J. Phys. Chem. 1994, 98, 13189-13194
13189
Structure and Bonding of Bulk and Surface @-Aluminafrom Periodic Hartree-Fock Calculations A. P. Borosy, B. Silvi,* and M. Allavena Laboratoire de Dynamique des Interactions Molkculaires (URP 271), Universitk Pierre et Marie Curie, 4, Place Jussieu, 75252 Paris Ckdex 05, France
P. Nortier Inorganic Synthesis Department, Rhone-Poulenc Aubewilliers Research Center, 52, rue de la Haie Coq, 93308 Aubewilliers Ckdex, France Received: May 30, 1994; In Final Form: September 19, 1994@
The structure and bonding in 8-A1203has been investigated at the ab initio level with the periodic HartreeFock method. The structure optimization reported is in very good agreement with the available crystallographic data. The energy of &A1203 is found to be 42 kJ mol-' higher than that of a-alumina. The bonding in 8-A1203is discussed from the electron density and electron localization (ELF) functions. The structure is found to be ionic; the main differences between tetrahedral and octahedral sites are due to the polarization of the oxygen dianion. Estimates of cationic radii for tetracoordinated and hexacoordinated A1 are found to be in good agreement with the recent compilation of Feth et al. (Feth, S.; Gibbs, G. V.; Boisen, M. B., Jr.; Myers, R. H. J. Phys. Chem. 1993,97, 11445). The electron density and localization of the [ 1001 and [OlO] surfaces are reported and compared to the bulk properties.
1. Introduction Aluminum oxides have many technological and industrial applications. Alumina occurs in various crystallographic modifications, among which a-, y-, &, and &polymorphs are the most important: a-alumina is a very inert and hard material, whereas y- and &alumina are widely used catalysts, not only as supports but also as typical acid-base catalyst^.^-^ In spite of the widespread interest in catalytic aluminas and of the great importance of their surface properties in determining the structure of the supported active phases, there is still only a limited understanding about the real nature of aluminum oxide^,^.^ which is testified to by the small number of publications reporting quantum chemical studies on these oxides. Corundum, a-alumina, has a rather simple and symmetric structure. It belongs to the R3c space group, and all the aluminum atoms are hexacoordinated. Its as well as its s u r f a ~ e properties ~~'~ have been investigated at the HartreeFock level whereas a possible high-pressure phase transition has been investigated within the density functional approach. The structures of the other polymorphs are much more intricate with larger unit cells of low symmetry containing many atoms and often vacancies, and therefore periodic calculations are expected to be either impossible or out of reach at the present state of the art. However, in the case of &alumina, though the symmetry is low, there are only 10 atoms in the unit cell. The 8-alumina structure can be described as a distorted cubic closestpacked array of oxygen anions in which the aluminum cations occupy one-eighth of the tetrahedral interstices and half of the octahedral ones. The structure of the bulk &alumina has been studied by X-ray diffraction and electron diffraction spect r o s c ~ p i e s , ~and ~ , ' ~the [OlO] and [loo] surfaces have been studied by electron diffra~tion.'~The space group is C2/m. There is good agreement between the two sets of experimental data for the lattice parameter and the B angle whereas rather large uncertainties occur for the atomic fractional coordinates and therefore for the A1-0 bond lengths.
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Abstract published in Advance ACS Abstracts, November 1, 1994.
0022-365419412098-13189$04.50/0
From the point of view of structural chemistry, &alumina is a very interesting structure because it contains both hexa- and tetracoordinated aluminums. In the present paper, we present the results of a series of periodic Hartree-Fock calculations performed on bulk &alumina and on slabs parallel to the [OlO] and [ 1001 planes. The main objective of this work is qualitative rather than quantitative. We intend to characterizethe chemical bonds between aluminum and oxygen in the bulk and on the surfaces. In fact, we wish to provide answers to the question of the bonding differences between tetrahedral and octahedral aluminum centers.
2. Method of Calculation The calculations have been performed at the periodic Hartree-Fock level with the CRYSTAL program developed in Turin.14 A documented description of the method has been published by Pisani et al.15 This method works with a single determinant approximation of the wave function in which the crystalline orbitals are expressed as linear combinations of Bloch functions which are themselves expressed in terms of the localized atomic basis functions of each unit cell. As input the program requires geometry and basis set information together with a set of thresholds which control the truncation of infinite sums and a set of shrinking factors which determines the sampling k-points over the Brillouin zone. As output it provides the unit cell energy, the wave function, and related one-electron properties. The program works at the all-electron (AE) level as well as with effective core pseudopotentials (ECP). Previous periodic Hartree-Fock calculations performed on corundum' have shown that minimal basis sets (Le. the popular STO-3G) yield unphysical or low-quality results because they are not flexible enough to allow an ionic description of the crystal. Reliable results8 have been obtained with the 6-21G16 basis set, and it was shown that 3d-polarization functions do not provide any significant improvement of the calculated properties but noticeably increase the cost of the calculation. The present calculations on the &phase have been performed 0 1994 American Chemical Society
Borosy et al.
13190 J. Phys. Chem., Vol. 98, No. 50, 1994 8 Alumina
TABLE 1: Structure Parameters of @-A1203 expl* exp13 11.795 000 2.9100 5.6212 103.79 0.101 340 0.794 40 0.352 35 0.687 39 0.162 68 0.122 76 0.489 45 0.261 27 0.829 95 0.438 63
11.813 2.906 5.625 104.1 0.0843 0.7929 0.3400 0.6832 0.1614 0.1071 0.4980 0.2602 0.8326 0.4404
calc 11.694 2.912 5.621 103.9 0.0959 0.7986 0.3448 0.6734 0.1631 0.1151 0.4915 0.2544 0.8268 0.4376
Tetrahedron All-01 [A] All-02 [A] All-03 [A]
1.78 1.73 1.82
[A] [A]
1.87 1.86 1.99 1.98
A12-01 A12-02
[AB,
d2-03111 1 A12-03
1.764 1.762 1.776
1.815 1.789 1.697
Octahedron 1.838 1.819 1.991 2.097
1.896 1.859 1.946 1.954
r z T R S y r X Figure 1. Band structure of e-Al203. The origin of the energy scale is taken at the Fermi level. 8 Alumina AlXU
TABLE 2: Mulliken Population Analysis of a-Al~O3and 8-A1203
e
a
Net Charges 1.99 2.11 -1.36 -1.38 -1.37
Al'V
AlV'
2.07 -1.38
01 02 03
Bond Population AIrV-0 1 Al'V-02 AlrV-03 AlV'- 0 1 A1"-02 A1"'-03
0.095
0.134 0.122 0.116 0.107 0.105 0.093
TABLE 3: Comparison of the Mulliken Net Charges of Surface Atoms in the [loo] and [OlO] Slabs and Bulk 8-Alumina bulk 0 AIIV AIV'
-1.157 1.990 2.11
[1001 -1.193 1.759 2.05
[O 101 - 1.379 1.669 1.906
with this same 6-21G basis set. The external exponents of the sp shells have been rescaled for solid state calculations. They are 0.35, 0.16, and 0.20 for oxygen and tetracoordinated and hexacoordinated aluminums, respectively. The reciprocal space integration is performed using a commensurate net, the meshes of which are determined by the shrinking factor S. S = 4, corresponding to 24 k-points, has been used for the present calculations; when S = 8 is used, the energy change is less than the SCF convergence threshold
3. Results and Discussion 3.1. Optimized Structure. The structure optimization has been automatically carried out with overhead conjugate gradient routines to the CRYSTAL 92 program written by Ph. D ' A r ~ 0 . l ~ In this facility, the coordinates can be either standard crystallographic coordinates (i.e. lattice parameters and atomic fractional coordinates), internal coordinates (bond lengths and bond angles), or a mixing of both.
I
-30.0
-25.0
I
-20.0
-15.0
I
-10.0 ENERGY (eV.)
1
I
-5.0
0.0
5.0
Figure 2. Projected density of states of e-Al203. The C2lm lattice of &alumina is defined by three lattice parameters, namely a, b, and c, and by the angle between the a and c lattice direction p. The positions of the atoms within the unit cell are defined by 10 fractional coordinates. Starting from the experimental values of Repelin and H u ~ s o n ,the ' ~ 14 structural parameters have been simultaneously varied until a au) has been achieved. The optimized good convergence ( structural parameters which correspond to the location of the minimum of the 14-dimensional Born-Oppenheimer energy surface are listed in Table 1 and compared to the corresponding experimental data of Yamaguchi et a1.12 and of Repelin and Husson.13 The agreement between the calculated and experimental lattice parameters is very good. The largest absolute deviation occurs for c, which is underestimated by 0.1 A, which corresponds to a relative uncertainty less than 1% whereas for the other parameters the accuracy is better than 0.1%. It is not surprising to find larger discrepancies for the atomic fractional coordinates, since there are some rather large differences between the two sets of experimental data. In the case of a good agreement between the experimental results, the calculated values are rather close to them whereas in the other cases they are in better agreement with the results of Yamaguchi et ~ 1 . ' ~ than with those of Repelin and Husson.13 Note that the
Structure and Bonding of 8-Alumina
J. Phys. Chem., Vol. 98, No. 50, 1994 13191
t
Figure 3. ELF isosurface (ELF = 0.8) in &alumina. The green nets correspond to isodensity surfaces (0= 0.05 e The topology of the ELF isosurfaces is related to that of free ions. The distorsions which mainly occur for oxygens are due to the Pauli repulsion arising from the A1 L shells.
distorsion of the polyhedra is much more important in this latter and in the calculated from the two sets of pancies which rise up - 0 3 bond. For the most controversial distance, the d bond length lies between the experimental ones. The averaged A1-0 distances in the tetrahedra are 1.765, 1.772, and 1.776 A from experiment12.13 and calculation, respectively, whereas in octahedra they are 1.926, 1.929, and 1.916 A. The A1-0 bond length in a-alumina, 1.855 A, appears to be intermediate. This ordering is very well accounted for by the Hartree-Fock calculations. 3.1 .I. Binding Energies. The calculated binding energies per A1203 unit of a-and O-aluminas are -0.848 and -0.832 au, respectively. In the case of a-alumina, for which experimental thermodynamic data are available,l8 the calculated % of the experimental one. contribution to the binding energy is mainly this discrepancy, since the correlation energies ecies are often very different. An estimate
(i.e. lo%), which can be
Hartree-F
ardly geometry dependent;
therefore, it is expected neither to alter the locations of the Born-Oppenheimer energy surface minima nor to significantly modify the energy difference between the polymorphs. With respect to the a-phase, the energy of &alumina is found to be higher by 0.016 au per A1203 (i.e. 42 kJ mol-'). This calculated difference qualitatively accounts for the relative stabilities of the two phases. 3.1.2. Bonding Properties. The nature of the bonding in crystals (or in molecules) can be investiga techniques. Most of them, such analysis, band structure, or project the approximations (Le. LCAO expansion, single-determinatal approximation) made in order to carry out the calculation. They are therefore unable to provide an objective answer. Other techniques, based on the topology of local functions, do not depend in their principles upon the approximations. Bader's theory of atoms in molecules20 allows a sound partition of the charge density. In this theory, atomic basins are defined as parts of the space containing an electron attractor (nucleus) and limited by surfaces of zero flux in the gradient vectors of the charge density. Moreover, the topology of the charge density allows one to define bond paths and critical points. The intersection of a zero-flux surface and a bond path corresponds to a bond critical point. The distance between a nucleus and the nearest bond critical point is recognized as the atomic (or crystal) radius. Recently, Becke and Edgecombe21 have pro-
13192 J. Phys. Chem., Vol. 98, No. 50, 1994
Borosy et al.
I Figure 4. Electron density and ELF in color scale.
posed an electron localization function (ELF) be very attractive to discuss the bonding. as
ELF =
1
in which D, and Di represent pair density for electrons of iden respectively the actual s with the same densi
ructure of &A1203 along in zone. The top parts e 2. The intermediate part and -7.0 eV, corresponds to the oxygen 2p bonding states, and the oxygen 2s states are
the main component of the lower part located between -20.0 and -25.0 eV. The overall topology of the band structure reflects the ionic nature of the bondings in &alumina. The projected densities of states, in Figure 2, are consistent with this picture because the aluminum contribution is negligible. Table 2 presents the Mulliken population analysis of the aand O-aluminas. The oxygen net charge is almost constant in the series. However the net charge of A1 depends upon the coordination; it is lower for the tetrahedral coordination (AIw) than for the octahedral one (Alw). The bond population between A1 and 0 is always very weak, and it decreases when the coordination increases. These three methods characterize 8-Al203, similar to corundum,8 as a highly ionic insulator. Figure 3 displays a 3-D representation of the electron localization function. The isosurfaces correspond to ELF = 0.8. Each isosurface encapsulates an aluminum or an oxygen center. When the ELF value is lowered enough, these atomic surfaces collapse and there is no evidence of any interatomic maximum of the localization function. Such a situation appears to be typical of closed shell interaction; Le., there is no indication of electron sharing. The aluminum envelops are almost spherical whereas for oxygen there is a significant distorsion of the surfaces facing the aluminum, which is due to the Pauli repulsion. Slices of the ELF in the [OlO] and [040] planes, in Figures 4 and 5 , corroborate the preceeding conclusions and also indicate that the electron density between anions and cations is rather low. Figures 6 and 7 display the total electron density, the norm of the electron density gradient, and the ELF along the A1-0 bond paths for tetracoordinatedand hexacoordinated
J. Phys. Chem., Vol. 98, No. 50, 1994 13193
Structure and Bonding of @Alumina
-1.0
- 0.75
,‘
I
8.0-
/
II -
1
- 6.0- \ -
l
i
-0.5
I
rp
-0.75
I I II
II
I
-
- II ;I
h
-0.5
1 1
Q
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-0.25
-
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I I I t I tI
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0.5 1.o 1.5 INTERNUCLEAR DISTANCE (A)
0.0
0.5 1.o 1.5 INTERNUCLEAR DISTANCE (A)
(eA-3, solid line), electron density gradient norm (eA-4, dotted line), and ELF along the A1-0 bonds in tetrahedra (dashed line).
Figure 7. Total electron density g (eA-3, solid line), electron density gradient norm (eA-4, dotted line), and ELF along the A1-0 bonds in
aluminum centers, respectively. The ELF curves clearly reproduce the shell structure of the ions. Moreover the bond critical points (i.e. the minimum of [Vel) almost coincide with the ELF minima, which is another indication of ionic bonding. Graphs such as those in Figures 6 and 7 can be used to estimate the atomic (crystal) and ionic radii. According to Bader’s partition scheme?O the atomic radius corresponds to the distance between a given nucleus and the nearest bond critical point. The minimum of the ELF along a bond path is located at the border of the cation core shell and anion valence shell and therefore will be used to estimate the ionic radius. Accordingly defined crystal radii are found to be 0.76 and 0.79
8, for tetracoordinated and hexacoordinated aluminum, respectively, whereas the ionic radii are 0.69 and 0.72 8,. These values
Figure 6. Total electron density
octahedra (dashed line).
are in quite satisfactory agreement with the promolecule radii published by Feth et aZ.:l 0.78 and 0.83 8, for tetracoordinated and hexacoordinated A1 cations, respectively. They are about 0.2 8, larger than those reported in Shannon’s ~ o m p i l a t i o n . ~ ~ 3.2. Study of the [loo] and [OlO] Surfaces. In order to have some insight into the bonding properties of the &alumina surfaces, two single layers parallel to the [lo01 and [OlO] planes have been calculated. There are 10 and 2 atomic layers in the [lo03 and [OlO] slabs, respectively. The study of the relaxation of surface atoms is hampered by the size of the unit cells and
Borosy et al.
13194 J. Phys. Chem., Vol. 98, No. 50, 1994
the surface oxygen, which is an indication of the nucleophilic character of the center. In the case of the [OlO] slab, both oxygen and aluminum are surface atoms. The drawing of the electron density difference in the surface plane (Figure 9) indicates an electron loss in the bond directions which correspond to a transfer toward the oxygen lone pairs.
0.02500000
contour line interval
Figure 8. Electron density difference (slab - bulk) for the [lo01 plane. The cutting plane is erpendicular to the surface. The contour line interval is 0.025 e Solid, dotted, and dashed lines represent positive, zero, and negative differences.
I-’. to101
I
1
,
I
4. Conclusions The results presented here constitute a set of preliminary results which intend to provide a better understanding of the bonding in aluminum oxides and of the properties of their surfaces. The bonding in 8-alumina as well as in corundum appears to be ionic. The differences between tetrahedral and octahedral sites are mostly due to the local polarizations of the oxygen dianions. The formation of a surface is characterized by an electron transfer toward the surface which can be the dominant origin of the catalytic properties. Moreover, in the case of the [loo] surface, there is a noticeable polarization of the oxygen dianions, which can magnify their nucleophilic character. Acknowledgment. This project was supported by CNRSRhGne Poulenc Contract No. 520515. The authors thank Dr. L.-H. Jolly for computational assistance. The data analyzer software S C ~ Awas ~ *used ~ to produce Figure 3. References and Notes
contour line interval
0.02500000
Figure 9. Electron density difference (slab - bulk) for the [OlO] surface plane. Same spacing and conventions as Figure 8.
by the number of degrees of freedom. In the [loo] slab the upper atomic layer is made of oxygen atoms bonded to tetracoordinated aluminums whereas in the [OlO] case the surface contains both oxygens and tetra- and hexacoordinated aluminums. The results of this calculation must be understood as qualitative rather than quantitative because these slabs are not thick enough to minimize the size effects. Moreover, relaxation of surface atoms has not been accounted for. For the [loo] and [OlO] surfaces the surface formation energies are respectively 0.00249 and 0.00283 au bobp2, which are lower than those calculated by Causa et al., for the [OOl] and [OTO] faces of ~ o r u n d u m . In ~ this latter calculation relaxation of surface atoms was allowed. From our values, it can be expected that relaxation does not play an important role in surface formation of 8-alumina. Moreover, it appears that the [lo01 surface is energetically favored. The change in the bonding can be investigated either for electron density differences or from the Mulliken population analysis. The Mulliken population analysis indicates a slight electron transfer toward the surface atoms. Figure 8 displays the electron density differences between the [loo] slab and the bulk in a plane perpendicular to the surface and containing the surface oxygen. This picture shows a strong polarization of
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