Study of the physisorption of CO on the MgO(100) surface using the approach of Kohn-Sham equations with constrained electron density 1

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1 Journal of Molecular Structure (Theochem) 458 (1999) Study of the physisorption of CO on the MgO(100) surface using the approach of Kohn-Sham equations with constrained electron density 1 Tomasz Adam Wesolowski*, Nathalie Vulliermet, Jacques Weber Université de Genève, Département de Chimie Physique, 30, quai Ernest-Ansermet, CH-1211 Geneva 4, Switzerland Received 19 January 1998; accepted 11 May 1998 Abstract The structure and stretching frequency of the CO molecule physisorbed on the MgO(100) surface were investigated using the recently developed formalism of Kohn-Sham equations with constrained electron density (KSCED). The KSCED method makes it possible to divide a large system into two subsystems and to study one of them using Kohn-Sham-like equations in which the effective potential takes into account the interactions between subsystems. Compared to the standard Kohn-Sham formalism, the KSCED method involves an additional functional due to the non-additivity of the kinetic energy. The surface was represented using a cluster ((MgO 5 ) 8 or Mg 9 O 9 ) embedded in an array of electric point-charges. The KSCED calculations led to a blue-shift of the stretching frequency of the C-down adsorbed CO molecule amounting to cm 1 depending on the distance from the surface. At the C Mg distance of 2.42 Å, which corresponds to a typical minimum of the potential energy curve derived from supermolecule Kohn-Sham calculations applying gradient-corrected functionals, the KSCED frequency shift amounts to 35 cm 1 in excellent agreement with the most recent experiments. The CO stretching frequency of the O-down adsorbed CO molecule is red-shifted. The effects of cluster size and choice of the functionals on the KSCED frequencies, geometries and energies were analyzed. For C Mg distances varying between 2.3 and 3.0 Å, changing the cluster size affects the frequencies by less than 4 cm 1 and the CO bond length by less than Å. At C Mg distances larger than 2.4 Å, the change of the cluster size negligibly affects the KSCED interaction energies. The KSCED formalism makes it possible to study directly the effects associated with relaxation of the surface s electron density upon adsorbing CO. It is shown that these effects might contribute up to 30% of the KSCED interaction energy, but that they do not result in significant changes of either the geometries or frequencies Elsevier Science B.V. All rights reserved. Keywords: Physisorption; Density functional theory; Kinetic energy functional 1. Introduction The recently developed formalism of Kohn-Sham equations with constrained electron density (KSCED) makes it possible to divide a large system into two 1 Dedicated to Professor Paul S. Bagus on the occasion of his 60th birthday. * Corresponding author. subsystems and to study one of them using Kohn- Sham-like equations in which the effective potential takes into account the interactions between subsystems [1, 2]. The formalism can be used to obtain the electron density and energy of the total system via the freeze-and-thaw cycle [3] or to obtain the electron density of one subsystem keeping the electron density of another one frozen [4 6]. In the latter case, the KSCED formalism can be seen as a theoretical /99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S (98)

2 152 T.A. Wesolowski et al. / Journal of Molecular Structure (Theochem) 458 (1999) framework for designing embedding potentials based on density functional theory. Similarly to the conventional formalism of Kohn and Sham (KS) [7], KSCED uses approximate exchange-correlation functionals (E xc ). The KSCED formalism also introduces an additional functional ) arising from non-additivity of the kinetic energy. Since the exact analytic form of used functionals is not known, any practical application of the formalism involves approximations. The applied approximations were previously tested on several weakly bound systems including hydrogen-bonded and van der Waals complexes [8 10]. These studies showed that: (T nadd s KSCED calculations using relatively simple expressions for the non-additive kinetic energy functional lead to electron densities which are very similar to the electron densities derived from supermolecule Kohn-Sham calculations [8, 9]; KSCED results are generally less sensitive to the choice of the approximate functionals than the KS ones owing to the fact that they depend on the accuracy of the sum of approximate functionals used (E xc and Ts nadd ) [10]. In this paper, we explore the possibility offered by the KSCED formalism to introduce additional approximations concerning the electron density. Aiming at reducing the computational effort needed to study physisorbed molecules on chemically inert surfaces, we treat separately the electron density of the adsorbate and that of the surface. This allows us to investigate the effect of the relaxation of the surface electron density upon forming the complex on the structure and on the vibrational frequencies of the adsorbed molecule. The system under investigation consists of the vertically oriented CO molecule adsorbed at the Mg-site of the MgO(100) surface. Experimental measurements [12 15] and different theoretical models (for comparison of different theoretical methods, see Ref. [16]; for ab initio results, see Refs. [17 20]; for density functional theory results, see Refs. [21 25]) indicate that CO binds weakly to the MgO(100) surface. In particular, the studies by Pacchioni et al. [17] showed that the dominant effect on the CO-stretching frequency originates from the wall effect and not from orbital interactions. According to the experience gained in studies of weak intermolecular interactions, this is an adequate system to be modeled using the KSCED formalism. 2. Methods Two different formalisms were applied. The standard Kohn-Sham method [7] and that of Kohn-Sham equations with constrained electron density (KSCED) [1, 2]. The description of the KSCED formalism and the details of its computer implementation can be found elsewhere (see Ref. [9] and references therein). Only its main features will be given below The Kohn-Sham equations with constrained electron density (KSCED) The electron density (r 1 ) of a fragment which interacts with an other subsystem comprising nuclear charges and a frozen electron density (r 2 ) is obtained using Kohn-Sham-like equations with the following effective potential [1, 2]: V eff KSCED ˆ X A Z Z A jr R A j r2 r 0 jr 0 rj dr 0 Z r1 r 0 jr 0 rj dr0 V xc r 1 r r 2 r dtnadd s r 1 ; r 2 Š 1 dr 1 where r 1 ˆ S N 1 i n 1 i jw 1 i j 2 ; V xc is conventional exchange-correlation potential V xc ˆ de xc =dr, Ts nadd denotes the non-additive kinetic energy Ts nadd r 1 ; r 2 ŠˆT s r 1 r 2 Š T s r 1 Š T s r 2 Š, and the index A runs through the nuclei of subsystem 1 and subsystem 2. The effective potential defined in Eq. (1) leads to the electron density r 1 which minimizes the total energy functional. It is worthwhile to note that Eq. (1) is valid regardless whether the fragments densities (r 1 and r 2 ) do or do not overlap, and that it makes it possible to minimize the total energy with respect to variations of r 1 without any knowledge of orbital representation of r 2. Any practical implementation of Eq. (1) involves

3 T.A. Wesolowski et al. / Journal of Molecular Structure (Theochem) 458 (1999) approximating Ts nadd r 1 ; r 2 Š. In the present paper, a particular subset of possible approximations of Ts nadd is considered. Firstly, the Ts nadd r 1 ; r 2 Š is approximated as: T nadd appr s r 1 ; r 2 ŠˆT appr s r 1 r 2 Š Ts appr r 1 Š T appr s r 2 Š 2 Secondly, the approximate kinetic energy functionals assume the following analytical form: T appr s rš ˆ p2 Š 2=3 Z r 5=3 F s dr 3 where s ˆj7rj=rk F (k F being the Fermi vector). Lee et al. [26] proposed to use the same analytical form of F(s) as the one used in the conventional form of the exchange functional (E x ). E appr x rš ˆ 3 4 3p2 Š 1=3 Z r 4=3 F s dr 4 Therefore, following the suggestion of Lee et al., it is possible to construct an approximate kinetic energy functional from a given approximate exchange energy functional. The particular case of F(s) ˆ 1 corresponds to the Dirac expression for the exchange energy and Thomas-Fermi (TF) expression for the kinetic energy. The following gradient-dependent kinetic energy functionals were used: LLP derived from the exchange functional proposed by Becke (B88) [28], PW86K derived from the exchange functional proposed by Perdew and Wang in 1986 (PW86) [29], and PW91K derived from the exchange functional proposed by Perdew and Wang in 1991 (PW91) [30]. Throughout this paper, the approximate kinetic energy functionals used to derive Ts nadd employ the same F(s) as the applied exchange energy functional. In practical applications of the KSCED equations, the use of a kinetic energy functional which is associated with the applied exchange energy functional offers a significant advantage, namely, the KSCED method does not involve any additional parameters as compared with the standard Kohn-Sham calculations. The total energy of a system comprising two subsystems is expressed as: E rš ˆE r 1 r 2 ŠˆT s r 1 Š T s r 2 Š Ts nadd r 1 ; r 2 Š 1 ZZ r1 r 0 r 2 r r 1 r 0 r 2 r 2 jr 0 dr 0 dr rj X Z Z A r jr R 1 r r 2 r dr E xc r 1 r 2 Š A A 5 To study adsorption phenomena on chemically inert surfaces Eqs. (1) and (5) provide a convenient theoretical framework. In the absence of strong chemical interactions between the adsorbate and the surface, it is natural to use the electron density of the adsorbate as r 1 and electron density of the surface as r 2. With such a partition of the electron density, one can consider the adsorbed molecule as a quantum mechanical system in an effective potential defined as in Eq. (1). The electron density r 2, the positions and charges of atomic nuclei of the surface enter as parameters into the expression of the effective potential. It is possible, therefore, to calculate the total energy of the system assuming given positions of the surface atoms as well as a given electron density of the surface. Such calculations in which r 2 does not depend on the position of the adsorbed molecule introduce an additional approximation, namely, neglecting the relaxation of the surface electron density. In the present work, the importance of this approximation is investigated. The KSCED results obtained using a r 2 density calculated for the surface without adsorbed molecules will be referred to as KSCED (1) or frozen-surface calculations. The KSCED formalism also makes it possible to relax the surface electron density by means of the freeze-and-thaw cycle in which the electron densities of the surface and of the adsorbed molecule invert their roles in a series of subsequent KSCED calculations [3]. Such calculations will be referred to as KSCED (conv) to denote that they correspond to the converged freeze-and-thaw cycle. The following functionals were used: SVWN (Slater expression for E x and Vosko et al. [27] expression for E c ), B88-P86 (E x proposed by Becke [28] and E c by Perdew [31]),

4 154 T.A. Wesolowski et al. / Journal of Molecular Structure (Theochem) 458 (1999) Fig. 1. Total KSCED(PW91/PW91K) energy of the complex made of the CO molecule (R C O ˆ Å) and the embedded cluster model of the MgO(100) surface ((MgO 5 ) 8, array of point charges, C-down orientation, Mg binding site) calculated for several R C Mg distances. The dashed line indicates the energy at infinite separation. PW86-P86 (E x proposed by Perdew and Wang [29] and E c by Perdew [31]), PW91 (E x and E c proposed by Perdew and Wang [30]). The interaction between the CO molecule and the MgO(100) surface is weak and the numerical experience shows that some implementations of the Kohn- Sham formalism are not applicable for such complexes [32, 33]. In fact, Harris showed as early as in 1975 [34] that the asymptotic r 6 behavior (or r 3 for charged molecules) of the interacting neutral molecules can be derived using ground-state functionals allowing him to conclude: density-functional theory provides a way of determining the asymptotic behavior of interacting electronic systems from a knowledge of the mean field densities of the components. More recent numerical studies [35 37] showed that difficulties with applying the Kohn- Sham theory to study weakly interacting systems originate not from the underlying assumptions of the density functional theory but from the use of inadequate functionals. For a more detailed discussion see Ref. [36] Numerical details Gaussian basis sets were used to expand one-electron orbitals and to fit (auxiliary functions) the electrostatic and exchange-correlation potentials. The orbitals were constructed using atomic basis sets with the following contraction pattern: (721/51/1*) for carbon [39], (621/41/1*) for oxygen [39], and (6321/411/1*) for magnesium [40], which were developed specifically for Kohn-Sham calculations. The coefficients of the auxiliary functions were taken from Ref. [39] (4,3;4,3) for carbon and oxygen [39], and from Ref. [40] (5,4;5,4) for magnesium. The expansion of the electron density of CO using atom-centered Gaussians involved only CO atoms. Similarly, the expansion of the surface electron density involved only Gaussians centered on surface atoms. Such an expansion of the electron density, referred to as KSCED(m) in Ref. [9] where m stands for monomer implies that the expansion of the electron density of a given fragment uses the same basis functions regardless of the mutual orientation of interacting fragments. It is therefore free from the basis set

5 T.A. Wesolowski et al. / Journal of Molecular Structure (Theochem) 458 (1999) Fig. 2. The surface relaxation effect on the KSCED(PW91/PW91K) energy calculated at several C Mg distances. For details, see the caption to Fig. 1. superposition error of the quantum mechanical results derived from the supermolecule calculations. Throughout the paper, the following notation will be used KSCED (a)(b/c), where a specifies the number of the freeze-and-thaw iterations, b the exchange-correlation functional, and c the exchange functional used to derive Ts nadd. The calculations were made using a small CO (MgO 5 ) 8 cluster model of the system. To study the cluster size effect on the obtained results, some calculations were made for a larger system comprising MgO(3 3 2) atoms (CO Mg 9 O 9 cluster). The cluster was embedded in an array ( ) of point charges ( 2e on Mg and 2e on O atoms, respectively) localized at the nuclei positions in the ideal crystal. The assumed Mg O distance amounting to Å corresponds to the structure of the MgO crystal determined experimentally at 21 C [41]. The choice of the cluster model and the embedding potential was such as to facilitate comparisons with previously published theoretical results [17]. At all considered geometries of the adsorbed CO molecule, the CO stretching frequency was calculated by fitting a third-order polynomial to the potential energy curve corresponding to the atomic displacements as in the isolated CO stretching vibrational mode. 3. Results 3.1. KS and KSCED energies Fig. 1 displays the KSCED (PW91/PW91K) energies for C-down orientation of a rigid CO molecule (R CO ˆ Å) on the (MgO 5 ) 8 cluster at several C Mg distances. The interaction energy amounts to 0.43 ev at the R C Mg distance of 2.33 Å. The freezeand-thaw cycle converges very fast and the second iteration already leads to energies differing less than ev from the converged ones. The energies obtained in the first freeze-and-thaw iteration, i.e. calculations with frozen, unperturbed surface, are higher than the converged ones. The difference between the KSCED(1) and the KSCED(conv) energies results from the effect of the surface relaxation upon forming the complex. The surface relaxation

6 156 T.A. Wesolowski et al. / Journal of Molecular Structure (Theochem) 458 (1999) Table 1 Intermolecular distance (R C Mg ) and binding energy (E int ) calculated for the CO molecule (R C O ˆ Å) and the embedded cluster model of the MgO(100) surface ((MgO 5 ) 8, array of point charges, vertical orientation, C-down, Mg binding site). For description of applied methods and functional parametrizations, see text Method R C Mg (Å) E int (ev) KSCED(SVWN/TF) KSCED(B88-P86/LLP) KSCED(PW91/PW91K) KS(SVWN) KS(B88-P86) KS(PW91) KS(SVWN)(BSSE) KS(B88-P86)(BSSE) KS(PW91)(BSSE) effect amounts to 30% of the interaction energy at R C Mg ˆ 2.35 Å, and it decreases with increasing C Mg distance (see Fig. 2). The minimum energy C Mg distance does not change significantly in subsequent freeze-and-thaw iterations (R C Mg ˆ 2.35 Å, and 2.33 Å, for KSCED(1) and for KSCED(conv), respectively). To investigate the effect of the functional parametrizations on the KSCED results, the potential energy curves were calculated using another set of gradientdependent functionals (B88-P86/LLP) and gradientless ones (SVWN/TF) instead of the PW91 functionals. The LDA kinetic energy functional (TF) was shown previously not to be sufficiently accurate Table 2 Intermolecular distance (R O Mg ) and binding energy (E int ) calculated for the CO molecule (R C O ˆ Å) and the embedded cluster model of the MgO(100) surface ((MgO 5 ) 8, array of point charges, vertical orientation, O-down, Mg binding site). For description of applied methods and functional parametrizations, see text Method R O Mg (Å) E int (ev) KSCED(SVWN/TF) KSCED(B88-P86/LLP) KSCED(PW91/PW91K) KS(SVWN) KS(B88-P86) KS(PW91) KS(SVWN)(BSSE) KS(B88-P86)(BSSE) KS(PW91)(BSSE) No minimum No minimum to be applied in KSCED calculations of interaction energies [9]. The LDA results (KS(SVWN) and KSCED(SVWN/TF)) are presented for reference purposes. The interaction energies and equilibrium C Mg distances derived from the KSCED calculations are collected in Table 1 together with corresponding results derived from the supermolecule Kohn-Sham calculations. It may be seen there that both the KSCED and the KS results depend on the choice of the functionals. The LDA results differ significantly from the ones obtained using gradient-corrected functionals. The KSCED energies depend less than the KS ones upon the choice of the functional. Indeed, the KSCED interaction energies fall within a 0.15-eV range, whereas the KS ones fall within a 0.36-eV range. If the analysis of the functional importance is restricted to gradientdependent functionals, it can be seen that the KSCED energies depend slightly less on the functional choice than the KS ones. These tendencies are in line with our previously obtained results for van der Waals complexes made of the benzene and O 2,N 2 or CO molecules [10]. The potential energy curves were also calculated for O-down orientation of the CO molecule and the (MgO 5 ) 8 cluster. The equilibrium distances and the interaction energies derived from the KS and the KSCED calculations are collected in Table 2. Results derived from both the KS and the KSCED calculations indicate that at the O-down orientation the CO molecule binds less strongly to the surface than at the C-down orientation. As in the case of the C-down orientation, the KSCED results are less sensitive to the choice of the functional than the KS ones. The basis set superposition error (BSSE), as estimated using the counterpoise method of Boys and Bernardi [11], significantly affects the supermolecule Kohn-Sham energies for both orientations of the CO molecule. Except for the LDA calculations, the KSCED interaction energies fall between the BSSE corrected and non-corrected supermolecule Kohn- Sham ones. Kohn-Sham results derived using different functionals are in line with the ones which can be found in the literature [16, 23, 24]. The KS(SVWN) interaction energies amount to 0.50 ev ((MgO 5 ) 8, this work), 0.56 ev (Mg 9 O 9 cluster [16]). The KS(SVWN) equilibrium C Mg distance amounting to 2.28 Å is

7 T.A. Wesolowski et al. / Journal of Molecular Structure (Theochem) 458 (1999) Fig. 3. The KSCED(PW91/PW91K) CO stretching frequency shift (Dv CO [cm 1 ]) relative to the free-co as a function of the C Mg distance (in Å). The free-co stretching frequency derived from KS(PW91) calculations amounts to cm 1. similar to that obtained for larger clusters (2.18 Å, Ref. [23]; 2.22 Å, Ref. [16]). Our calculations and the ones reported for larger clusters [16, 24] indicate that introduction of gradient corrections to the exchange-correlation functional significantly affects the Kohn-Sham results; the binding energy is reduced by about 50% and the equilibrium C Mg distance increases by Å depending on the analytical form of the gradient corrections. To investigate the effect of the cluster size on the KSCED energies, the CO MgO(100) interaction energy was calculated using a larger cluster model of the surface (Mg 9 O 9 cluster embedded in the array of point charges). The KSCED (PW91/ PW91K) interaction energies were calculated for the C-down orientation of the CO molecule at selected distances. The changes of the interaction energy amount to 0.015, 0.007, 0.005, and ev for R C Mg ˆ 2.375, 2.45, 2.60, and 2.80 Å, respectively. These values are small compared to the values of the interaction energy. At shorter C Mg distances, the difference between the KSCED energies derived using different clusters increases CO-stretching frequency At several C Mg distances of the C-down oriented CO molecule, the CO stretching frequency was determined by means of the KSCED(PW91/PW91K) calculations and compared with the free-co stretching frequency derived from KS(PW91) calculations ( cm 1 ). Fig. 3 displays the frequency shift (Dv CO ) as a function of the C Mg distance. It can be seen that Dv CO is positive (blue-shift) and it decreases with increasing C Mg distance. The Dv CO shifts obtained with frozen surface density (KSCED(1))(PW91/PW91K) agree within 3 cm 1 with the ones obtained with converged freeze-andthaw cycle (KSCED(conv)(PW91/PW91K)). At the KSCED(conv)(PW91/PW91K) energy minimum (R C Mg ˆ 2.35 Å), Dv CO amounts to 45 and 47 cm 1 for KSCED(1) and KSCED(conv), respectively. The KSCED calculations with frozen electron density of the surface make it possible to evaluate the vibrational frequencies of an adsorbed molecule as a function of the position of the adsorbate relative to the surface. Depending on the cluster size and the embedding approach, previously published

8 158 T.A. Wesolowski et al. / Journal of Molecular Structure (Theochem) 458 (1999) Fig. 4. CO bond length (in Å) calculated as a function of the C Mg distance (in Å) derived from KSCED(PW91/PW91K) calculations. theoretical estimates of the equilibrium C Mg distance vary between Å (KS calculations with gradient-corrected functionals) [16, 24] and Å (ab initio) [16, 17, 19]. Within the range of R C Mg distances reported in the literature ( Å), the KSCED(1) (PW91/PW91K) frequency shifts vary between 15 and 48 cm 1.The experimental value of Dv CO, amounting to 35 cm 1 [13], falls within this range. In particular, the shifts of R C Mg ˆ 2.42 Å, which corresponds to the minimum of the KS(PW91) potential energy curve, amount to 35 cm 1 (KSCED(1)) and 36 cm 1 (KSCED(conv)), in excellent agreement with experiment. Pacchioni et al. [17] also used the (MgO 5 ) 8 cluster as a model of the surface in Hartree-Fock calculations. They obtained the value of Dv CO ˆ 31 cm 1 at the R C Mg distance of 2.60 Å. At the same distance, the KSCED(PW91/PW91K) calculations led to a smaller value of Dv CO amounting to 20 cm 1. The CO bond lengths derived from the KSCED calculations are shorter than the free-co bond length amounting to Å (see Fig. 4). The CO bond length shortens by about Å at R C Mg ˆ 2.42 Å. To investigate the effect of cluster size on KSCED frequencies, the CO stretching frequency was calculated using a larger cluster model of the surface (Mg 9 O 9 cluster embedded in the array of point charges). The KSCED(1)(PW91/PW91K) frequencies calculated for the C-down orientation of CO at selected distances. The effect of change of cluster size on frequency amounts to 0.0, 1.5, 2.4, 5.2, and 4.1 cm 1 for R C Mg ˆ 2.3, 2.375, 2.45, 2.60, and 2.80 Å, respectively. The cluster size effect on CO stretching frequency is small especially for short CO-surface distances. It is comparable to our estimated accuracy of the procedure used to determine the frequencies. To study the effect of the functionals parametrizations on frequency shifts, Dv CO was derived from KSCED(B88-P86/LLP) calculations for C-down oriented CO molecule in (MgO 5 ) 8 system. The KSCED(PW91/PW91K), KSCED(B88-P86/LLP), and KSCED(PW86-P86/PW86K) frequencies agree within 4 cm 1 as calculated at R C Mg ˆ 2.35 Å and R C Mg ˆ 2.5 Å. Additionally, the CO-stretching frequency was calculated for O-down orientation of the CO molecule. The KS(PW91) and the KSCED(conv)(PW91/ PW91K) calculations predicted the equilibrium O Mg distance to lie at 2.35 and 2.36 Å, respectively.

9 T.A. Wesolowski et al. / Journal of Molecular Structure (Theochem) 458 (1999) At a O Mg distance amounting to 2.35 Å, the KSCED(PW91/PW91K) calculations predicted a negative shift amounting to 9 cm 1 in agreement with other theoretical estimates [17, 23]. 4. Conclusions The freeze-and-thaw cycle of KSCED calculations with gradient-dependent functionals leads to interaction energies which fall between the BSSE corrected energies derived from supermolecule Kohn-Sham calculations. The KSCED energies appear to be less dependent on the functional choice than the ones derived from the supermolecule Kohn- Sham calculations. The surface relaxation effects on the KSCED energies, as measured by the difference between the energies obtained with frozen and relaxed electron densities of the surface, are significant. These effects contribute up to 30% of the interaction energy. However, the surface relaxation effects do not significantly affect neither the geometry of the complex nor the stretching frequency of the adsorbed CO molecule. Therefore, the frozen surface KSCED calculations, which are substantially less demanding in computer time as compared to the supermolecule Kohn-Sham cluster calculations, provide a valuable alternative to investigate properties of adsorbates physisorbed on chemically inert similar surfaces. The computational advantages originate from the fact that the frozen surface KSCED calculations may be seen as Kohn-Sham computations for the adsorbed molecule in which the isolated adsorbate effective potential is supplemented with terms responsible for the interactions with the surface. The interaction energies as well as the CO-stretching frequency for the C-down orientation of the CO molecule are not significantly affected by the change of the cluster size used to model the MgO(100) surface. Acknowledgements The authors are greatly indebted to Prof. Paul Bagus for stimulating discussions which inspired this work. The authors are grateful to Prof. D.R. Salahub for providing a copy of the demon program [38], which was used to perform the Kohn-Sham calculations and into which the KSCED formalism was implemented. Financial support by the Federal Office for Education and Science, acting as Swiss COST office, is greatly acknowledged. This work is also a part of Project of the Swiss National Science Foundation. References [1] P. Cortona, Phys. Rev. B 44 (1991) [2] T.A. Wesolowski, A. Warshel, J. Phys. Chem. 98 (1993) [3] T.A. Wesolowski, J. Weber, Chem. Phys. Lett. 248 (1996) 71. [4] T.A. Wesolowski, A. Warshel, J. Phys. Chem. 98 (1994) [5] E.V. Stefanovitch, T.N. Truong, J. Chem. Phys. 104 (1996) [6] T.A. Wesolowski, R. Muller, A. Warshel, J. Phys. Chem. 100 (1996) [7] W. Kohn, L.J. Sham, Phys. Rev. 140A (1965) [8] T.A. Wesolowski, Chermette, J. Weber, J. Chem. Phys. 105 (1996) [9] T.A. Wesolowski, J. Chem. Phys. 106 (1997) [10] T.A. Wesolowski, Y. Ellinger, J. Weber, J. Chem. Phys. 108 (1998) [11] S.F. Boys, F. Bernardi, Mol. Phys. 19 (1970) 553. [12] C.R. Henry, C. Chapon, C. Duriez, J. Chem. Phys. 95 (1991) 700. [13] J.-W. He, C.A. Estrada, J.S. Corneille, M.C. Wu, D.W. Goodman, Surf. Sci., 261 (1992) 164. The value of Dv CO ˆ 35 cm 1 was confirmed in recent IR experiments by F. Vanolli, Ph.D Thesis, Université de Lausanne, [14] D. Scarano, G. Spoto, S. Bordiga, S. Coluccia, A. Zecchina, J. Chem. Soc. Faraday Trans. 88 (1992) 291. [15] L. Marchese, S. Coluccia, G. Martra, A. Zecchina, Surf. Sci. 269 (1992) 135. [16] K. Jug, G. Geudtner, J. Chem. Phys. 105 (1996) [17] G. Pacchioni, G. Cogliandro, P.S. Bagus, Int. J. Quant. Chem. 42 (1992) [18] J.A. Mejias, A.M. Marquez, J. Fernandez-Sanz, M. Fernandez-Garcia, J.M. Ricart, C. Sausa, F. Illias, Surf. Sci. 327 (1995) 59. [19] M. Nygren, L.G.M. Petterson, J. Chem. Phys. 105 (1996) [20] G. Pacchioni, A.M. Ferrari, A.M. Marquez, F. Illias, J. Comp. Chem. 18 (1997) 617. [21] K.M. Neyman, N. Rosch, Berl. Bunsengess., Phys. Chem. 96 (1992) [22] K.M. Neyman, N. Rosch, Chem. Phys. 168 (1992) 267. [23] K.M. Neyman, N. Rosch, Chem. Phys. 177 (1993) 561. [24] K.M. Neyman, S.P. Ruzankin, N. Rosch, Chem. Phys. Lett. 246 (1995) 546. [25] H. Kobayashi, D.R. Salahub, T. Ito, Catalysis Today 23 (1995) 357.

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