose to determine the possible piezoelectric properties of PrMnO3, the theoretical results were divided in two parts. Initially, to verify the absence of centersymmetry in the PrMnO3, the dipole moment, the total energy and the atomic charges are computed after optimization of the geometry, in the C1 symmetry, 3A electronic state to the [PrMnO3]2 fragment with the manganese atoms in three positions of the spatial coordinators: (a) the manganese atoms are in the geometric center of the fragment, (b) the manganese atoms ˚ displaced and (c) the manganese atoms are are þ 0.005 A ˚ 2 0.005 A displaced, respectively, to the X-axis of the spatial coordinators. The others atoms (Pr and O) are kept freeze in their positions. The total energy results (Table 3) indicated that the [PrMnO3]2 fragment is more stable at position (a) than at positions (b) and (c), but do not indicating the possibility of being uncentersymmetric. Finally, as well as known, the piezoelectric materials show electric charges when being under pressure [22]. For that reason, the dipole moment, the total energy and the total atomic charges were estimate to the [PrMnO3]2 fragment having the same geometry but with Mn1 –O2, Mn2 –O1, ˚ smaller then Mn2 –O2 and Pr2 – O3 bond length 0.005 A normal (d). The results are listed in Table 4. The increase of the dipole moment in the main axis of C1 symmetry shows
Table 4 Total atomic charges of [PrMnO3]2 fragment Atom
Mn1 Mn2 Pr1 Pr2 O1 O2 O3 O4 O5 O6
Atom position (a)
(b)
(c)
(d)
0.511 0.546 2.59 2.43 20.497 20.356 21.65 21.44 21.15 20.984
1.06 0.516 2.36 2.36 20.291 20.845 21.30 21.42 21.44 21.00
1.02 0.694 1.73 2.39 20.319 20.978 21.318 21.431 20.788 21.00
0.780 0.521 2.56 2.03 20.408 20.362 21.54 21.45 21.14 20.991
the fragment (d) presented a spontaneous polarization than in (a). The atomic charges at position (a) and (d) show a small atomic charge migration has occurred between the Pr2, Mn1, and O5 atoms from position (a) to position (d). Besides the fact of PrMnO3 presents centesymmetric characteristics for the Mn central atoms, this small charge migration lead us to suggest that PrMnO3 does not present piezoelectric properties.
4. Conclusions Among the electronic structure of [PrMnO3]2 fragment in the C1 symmetry, 3A electronic state, the used atomic basis set indicates, more explicitly, a very good bond length optimization of [PrMnO3]2 fragment. Obtained values of the dipole moment, the total energy and the total atomic charges show that is reasonable to believe that PrMnO3 does not present behavior of piezoelectric material. The atomic basis developed by the GCHF method is an effective alternative to investigate electronic theoretical properties of PrMnO3.
Acknowledgements The authors are very grateful to Mrs M.C. da Costa Freitas at Universidade Federal do Para´ for valuable help in the manuscript preparation for the financial support by CNPq and FAPESP (Brazilian agencies). We employed computing facilities at IQ-UNESP, IBILCE-UNESP, CENAPAD-UNICAMP, and of the LQTC-DQ-UFPA.
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O. T. Filho et al. / Journal of Molecular Structure (Theochem) 668 (2004) 113–117 [14] T. Koga, S. Wathnabe, A.J. Thakkar, Int. J. Quant. Chem. 54 (1995) 261. [15] T.H. Dunning Jr., P.J. Hey, in: H.F. Schaefer III (Ed.), Methods of Electronic Structure Theory, Plenum, New York, 1977. [16] S.J. Chakravorty, G. Corongiu, J.R. Flores, V. Sonnad, E. Clementi, V. Carravetta, I. Cacelli, in: E. Clementi (Ed.), MOTEC—Modern Techniques in Computational Chemistry, ESCOM, Leiden, 1989. [17] C.C.J. Roothaan, Rev. Phys. Mod. 23 (1951) 69. [18] R.S. Mulliken, J. Chem. Phys. 23 (1955) 1833. [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. Roob, J.R. Cheeseman, T.A. Keith, G.A. Petersson, J.A.
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Montgomery, K. Raghavachari, M.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, J.P. Stewart, M. HeadGordon, C. Gonzalez, J.A. Pople, GAUSSIAN 94 (Revision D. 1). Gaussian, Inc., Pittsburgh, PA, 1995 [21] S. Quezel-Ambrunaz, Bulletin de la Societe Francaise de Mineralogie et de Cristallographie 91 (1968) 339. [22] B. Jaffe, W.R. Cook Jr., H. Jaffe, Piezoelectric Ceramics, Academic, New York, 1991.
GCHF basis sets and their application in the electronic structure study of PrMnO3 Oswaldo Treu Filhoa, Jose´ Cirı´aco Pinheirob,*, Roge´rio Toshiaki Kondoc, Miguel Jafelicci Jr.a a
Instituto de Quı´mica, UNESP, CP 335, 14801-140 Araraquara, SP, Brazil Laborato´rio de Quı´mica Teo´rica e Computacional, Departamento de Quı´mica, Centro de Cieˆncias Exatas e Naturais, Universidade Federal do Para´, CP 101101, 66075-110 Bele´m, PA, Amazoˆnia, Brazil c Sec¸a˜o de Suporte, Centro de Informa´tica de Sa˜o Carlos, Universidade de Sa˜o Paulo, 13560-970 Sa˜o Carlos, SP, Brazil b
Received 21 July 2003; accepted 20 October 2003
Abstract The approach called generator coordinate Hartree –Fock (GCHF) method is used in the selection of Gaussian basis set [25s18p for O (3P), 31s21p14d for Mn (6S), and 33s22p16d9f for Pr (4J)] for atoms. The role of the weight functions in the assessment of the numerical integration range of the GCHF equations is shown. These basis sets are contracted to (25s18p/9s5p), (31s21p14d/9s6p4d), and (33s22p16d9f/18s12p5d3f) by segmented contraction scheme of Dunning and they are utilized in calculations of Restricted-Open-HF (ROHF) Total and Orbital energies of the 3MnOþ1 and 1PrOþ1 fragments, to evaluate their quality in molecular studies. The addition of one d polarization function in the contracted (9s5p) basis set for O(3P) atom and their application with the contracted (9s6p4d), (18s21p5d3f) basis sets for Mn (6S) and Pr– Pr (4J) atoms lead to the electronic structure study of PrMnO3. The dipole moment, the total energy, and total atomic charges properties were calculated and were carried out at ROHF level with the [PrMnO3]2 fragment. The calculated values show that PrMnO3 does not present piezoelectric properties. q 2004 Elsevier B.V. All rights reserved. Keywords: Generator coordinate Hartree –Fock basis sets; Contracted basis sets; Electronic structure of PrMnO3; Piezoelectric
1. Introduction The generator coordinate Hartree –Fock (GCHF) method was developed in 1986 [1] and it has been used as a powerful technique in design of basis sets in atomic and molecular environments [2,3]. One of the early applications was in the construction of universal basis sets [2,4]. Also, the GCHF method was tested as a tool for choosing polarization functions to be used in Gaussian basis sets for the first- [5] and the second-row [6] atoms, and applied in the calculations of electronic properties and IR-spectrum of high trydimite [7]. In addition, the GCHF method is used in the generation of GTF basis sets, for ab initio calculations of electronic affinities of some enolates [8], and in the generation of contracted GTF basis sets to the theoretical interpretation of the Raman spectrum of hexaaquachromium(III) ion [9]. * Corresponding author. E-mail address: [email protected] (J.-C. Pinheiro). 0166-1280/$ - see front matter q 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2003.10.031
In previous reports we used the GCHF method as a tool to develop basis sets for calculations of the electronic structure of YMnO3 and perovskite LaMnO3 and to verify possible piezoelectric properties of those materials [10,11]. Now, in this article, we explore the strategy of the GCHF method in the selection of basis sets and their application in the electronic structure study of PrMnO3. Our intent is also to verify possible piezoelectric properties in the crystal structure of PrMnO3.
2. Theoretical methodology 2.1. GCHF basis sets The GCHF approach [1] is based in choosing the oneelectron functions as the continuous superposition, i.e. ð Ci ¼ ci ð1; aÞfi ðaÞda i ¼ 1; …; n ð1Þ
114
O. T. Filho et al. / Journal of Molecular Structure (Theochem) 668 (2004) 113–117
Fig. 1. The 2s, 3p, and 4f weight functions for O (3P), Mn (6S), and Pr (4J) obtained with 25s18p, 31s21p14d, and 33s22p16d9f GCHF basis sets, respectively.
where ci ð1; aÞ are the generator functions, the generator coordinate (Slater or Gaussian type orbitals or other functions), the fi ðaÞ are unknown weight functions and a is the generator coordinate, a parameter or set of parameters relevant to the description of the system. The application of the variational principle to the energy expectation value built with such one-electron functions leads to the Griffin – Wheeler HF (GWHF) equations ð ½Fða; bÞ 2 e i Sða; bÞfi ðbÞdb ¼ 0 i ¼ 1; …; m ð2Þ where the 1i 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 a technique called integral discretization (ID) [12]. This technique is implemented through a relabling of the generator coordinate space, i.e.
V ¼ ln a=A A . 1
ð3Þ
with A being a scaling parameter determined numerically. The new generator coordinate space, V; is discretized for each s, p, d, f, symmetry in an equally space mesh {Vi } so that
Vl ¼ Vmin þ ðq 2 1ÞDV q ¼ 1; 2; …; N
ð4Þ
In Eq. (4), N is an option that allows the definition of the size of the basis sets, and DV is the numerical integration interval. The values of Vmin (V lowest value) and Vmax (V highest value) ¼ Vmin þ ðN 2 1Þ DV are chosen so as to adequately encompass the integration range of f ðVÞ: This is clearly visualized by drawing the weight functions from a preliminary calculation with arbitrary discretization parameters in Fig. 1. It shows the 2s, 3p, and 4f weight functions for O (3P), Mn (6S), and Pr (4J), respectively. The lowest values at the left and the decay at the right indicate clearly the integration range relevant for the correct numerical integration of the GWHF equations. Plots of weight functions were thus used to build rather large GCHF basis sets for the referred atoms. The size (25 for s symmetry and 18 for p symmetry for O atom, 31 for s symmetry, 21 and 14 for p and d symmetries, respectively, for Mn atom, and 33 for s symmetry, 22, 16, and 9 for p, d, and f symmetries, respectively, for Pr atom), ensuring the error for the Total HF ground state energy, was 9.8 £ 1027, 1.7 £ 1025, and 1.35 £ 1023 hartree for O, Mn, and Pr atoms, respectively, compared to numerical HF values [13,14]. Table 1 shows the discretization parameters (which define the exponents) for the GCHF basis sets.
Table 1 Discretization parameters (basis set exponents) for O (3P), Mn (6S), and Pr (4J) atoms ðA ¼ 6:0Þ Sy‘mmetry
s p d f
O (3P)
Mn (6S)
Pr (4J)
Vmin
DV
N
Vmin
DV
N
Vmin
DV
N
20.47680 20.45420 – –
0.11977 0.10916 – –
25 18 – –
20.59427 20.28200 20.38600 –
0.11214 0.10327 0.11339 –
31 21 14 –
20.64383 20.40552 20.13618 20.26615
0.11021 0.10950 0.10430 0.12900
33 22 16 9
O. T. Filho et al. / Journal of Molecular Structure (Theochem) 668 (2004) 113–117 Table 2 Total and orbital energies of the highest occupied molecular orbitals (in hartree) 21HOMO 21HOMO21
Fragment GCHF basis sets
2ET
3
MnO1þ
(9s6p4d)/(9s5p)a,b 31s21p14d/25s18pa,b
1224.026402 0.20227 1224.05978 0.33251
0.20668 0.36292
1
PrO1þ
(18s12p5d3f)/(9s5p)b,c 8993.578112 0.40467 33s22p16d9f/25s18pb,c 8995.464772 0.40565
0.52845 0.57981
a b c
Contracted and uncontracted GCHF basis sets for Mn (6S) atom. Contracted and uncontracted GCHF basis sets for O (3P) atom. Contracted and uncontracted GCHF basis sets for Pr (4J) atom.
2.2. Contraction, evaluation of quality in molecular calculations, and polarization function of the GCHF basis sets For O atom, the 25s18p primitive GCHF basis set was contacted to (9s5p) through the contraction scheme 16,1,1,1,2,1,1,1,1/14,1,1,1,1 For the Mn atom the 31s21p14d primitive GCHF basis set was contracted to (9s6p4d) through the contraction scheme 14,3,3,1,1,1,3,4,1/15,2,1,1,1,1/11,1,1,1. For the Pr atom the 33s22p16d9f primitive GCHF basis set was contracted to (18s12p5d3f) through the contraction scheme 10,1,1,1,1,1,1,1,1,1,11,2,3,3,2,1,1/6,1,1,1,1,1,1,3,3,2,1,1/9,2,3,1,1/7,1,1. The total energy for O, Mn, and Pr atoms, obtained with contracted GCHF basis sets, were compared to the numerical HF values [13,14] and the differences were 2.9 £ 1024, 4.6 £ 1022, and 0.24 hartree, respectively. For the contraction of the GCHF basis sets, we used the contraction scheme proposed by Dunning and Hey [15]. The uncontracted and contracted GCHF basis sets were designed using a modified version of the ATOMSCF of MOTEC-89 [16]. The quality of the contracted basis sets in molecular studies was evaluated through of the application in calculations of the total energy, the highest occupied molecular orbital energy ð1HOMO Þ; and one level below to
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the highest occupied molecular orbital energy ð1HOMO21 Þ for the 3MnOþ1 and 1PrOþ1 fragments in the ROHF level with the HF-Roothaan approach [17]. In this case the calculations were compared with that obtained with uncontracted GCHF basis sets. In Table 2 the total, 1HOMO ; and 1HOMO21 energies are shown for 3MnOþ1 and 1PrOþ1. The discrepancies are 0.0333 and 1.89 hartree for the fragments studied, respectively. The values of 1HOMO energy obtained with contracted GCHF basis sets presented discrepancies of 0.1302 and 0.1562 hartree, respectively, when compared with the uncontracted GCHF basis sets. For 1HOMO21 energy this discrepancies are 9.8 £ 1024 and 0.05134 hartree, respectively. In order to describe better the electronic properties of the studied [PrMnO3]2 fragment we included one polarization function of d ða ¼ 0:4675168Þ symmetry in contracted GCHF basis set for O atom. This basis set along with the other contracted GCHF basis sets (Mn and Pr atoms) were used in the calculations of the dipole moment, the total energy, and the total atomic charges (obtained with Mulliken’s population analysis [18]) of fragment of our interest (see Section 3).
3. Results and discussion The atomic basis sets obtained by the Generated Coordinate Hartree – Fock method were used to calculate the dipole moment, the total energy and the atomic charges of [PrMnO3]2 fragment (Fig. 2). The results for the dipole moment and the total energy are shown in Table 3 and the total atomic charges are shown in Table 4. The geometry optimization was calculate by the ROHF level [17] using the Benry optimization algorithm of Schlegel [19]. The molecular calculations performed in this work were done by using the GAUSSIAN 94 routine [20]. The Mn – O and Pr –O bond length obtained after the optimization are 1.934
˚ displaced, (c) the Fig. 2. The [PrMnO3]2 fragment: (a) the manganese atoms are in the geometric center of the fragment, (b) the manganese atoms are þ 0.005 A ˚ displaced to the x-axis of the spatial coordinators and Pr and O atoms being kept freeze in their positions, (d) represents the manganese atoms are 20.005 A ˚ smaller then normal. fragment having the same geometry but with bond lengths Mn1 –O2, Mn2 – O1, Mn2 –O2, and Pr2 –O3 0.005 A
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Table 3 Dipole moments (in Debye) and total energy (in hartree) of [PrMnO3]2 fragment Mn atom position mx
my
mz
m
23.744 20.067 0.976 3.870 20.57 16.44 20.037 26.33 20.91 11.55 22.170 23.99 24.317 5.858 2.207 7.605
(a) (b) (c) (d)
2TE 20584.4267756 20584.3430380 20584.3326061 20584.4121449
˚ , while the experimental bond length [21], are and 2.687 A ˚ , respectively. The respective discrepan1.991 and 2.773 A ˚. cies are 0.057 and 0.087 A With the purpose to determine the possible piezoelectric properties of PrMnO3, the theoretical results were divided in two parts. Initially, to verify the absence of centersymmetry in the PrMnO3, the dipole moment, the total energy and the atomic charges are computed after optimization of the geometry, in the C1 symmetry, 3A electronic state to the [PrMnO3]2 fragment with the manganese atoms in three positions of the spatial coordinators: (a) the manganese atoms are in the geometric center of the fragment, (b) the manganese atoms ˚ displaced and (c) the manganese atoms are are þ 0.005 A ˚ 2 0.005 A displaced, respectively, to the X-axis of the spatial coordinators. The others atoms (Pr and O) are kept freeze in their positions. The total energy results (Table 3) indicated that the [PrMnO3]2 fragment is more stable at position (a) than at positions (b) and (c), but do not indicating the possibility of being uncentersymmetric. Finally, as well as known, the piezoelectric materials show electric charges when being under pressure [22]. For that reason, the dipole moment, the total energy and the total atomic charges were estimate to the [PrMnO3]2 fragment having the same geometry but with Mn1 –O2, Mn2 –O1, ˚ smaller then Mn2 –O2 and Pr2 – O3 bond length 0.005 A normal (d). The results are listed in Table 4. The increase of the dipole moment in the main axis of C1 symmetry shows
Table 4 Total atomic charges of [PrMnO3]2 fragment Atom
Mn1 Mn2 Pr1 Pr2 O1 O2 O3 O4 O5 O6
Atom position (a)
(b)
(c)
(d)
0.511 0.546 2.59 2.43 20.497 20.356 21.65 21.44 21.15 20.984
1.06 0.516 2.36 2.36 20.291 20.845 21.30 21.42 21.44 21.00
1.02 0.694 1.73 2.39 20.319 20.978 21.318 21.431 20.788 21.00
0.780 0.521 2.56 2.03 20.408 20.362 21.54 21.45 21.14 20.991
the fragment (d) presented a spontaneous polarization than in (a). The atomic charges at position (a) and (d) show a small atomic charge migration has occurred between the Pr2, Mn1, and O5 atoms from position (a) to position (d). Besides the fact of PrMnO3 presents centesymmetric characteristics for the Mn central atoms, this small charge migration lead us to suggest that PrMnO3 does not present piezoelectric properties.
4. Conclusions Among the electronic structure of [PrMnO3]2 fragment in the C1 symmetry, 3A electronic state, the used atomic basis set indicates, more explicitly, a very good bond length optimization of [PrMnO3]2 fragment. Obtained values of the dipole moment, the total energy and the total atomic charges show that is reasonable to believe that PrMnO3 does not present behavior of piezoelectric material. The atomic basis developed by the GCHF method is an effective alternative to investigate electronic theoretical properties of PrMnO3.
Acknowledgements The authors are very grateful to Mrs M.C. da Costa Freitas at Universidade Federal do Para´ for valuable help in the manuscript preparation for the financial support by CNPq and FAPESP (Brazilian agencies). We employed computing facilities at IQ-UNESP, IBILCE-UNESP, CENAPAD-UNICAMP, and of the LQTC-DQ-UFPA.
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