Journal of the Chilean Chemical Society
versión ISSN 0717-9707
J. Chil. Chem. Soc. vol.57 no.1 Concepción mar. 2012
J. Chil. Chem. Soc, 57, No 1 (2012); págs.: 1022-1028
THEORETICAL STUDY OF THE ADSORPTION OF OXYGEN ON SMALL BIMETALLIC K Cu (m, n ≤ 4 AND m, n=1, 12) CLUSTERS
FRANCISCO CÉSPEDES1 AND LUIS PADILLA-CAMPOS2
1Departamento de Física, Facultad de Ciencias, Universidad Católica del Norte Casilla 1280, Antofagasta, Chile.
2Departamento de Química, Facultad de Ciencias Básicas, Universidad de Antofagasta, Casilla 170, Antofagasta, Chile. *e-mail: email@example.com
A theoretical study of the adsorption of molecular oxygen on small bimetallic KmCun (m, n ≤ 4 and m, n=1,12) clusters was carried out using density functional methods, and compared with the adsorption of O2 on bimetallic LimCun (m, n ≤ 4) clusters. The study of the O2-KmCun system is important to understand the promotion effects of the alkali atoms on the copper surface participating in the catalytic processes. Adsorption energies ranging from 5.7 to 48.6 kcal/mol were found, which represented values slightly smaller than those calculated for the adsorption of O2 on LimCun clusters in a previous study. However, the global reactivity towards O2 was higher in KmCun than in LimCun clusters.
Keywords: alkali-copper cluster, O2 adsorption, quantum chemical calculations.
The study of small metal clusters has received considerable attention on the theoretical as well as the experimental level due to their potential application in the catalytic processes and electronic materials [1-3]. Clusters containing as little as a few to several thousand atoms have been studied to determine the evolution of the bulk matter properties from atomic properties . In addition, the basic properties of metal clusters are believed to be an important link in understanding the fundamental mechanism of catalysis and other chemical transformations [5,6]. Recently, the study of adsorption of gases on clusters of noble and transition metals was carried out [7-19]. The effect of catalytic promotion is an interesting, but unanswered question related to the properties of impurities, such as alkali atoms on metal surfaces. Thus, the study of bimetallic clusters is important as it could help clarify the properties of these impurities. Recently, a theoretical study concerning the adsorption of molecular oxygen on bimetallic lithium-copper clusters  was reported. The previous study demonstrated that the reactivity of O2 on bimetallic clusters was higher than that for copper clusters. In addition, an adsorption energy about 30 to 68% greater in comparison to the CunO2 (n ≤ 8) system was found in the LimCunO2 system. The above mentioned result was fundamentally related to the lithium atoms in bimetallic clusters for increasing the charge transfer towards the O2 molecule. The existence of a charge transfer was indicative of the chemisorptions' process.
Studies of surfaces demonstrate that lithium and potassium deposited on copper surfaces modify the surface properties, thereby presenting differences in the reactivity towards the adsorbates . Thus, it is believed that the manner in which lithium and potassium interact with the copper surfaces is different, whereby the characteristics have not yet been solved completely. Consequently, the cluster study can help to elucidate these aspects. Thus, the aim of the present work is to study the adsorption of the molecular oxygen on small bimetallic KmCun (m,n ≤ 4) clusters using functional density methods, and comparing their properties with the adsorption of O2 on LimCun clusters, as recently reported . In addition, the KCu12O2 system is also analyzed to study the behavior of the adsorption of oxygen on a bigger bimetallic cluster.
METHODOLOGY AND COMPUTATIONAL DETAILS
Energy and electronic properties have been calculated by solving the Kohn-Sham equations in an atomic basis set formed by Gaussian functions. The calculations have been done using the B3LYP [22-25] exchange correlation functional, which is of the hybrid type. This consists of a careful mixing of the Hartree-Fock exchange, calculated with Kohn-Sham orbitals, and the B88 exchange functional  plus the LYP correlation functional . The use of the B3LYP functional has been utilized in previous calculations with good agreement with the experimental data [17, 20]. The Stuttgart pseudopotential  with the corresponding basis set (8s7p6d) has been used for the copper atoms. The pseudopotential replaces the ten core electrons. Thus, nineteen valence electrons have been considered. A new Stuttgart pseudopotential  with the corresponding basis set (11s11p5d) has been used for the potassium atoms. The pseudopotential replaces the ten core electrons. Thus, nine valence electrons have been considered. The D95+(d) basis set [28,29] has been used for the oxygen atom. The calculated bond length and bond energy for the gas phase O2 were 1.22 A and 120.4 kcal/mol, respectively, which was in good agreement with the experimental values of 1.21 A and 120.6 kcal/ mol, as reported previously . The natural bond order (NBO) population analysis [30,31] was used for the discussion of the obtained results. Initial geometries of bimetallic KmCun clusters were those optimized in a previous study . The initial geometry for the KCu12 cluster was determined in the present work and is shown in Fig. 1. The interaction of molecular oxygen with the bimetallic clusters was fully optimized without symmetry restrictions in different adsorption sites (top, bridge and threefold hollow) and orientations on the bimetallic cluster. All the calculations were done of the unrestricted type using the Gaussian 98 package .
The binding energy for the KmCunO2 system was calculated as:
and the adsorption energy for the optimum process was calculated as:
E(KmCunO2), E(KmCun), E(O), E(K), E(Cu) and E(O2) being the energies of O2-bimetallic cluster, bare bimetallic cluster, oxygen atom, potassium atom, copper atom and molecular oxygen, respectively. Equations (1) and (2) are similar to the ones used to calculate the binding and adsorption energies in the adsorption of molecular oxygen on lithium-copper clusters in a previous study .
RESULTS AND DISCUSSION
Fig. 1 shows the optimal geometrical structures of the O2-bimetallic cluster systems with the corresponding geometrical parameters. The above mentioned structures were compared with those reported for the bare bimetallic clusters in a previous study . It was observed that when the interaction between the O2 molecule was realized for Cu-Cu or Cu-K bonds of the bimetallic cluster, the geometry of the cluster was modified weakly. However, when the interaction O2 was realized for a K-K bond, it produced a strong distortion in the geometry of the bimetallic clusters, thereby conducing to the rearrangement of the atoms of the bimetallic clusters, as observed for K2Cu2O2, K2Cu4O2 and K3Cu4O2 systems in the present study. In addition, the great distortion in the geometry of the bimetallic cluster produced the dissociation of the O2 molecule in K2Cu4O2 and K3Cu4O2 systems. The above mentioned latter systems would be discussed in more detail in the next section. Generally, distortions were observed in the local environment, that is, in the region of adsorption of the molecular oxygen. For the most stable systems, the average Cu-Cu and K-Cu distances in KmCunO2 and in bare bimetallic clusters are plotted in Fig. 2. It was clearly observed that the adsorption of molecular oxygen did not produce any significant changes in the Cu-Cu distance with the exception of the K2Cu2O2 system, where rearrangement took place, thereby producing a shortening of the Cu-Cu distance and formation of the Cu-Cu bond. Thus, for average Cu-Cu distances, a shortening of 0.02 to 0.12 A for KmCunO2 systems with m = 1, 2, 4 and n = 2 (m # 2), 3 (m # 1), 4 and a stretching of only 0.02 to 0.04 A for systems with m = 1, 3 and n = 3 (m # 1), 4 were produced. The maximum Cu-Cu distance was produced in systems when Cu-Cu was broken, i.e., when m = n. In general, with respect to the average distance of K-Cu, a stretching of 0.03 to 0.27 A was produced, which indicated that the K-Cu distance did not suffer a great distortion under the adsorption of O2. For KCu12O2 system the behavior is similar, i.e., under the adsorption process the average distance of Cu-Cu is not affected and the average distance K-Cu a stretching of only 0.14 A was produced. For the bare KCu12 is shown in Fig.1k. On the other hand, Fig. 1 showed that the adsorption of O2 occurred on a bridge site (2-fold coordination) with the O2 molecule approaching parallel to the adsorption site of each one of bimetallic clusters. This latter occurred in all structures except for bimetallic clusters that suffered rearrangement, K2Cu2O2(I) system, and the dissociation of O2 molecule, K2Cu2O2(I) and K3Cu4O2(I) systems. Generally, when the structure of bimetallic clusters did not suffer rearrangement, the oxygen molecule was oriented through the K-Cu bond (KCuO2(I), K2Cu2O2(II), KCu3O2(I), K2Cu3O2(I), K3Cu3O2, KCu4O2(I), K2Cu4O2(II), K3Cu4O2(II), K4Cu4O2(II) and KCu12O2(I)). The adsorbed O-O distance varied from 1.30 to 1.34 A, which was between 7% and 10% longer than the O-O distance in the gas phase (1.22 A). On the basis of the experimental evidence and other theoretical studies it was proposed that the O2 was adsorbed on a coordinate site. Thus, it would have a parallel orientation to the Cu(111) surface [34, 35], which was in agreement with the results obtained in the present study. A similar trend was found in a previous study for LimCunO2 systems . In all the adsorption systems, the O-Cu distance and the O-K distance varied between 1.91 Å and 2.08 Å and 2.51 Å to 2.63 Å, respectively.
ENERGIES AND ELECTRONIC PROPERTIES
The binding energy per atom, adsorption energy, LUMO-HOMO gap, and other properties for KmCunO2 systems are summarized in Table 1. First, the electronic state of each system is consequence of their structures electronic. In bare bimetallic clusters, KmCun, depending if n + m is an even or odd number of atoms, the number of total electrons in the cluster will also be even or odd . It is well known that the ground state for O2 molecule is a triplet. When the interaction takes place between the bimetallic cluster and the O2 molecule, the possible ground state for adsorption systems (KmCunO2) can be a doublet, triplet or of higher order. In the present study the lower energy structures of all adsorption systems only presented a doublet or triplet multiplicity. Thus, when m + n is odd the electronic ground state of adsorption system is a doublet and when m + n is even gives a triplet, as we can see in Table 1. Second, the binding energy per atom and the adsorption energy for the most stable systems are shown in Fig. 3. The adsorption energy for the systems, which do not show dissociation of the oxygen molecule have values ranging from 5.7 to 48.6 kcal/mol, with the values being maximum for the K2Cu3O2 system. In general, as shown in Fig. 3, the binding and the adsorption energies decrease progressively with the increase in the number of potassium atoms in the bimetallic cluster. However, KCu2O2, K2Cu3O2, KCu4O2 and KCu12O2 systems demonstrate the maximum binding energies and coincide with the systems that have the highest adsorption energy. It was also observed that the systems with a minimum binding energy coincided with the systems that had the lowest adsorption energy, i.e., for KCuO2, KCu3O2, K3Cu3O2, and K4Cu4O2 systems. Third, in general, a high reactivity towards the O2 molecule, was observed for KmCunO2 systems when m = n - 1, which presented high adsorption energy. The minor reactivity was observed for systems when m = n, which presented low adsorption energy and low binding energy. In addition, when m = n, the adsorption of O2 was preferably produced on the K-Cu bond, while the adsorption energy decreased with an increase in the size of the bimetallic clusters. The Cu-Cu bonds were broken in the above mentioned cluster. The K2Cu2 cluster was the exception, where the adsorption of O2 was preferable on the K-K bond as a product of the rearrangement of the bimetallic cluster, thereby producing the formation of the Cu-Cu bond as well as a significant increase of the adsorption energy. In addition, in the systems with n > m and the Cu-Cu bonds not broken, the adsorption of O2 was also preferable on the K-Cu bond, with the exception of K2Cu3O2, K2Cu4O2 and K3Cu4O2 systems, where the adsorption of O2 was preferably produced on the K-K bond. In the above mentioned latter systems, the adsorption energy increased with an increase in the size of the bimetallic cluster. The adsorption energy and charge transfer towards the O2 molecule are shown in Fig. 4. The charge transfer was calculated from NBO results, from the computation of the difference of net charges between components of the system, the bimetallic cluster and the O2 molecule, with and without interaction. In general, it was observed that there was a good correlation between the above mentioned properties. Thus, systems that presented high charge transfer also presented high adsorption energy. This behavior was observed for K2Cu3O2, KCu4 and K3Cu4O2 systems. The minor charge transfer was observed for systems with m = n, where the Cu-Cu bonds were broken, which presented the smallest adsorption energy. The exception was for the K2Cu2O2 system, in which the system suffered rearrangement as mentioned above, thereby producing formation of the Cu-Cu bond and a high charge transfer. Fourth, in general, the adsorption of O2 on the Cu-Cu bond was less effective in comparison to the adsorption on the K-Cu bond, in which a decrease between 23 to 56% of the adsorption energy was observed in KCu2O2, KCu3O2, KCu4O2 and KCu12O2 systems. Fig. 5 also demonstrated another good behavior when the values of the O-O distance and the charge transfer towards the O2 molecule were compared. It was indicated that the greater the charge transfer, the greater was the O-O distance. In addition, from Table I the O-O frequency of the adsorbed O2 molecule showed values ranging from 1122.3 cm-1 to 1234.4 cm-1, which was between 25% and 32% lower than the O-O frequency for O2 in the gas phase (1647.5 cm-1). The above mentioned behaviors explained that the charge transfer from bimetallic clusters to empty O2 π* orbitals weakened the O-O bond, thereby causing an elongation of up to 10% in comparison to its gas-phase length, as mentioned in the previous section, and also reduced its stretching frequency. On the other hand, values ranging from 0.61e to 0.94e for the charge transfer from bimetallic cluster towards the O2 molecule were found. Thus, above results are indicative with the formation of a superoxo-like specie (O2-) under the adsorption process .
The LUMO-HOMO gap is shown in Fig. 6 and is compared with the adsorption energy for KmCunO2 systems with respect to the number of potassium and copper atoms in the bimetallic cluster. The above mentioned gap is a measure of the molecular hardness  and it is commonly used to explain reactivity trends in molecules. In general, it was observed that systems presenting high adsorption energy, presented a large gap, i.e., the adsorption systems were inclined to increase the molecular hardness, thereby producing a high reactivity towards the O2 molecule. Fig. 7 also demonstrated the difference in gaps between KmCunO2 and KmCun systems and the adsorption energy. In general, it demonstrated a good behavior for the above mentioned two properties indicating that the greater the difference in the gap, the greater is the adsorption energy. The largest difference in the gap was observed for KCu2O2, K2Cu3O2, KCu4O2, K3Cu4O2 and KCu12O2 systems with m + n as the odd number of atoms, i.e., for systems with an odd number of electrons, thereby presenting high reactivity towards the molecular oxygen. In the remaining systems, the difference between gaps was presented as negative values, i.e., the gap (KmCunO2) was smaller than the gap (KmCun), thereby showing a low reactivity to the adsorption of molecular oxygen.
In base to the above results, we can see that the behavior for adsorption of O2 on a bigger bimetallic cluster (KCu12) is similar to smaller bimetallic clusters, with the difference that a bigger bimetallic cluster trend to be more stable under adsorption process, i.e., only smallest distortions in the geometry of the bimetallic cluster was found.
In continuation, the rearrangement process was analyzed in detail. In general, the adsorption on the K-K bond produced rearrangement of the atoms of the bimetallic cluster, thereby decreasing the distance of the K-K bond. Thus, the rearrangement of system I of K2Cu2O2 was produced for a decrease of K-K and formation of the Cu-Cu bond, as shown in Fig 8a. For the case system I of K2Cu4O2, the K atoms on and under the plane formed for the Cu atoms, moved towards a side of the Cu plane, thereby decreasing the K-K distance. This allowed an interaction more favorable with the O2 molecule and finally produced the dissociation, as shown in Fig 8b. The rearrangement of the system I of K3Cu4O2 was produced for the compression of the bimetallic cluster, in a way that favored the interaction of K atoms with the O2 molecule, thereby producing a decrease of the distance K-K. The rearrangement produced a strong interaction with the O2 molecule, conducing finally to the dissociation, as in the previous case. It was concluded that the rearrangement produced a great stability of the adsorption system, thereby presenting a higher adsorption energy and binding energy in comparison with the system without rearrangement.
COMPARISON BETWEEN THE K Cu O2 AND Li Cu O2 SYSTEMS
A theoretical study of the adsorption of molecular oxygen on lithium-copper clusters was recently reported . In the following section, the similarities and the differences between the KmCunO2 and LimCunO2 systems were analyzed. First, the adsorption energy in the KmCunO2 systems was slightly smaller in comparison with LimCunO2, in which the average values of 29.5 and 37.1 kcal/mol were found, respectively. However, in KmCunO2 a bigger probability of dissociation was produced, that is two systems in comparison with only one in the LimCunO2 system, with a significantly smaller amount of formation energy. For example, the maximum dissociation energy for the K3Cu4O2 system was 67.6 kcal/mol, which was almost half of the energy found for the Li2Cu3O2 system. Also, a bigger number of stables isomers were found in KmCunO2 systems in comparison to LimCunO2 systems, with a minor energy cost, i.e., minor adsorption energy. Thus, KmCun clusters would present a bigger reactivity toward the adsorption of oxygen in comparison with the LimCun clusters. Second, in both systems, KmCunO2 and LimCunO2, the interaction of molecular oxygen was parallel to the bimetallic cluster, with the oxygen molecule being adsorbed on a bridge site (2-fold coordination).
However, in the KmCunO2 systems, the O2 was preferably adsorbed on K-Cu or the K-K bonds, whereas in LimCunO2 systems the O2 was only preferably adsorbed on Li-Cu bonds. Third, in both systems, an effective charge transfer was produced from the cluster to the oxygen molecule, which was decisive for the most effective interaction between the cluster and the O2 molecule as well as the high reactivity observed for both types of clusters. However, in KmCunO2 systems a larger number of stable isomers were observed, which indicated that the reactivity towards the O2 molecule of KmCun clusters could be higher than LimCun clusters. Fourth, a good correlation was observed in both systems between the HOMO-LUMO gap and the difference in the gaps with the adsorption energy. This indicated that a most favorable interaction of O2 with each bimetallic cluster was produced when the adsorption system evolved towards an increase of the gap (or the difference of gaps), thereby producing high adsorption energy, i.e., a high reactivity towards the oxygen molecule. However, the difference of gaps in KmCunO2 systems is bigger than in LimCunO2 systems, thereby indicating that the adsorption of O2 on the KmCun clusters produced adsorption systems that are more stable than LimCunO2. Fifth, it was observed that in both systems the increased reactivity towards the oxygen molecule was produced for the cluster with m + n equal was an odd number of atoms, while the minor reactivity towards the oxygen molecule was produced for the cluster when n + m was an even number of atoms. An odd number of atoms implied an electronic structure of the open shell. Therefore, the bimetallic clusters presenting this electronic structure favored the reactivity and increased the charge transfer towards the O2 molecule. The presence of the charge transfer in both systems indicated that the interaction between O2 and the bimetallic cluster occurred by a chemisorption process. Sixth, in KmCunO2 systems a greater rearrangement was observed in comparison to LimCunO2 systems, with a tendency to produce the dissociation of the O2 molecule. This indicated that KmCun clusters were less stable than LimCun clusters. However, the rearrangement mechanism was used in both bimetallic clusters, LimCun and KmCun, to produce a more stable adsorption system. Finally, in both systems a decrease of the O-O frequency of the order of 28% with respect to the O-O frequency for O2 in the gas phase (1647.5 cm-1) and an increase of the O-O distance of about 10% were observed. This was indicative that the adsorption on the bimetallic cluster favored the dissociation to the O2 molecule. Also, for both systems, KmCunO2 and LimCunO2, from the analysis of the O-O frequency of the adsorbed O2 molecule and charge transfer from bimetallic cluster towards the O2 molecule, the results are consistent with the formation of a superoxo-like specie (O2-) under the adsorption process . Thus, the obtained results could be imported to future applications of the bimetallic clusters in the catalytic processes.
The adsorption of molecular oxygen on small bimetallic KmCun clusters was theoretically studied, and their properties were analyzed and compared with those found for the adsorption of O2 on bimetallic LimCun clusters, as reported in a previous study . It was found that the reactivity of O2 on KmCun clusters was greater than that of LimCun clusters, as a base to a bigger number of isomers found for the adsorption system and the minor formation energy to produce the rearrangement of the atoms in the bimetallic cluster. In both systems, KmCunO2 and LimCunO2, the alkali atoms in bimetallic clusters had the finality to increase the charge of the copper atoms and produce a more effective charge transfer towards the O2 molecule. The presence of the charge transfer indicated that the adsorption was produced by a chemisorption process. Thus, the interaction of O2 with the bimetallic cluster was of a local nature. In general, the optimum interaction of the O2 molecule with XmCun (X = Li, K) in both systems was produced when the number of alkali atoms was m = n - 1. When m = n - 1 the Cu-Cu bonds are not broken and the adsorption system present the higher charge transfer, having as consequence the higher reactivity of bimetallic clusters towards the O2 molecule. On the other hand, when m = n the Cu-Cu bond were broken and the adsorption system presented the minor charge transfer, producing a minor reactivity towards O2 molecule. The molecular oxygen was adsorbed on bridge sites (2-fold coordination), parallel to the adsorption site, preferably along K-Cu or K-K bonds in KmCun clusters and along Li-Cu bonds in LimCun clusters.
This work was supported by Proyectos de la Dirección de Investigación (U. de Antofagasta), Grant DI-CODEI-2010-01. F. Céspedes thanks to CONICYT-FIC-R for the Magister fellowship. Also, the authors are most grateful to Mrs. Viviana Guerrero for the constant support to the realization of this investigation.
(Received: July 1, 2011 - Accepted: October 19, 2011)
1. G. Mills, M.S. Gordon, H. Metiu. Chem.Phys.Letters. 359, 493, (2002). [ Links ]
2. A.K. Santra, D.W. Goodman, J.Phys:Cond.Matter. 15, R31, (2003). [ Links ]
3. R.R. Zope, T. Baruah, Phys.Rev.A. 64, 053202, (2001). [ Links ]
4. W. Ekardt, Metal Clusters, Wiley, U.S.A., 1999. [ Links ]
5. W. Eberhardt, Surf.Sci. 500, 242, (2002). [ Links ]
6. H.H. Kung, M.C Kung, Catal.Today. 97, 219, (2004). [ Links ]
7. X.L. Ding, Z.Y. Li, J.L. Yang, J.G. Hou, Q.S. Zhu, J.Chem.Phys. 12, 2558, (2004). [ Links ]
8. M. Schmidt, A. Masson, C. Brechignac, Phys.Rev.Lett. 91, 243401, (2004) [ Links ].
9. H.J. Zhai, L.S. Wang, J.Chem.Phys. 122, 051101, (2005). [ Links ]
10. G.H. Guvelioglu, P. Ma, X. He, R.C. Forrey, H. Cheng, Phys.Rev.Lett. 94, 026103, (2005). [ Links ]
11. M. Boyukata, Z.B. Guvenc, S. Ozcelik, P. Durmus, J. Jellinek, Int.J.Mod. Phys.C. 16, 295, (2005). [ Links ]
12. F.Y. Liu, P.B. Armentrout, J. Chem. Phys. 122, 194320, (2005). [ Links ]
13. S. Zhao, Z.H. Li, W.N. Wang, K.N. Fan, J. Chem. Phys. 122, 144701, (2005) [ Links ].
14. H. Gronbeck, A. Rosen, Chem. Phys. Lett. 227, 149, (1994) . [ Links ]
15. H. Gronbeck, A. Rosen, Surf. Rev.and Lett. 3, 687, (1996). [ Links ]
16. E. Florez, W. Tiznado, F. Mondragón, P. Fuentealba, J. Chem. Phys. A. 109, 7815, (2005) . [ Links ]
17. L. Padilla-Campos, J. Chi. Chem. Soc., 50, 745, (2005). [ Links ]
18. L. Padilla-Campos, J. Molec. Struct. (Theochem). 815, 63, (2007). [ Links ]
19. L. Padilla-Campos, J. Molec. Struct. (Theochem). 851, 15, (2008). [ Links ]
20. L. Padilla-Campos, Int. J. Quant. Chem. 109, 1357, (2009). [ Links ]
21. P. Fuentealba, L. Padilla-Campos, Theor. Chem. Acc. 414, 110, (2003). [ Links ]
22. A.D. Becke, J. Chem. Phys. 98, 5648, (1993). [ Links ]
23. A.D. Becke, Phys. Rev. A. 38, 3098, (1988). [ Links ]
24. B. Miehlich, A. Savin, H. Stoll, H. Preuss, Chem. Phys. lett. 157, 200, (1989). [ Links ]
25. C. Lee, W. Yang, R.G. Parr, Phys.Rev.B. 37, 785, (1988). [ Links ]
26. M. Dolg, U. Wedig, H. Stoll, H. Preuss, J.Chem.Phys. 86, 866, (1986). [ Links ]
27. I.S. Lim, P. Schwerdtfeger, B. Metz, J. Stoll, J.Chem.Phys. 122, 104103, (2005). [ Links ]
28. T.H. Dunning, P. Hay, In Modern Theoretical Chemistry, H.H. Schaefer, Ed, New York, 1976, Vol. 3, p.1. [ Links ]
29. M.J Frisch, J.A. Pople, J.S. Binkley, J. Chem. Phys. 80, 3265, (1984). [ Links ]
30. J.E. Carpenter, F. Weinhold, J. Mol. Struct. (THEOCHEM). 169, 41, (1988). [ Links ]
31. F. Weinhold, J.E. Carpenter, J.E. The Structure of Small Molecules and Ions, Plenum: New Yok, 1988. [ Links ]
32. L. Padilla-Campos, E. Chavez, J. Molec. Struct. (Theochem). 958, 92, (2010). [ Links ]
33. M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, V. G. Zakrzewski, J. A. Montgomery, Jr., R. E. Stratmann, J. C. Burant, S. Dapprich, J. M. Millam, A. D. Daniels, K. N. Kudin, M. C. Strain, O. Farkas, J. Tomasi, V. Barone, M. Cossi, R. Cammi, B. Mennucci, C. Pomelli, C. Adamo, S. Clifford, J. Ochterski, G. A. Petersson, P. Y. Ayala, Q. Cui, K. Morokuma, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. Cioslowski, J. V. Ortiz, A. G. Baboul, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. Gomperts, R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, M. Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M. W. Wong, J. L. Andres, C. Gonzalez, M. Head-Gordon, E. S. Replogle, J. A. Pople, Gaussian 98, Revision A.9, Gaussian, Pittsburgh, PA, 1998. [ Links ]
34. Y. Xu, M. Mavrikakis, Surf.Sci. 494, 13, (2001). [ Links ]
35. T. Sueyoshi, T. Sasaki, Y. Iwasawa Surf. Sci. 365, 310, (1996). [ Links ]
36. R.G. Parr, W. Yang. Density Functional Theory of Atoms and Molecules, Oxford Press, UK, 1989. [ Links ]