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Electrostatic potential from embedded clusters
23May 1997
CHEMICAL PHYSICS LETTERS
ELSEVIER
Chemical Physics Letters 270 (1997) 351-356
Electrostatic potential from embedded clusters Jos6 Luis Pascual 1, Lars G.M. Pettersson FYS1KUM, University of Stockholm, Box 6730, S-113 85 Stockholm, Sweden
Received 1 July 1996; in final form 24 March 1997
Abstract
We have calculated the electrostatic potential above the (100) surface of MgO using different cluster models, embedded in a representation of the surrounding lattice that takes into account quantum mechanical interactions as well as the full Madelung potential of the lattice. The use of embedded clusters results in electrostatic potentials, for both the Mg2+ and 02- sites, similar to those obtained by periodic calculations, independently of cluster size. The results are also similar for both stoichiometric and non-stoichiometric clusters. Relatively small clusters, if properly embedded, provide a good representation of the electrostatics above an ionic surface. (~) 1997 Elsevier Science B.V.
1. Introduction
Chemisorption and reactions on the surfaces of highly ionic metal oxides such as MgO and NiO are believed to be controlled to a large extent by the fields present at regular, low-coordinated and other defective sites. Theoretical modelling of these substrates thus requires an accurate treatment of both long- and short-range electrostatic effects; for the long-range contributions the Madelung potential of the infinite crystal needs to be accounted for, while the shortrange contributions will be more dependent on the actual extent of the ions constituting the crystal. Recently Ferrari and Pacchioni [ 1 ] reported on the computed electrostatic potential above cluster models representing both regular and defective sites of the MgO(100) surface. They found a strong dependence on both the size and shape of the cluster models employed and, in particular, oscillations in the results when various non-stoichiometric clusters were i Permanentaddress: DepartmentQufmicaFisica Aplicada,C-14 Universidad Aut6noma de Madrid, 28049 Madrid, Spain.
used. The embedding of the clusters was only based on naked point charges, however, which may lead to artificial polarisation of the anions. Problems with using non-stoichiometric clusters have been reported previously by Neyman and Rtsch [2] who claim that e.g. a cluster model such as MgO~- (charge-neutral in combination with crystal charges) could not give an accurate representation of the surface properties; rather, a stoichiometric model (in the MgO example this would be Mg505 ) must be used. This was based on the results of the surrounding stoichiometric cluster models with point charges only or by including a rather limited set of neighbors in the case of the non-stoichiometric models. We feel that the importance of a proper embedding of the cluster models in the description of strongly ionic systems has been underestimated in previous work. Thus, in this Letter we will demonstrate that the oscillations found in the electrostatic potential are indeed due to the excessive polarisation of the anions located at the cluster boundaries and that this effect is eliminated once the cluster is properly embedded.
0009-2614/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved. PH S0009-26 14(97)00378-3
352
J.L. Pascual, L.G.M. Pettersson/ Chemical Physics Letters 270 (1997) 351-356
Furthermore, once the cluster is embedded both stochiometric and non-stochiometric clusters show similar electrostatic potentials and properties. This leads to a more rapid convergence of the electrostatics towards that of the extended system than what is obtained with clusters only embedded in point charges. Several approaches exist in the literature for treating the interaction with the remaining crystal and a main subdivision may be made between periodic calculations using a supercell technique for the study of chemisorption and a cluster approach, where a small piece of the crystal is cut out to represent the local active site. In the latter case one may neglect the remaining crystal entirely and model the surface with a gas-phase cluster, which has the advantage that all theoretical techniques that have been developed for gasphase systems are immediately available. The cluster in this approach may become too flexible allowing too large relaxations of the substrate, however, and the complete neglect of the interaction with the crystal is an approximation that needs to be investigated from case to case [3,4]. In most studies some type of embedding scheme is thus employed where at least the Madelung potential from the remaining crystal is accounted for, either through a finite point charge expansion or a complete Ewald summation [5-8]. In the case of the chemisorption of CO on MgO [9] and NiO [ 10] it has been shown, however, that an embedding of the cluster that also includes approximately quantum mechanical effects through, e.g., effective core potentials or by including the next neighbors leads to significant changes in the computed properties of the cluster: CO becomes unbound at the SCF level and the small computed chemisorption energy is obtained only through the inclusion of dynamical correlation. In Ref. [ 9 ] the origin of this effect was traced to an unphysical polarisation of the charge on the O zanions towards the surrounding +2 point charges. Several different embedding schemes exist; an embedding potential, including orthogonality effects on the cluster wavefunction from the surrounding ions, may be extracted from the fully periodic solution for the bare surface [ 11,12]. A simpler description is obtained by using effective core potentials based on the frozen ionic charge distribution [ 13-15] to describe the nearest neighbors to the cluster. These static potentials give an approximate description of the quantum mechanical effects, including orthogonality and
exchange, from the frozen charge distributions on the cluster without explicitly including either the electrons on the surrounding ions or the basis functions to describe them. This is the approach taken in the present work.
2. Quantum chemical calculations The electrostatic potential on the (100) surface of MgO has been computed using several different models (Fig. I): a single ion ( 0 2 - or Mg2+), a nonstochiometric cluster (OMg 8+ or MgO8-), a stoichiometric cluster (Mg505 with either a central oxygen or magnesium) and finally a larger Mg909 cluster. The smaller clusters were embedded using either the infinite set of point charges only or with the neighboring ions (out to a radius of 14.5 bohr from any ion in the cluster) described by total-ion ab initio model potentials (AIMP) [ 13,14]. These were optimised for Mg 2+ and 0 2 - ions in the MgO bulk [9] and located in the positions of the undistorted crystal lattice using the experimental lattice parameter of 2.105/~ [ 16 ]. Finally, the electrostatic potentials associated with gasphase Mg505 and Mg909 cluster models were computed as well. The AIMP formalism [13,14] includes all main quantum mechanical effects of the (frozen) ions making up the environment. Long-range Coulomb interactions are given through the interaction with the ionic charge, deviations from complete screening are given by a fit in spherically symmetric Gaussians to the potential generated by the actual charge distribution of the ion in the appropriate environment. Core-valence orthogonality is enforced through the use of projection operators built from the ionic core-orbitals. Finally, the exchange interaction with the frozen ion is approximated by a spectral resolution of the atomic exchange operator. For further details see Refs. [ 13,14]. For the cluster ions the same basis sets as in the previous work [9] was used. This is a large [5s4p] basis for the oxygen anions and a [4s2p] contracted basis for the Mg 2+ cations. The charge densities were determined at the closed-shell restricted Hartree-Fock SCF level. The Madelung potential was included through a direct evaluation of the Gaussian type integrals over the full infinite Ewald sum [5-7] generated by the ions in
J.L. Pascual, L.G.M. Pettersson/ Chemical Physics Letters 270 (1997) 351-356
(a)
353
screening at short distances) and the contribution to the electrostatic potential from the remaining crystal were all included.
3. Results and discussion
(b)
Fig. I. MgsO5 and MggO9 cluster models used in the present work. the repeated surface unit cell using the ECPAIMP integral program [ 17] ; the full ionic charge, i.e. -t-2, was used. In the evaluation of the electrostatic potential at a given point above the surface, standard procedures as described in Ref. [ 1] were used with the modification that the potential from the infinite array of point charges was included through a complete Ewald summation. Thus, the nuclear and electronic contributions from the cluster, the contribution from the AIMP embedding (which takes into account the incomplete
The electrostatic potentials above the cluster models with a central Mg 2+ ion are given in Figs. 2 and 3a, where the step between equipotential lines in Figs. 2 is 0.2 eV. The Mg 2+ cation gives rise to a potential (Fig. 2a) similar to that from a point charge and the resulting potential with only a point charge embedding is symmetrical both for a given ion and also between anion and cation positions. When the cluster is extended to a non-stoichiometric MgO~- cluster (charge neutral when the point charge embedding is included), the effects of polarisation of the 0 2- anions are clearly visible (Fig. 2b) in the electrostatic potential surface. The charge-density is polarised towards the surrounding +2 point charges, which results in a severely distorted equipotential surface; the reduction of charge density in the central region of the cluster was earlier demonstrated in the case of CO/MgO(100) [9]. Inclusion of the/LIMP embedding, or equally well the next layer of cations, results in a potential surface much more similar to that of the extended surface (Fig. 2c). Some slight asymmetry persists, but at a much lower level. Indeed, at this point inclusion of the next four Mg 2+ cations needed for a stoichiometric cluster has essentially no effect on the computed properties in this highly ionic system. This is contrary to what has been reported for unembedded clusters where it has been argued that the cluster models must always be stoichiometric [2] ; this is thus not a requirement once a "quantum mechanical embedding" has been introduced. In fact, in a recent study of CO/MgO(100) [ 18 ] the computed binding energy of CO to an embedded Mg505 cluster was found to differ by less than 0.01 eV from that computed using a non-stoichiometric MgO~- embedded cluster [9]. The most important difference was found to be the reduction in basis set superposition error due to the presence of additional basis sets in the description of the cluster model [ 18 ]. Following Ferrari and Pacchioni [1 ] we give the variation of the electrostatic potential with increasing distance above an Mg 2+ cation (Fig. 3a) for the dif-
354
J.L. Pascual, L.G.M. Pettersson/ Chemical Physics Letters 270 (1997) 351-356 mg
•~ 20
mgo5
(a)
N 18
18
16
16
14
14
12
12
f--.~
10
10
8
6 /" - " Z - - -
"x 4
2
2
10
8
-6
-2
~
0
5
4
8
~0 X oxis
--10
--8
--6
--4
2
0
2
4
6
8
10 X O×iS
mgo5.aimp
~o r~4
:(c)
18
16
8
~10
8
-6
~4
-2
,,J,, 0
2
4
6
,,1,, B
1
10 X cxis
Fig. 2. Electrostatic potential (eV) felt by a positive charge above (a) an Mg2+ cation (at the origin) in an infinite array of 4-2 point charges; (b) an MgO58- unembedded cluster model surrounded by an infinite array of 4-2 point charges and (c) an MgO58- cluster model embedded in AIMPs out to 14.5 bohr and surrounded by an infinite array of 4-2 point charges. The plane cuts through the ion positions as shown. The plots are drawn in intervals of 0.2 eV each, from -1.2 to 1.0 eV. Positive contours are drawn with full lines, negative contours are drawn using dashed lines. ferent clusters and embeddings. It is clearly seen that, with the AIMP embedding included, stable results for the electrostatic potential are obtained and all curves are close to the fully periodic, three-layer slab calculation reported in Ref. [ 1 ]. The electrostatic potentials given by the unembedded cluster models, both sto-
ichiometric and non-stoichiometric, show a substantially larger curvature at large distances, thus giving rise to spurious field-effects. It is clear that the stoichiometric model shows less o f this artefact, but it is still far from the embedded cluster and periodic Hartree-Fock result. As a point o f interest one may
J.L. Pascual, L.G.M. Penersson/Chemical Physics Letters 270 (1997) 351-356
Electrostatic potential
'~ 2o N
Mg 2* site
18
Mg~O s airnp
Mo,O, MgO5aimp MgO~ Mg abmp Mg
1.7
355
mg9o9.gos
16 14
1.2
12 0.7
10 8
0.2 6 -0.3 L-0.0
5.0
10.0 R (bohr)
15.0
20.0 2
Electrostatic potential
10
O 2 site
0.5 ~6i-
F-
8
6
4
-2
0
2
4
6
8
10 × cxis
Fig. 4. Electrostatic potential (eV) above the Mg909 gas-phase cluster model (see caption to Fig. 2).
0.0
-0.5
-1.0
"1"50,0
i
'
5 i0
OiMg~aimp O,Mg= OMg, aimp
~ 0
10.0 '
aimp
15.0 '
20.0
R (bohr)
Fig. 3. Dependence of electrostatic potential felt by a positive charge (in eV) on the height (Bohr) above a central (a) Mg2+ ion and (b) 02- ion using differentclustermodelsand embedding schemes (see text). note the rapid decrease in electrostatic potential with increasing height above the surface; this is true for these simple non-polar surfaces and shows that electrostatic effects will have a negligible contribution at distances longer than about 5 bohr from the regular site. Furthermore, one may note the rapid increase in electrostatic potential at distances shorter than 5 bohr. The CO molecule is found from model calculations to be chemisorbed with the carbon at about 5.1 bohr above an Mg 2+ cation at a regular site [9,18]. The chemisorption distance is mainly determined by the repulsion against the charge density of the surface ions; clearly, small errors in determining this repulsion could have a significant effect on the electrostatic
interaction felt by the incoming CO molecule. The electrostatic potential above cluster models with a central 0 2- anion show a similar behavior as reported above, and we will only show the electrostatic potential as a function of the distance above the central 0 2- ion (Fig. 3b). Here it is clearly demonstrated that problems arise due to the presence of an unscreened positive charge as nearest neighbor of an oxygen anion; the stoichiometric MgsO5 now shows a much poorer representation of the potential than the non-stoichiometric OMg 8+ cluster. In the latter case the 0 2- ion is fully embedded using a correct description (the actual ions) of the interaction with the nearest neighbors, while the MgsO5 cluster model shows edge-effects due to the interaction of anions with the unscreened + 2 point charges. The properly embedded clusters again show a consistent picture with electrostatic potentials close to that of the infinite surface. Larger cluster models have been considered by Ferrari and Pacchioni [ 1 ], and it is clear that by increasing the cluster size the edge effects due to the polarisation of anions at some point will no longer affect the electrostatic potential measured in the central parts of the cluster. It is also clear that, by including the embedding, the convergence is much more rapid and from a purely electrostatic point of view it is not necessary to go to larger cluster sizes. Finally, it is of interest to perform the analysis of
356
J.L. Pascual, L.G.M. Pettersson/ Chemical Physics Letters 270 (1997) 351-356
the electrostatic potential also in the case of a gasphase cluster model and for this purpose we use a twolayer Mg909 model which contains a complete set of nearest neighbors to the central ion (top: Mg 2+, bottom: 0 2 - ) . The electrostatic potential obtained from this model (Fig. 4) is, generally speaking, different from the one obtained from embedded calculations. As pointed out by Ferrari and Pacchioni [ 1 ], who studied a stoichiometric Mg13O13 cluster, there is a small region localized around the center of the cluster, in which the potential is similar to the one calculated by the slab-model or the embedded cluster calculations. However, the presence of a net positive charge on the first layer (which contains five Mg 2+ ions and four 0 2 - ions) makes the potential positive in a large region above the surface. Fortuitously, the electric field above the central Mg 2+ ion does not have a large value in the direction normal to the surface, allowing the calculation of adsorption properties. For sites around the central cation, however, the anisotropy in the potential is evident and it is clear that here spurious electric fields will exist, that may or may not significantly affect the properties of interest. Thus, the use of gas-phase clusters as models of the extended ionic substrate in adsorption studies should be applied with caution in view of the results of Fig. 4 and Ref. [ 1 ].
4. Conclusions In the present study, we have calculated the electrostatic potential above the (100) surface of MgO using the cluster approach. We have used different cluster models, representing both the Mg 2+ and 0 2 sites, ranging from a single ion to Mg909 clusters. The clusters are embedded in a representation of the surrounding lattice that takes into account electrostatic interactions (actual Madelung potential and incomplete screening) as well as quantum effects (ion size) of the lattice ions. The electrostatic potentials calculated using the embedded clusters show a good agreement with those obtained from periodic calculations on slab models, even for the small (properly embedded) MgO58- cluster. The results for the electrostatic potential do not depend to any greater extent on the size of the cluster or on the use of stoichiometric/non-stoichiometric clusters,
neither for Mg 2+ nor for 0 2 - sites. The oscillatory behavior found in calculations of the electrostatic potential of clusters embedded in naked point charges, with respect to cluster size or stoichiometry, is clearly due to the simplification of the embedding potentials, that do not take into account the size of the ions around the central cluster.
Acknowledgements This work was partly supported by funds from the Swedish Consortium on Oxidic Overlayers.
References [11 A.M. Ferrari and G. Pacchioni, Int. J. Quantum Chem. 58 (1996) 241. [2l K.M. Neyman and N. R6sch, Surf. Sci. 297 (1993) 223. [31 K. Sawabe, N. Koga, K. Morokuma and Y. lwasawa, J. Chem. Phys. 97 (1992) 6871. [4J K. Sawabe, N. Koga, K. Morokuma and Y. Iwasawa, J. Chem. Phys. 101 (1994) 4819. [5] P.P. Ewald, Ann. Phys. 64 (1921) 253. [61 D.E. Parry, Surf. Sci. 49 (1975) 433. [7] D.E. Parry, Surf. Sci. 54 (1976) 195. [81 R. Orlando, R. Dovesi, C. Roetti and V.R. Saunders, Chem. Phys. Lett. 228 (1994) 225. [9] M.A. Nygmn, L.G.M. Pettersson, Z. Barandiantn and L. Seijo, J. Chem. Phys. 100 (1994) 2010. [101 M. Pthlchen and V. Staemmler, J. Chem. Phys. 97 (1993) 2583. [11 ] C. Pisani, R. Dovesi, R. Nada and S. Tamiro, Surf. Sci. 216 (1989) 489. [12] C. Pisani, R. Dovesi, R. Nada and L.N. Kantorovich, J. Chem. Phys. 92 (1990) 7448. [ 13] Z. Barandiar~ and L. Seijo. J. Chem. Phys. 89 (1988) 5739. [14] Z. Barandiaffm and L. Seijo, in: Studies in Physical and Theoretical Chemistry, Vol. 77 B, Computational Chemistry: Structure, Interactions and Reactivity, ed. S. Fraga (Elsevier, Amsterdam, 1992) p. 435. [15] J.A. Mejfas and J. Fem~lndez Sanz, J. Chem. Phys. 102 (1995) 327. [161 R.W.G. Wyckoff, Crystal Structures (Wiley, New York, 1963). [ 17] L.G.M. Pettersson, L. Seijo and M.A. Nygren, ECPAIMP integral program for ECP and AIMP calculations. [18] M.A. Nygren and L.G.M. Pettersson, J. Chem. Phys. 105 (1996) 9339.
CHEMICAL PHYSICS LETTERS
ELSEVIER
Chemical Physics Letters 270 (1997) 351-356
Electrostatic potential from embedded clusters Jos6 Luis Pascual 1, Lars G.M. Pettersson FYS1KUM, University of Stockholm, Box 6730, S-113 85 Stockholm, Sweden
Received 1 July 1996; in final form 24 March 1997
Abstract
We have calculated the electrostatic potential above the (100) surface of MgO using different cluster models, embedded in a representation of the surrounding lattice that takes into account quantum mechanical interactions as well as the full Madelung potential of the lattice. The use of embedded clusters results in electrostatic potentials, for both the Mg2+ and 02- sites, similar to those obtained by periodic calculations, independently of cluster size. The results are also similar for both stoichiometric and non-stoichiometric clusters. Relatively small clusters, if properly embedded, provide a good representation of the electrostatics above an ionic surface. (~) 1997 Elsevier Science B.V.
1. Introduction
Chemisorption and reactions on the surfaces of highly ionic metal oxides such as MgO and NiO are believed to be controlled to a large extent by the fields present at regular, low-coordinated and other defective sites. Theoretical modelling of these substrates thus requires an accurate treatment of both long- and short-range electrostatic effects; for the long-range contributions the Madelung potential of the infinite crystal needs to be accounted for, while the shortrange contributions will be more dependent on the actual extent of the ions constituting the crystal. Recently Ferrari and Pacchioni [ 1 ] reported on the computed electrostatic potential above cluster models representing both regular and defective sites of the MgO(100) surface. They found a strong dependence on both the size and shape of the cluster models employed and, in particular, oscillations in the results when various non-stoichiometric clusters were i Permanentaddress: DepartmentQufmicaFisica Aplicada,C-14 Universidad Aut6noma de Madrid, 28049 Madrid, Spain.
used. The embedding of the clusters was only based on naked point charges, however, which may lead to artificial polarisation of the anions. Problems with using non-stoichiometric clusters have been reported previously by Neyman and Rtsch [2] who claim that e.g. a cluster model such as MgO~- (charge-neutral in combination with crystal charges) could not give an accurate representation of the surface properties; rather, a stoichiometric model (in the MgO example this would be Mg505 ) must be used. This was based on the results of the surrounding stoichiometric cluster models with point charges only or by including a rather limited set of neighbors in the case of the non-stoichiometric models. We feel that the importance of a proper embedding of the cluster models in the description of strongly ionic systems has been underestimated in previous work. Thus, in this Letter we will demonstrate that the oscillations found in the electrostatic potential are indeed due to the excessive polarisation of the anions located at the cluster boundaries and that this effect is eliminated once the cluster is properly embedded.
0009-2614/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved. PH S0009-26 14(97)00378-3
352
J.L. Pascual, L.G.M. Pettersson/ Chemical Physics Letters 270 (1997) 351-356
Furthermore, once the cluster is embedded both stochiometric and non-stochiometric clusters show similar electrostatic potentials and properties. This leads to a more rapid convergence of the electrostatics towards that of the extended system than what is obtained with clusters only embedded in point charges. Several approaches exist in the literature for treating the interaction with the remaining crystal and a main subdivision may be made between periodic calculations using a supercell technique for the study of chemisorption and a cluster approach, where a small piece of the crystal is cut out to represent the local active site. In the latter case one may neglect the remaining crystal entirely and model the surface with a gas-phase cluster, which has the advantage that all theoretical techniques that have been developed for gasphase systems are immediately available. The cluster in this approach may become too flexible allowing too large relaxations of the substrate, however, and the complete neglect of the interaction with the crystal is an approximation that needs to be investigated from case to case [3,4]. In most studies some type of embedding scheme is thus employed where at least the Madelung potential from the remaining crystal is accounted for, either through a finite point charge expansion or a complete Ewald summation [5-8]. In the case of the chemisorption of CO on MgO [9] and NiO [ 10] it has been shown, however, that an embedding of the cluster that also includes approximately quantum mechanical effects through, e.g., effective core potentials or by including the next neighbors leads to significant changes in the computed properties of the cluster: CO becomes unbound at the SCF level and the small computed chemisorption energy is obtained only through the inclusion of dynamical correlation. In Ref. [ 9 ] the origin of this effect was traced to an unphysical polarisation of the charge on the O zanions towards the surrounding +2 point charges. Several different embedding schemes exist; an embedding potential, including orthogonality effects on the cluster wavefunction from the surrounding ions, may be extracted from the fully periodic solution for the bare surface [ 11,12]. A simpler description is obtained by using effective core potentials based on the frozen ionic charge distribution [ 13-15] to describe the nearest neighbors to the cluster. These static potentials give an approximate description of the quantum mechanical effects, including orthogonality and
exchange, from the frozen charge distributions on the cluster without explicitly including either the electrons on the surrounding ions or the basis functions to describe them. This is the approach taken in the present work.
2. Quantum chemical calculations The electrostatic potential on the (100) surface of MgO has been computed using several different models (Fig. I): a single ion ( 0 2 - or Mg2+), a nonstochiometric cluster (OMg 8+ or MgO8-), a stoichiometric cluster (Mg505 with either a central oxygen or magnesium) and finally a larger Mg909 cluster. The smaller clusters were embedded using either the infinite set of point charges only or with the neighboring ions (out to a radius of 14.5 bohr from any ion in the cluster) described by total-ion ab initio model potentials (AIMP) [ 13,14]. These were optimised for Mg 2+ and 0 2 - ions in the MgO bulk [9] and located in the positions of the undistorted crystal lattice using the experimental lattice parameter of 2.105/~ [ 16 ]. Finally, the electrostatic potentials associated with gasphase Mg505 and Mg909 cluster models were computed as well. The AIMP formalism [13,14] includes all main quantum mechanical effects of the (frozen) ions making up the environment. Long-range Coulomb interactions are given through the interaction with the ionic charge, deviations from complete screening are given by a fit in spherically symmetric Gaussians to the potential generated by the actual charge distribution of the ion in the appropriate environment. Core-valence orthogonality is enforced through the use of projection operators built from the ionic core-orbitals. Finally, the exchange interaction with the frozen ion is approximated by a spectral resolution of the atomic exchange operator. For further details see Refs. [ 13,14]. For the cluster ions the same basis sets as in the previous work [9] was used. This is a large [5s4p] basis for the oxygen anions and a [4s2p] contracted basis for the Mg 2+ cations. The charge densities were determined at the closed-shell restricted Hartree-Fock SCF level. The Madelung potential was included through a direct evaluation of the Gaussian type integrals over the full infinite Ewald sum [5-7] generated by the ions in
J.L. Pascual, L.G.M. Pettersson/ Chemical Physics Letters 270 (1997) 351-356
(a)
353
screening at short distances) and the contribution to the electrostatic potential from the remaining crystal were all included.
3. Results and discussion
(b)
Fig. I. MgsO5 and MggO9 cluster models used in the present work. the repeated surface unit cell using the ECPAIMP integral program [ 17] ; the full ionic charge, i.e. -t-2, was used. In the evaluation of the electrostatic potential at a given point above the surface, standard procedures as described in Ref. [ 1] were used with the modification that the potential from the infinite array of point charges was included through a complete Ewald summation. Thus, the nuclear and electronic contributions from the cluster, the contribution from the AIMP embedding (which takes into account the incomplete
The electrostatic potentials above the cluster models with a central Mg 2+ ion are given in Figs. 2 and 3a, where the step between equipotential lines in Figs. 2 is 0.2 eV. The Mg 2+ cation gives rise to a potential (Fig. 2a) similar to that from a point charge and the resulting potential with only a point charge embedding is symmetrical both for a given ion and also between anion and cation positions. When the cluster is extended to a non-stoichiometric MgO~- cluster (charge neutral when the point charge embedding is included), the effects of polarisation of the 0 2- anions are clearly visible (Fig. 2b) in the electrostatic potential surface. The charge-density is polarised towards the surrounding +2 point charges, which results in a severely distorted equipotential surface; the reduction of charge density in the central region of the cluster was earlier demonstrated in the case of CO/MgO(100) [9]. Inclusion of the/LIMP embedding, or equally well the next layer of cations, results in a potential surface much more similar to that of the extended surface (Fig. 2c). Some slight asymmetry persists, but at a much lower level. Indeed, at this point inclusion of the next four Mg 2+ cations needed for a stoichiometric cluster has essentially no effect on the computed properties in this highly ionic system. This is contrary to what has been reported for unembedded clusters where it has been argued that the cluster models must always be stoichiometric [2] ; this is thus not a requirement once a "quantum mechanical embedding" has been introduced. In fact, in a recent study of CO/MgO(100) [ 18 ] the computed binding energy of CO to an embedded Mg505 cluster was found to differ by less than 0.01 eV from that computed using a non-stoichiometric MgO~- embedded cluster [9]. The most important difference was found to be the reduction in basis set superposition error due to the presence of additional basis sets in the description of the cluster model [ 18 ]. Following Ferrari and Pacchioni [1 ] we give the variation of the electrostatic potential with increasing distance above an Mg 2+ cation (Fig. 3a) for the dif-
354
J.L. Pascual, L.G.M. Pettersson/ Chemical Physics Letters 270 (1997) 351-356 mg
•~ 20
mgo5
(a)
N 18
18
16
16
14
14
12
12
f--.~
10
10
8
6 /" - " Z - - -
"x 4
2
2
10
8
-6
-2
~
0
5
4
8
~0 X oxis
--10
--8
--6
--4
2
0
2
4
6
8
10 X O×iS
mgo5.aimp
~o r~4
:(c)
18
16
8
~10
8
-6
~4
-2
,,J,, 0
2
4
6
,,1,, B
1
10 X cxis
Fig. 2. Electrostatic potential (eV) felt by a positive charge above (a) an Mg2+ cation (at the origin) in an infinite array of 4-2 point charges; (b) an MgO58- unembedded cluster model surrounded by an infinite array of 4-2 point charges and (c) an MgO58- cluster model embedded in AIMPs out to 14.5 bohr and surrounded by an infinite array of 4-2 point charges. The plane cuts through the ion positions as shown. The plots are drawn in intervals of 0.2 eV each, from -1.2 to 1.0 eV. Positive contours are drawn with full lines, negative contours are drawn using dashed lines. ferent clusters and embeddings. It is clearly seen that, with the AIMP embedding included, stable results for the electrostatic potential are obtained and all curves are close to the fully periodic, three-layer slab calculation reported in Ref. [ 1 ]. The electrostatic potentials given by the unembedded cluster models, both sto-
ichiometric and non-stoichiometric, show a substantially larger curvature at large distances, thus giving rise to spurious field-effects. It is clear that the stoichiometric model shows less o f this artefact, but it is still far from the embedded cluster and periodic Hartree-Fock result. As a point o f interest one may
J.L. Pascual, L.G.M. Penersson/Chemical Physics Letters 270 (1997) 351-356
Electrostatic potential
'~ 2o N
Mg 2* site
18
Mg~O s airnp
Mo,O, MgO5aimp MgO~ Mg abmp Mg
1.7
355
mg9o9.gos
16 14
1.2
12 0.7
10 8
0.2 6 -0.3 L-0.0
5.0
10.0 R (bohr)
15.0
20.0 2
Electrostatic potential
10
O 2 site
0.5 ~6i-
F-
8
6
4
-2
0
2
4
6
8
10 × cxis
Fig. 4. Electrostatic potential (eV) above the Mg909 gas-phase cluster model (see caption to Fig. 2).
0.0
-0.5
-1.0
"1"50,0
i
'
5 i0
OiMg~aimp O,Mg= OMg, aimp
~ 0
10.0 '
aimp
15.0 '
20.0
R (bohr)
Fig. 3. Dependence of electrostatic potential felt by a positive charge (in eV) on the height (Bohr) above a central (a) Mg2+ ion and (b) 02- ion using differentclustermodelsand embedding schemes (see text). note the rapid decrease in electrostatic potential with increasing height above the surface; this is true for these simple non-polar surfaces and shows that electrostatic effects will have a negligible contribution at distances longer than about 5 bohr from the regular site. Furthermore, one may note the rapid increase in electrostatic potential at distances shorter than 5 bohr. The CO molecule is found from model calculations to be chemisorbed with the carbon at about 5.1 bohr above an Mg 2+ cation at a regular site [9,18]. The chemisorption distance is mainly determined by the repulsion against the charge density of the surface ions; clearly, small errors in determining this repulsion could have a significant effect on the electrostatic
interaction felt by the incoming CO molecule. The electrostatic potential above cluster models with a central 0 2- anion show a similar behavior as reported above, and we will only show the electrostatic potential as a function of the distance above the central 0 2- ion (Fig. 3b). Here it is clearly demonstrated that problems arise due to the presence of an unscreened positive charge as nearest neighbor of an oxygen anion; the stoichiometric MgsO5 now shows a much poorer representation of the potential than the non-stoichiometric OMg 8+ cluster. In the latter case the 0 2- ion is fully embedded using a correct description (the actual ions) of the interaction with the nearest neighbors, while the MgsO5 cluster model shows edge-effects due to the interaction of anions with the unscreened + 2 point charges. The properly embedded clusters again show a consistent picture with electrostatic potentials close to that of the infinite surface. Larger cluster models have been considered by Ferrari and Pacchioni [ 1 ], and it is clear that by increasing the cluster size the edge effects due to the polarisation of anions at some point will no longer affect the electrostatic potential measured in the central parts of the cluster. It is also clear that, by including the embedding, the convergence is much more rapid and from a purely electrostatic point of view it is not necessary to go to larger cluster sizes. Finally, it is of interest to perform the analysis of
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the electrostatic potential also in the case of a gasphase cluster model and for this purpose we use a twolayer Mg909 model which contains a complete set of nearest neighbors to the central ion (top: Mg 2+, bottom: 0 2 - ) . The electrostatic potential obtained from this model (Fig. 4) is, generally speaking, different from the one obtained from embedded calculations. As pointed out by Ferrari and Pacchioni [ 1 ], who studied a stoichiometric Mg13O13 cluster, there is a small region localized around the center of the cluster, in which the potential is similar to the one calculated by the slab-model or the embedded cluster calculations. However, the presence of a net positive charge on the first layer (which contains five Mg 2+ ions and four 0 2 - ions) makes the potential positive in a large region above the surface. Fortuitously, the electric field above the central Mg 2+ ion does not have a large value in the direction normal to the surface, allowing the calculation of adsorption properties. For sites around the central cation, however, the anisotropy in the potential is evident and it is clear that here spurious electric fields will exist, that may or may not significantly affect the properties of interest. Thus, the use of gas-phase clusters as models of the extended ionic substrate in adsorption studies should be applied with caution in view of the results of Fig. 4 and Ref. [ 1 ].
4. Conclusions In the present study, we have calculated the electrostatic potential above the (100) surface of MgO using the cluster approach. We have used different cluster models, representing both the Mg 2+ and 0 2 sites, ranging from a single ion to Mg909 clusters. The clusters are embedded in a representation of the surrounding lattice that takes into account electrostatic interactions (actual Madelung potential and incomplete screening) as well as quantum effects (ion size) of the lattice ions. The electrostatic potentials calculated using the embedded clusters show a good agreement with those obtained from periodic calculations on slab models, even for the small (properly embedded) MgO58- cluster. The results for the electrostatic potential do not depend to any greater extent on the size of the cluster or on the use of stoichiometric/non-stoichiometric clusters,
neither for Mg 2+ nor for 0 2 - sites. The oscillatory behavior found in calculations of the electrostatic potential of clusters embedded in naked point charges, with respect to cluster size or stoichiometry, is clearly due to the simplification of the embedding potentials, that do not take into account the size of the ions around the central cluster.
Acknowledgements This work was partly supported by funds from the Swedish Consortium on Oxidic Overlayers.
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