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Ab initio Hartree-Fock extended basis set calculation of the electronic structure of crystalline lithium oxide
Solid State Communications, Vol. 54, No. 2, pp. 183-185, 1985. Printed in Great Britain.
0038-1098/85 $3.00 + .00 Pergamon Press Ltd.
AB INITIO HARTREE-FOCK EXTENDED BASIS SET CALCULATION OF THE ELECTRONIC STRUCTURE OF CRYSTALLINE LITHIUM OXIDE
R. Dovesi Institute of Theoretical Chemistry, University of Torino, Via P. Giuria 5, I-10125 Torino, Italy (Received 19 November 1984 by F. Bassani)
The electronic structure of lithium oxide is investigated at an ab initio all electron Hartree-Fock level, using an extended basis set. Calculated total energy, lattice parameter, bulk modulus, electron charge and momentum data are reported. The totally ionic character of the compound and the distortion of the oxygen anion are documented. I. INTRODUCTION VERY ACCURATE COMPUTATIONAL schemes are available for the ab initio self-consistent calculation of the electronic structure of simple metallic and covalent crystals, both in Hartree-Fock LCAO [1, 2] and LocalDensity Functional Pseudopotential Plane waves [3] framework; on the contrary the ionic or partially ionic compounds are usually studied with approximate non self-consistent and/or non variational techniques. The most common of those models, the "minimal closed shell" one [4], defines the wave function of the solid simply by orthogonalizing the Bloch waves built up in terms of the wave functions of the isolated ions. In this way the purely ionic character of the compound is imposed a priori and no answer can be obtained to the question concerning the real nature of the bonds in the crystal. Recently we proposed a general approach to the study of ionic systems, by extending the applicability of our ab initio Hartree-Fock (HF) all-electron SCF program (CRYSTAL), previously used to study metallic and covalent systems periodic in three [5], two [6] and one dimension [7]. The essential difference in the treatment of the ionic compounds with respect to the non ionic ones is due to the fact that one has to take into account the long range Coulomb interactions both in the evaluation of the total energy and of the Fock matrix elements; whereas the former is a classical problem, the latter requires the numerical evaluation of a non standard type of integrals. An account of the technique we used for the solution of this problem has been given in a previous paper [8], to which we refer for the details of the calculation. Presently, our computational scheme has been applied to two simple ionic compounds, LiH [8] and Li3N [9], both studied with an extended basis set. In this paper we present an ab initio HF study of cubic Li20, to which theoretical and experimental
attention has been recently devoted (see ref. [10] and references therein) for its technological importance in the construction of Controlled Thermonuclear Reactors [11]. To our knowledge this is the first ab initio self consistent calculation of Li20. In a sense it represents a continuation and a complement of the studies devoted to LiH and Li3N, making it possible to compare the behavior of lithium in ionic compounds with different symmetry and in the presence of different anions. The principal aim was however to gain information on the oxygen anion. Li2 O is in fact the simplest member of the very large and important family of the oxides. In the following section the choice of the basis set will be shortly commented; results concerning total and kinetic energy, minimal energy lattice parameter, bulk modulus, population analysis, band structure, charge and momentum distribution will then be presented. II. CHOICE OF THE BASIS SET A large number of computations (about 30) have been performed in order to find an accurate basis set (BS); This BS, reported in Table 1, has been used in the calculation of the crystal properties. For oxygen we started from a 6G BS (12); we then introduced two more functions in the Is atomic orbital and reoptimized the coefficients of the contraction. For the valence shell first we optimized the exponent of the two outer independent Gaussians, and then the coefficients of the four contracted functions (shell 2 in Table 1). Whereas the exponent of the largest Gaussian is slightly smaller than in the isolated atom (t~ = 0.186 and 0.246 (a.u.) -2 , respectively), the rest of the O - - valence shell results to be unchanged with respect to the atomic situation. In the optimization of the exponents we had no linear dependence problems of the kind we had in the case of Li3 N [9], where however the three extra electrons on nitrogen caused a dramatic increase of the valence shell size, with an outer Gaussian exponent as
183
184
THE ELECTRONIC STRUCTURE OF CRYSTALLINE LITHIUM OXIDE
Vol. 54, No. 2
Table 1. Exponents [in a.u.) and coefficients o f the Gaussian functions used in the present calculation. The contraction coefficients multiply normalized individual Gaussians Atom Oxygen
Lithium
Shell number
Shell type
Exponent
1
s
2
sp
3 4
sp sp
4000. 1355.58 248.545 69.5339 23.8868 9.27593 3.82034 1.23514 52.1878 10.3293 3.21034 1.23514 0.536 0.186
1
s
2
sp
Table 2. Calculated energy data Total energy (a.u.) Kinetic energy (a.u.) Virial coefficient H.F. Cohesive energy (a.u.) Lattice parameter (A) Bulk modulus (dyn cm -2 )
-- 89.954 89.905 0.9997 0.298 4.62
1.05.1012
small as 0.08 (a.u.) -2 . The introduction of d functions on oxygen caused only a minor improvement in the energy ( 5 - 1 0 -4 a.u. cell -1, with a population 0.02 electrons cell -1 ). As far as lithium is concerned, the optimized exponent of the outer s and p shells resulted to coincide with the ones used for LiH and Li3N. As in previous calculations the p orbitals give a small contribution to the total energy (6" 10 -3 a.u., with a population of 0.04el cell -1 ). III. RESULTS AND DISCUSSION The most relevant energy data of cubic Li20 (space group Fm3m) are reported in Table 2; the HF cohesive energy has been obtained by subtracting the HF total energy of the constituent atoms (7.433 and 74.790 a.u. for Li and O respectively) from the crystal energy. The
700. 200. 70. 20. 5. 1.5 0.5
Coefficients s 0.00144 0.00764 0.05370 0.16818 0.36039 0.38612 0.14712 0.00711 --0.00873 --0.08979 --0.04079 0.37666 1. 1. 0.001421 0.003973 0.016390 0.089954 0.315646 0.494595 1.
p
0.00922 0.07068 0.20433 0.34958 1. 1.
1.
experimental cohesive energy (0.432 a.u. cell -1 ) has been obtained from the thermochemical data reported in ref. [10] and collected from different sources. The difference between the experimental and the theoretical cohesive energy (0.134a.u. cell - l ) is nearly totally due to the difference of correlation energy between O - - and O; this difference for the isolated atom is 0.148a.u. [13]. The virial coefficient is very near to unity, confirming the good quality of the solution; the calculated lattice parameter coincides with the experimental one. The bulk modulus has been calculated numerically by best fit of 5 energy points in the range 4.32-4.92.& of the lattice parameter. The band structure of Li20 is very simple; there is no appreciable contribution of lithium to the valence bands; the oxygen 2s band lies at --1.15 a.u., and is quite narrow (0.017 a.u.); the 2p oxygen bands are wider (0.11 a.u.) with a maximum splitting (/XE = 0.10 a.u.) at the L point; the maximum of the 2p valence bands is at F (--0.381 a.u.); the minimum is at L (--0.492 a.u.). A MuUiken analysis of the charge distribution attributes to oxygen a net charge o f - 1.95 electrons; the overlap populations between the nearest O - L i , O - O and L i - L i neighbours are very small and negative (--0.010, --0.003, 0.000 respectively). The lack of any covalent character of the bonds in
Vol. 54, No. 2
THE ELECTRONIC STRUCTURE OF CRYSTALLINE LITHIUM OXIDE
185
the EMD along the three main crystallographic directions is reported. The anisotropy in all the valence 0.8 region (q < 2.5a.u.) is far from negligible. From an Z [iii] analysis of the different contributions to the EMD it L~J ',,)\ [110] 0.6 results that all the anisotropy is to be attributed to oxygen. It is interesting to notice that the EMD distributions in the directions corresponding to first and 0.~ second neighbors oxygen atoms, (110) and (100) respectively, show high momentum (1.5 < q < 2 . 5 a . u . ) 0.2 components, due to O-O antibonding, much larger than in the O-Li (I 11) direction. 0.0 1 2 In conclusion it has been shown that with our q (el. u. ) approach it is possible to perform an accurate study of Fig. 1. Electron Momentum Distribution (EMD; in the ground state properties of simple oxides. Using atomic units). different tools (energy data, population analysis, charge and momentum densities) one can obtain a full characteilzation of the electronic structure of the system. Li20 is confirmed by the electron density, whose minima along the O-Li, O - O and Li-Li directions are Acknowledgements - This work has been partially respectively 0.028, 0.011 and 0.001el(a.u.) -3, to be supported by the Italian Ministero della Pubblica compared with the mean value of the valence density, Istruzione. Financial support by Consorzio per il Sistema Informativo CSI-Piemonte for the calculations which is 0.048 el (a.u.) -3 . The calculated charge density here reported is also gratefully acknowledged. along the first Li-O neighbors is smaller than this value not only in the minimum (located at 2.37a.u. from REFERENCES oxygen and 1.41 from lithium), but for about one fourth of the total atom-atom distance. 1. C. Pisani & R. Dovesi, Int. J. Quantum Chem. 17, In a recent paper Nasu and Takeshita [10] per501 (1980). 2. R. Dovesi, C. Pisani, C. Roetti & V.R. Saunders, formed a "term by term" calculation of the cohesive Phys. Rev. B28, 5781 (1983). energy of Li20; using the ionic charge as a parameter to 3. J. Ihm & M.L. Cohen, Phys. Rev. B21, 1527 reproduce the experimental cohesive energy they (1980). obtained Li ÷°'4 and O --°'a. So large a discrepancy with 4. O. Aikala, J. Phys. C9, L131 .(1976). S. Ameri, respect to our results and to what one can argue from G. Grosso & G. Pastori Parravicini, Phys. Rev. B23, 4242 (1981). simple chemical arguments must probably be explained 5. R. Dovesi, C. Pisani, F. Ricca & C. Roetti, Phys. with the poorness of the input data and with the roughRev. B25, 3731 (1982). ness of their model. 6. G. Angonoa, J. Kouteck'y, A.N. Ermoshkin & Once the purely ionic character of the system conC. Pisani, Surface Sci. 138, 51 (1984). firmed, one can try to answer the question concerning 7. R. Dovesi, lnt. J. Quantum Chem. 26, 197 (1984). 8. R. Dovesi, C. Ermondi, E. Ferrero, C. Pisani & the degree of distortion of the anion. To this er/d, total C. Roetti, Phys. Rev. B29, 3591 (1984). and differential electron charge and momentum density 9. R. Dovesi, C. Pisani, F. Ricca, C. Roetti & V.R. plots along different directions, as well as an analysis of Saunders, Phys. Rev. B30, 972 (1984). the electron charge distribution in terms of multipole 10. S. Nasu & H. Takeshita, Z NucL Mat. 75, 110 momenta [2] of the ions might be helpful. Here we limit (1978). ourselves to an example (Electron Momentum 11. K. Sako, M. Ohta, Y. Seki, H. Yamoto, T. Hiraoka, K. Tanaka, N. Asami & S. M0il, JAERI-M 5502 Distribution (EMD) anisotropy), and delay a deeper (1973). discussion of the point to a further paper, when a com- 12. W.J. Hehre, R.F. Stewart & J.A. Pople, J. Chem. parison between oxygen anions in different crystalline Phys. 51, 2657 (1969). situations (Li20, MgO, BeO) will be possible. In Fig. 1 13. E. Clementi, J. Chem. Phys. 38, 2248 (1963).
0038-1098/85 $3.00 + .00 Pergamon Press Ltd.
AB INITIO HARTREE-FOCK EXTENDED BASIS SET CALCULATION OF THE ELECTRONIC STRUCTURE OF CRYSTALLINE LITHIUM OXIDE
R. Dovesi Institute of Theoretical Chemistry, University of Torino, Via P. Giuria 5, I-10125 Torino, Italy (Received 19 November 1984 by F. Bassani)
The electronic structure of lithium oxide is investigated at an ab initio all electron Hartree-Fock level, using an extended basis set. Calculated total energy, lattice parameter, bulk modulus, electron charge and momentum data are reported. The totally ionic character of the compound and the distortion of the oxygen anion are documented. I. INTRODUCTION VERY ACCURATE COMPUTATIONAL schemes are available for the ab initio self-consistent calculation of the electronic structure of simple metallic and covalent crystals, both in Hartree-Fock LCAO [1, 2] and LocalDensity Functional Pseudopotential Plane waves [3] framework; on the contrary the ionic or partially ionic compounds are usually studied with approximate non self-consistent and/or non variational techniques. The most common of those models, the "minimal closed shell" one [4], defines the wave function of the solid simply by orthogonalizing the Bloch waves built up in terms of the wave functions of the isolated ions. In this way the purely ionic character of the compound is imposed a priori and no answer can be obtained to the question concerning the real nature of the bonds in the crystal. Recently we proposed a general approach to the study of ionic systems, by extending the applicability of our ab initio Hartree-Fock (HF) all-electron SCF program (CRYSTAL), previously used to study metallic and covalent systems periodic in three [5], two [6] and one dimension [7]. The essential difference in the treatment of the ionic compounds with respect to the non ionic ones is due to the fact that one has to take into account the long range Coulomb interactions both in the evaluation of the total energy and of the Fock matrix elements; whereas the former is a classical problem, the latter requires the numerical evaluation of a non standard type of integrals. An account of the technique we used for the solution of this problem has been given in a previous paper [8], to which we refer for the details of the calculation. Presently, our computational scheme has been applied to two simple ionic compounds, LiH [8] and Li3N [9], both studied with an extended basis set. In this paper we present an ab initio HF study of cubic Li20, to which theoretical and experimental
attention has been recently devoted (see ref. [10] and references therein) for its technological importance in the construction of Controlled Thermonuclear Reactors [11]. To our knowledge this is the first ab initio self consistent calculation of Li20. In a sense it represents a continuation and a complement of the studies devoted to LiH and Li3N, making it possible to compare the behavior of lithium in ionic compounds with different symmetry and in the presence of different anions. The principal aim was however to gain information on the oxygen anion. Li2 O is in fact the simplest member of the very large and important family of the oxides. In the following section the choice of the basis set will be shortly commented; results concerning total and kinetic energy, minimal energy lattice parameter, bulk modulus, population analysis, band structure, charge and momentum distribution will then be presented. II. CHOICE OF THE BASIS SET A large number of computations (about 30) have been performed in order to find an accurate basis set (BS); This BS, reported in Table 1, has been used in the calculation of the crystal properties. For oxygen we started from a 6G BS (12); we then introduced two more functions in the Is atomic orbital and reoptimized the coefficients of the contraction. For the valence shell first we optimized the exponent of the two outer independent Gaussians, and then the coefficients of the four contracted functions (shell 2 in Table 1). Whereas the exponent of the largest Gaussian is slightly smaller than in the isolated atom (t~ = 0.186 and 0.246 (a.u.) -2 , respectively), the rest of the O - - valence shell results to be unchanged with respect to the atomic situation. In the optimization of the exponents we had no linear dependence problems of the kind we had in the case of Li3 N [9], where however the three extra electrons on nitrogen caused a dramatic increase of the valence shell size, with an outer Gaussian exponent as
183
184
THE ELECTRONIC STRUCTURE OF CRYSTALLINE LITHIUM OXIDE
Vol. 54, No. 2
Table 1. Exponents [in a.u.) and coefficients o f the Gaussian functions used in the present calculation. The contraction coefficients multiply normalized individual Gaussians Atom Oxygen
Lithium
Shell number
Shell type
Exponent
1
s
2
sp
3 4
sp sp
4000. 1355.58 248.545 69.5339 23.8868 9.27593 3.82034 1.23514 52.1878 10.3293 3.21034 1.23514 0.536 0.186
1
s
2
sp
Table 2. Calculated energy data Total energy (a.u.) Kinetic energy (a.u.) Virial coefficient H.F. Cohesive energy (a.u.) Lattice parameter (A) Bulk modulus (dyn cm -2 )
-- 89.954 89.905 0.9997 0.298 4.62
1.05.1012
small as 0.08 (a.u.) -2 . The introduction of d functions on oxygen caused only a minor improvement in the energy ( 5 - 1 0 -4 a.u. cell -1, with a population 0.02 electrons cell -1 ). As far as lithium is concerned, the optimized exponent of the outer s and p shells resulted to coincide with the ones used for LiH and Li3N. As in previous calculations the p orbitals give a small contribution to the total energy (6" 10 -3 a.u., with a population of 0.04el cell -1 ). III. RESULTS AND DISCUSSION The most relevant energy data of cubic Li20 (space group Fm3m) are reported in Table 2; the HF cohesive energy has been obtained by subtracting the HF total energy of the constituent atoms (7.433 and 74.790 a.u. for Li and O respectively) from the crystal energy. The
700. 200. 70. 20. 5. 1.5 0.5
Coefficients s 0.00144 0.00764 0.05370 0.16818 0.36039 0.38612 0.14712 0.00711 --0.00873 --0.08979 --0.04079 0.37666 1. 1. 0.001421 0.003973 0.016390 0.089954 0.315646 0.494595 1.
p
0.00922 0.07068 0.20433 0.34958 1. 1.
1.
experimental cohesive energy (0.432 a.u. cell -1 ) has been obtained from the thermochemical data reported in ref. [10] and collected from different sources. The difference between the experimental and the theoretical cohesive energy (0.134a.u. cell - l ) is nearly totally due to the difference of correlation energy between O - - and O; this difference for the isolated atom is 0.148a.u. [13]. The virial coefficient is very near to unity, confirming the good quality of the solution; the calculated lattice parameter coincides with the experimental one. The bulk modulus has been calculated numerically by best fit of 5 energy points in the range 4.32-4.92.& of the lattice parameter. The band structure of Li20 is very simple; there is no appreciable contribution of lithium to the valence bands; the oxygen 2s band lies at --1.15 a.u., and is quite narrow (0.017 a.u.); the 2p oxygen bands are wider (0.11 a.u.) with a maximum splitting (/XE = 0.10 a.u.) at the L point; the maximum of the 2p valence bands is at F (--0.381 a.u.); the minimum is at L (--0.492 a.u.). A MuUiken analysis of the charge distribution attributes to oxygen a net charge o f - 1.95 electrons; the overlap populations between the nearest O - L i , O - O and L i - L i neighbours are very small and negative (--0.010, --0.003, 0.000 respectively). The lack of any covalent character of the bonds in
Vol. 54, No. 2
THE ELECTRONIC STRUCTURE OF CRYSTALLINE LITHIUM OXIDE
185
the EMD along the three main crystallographic directions is reported. The anisotropy in all the valence 0.8 region (q < 2.5a.u.) is far from negligible. From an Z [iii] analysis of the different contributions to the EMD it L~J ',,)\ [110] 0.6 results that all the anisotropy is to be attributed to oxygen. It is interesting to notice that the EMD distributions in the directions corresponding to first and 0.~ second neighbors oxygen atoms, (110) and (100) respectively, show high momentum (1.5 < q < 2 . 5 a . u . ) 0.2 components, due to O-O antibonding, much larger than in the O-Li (I 11) direction. 0.0 1 2 In conclusion it has been shown that with our q (el. u. ) approach it is possible to perform an accurate study of Fig. 1. Electron Momentum Distribution (EMD; in the ground state properties of simple oxides. Using atomic units). different tools (energy data, population analysis, charge and momentum densities) one can obtain a full characteilzation of the electronic structure of the system. Li20 is confirmed by the electron density, whose minima along the O-Li, O - O and Li-Li directions are Acknowledgements - This work has been partially respectively 0.028, 0.011 and 0.001el(a.u.) -3, to be supported by the Italian Ministero della Pubblica compared with the mean value of the valence density, Istruzione. Financial support by Consorzio per il Sistema Informativo CSI-Piemonte for the calculations which is 0.048 el (a.u.) -3 . The calculated charge density here reported is also gratefully acknowledged. along the first Li-O neighbors is smaller than this value not only in the minimum (located at 2.37a.u. from REFERENCES oxygen and 1.41 from lithium), but for about one fourth of the total atom-atom distance. 1. C. Pisani & R. Dovesi, Int. J. Quantum Chem. 17, In a recent paper Nasu and Takeshita [10] per501 (1980). 2. R. Dovesi, C. Pisani, C. Roetti & V.R. Saunders, formed a "term by term" calculation of the cohesive Phys. Rev. B28, 5781 (1983). energy of Li20; using the ionic charge as a parameter to 3. J. Ihm & M.L. Cohen, Phys. Rev. B21, 1527 reproduce the experimental cohesive energy they (1980). obtained Li ÷°'4 and O --°'a. So large a discrepancy with 4. O. Aikala, J. Phys. C9, L131 .(1976). S. Ameri, respect to our results and to what one can argue from G. Grosso & G. Pastori Parravicini, Phys. Rev. B23, 4242 (1981). simple chemical arguments must probably be explained 5. R. Dovesi, C. Pisani, F. Ricca & C. Roetti, Phys. with the poorness of the input data and with the roughRev. B25, 3731 (1982). ness of their model. 6. G. Angonoa, J. Kouteck'y, A.N. Ermoshkin & Once the purely ionic character of the system conC. Pisani, Surface Sci. 138, 51 (1984). firmed, one can try to answer the question concerning 7. R. Dovesi, lnt. J. Quantum Chem. 26, 197 (1984). 8. R. Dovesi, C. Ermondi, E. Ferrero, C. Pisani & the degree of distortion of the anion. To this er/d, total C. Roetti, Phys. Rev. B29, 3591 (1984). and differential electron charge and momentum density 9. R. Dovesi, C. Pisani, F. Ricca, C. Roetti & V.R. plots along different directions, as well as an analysis of Saunders, Phys. Rev. B30, 972 (1984). the electron charge distribution in terms of multipole 10. S. Nasu & H. Takeshita, Z NucL Mat. 75, 110 momenta [2] of the ions might be helpful. Here we limit (1978). ourselves to an example (Electron Momentum 11. K. Sako, M. Ohta, Y. Seki, H. Yamoto, T. Hiraoka, K. Tanaka, N. Asami & S. M0il, JAERI-M 5502 Distribution (EMD) anisotropy), and delay a deeper (1973). discussion of the point to a further paper, when a com- 12. W.J. Hehre, R.F. Stewart & J.A. Pople, J. Chem. parison between oxygen anions in different crystalline Phys. 51, 2657 (1969). situations (Li20, MgO, BeO) will be possible. In Fig. 1 13. E. Clementi, J. Chem. Phys. 38, 2248 (1963).