- Email: [email protected]
Cation polarizability from first-principles: Sn2+
Computational Materials Science 22 (2001) 94±98
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Cation polarizability from ®rst-principles: Sn2 Leonardo Bernasconi *, Mark Wilson, Paul A. Madden Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OXI 3QZ, UK Accepted 2 April 2001
Abstract Generalized Gradient Corrected Density Functional Theory calculations have been performed on SnO in the gas phase. The total charge density of the molecular system has been partitioned between Sn and O using a Wannier localization transformation of the Kohn Sham eigenvectors, and the single-ion dipole moment of Sn2 at the optimized bond-distance has been estimated in terms of the position of the resulting Wannier function centers. This analysis has been extended over a wide range of ionic separations in order to monitor the dependence of the Sn2 dipole on both Coulombic and short-range interactions with O2 . The mechanism responsible for Sn2 polarization proves to be easily explained in terms of the non-bonding orbital center distance from the nucleus, without any major contributions deriving from mixing with orbitals centered on O. The Sn2 polarizability in the molecular system at the optimized bond distance (15.23 a.u.) is intermediate between the value for the free-ion (14.50 a.u.) and the estimate for crystalline SnO in the rocksalt structure (15.83 a.u.). The bond-length dependent polarizability at large ionic separations shows excellent agreement with the available Hartree Fock free-ion value. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction The ab initio investigation of ionic polarization in condensed phase systems has been the subject of considerable interest in the last years. Indeed, it has been shown that many of the anomalous structural properties of ionic materials, usually attributed to the manifestation of ``covalent'' effects, can be accounted for in terms of ion polarization phenomena and reproduced with considerable degree of success by a polarizable ionic prescription [1]. Although the polarizability of an ion in the gas phase can be predicted via conventional ®rst-principle methods, a thorough
*
Corresponding author. E-mail address: [email protected] (L. Bernasconi).
description of the polarization mechanism when this is part of a true physical system is rather more complicated. In addition to the purely Coulombic perturbation, the con®ning potential aecting an ion in the condensed-phase coordination environment is shaped by short-range repulsive charge density overlap with neighbouring ions [2]. In a previous paper [3], we have shown that density functional theory (DFT) calculations are indeed able to describe, at a quantitative level, the development of single-ion dipole moments in molecular systems. In particular, we concentrated on alkali metal ¯uorides and we were able to single out the relative weight of Coulombic and shortrange eects in determining the total ionic dipoles, by monitoring the dipole dependence on the ionic separation. An essential ingredient of our procedure was that of using a Wannier localization transformation [4] to partition the total electronic
0927-0256/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 1 ) 0 0 1 7 3 - 2
L. Bernasconi et al. / Computational Materials Science 22 (2001) 94±98
charge density, given by the squared modulus of the Kohn Sham eigenvectors, into that of the single ions. In this paper we will show that a similar approach can be generalized with no major changes to non-closed-shell species. In particular, we will concentrate on the calculation of the polarizability of Sn2 . This ion is known to have a very large polarizability, which can be attributed to intravalence s ! p excitations out of its 5s2 ground state. Sn(II) compounds in the condensed phase exhibit unusual crystal structures which are often attributed to the ``stereochemical requirements of the lonepair'' [5]. However, it has been suggested that such eects can also be explained as the consequence of polarization of the Sn2 cation [16], which favours the occupation of low symmetry sites. To fully examine this suggestion, an understanding of the short-range contributions to the cation polarization is required.
2. Method and computational details Generalized Gradient Corrected [6] Kohn Sham Density Functional Theory [7] calculations 1 were performed on a model system constituted of one Sn and one O atom in a cubic box of size L 10 A, periodically repeated in space. Norm-conserving ab initio pseudopotentials [8] were used to model the ion±electron interactions. The Kohn Sham eigenvectors were expanded in plane-waves up to an energy cuto of 330 eV, with only the C point included in the Brillouin zone sampling. All-electron minimizations were performed using a conjugate gradient minimization algorithm [9] in conjunction with a Pulay charge density mixing scheme [10]. The atomic separation for the SnO molecular system was optimized using a BFGS minimization algorithm, until all forces were lower than yielding an equilibrium bondlength 7 10 5 eV=A, of 1.837 A. The atomic separation was then varied including the equilibrium within a range (1.6±2.5 A) bondlength. The Kohn Sham eigenvectors were calculated in several points within this range, for 1 We used the code CASTEP 3.9 academic version, licensed under UKCP-MSI Agreement, 1999.
95
each of which a set of maximally localized Wannier functions (MLWF) was determined by a unitary transformation of the initial Kohn Sham orbitals. X Umn jwm i: 1 wn r m
The unitary matrix U was determined by iteratively minimizing the total spread of the resulting Wannier functions: X
J X n1
hr2 in
hri2n ;
2
where the sum runs over all the electronic bands, following the procedure outlined in [11] for singlepoint Brillouin Zone sampling. The Wannier function centers (WFC) were computed according to the expression for the position operator in extended systems [12]
L a rna Im ln wn e i 2p=Lr wn ; a x; y; z: 2p 3 3. Results and discussion Ten valence electrons were explicitly considered in the calculations, yielding 5 doubly occupied Kohn Sham states, from which 5 MLWF were determined. The total electronic density was then partitioned between O and Sn by considering the relative distance of each Wannier function center from the O and Sn position. The molecular dipole moment was computed from the position of the WFC (each carrying a formal charge ZWFC 2) and of the ionic pseudopotential cores (ZSn 4 and ZO 6): lSnO ' ZSn RSn ZO RO
J X
ZWFC rj :
4
j1
A value of 1.92 a.u. was calculated for the molecular dipole moment at the equilibrium bondlength. The nature of the resulting MLWF turned out to be readily interpretable in terms of their shape and of the position of their centres. In particular, one WFC turned out to be closer to Sn in the whole range of SnO separations, its distance
96
L. Bernasconi et al. / Computational Materials Science 22 (2001) 94±98
Fig. 1. Sn-centered MLWF plotted in the plane containing the SnO bond. The phase of each lobe has been indicated for clarity. A dot and an empty circle represent the approximate positions of Sn and O, respectively. The star indicates the WFC position.
from the atomic center varying between 1.5 and as the atomic separation was changed from 0.61 A The shape of the corresponding 1.6 to 2.5 A. MLWF is consistent with its non-bonding (``lonepair'') nature (Fig. 1). From the total molecular dipole moment we estimated the atomic polar tensor (APT), de®ned as the change of dipole moment in direction b linearly induced by a small displacement of atom j in the direction a sj;a [13]: Zj;ab
olb ; osj;a
5
which, in our speci®c case, reduces to a scalar. For around the optimized displacements of 0.01 A distance, we found ZSn 2:29, to be compared with the nominal ionic limit ZSn 2:00. In order to identify the origin of the non-neg ligible anomalous contribution to ZSn , we analysed the localization properties of the Wannier orbitals, as measured by their spread xn hwn jr2 jwn i 2 1=2 hwn jrjwn i (Fig. 2). The absence of variations in the Sn-lonepair MLWF spread (and, similarly, in the O 2s orbital) is indicative of the absence of
Fig. 2. Band by band decomposition of the total MLWF spread. The arrow indicates the optimized Sn±O distance. The Sn±O bond is oriented along z. Note 2 of the three 2p bands of O remain nearly degenerate across the whole range of ionic separation. Line are just a guide to the eye.
changes in its hybridization when the bond with O is begin shrunk or stretched. On the contrary, the spread of the O 2p states increases rapidly with the SnO distance, though one of them (directed along
L. Bernasconi et al. / Computational Materials Science 22 (2001) 94±98
the SnO bond) is always lower than the other two (corresponding to the orthogonal orbitals). This is consistent with their involvement in orbital hybridization with empty Sn2 5p states. Increasing the Sn±O distance corresponds to inducing a transfer of electronic charge towards O2 , which results in a larger spread of the target orbitals and increases ZSn to more than its nominal ionic value. The magnitude of the charge transfer (related to the slope of the x vs. R curve) appears to be the same for each of the three 2p orbitals. The O 2s orbital is unaected, as is the lonepair on Sn, since no on-site hybridization is involved in this mechanism. This model is supported by examination of the shape of the MLWF corresponding to the O 2p orbitals, which show evident deformations with respects to the pure p symmetry, consistent with the picture outlined above, and con®rms that the physical mechanism responsible for the development of a dipole moment in Sn2 is particularly simple, only implying a displacement of the center of the lonepair and no variations in its orbital hybridization (i.e., shape or spread). Moreover, the moderately low magnitude of the anomalous contribution to ZSn is consistent with a prevailing ionic character of the Sn±O interactions, and justi®es a ``corrected'' (polarizable) ionic description of the bonding properties. The dipole moment of Sn2 was thus computed as in Eq. (4), with only the lonepair WFC included in the sum: Z lSn2 ZSn RSn ZWFC drqlp r r X
' ZSn RSn ZWFC rlp ;
6
where qlp is the square modulus of the MLWF centered on Sn, rlp its WFC position and X is the volume of the unit cell. The Sn2 dipole moment thus computed was found to vary between 6.34 and 3.69 a.u. in the examined range of ionic distances. The dependence of the Sn2 dipole on the Sn±O distance remains linear across the whole range of ionic separations, and can be approximated to good accuracy by the relation l
0:53R 10:50;
7
where l and R are both expressed in atomic units.
97
The ionic dipole is related to a weak electrostatic ®eld by [14] l ' aE:
8
Short-range charge density overlap eects between ions modify the purely Coulombic dipole (Eq. (8)), determining, for a cation, an enhanced overall dipole [15]. Our calculations indicate that, at the equilibrium bondlength, short-range eects account for about 5% of the total ionic dipole. The external Coulombic ®eld is determined by the bare-charge O pseudopotential core screened by the 8 O2 valence electron. The polarizability of Sn2 can thus be estimated as jaSn2 Rj jlSn2 Rj=jE Rj
9
with jE Rj
ZO2 2e R2 J
Z X
dr
qO2 r jrO2
rSn2 j
O X Z 2 2e ' O2 ; 2 R rSn2 j m1 jrO2 m
2
2
10
where the sum runs over the 4 O2 WFC. In Fig. 3 the calculated value of the cation polarizability is plotted as a function of the Sn±O separation. The upper curve represents the value which would be obtained in conditions of perfect screening of the O2 core (by imposing all the WFC belonging to O to be collapsed on the nucleus), while the lower curve is the polarizability calculated according to the electrostatic ®eld given by Eq. (10). Only in the latter case is the correct long-range behaviour recovered, with a slow convergence rate toward the free-ion value. The latter fact shows that the mapping of the O2 electronic density onto a set of point charges is indeed able to fully reproduce the correct electrostatic ®eld generated by the nonspherically symmetric anion. The value of the free-ion polarizability was estimated by extending the a vs. dSn±O curve toward larger ionic separations and imposing the resulting curve to behave smoothly. This yielded an asymptotic limit of 14:48 1:05 a:u:, to be compared with the Hartree Fock estimate for the freeion polarizability of 14.40±14.50 a.u. [16].
98
L. Bernasconi et al. / Computational Materials Science 22 (2001) 94±98
Fig. 3. Sn2 polarizability as a function of the Sn±O separation, compared with the free-ion Hartree Fock value and with the estimated value for Sn2 in the rocksalt structure in the absence of overlap compression eects. The upper curve has been calculated by considering O as a point charge of nominal value )2, while the presence of a non-isotropic distribution of the O electron density has been taken into account for the lower curve by mapping it onto a set of point-charges located in the WFC positions. Lines are just a guide to the eye.
The value of the molecular Sn2 polarizability at the optimized bondlength is 15.23 a.u., which, though slightly larger than the free-ion value, remains smaller than the estimate for the ion in the rocksalt SnO structure (15.83 a.u.) [16], computed in the absence of overlap compression eects, which provides an upper bound to the Sn2 incrystal polarizability. This ®nding extends to the molecular case the qualitative arguments outlined in [16] on the basis of Hartree Fock calculations, which seem to suggest that the Sn2 polarizability does not vary signi®cantly when passing from the free ion to the condensed phase coordination environment.
4. Summary Single-ion dipole moments have been calculated in molecular SnO from a Wannier decomposition of the total electronic density obtained by conventional Kohn Sham DFT computations. The dependence of the Sn2 dipole on the distance has
been monitored over a wide range of ionic separations and has been found to satisfy a simple linear relation with the Sn±O distance. This could be helpful in the modelling of accurate potentials for molecular dynamics simulations which aim at taking ionic polarization phenomena into account. A small anomalous contribution to the Sn2 dynamical charge with regard to the ionic limit has been calculated, which appears to be determined merely by electronic charge transfers between 2p states of O2 and empty 5p states of Sn2 , without involving any orbital hybridization with the Sn2 lonepair orbital, which is thus alone responsible for the development of a dipole by Sn2 . A distance dependent polarizability has been estimated which compares quite accurately with the Hartree Fock value for the free ion, while the calculated value for the molecular Sn2 polarizability appears to be quite close to the estimate for the condensed phase environment, suggesting that no major changes take place in the Sn±O interactions when passing from the molecular to the solid phase of SnO. References [1] P.A. Madden, M. Wilson, Chem. Soc. Rev. 25 (1996) 399. [2] A.J. Stone, The Theory of Intermolecular Forces, Clarendon, Oxford, 1996. [3] L. Bernasconi, M. Wilson, P.A. Madden, to be published. [4] N. Marzari, D. Vanderbilt, Phys. Rev. B 56 (1997) 12847. [5] N.N. Greenwood, A. Earnshaw, Chemistry of the Elements, Pergamon Press, Oxford, 1984. [6] J.P. Perdew, Y. Wang, Phys. Rev. B 46 (1992) 6671. [7] R.G. Parr, W. Yang, Density Functional Theory of Atoms and Molecules, Oxford University Press, Oxford, 1989. [8] J.S. Lin, A. Qteish, M.C. Payne, V. Heine, Phys. Rev. B 47 (1993) 4174. [9] M.C. Payne, M.P. Teter, D.C. Allan, T.A. Arias, J.D. Joannopoulos, Rev. Mod. Phys. 64 (1992) 1045. [10] G. Kresse, J. Furthm uller, Phys. Rev. B 54 (1996) 11169. [11] P.L. Silvestrelli, N. Marzari, D. Vanderbilt, M. Parrinello, Solid State Commun. 107 (1998) 7. [12] R. Resta, J. Phys.: Condens. Matter 12 (2000) R107. [13] Ph. Ghosez, J.-P. Michenaud, X. Gonze, Phys. Rev. B 58 (1998) 6224. [14] G.D. Mahan, K.R. Subbaswamy, Local Density Theory of Polarizability, Plenum Press, New York, 1990. [15] M. Wilson, B.J.C. Cabral, P.A. Madden, J. Phys. Chem. 100 (1996) 1227. [16] M. Wilson, P.A. Madden, S.A. Peebles, P.W. Fowler, Molec. Phys. 88 (1996) 1143.
www.elsevier.com/locate/commatsci
Cation polarizability from ®rst-principles: Sn2 Leonardo Bernasconi *, Mark Wilson, Paul A. Madden Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OXI 3QZ, UK Accepted 2 April 2001
Abstract Generalized Gradient Corrected Density Functional Theory calculations have been performed on SnO in the gas phase. The total charge density of the molecular system has been partitioned between Sn and O using a Wannier localization transformation of the Kohn Sham eigenvectors, and the single-ion dipole moment of Sn2 at the optimized bond-distance has been estimated in terms of the position of the resulting Wannier function centers. This analysis has been extended over a wide range of ionic separations in order to monitor the dependence of the Sn2 dipole on both Coulombic and short-range interactions with O2 . The mechanism responsible for Sn2 polarization proves to be easily explained in terms of the non-bonding orbital center distance from the nucleus, without any major contributions deriving from mixing with orbitals centered on O. The Sn2 polarizability in the molecular system at the optimized bond distance (15.23 a.u.) is intermediate between the value for the free-ion (14.50 a.u.) and the estimate for crystalline SnO in the rocksalt structure (15.83 a.u.). The bond-length dependent polarizability at large ionic separations shows excellent agreement with the available Hartree Fock free-ion value. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction The ab initio investigation of ionic polarization in condensed phase systems has been the subject of considerable interest in the last years. Indeed, it has been shown that many of the anomalous structural properties of ionic materials, usually attributed to the manifestation of ``covalent'' effects, can be accounted for in terms of ion polarization phenomena and reproduced with considerable degree of success by a polarizable ionic prescription [1]. Although the polarizability of an ion in the gas phase can be predicted via conventional ®rst-principle methods, a thorough
*
Corresponding author. E-mail address: [email protected] (L. Bernasconi).
description of the polarization mechanism when this is part of a true physical system is rather more complicated. In addition to the purely Coulombic perturbation, the con®ning potential aecting an ion in the condensed-phase coordination environment is shaped by short-range repulsive charge density overlap with neighbouring ions [2]. In a previous paper [3], we have shown that density functional theory (DFT) calculations are indeed able to describe, at a quantitative level, the development of single-ion dipole moments in molecular systems. In particular, we concentrated on alkali metal ¯uorides and we were able to single out the relative weight of Coulombic and shortrange eects in determining the total ionic dipoles, by monitoring the dipole dependence on the ionic separation. An essential ingredient of our procedure was that of using a Wannier localization transformation [4] to partition the total electronic
0927-0256/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 1 ) 0 0 1 7 3 - 2
L. Bernasconi et al. / Computational Materials Science 22 (2001) 94±98
charge density, given by the squared modulus of the Kohn Sham eigenvectors, into that of the single ions. In this paper we will show that a similar approach can be generalized with no major changes to non-closed-shell species. In particular, we will concentrate on the calculation of the polarizability of Sn2 . This ion is known to have a very large polarizability, which can be attributed to intravalence s ! p excitations out of its 5s2 ground state. Sn(II) compounds in the condensed phase exhibit unusual crystal structures which are often attributed to the ``stereochemical requirements of the lonepair'' [5]. However, it has been suggested that such eects can also be explained as the consequence of polarization of the Sn2 cation [16], which favours the occupation of low symmetry sites. To fully examine this suggestion, an understanding of the short-range contributions to the cation polarization is required.
2. Method and computational details Generalized Gradient Corrected [6] Kohn Sham Density Functional Theory [7] calculations 1 were performed on a model system constituted of one Sn and one O atom in a cubic box of size L 10 A, periodically repeated in space. Norm-conserving ab initio pseudopotentials [8] were used to model the ion±electron interactions. The Kohn Sham eigenvectors were expanded in plane-waves up to an energy cuto of 330 eV, with only the C point included in the Brillouin zone sampling. All-electron minimizations were performed using a conjugate gradient minimization algorithm [9] in conjunction with a Pulay charge density mixing scheme [10]. The atomic separation for the SnO molecular system was optimized using a BFGS minimization algorithm, until all forces were lower than yielding an equilibrium bondlength 7 10 5 eV=A, of 1.837 A. The atomic separation was then varied including the equilibrium within a range (1.6±2.5 A) bondlength. The Kohn Sham eigenvectors were calculated in several points within this range, for 1 We used the code CASTEP 3.9 academic version, licensed under UKCP-MSI Agreement, 1999.
95
each of which a set of maximally localized Wannier functions (MLWF) was determined by a unitary transformation of the initial Kohn Sham orbitals. X Umn jwm i: 1 wn r m
The unitary matrix U was determined by iteratively minimizing the total spread of the resulting Wannier functions: X
J X n1
hr2 in
hri2n ;
2
where the sum runs over all the electronic bands, following the procedure outlined in [11] for singlepoint Brillouin Zone sampling. The Wannier function centers (WFC) were computed according to the expression for the position operator in extended systems [12]
L a rna Im ln wn e i 2p=Lr wn ; a x; y; z: 2p 3 3. Results and discussion Ten valence electrons were explicitly considered in the calculations, yielding 5 doubly occupied Kohn Sham states, from which 5 MLWF were determined. The total electronic density was then partitioned between O and Sn by considering the relative distance of each Wannier function center from the O and Sn position. The molecular dipole moment was computed from the position of the WFC (each carrying a formal charge ZWFC 2) and of the ionic pseudopotential cores (ZSn 4 and ZO 6): lSnO ' ZSn RSn ZO RO
J X
ZWFC rj :
4
j1
A value of 1.92 a.u. was calculated for the molecular dipole moment at the equilibrium bondlength. The nature of the resulting MLWF turned out to be readily interpretable in terms of their shape and of the position of their centres. In particular, one WFC turned out to be closer to Sn in the whole range of SnO separations, its distance
96
L. Bernasconi et al. / Computational Materials Science 22 (2001) 94±98
Fig. 1. Sn-centered MLWF plotted in the plane containing the SnO bond. The phase of each lobe has been indicated for clarity. A dot and an empty circle represent the approximate positions of Sn and O, respectively. The star indicates the WFC position.
from the atomic center varying between 1.5 and as the atomic separation was changed from 0.61 A The shape of the corresponding 1.6 to 2.5 A. MLWF is consistent with its non-bonding (``lonepair'') nature (Fig. 1). From the total molecular dipole moment we estimated the atomic polar tensor (APT), de®ned as the change of dipole moment in direction b linearly induced by a small displacement of atom j in the direction a sj;a [13]: Zj;ab
olb ; osj;a
5
which, in our speci®c case, reduces to a scalar. For around the optimized displacements of 0.01 A distance, we found ZSn 2:29, to be compared with the nominal ionic limit ZSn 2:00. In order to identify the origin of the non-neg ligible anomalous contribution to ZSn , we analysed the localization properties of the Wannier orbitals, as measured by their spread xn hwn jr2 jwn i 2 1=2 hwn jrjwn i (Fig. 2). The absence of variations in the Sn-lonepair MLWF spread (and, similarly, in the O 2s orbital) is indicative of the absence of
Fig. 2. Band by band decomposition of the total MLWF spread. The arrow indicates the optimized Sn±O distance. The Sn±O bond is oriented along z. Note 2 of the three 2p bands of O remain nearly degenerate across the whole range of ionic separation. Line are just a guide to the eye.
changes in its hybridization when the bond with O is begin shrunk or stretched. On the contrary, the spread of the O 2p states increases rapidly with the SnO distance, though one of them (directed along
L. Bernasconi et al. / Computational Materials Science 22 (2001) 94±98
the SnO bond) is always lower than the other two (corresponding to the orthogonal orbitals). This is consistent with their involvement in orbital hybridization with empty Sn2 5p states. Increasing the Sn±O distance corresponds to inducing a transfer of electronic charge towards O2 , which results in a larger spread of the target orbitals and increases ZSn to more than its nominal ionic value. The magnitude of the charge transfer (related to the slope of the x vs. R curve) appears to be the same for each of the three 2p orbitals. The O 2s orbital is unaected, as is the lonepair on Sn, since no on-site hybridization is involved in this mechanism. This model is supported by examination of the shape of the MLWF corresponding to the O 2p orbitals, which show evident deformations with respects to the pure p symmetry, consistent with the picture outlined above, and con®rms that the physical mechanism responsible for the development of a dipole moment in Sn2 is particularly simple, only implying a displacement of the center of the lonepair and no variations in its orbital hybridization (i.e., shape or spread). Moreover, the moderately low magnitude of the anomalous contribution to ZSn is consistent with a prevailing ionic character of the Sn±O interactions, and justi®es a ``corrected'' (polarizable) ionic description of the bonding properties. The dipole moment of Sn2 was thus computed as in Eq. (4), with only the lonepair WFC included in the sum: Z lSn2 ZSn RSn ZWFC drqlp r r X
' ZSn RSn ZWFC rlp ;
6
where qlp is the square modulus of the MLWF centered on Sn, rlp its WFC position and X is the volume of the unit cell. The Sn2 dipole moment thus computed was found to vary between 6.34 and 3.69 a.u. in the examined range of ionic distances. The dependence of the Sn2 dipole on the Sn±O distance remains linear across the whole range of ionic separations, and can be approximated to good accuracy by the relation l
0:53R 10:50;
7
where l and R are both expressed in atomic units.
97
The ionic dipole is related to a weak electrostatic ®eld by [14] l ' aE:
8
Short-range charge density overlap eects between ions modify the purely Coulombic dipole (Eq. (8)), determining, for a cation, an enhanced overall dipole [15]. Our calculations indicate that, at the equilibrium bondlength, short-range eects account for about 5% of the total ionic dipole. The external Coulombic ®eld is determined by the bare-charge O pseudopotential core screened by the 8 O2 valence electron. The polarizability of Sn2 can thus be estimated as jaSn2 Rj jlSn2 Rj=jE Rj
9
with jE Rj
ZO2 2e R2 J
Z X
dr
qO2 r jrO2
rSn2 j
O X Z 2 2e ' O2 ; 2 R rSn2 j m1 jrO2 m
2
2
10
where the sum runs over the 4 O2 WFC. In Fig. 3 the calculated value of the cation polarizability is plotted as a function of the Sn±O separation. The upper curve represents the value which would be obtained in conditions of perfect screening of the O2 core (by imposing all the WFC belonging to O to be collapsed on the nucleus), while the lower curve is the polarizability calculated according to the electrostatic ®eld given by Eq. (10). Only in the latter case is the correct long-range behaviour recovered, with a slow convergence rate toward the free-ion value. The latter fact shows that the mapping of the O2 electronic density onto a set of point charges is indeed able to fully reproduce the correct electrostatic ®eld generated by the nonspherically symmetric anion. The value of the free-ion polarizability was estimated by extending the a vs. dSn±O curve toward larger ionic separations and imposing the resulting curve to behave smoothly. This yielded an asymptotic limit of 14:48 1:05 a:u:, to be compared with the Hartree Fock estimate for the freeion polarizability of 14.40±14.50 a.u. [16].
98
L. Bernasconi et al. / Computational Materials Science 22 (2001) 94±98
Fig. 3. Sn2 polarizability as a function of the Sn±O separation, compared with the free-ion Hartree Fock value and with the estimated value for Sn2 in the rocksalt structure in the absence of overlap compression eects. The upper curve has been calculated by considering O as a point charge of nominal value )2, while the presence of a non-isotropic distribution of the O electron density has been taken into account for the lower curve by mapping it onto a set of point-charges located in the WFC positions. Lines are just a guide to the eye.
The value of the molecular Sn2 polarizability at the optimized bondlength is 15.23 a.u., which, though slightly larger than the free-ion value, remains smaller than the estimate for the ion in the rocksalt SnO structure (15.83 a.u.) [16], computed in the absence of overlap compression eects, which provides an upper bound to the Sn2 incrystal polarizability. This ®nding extends to the molecular case the qualitative arguments outlined in [16] on the basis of Hartree Fock calculations, which seem to suggest that the Sn2 polarizability does not vary signi®cantly when passing from the free ion to the condensed phase coordination environment.
4. Summary Single-ion dipole moments have been calculated in molecular SnO from a Wannier decomposition of the total electronic density obtained by conventional Kohn Sham DFT computations. The dependence of the Sn2 dipole on the distance has
been monitored over a wide range of ionic separations and has been found to satisfy a simple linear relation with the Sn±O distance. This could be helpful in the modelling of accurate potentials for molecular dynamics simulations which aim at taking ionic polarization phenomena into account. A small anomalous contribution to the Sn2 dynamical charge with regard to the ionic limit has been calculated, which appears to be determined merely by electronic charge transfers between 2p states of O2 and empty 5p states of Sn2 , without involving any orbital hybridization with the Sn2 lonepair orbital, which is thus alone responsible for the development of a dipole by Sn2 . A distance dependent polarizability has been estimated which compares quite accurately with the Hartree Fock value for the free ion, while the calculated value for the molecular Sn2 polarizability appears to be quite close to the estimate for the condensed phase environment, suggesting that no major changes take place in the Sn±O interactions when passing from the molecular to the solid phase of SnO. References [1] P.A. Madden, M. Wilson, Chem. Soc. Rev. 25 (1996) 399. [2] A.J. Stone, The Theory of Intermolecular Forces, Clarendon, Oxford, 1996. [3] L. Bernasconi, M. Wilson, P.A. Madden, to be published. [4] N. Marzari, D. Vanderbilt, Phys. Rev. B 56 (1997) 12847. [5] N.N. Greenwood, A. Earnshaw, Chemistry of the Elements, Pergamon Press, Oxford, 1984. [6] J.P. Perdew, Y. Wang, Phys. Rev. B 46 (1992) 6671. [7] R.G. Parr, W. Yang, Density Functional Theory of Atoms and Molecules, Oxford University Press, Oxford, 1989. [8] J.S. Lin, A. Qteish, M.C. Payne, V. Heine, Phys. Rev. B 47 (1993) 4174. [9] M.C. Payne, M.P. Teter, D.C. Allan, T.A. Arias, J.D. Joannopoulos, Rev. Mod. Phys. 64 (1992) 1045. [10] G. Kresse, J. Furthm uller, Phys. Rev. B 54 (1996) 11169. [11] P.L. Silvestrelli, N. Marzari, D. Vanderbilt, M. Parrinello, Solid State Commun. 107 (1998) 7. [12] R. Resta, J. Phys.: Condens. Matter 12 (2000) R107. [13] Ph. Ghosez, J.-P. Michenaud, X. Gonze, Phys. Rev. B 58 (1998) 6224. [14] G.D. Mahan, K.R. Subbaswamy, Local Density Theory of Polarizability, Plenum Press, New York, 1990. [15] M. Wilson, B.J.C. Cabral, P.A. Madden, J. Phys. Chem. 100 (1996) 1227. [16] M. Wilson, P.A. Madden, S.A. Peebles, P.W. Fowler, Molec. Phys. 88 (1996) 1143.