Electronic Structure of Transition Metal Clusters from Density

J. Phys. Chem. 1996, 100, 565-572

565

Electronic Structure of Transition Metal Clusters from Density Functional Theory. 1. Transition Metal Dimers Masafumi Harada*,†,‡ and Herve´ Dexpert Laboratoire pour l’Utilisation du Rayonnement Electromagne´ tique (CNRS, CEA, MESR), Baˆ timent 209d, UniVersite´ de Paris-Sud, 91405 Orsay Cedex, France ReceiVed: March 30, 1995; In Final Form: October 4, 1995X

The electronic structure of transition metal clusters is of great interest for a wide range of investigations including the properties of catalytic systems. In order to determine the electronic structure of transition metal dimers, calculations of total energies and geometry optimizations have been carried out on Pd2, Rh2, Ru2, Au2, and Pt2 dimers as well as PdPt, RhPt, RuPt, and PdAu dimers using a method based on density functional theory (DFT) with nonlocal (NLDFT) correction. The typical applications of DFT for the determination of the general features of electronic structure of Pd4, Rh4, Au4, and Pt4 clusters have also been performed at the level of NLDFT using the norm-conserving pseudopotential method (NCPP). The results of these calculations are in good qualitative agreement with those of the previous DFT calculations. The Mulliken charge analysis has indicated that a charge transfer occurs between the atoms of Pd and Pt, as well as between Rh and Pt atoms, Ru and Pt atoms, and Pd and Au atoms in the corresponding bimetallic dimers. Finally, this DFT method is expected to be very helpful to predict the electronic structure of small metal cluster systems.

Introduction Nanometer scale metal particles or metal clusters are of great interest to many scientists interested in designing new materials at the molecular level and in assessing various strategies for producing the materials required for these microscopic structures. In a variety of industrially important reaction processes, small metal particles or metal clusters are widely used as effective catalysts. In particular, catalytic properties, such as activity, selectivity, and stability of metal clusters, have been investigated for a long time in order to increase the advanced understanding of the catalytic reaction by metal clusters. Determining the electronic structure of metal clusters is also very important for researchers who recognize that the surface structure of metal clusters highly influences their catalytic properties since electronic and conformational properties of metallic components have been attributed to their surface electronic structures. The concept of “magic” numbers is well-known to be very significant for the study of metal cluster chemistry. The magic numbers N ) 13, 55, 147, ... for cuboctahedral and icosahedral structures have been intensively studied utilizing various methods.1 For example, there is a controversy on the structure of metal clusters with a high nuclearity. The two main structures are often described in the literature: one is a vertex-sharing icosahedral structure, which builds upon each unit resulting in forming highly symmetric supraclusters;2 the other is a cubic close packed (ccp) or hexagonal close packed (hcp) stacking of polyhedra with the outer geometry of a cuboctahedral structure.3 On the other hand, bimetallic clusters4,5 have been extensively investigated in order to improve their preparative procedures as well as to control the activity, selectivity, and stability in catalysis. Especially, from the practical and theoretical points6 of view, the additive effect of the second metallic component †

Postdoctoral fellow in L.U.R.E.-C.N.R.S. Present address: Catalysis Research Center, Hokkaido University, Sapporo 060, Japan. * To whom correspondence should be addressed. X Abstract published in AdVance ACS Abstracts, December 1, 1995. ‡

0022-3654/96/20100-0565$12.00/0

in bimetallic systems is playing an important role in catalytic properties, surface segregations, and shape stabilities of the clusters. In recent years the advantages of utilizing EXAFS (extended X-ray absorption fine structure) as well as XANES (X-ray absorption near edge structure) have been very well recognized for structural analysis of small metal clusters.7,8 Detailed information on electronic properties can be obtained from the region of XANES. For example, Sinfelt et al.9 made extensive investigations to characterize the structural and electronic properties of a series of bimetallic clusters. Moreover, the size of the individual monocrystalline domains, which are conglomerated into large particles protected by polymers, has been successfully proved by both EXAFS and HRTEM (highresolution transmission electron microscopy) measurements.10 The need for HRTEM observations comes from the fact that the interactions between each monocrystalline domain give little influence to the EXAFS signal of the first coordination shell, HRTEM providing direct figures of the conglomerates. The density functional theory (DFT) is a first principle quantum chemical method based on a theorem that the ground state energy of an electronic system can be expressed as a unique function of the electron density. A conventional technique for solving the minimization of this energy originates from the Kohn-Sham formalism,11 which leads to a one-electron Schro¨dinger equation with a density-dependent effective local potential. The most common formula for the total energy expression is constituted from three terms: a kinetic energy contribution, the electrostatic Coulomb interaction energy between all the electrons and the nuclei, and the exchangecorrelation energy of the system. In order to treat the exchangecorrelation energy term, various approximations have been carried out. This approach is called the local density approximation (LDA). Here it should be noted that, at this level, the LDA provides good results for bond distances and vibrational frequencies. However, the values for binding energies are generally obtained with some errors. One technique to improve the quality of the calculations is to include nonlocal corrections12 which are more sophisticated forms for the © 1996 American Chemical Society

566 J. Phys. Chem., Vol. 100, No. 2, 1996 expression of exchange-correlation energy term in using gradients of the electronic density. Moreover, the local spin density (LSD) approximation is considered as an extension of the LDA method, taking into account of the spin-polarized case where different densities are defined for electrons with up and down spins. LSD is also refined by the nonlocal corrections, with which nonlocal LSD (NLSD) can be introduced under the form of density gradient expansions. In recent years, further developments of DFT-based methods have been made by including pseudopotentials and relativistic effects,13,14 as well as analytic energy gradients for geometry optimization. The various computational approaches based on DFT method are a very fascinating alternative to conventional ab initio methods, especially for the study of large metal clusters since the computational effort increases with the number of basis functions as approximately N3, instead of N4 in the case of the HF method. They allow an accurate and efficient treatment of solving the electronic structure problem for relatively large systems of transition metal cluster where the standard HF method is not easily applicable. However, there have been relatively few studies focusing on the electronic properties of transition metal clusters15-19 or bimetallic dimers20,21 by means of calculations with density functional theory. Quantum chemical calculations, in general, allow many researchers to obtain detailed geometries and electronic properties of mono- and/or bimetallic clusters as well as to determine the structure of small conglomerates (or aggregates). The determination of electronic structure using quantum chemical calculations can also provide us with the prediction of the magnetic properties of small metal clusters and of the segregation site of small bimetallic clusters. Zerner et al.19 have recently reported the results of structural and magnetic characteristics of 13-atom transition metal clusters, i.e., Rh13,19a Pd13,19b and Ni1319c clusters, by means of intermediate neglect of differential overlap (INDO) calculations. Goursot et al.6b have also studied the electronic properties and relative stabilities of small bimetallic RuGe and RuSn aggregates both at a fixed bulk geometry and after geometry optimization by means of the LCGTO-MCPDF (linear combination of Gaussian type orbitals-model core potential-density functional) method. They performed the DFT calculations of Ru8Ge and Ru8Sn clusters as starting models and concluded that a central Sn atom costs a large amount of energy to distort the cluster, resulting in weakening all bonds in each layer and leading to a large preference for Sn atom to segregate at corner sites. On the contrary, the relaxation which occurs in the presence of a Ge atom stabilizes the particles at both central and corner sites, leading to a small preference for the site location of Ge atom. In the present paper we have studied the electronic and geometrical structures of transition metal dimers, such as Pd2, Rh2, Ru2, Au2, and Pt2, as well as PdPt, RhPt, RuPt, and PdAu dimers, using the Dgauss program based on the density functional theory with nonlocal corrections in order to apply, in our future investigations, DFT calculations for the monometallic and/or bimetallic clusters which are composed of atoms with large magic numbers. In addition, the DFT calculations of Pd4, Rh4, Au4, and Pt4 clusters have been also carried out, mainly for the evaluation of the Dgauss program. Finally, we have tried to predict the charge transfer between atoms of Pd and Pt, as well as Rh and Pt, Ru and Pt, and Pd and Au atoms in the corresponding bimetallic dimers, by means of Mulliken charge analysis. This objective is to consider the relationship between the catalytic activities and the partial charges localized on surface atoms of Pd/Pt, Rh/Pt, Ru/Pt, and Pd/Au bimetallic cluster systems.

Harada and Dexpert TABLE 1: Total Energies (au) for Pd2, Rh2, and Ru2 Dimers dimer

basisa

spin

energy (au)

Pd

Gaussian Gaussian Gaussian (∞) ECP ECP ECP (∞) Gaussian Gaussian Gaussian Gaussian (∞) ECP ECP (∞) Gaussian Gaussian Gaussian Gaussian (∞) ECP ECP (∞)

1 3 1b,e 1 3 1b,e 1 3 5 4c,e 5 4c,e 1 5 7 5d,e 7 5d,e

-9880.405 774 -9880.405 450 -9880.384 989 -58.225 633 -58.241 994 -58.203 387 -9376.160 847 -9376.219 977 -9376.228 036 -9376.151 389 -43.222 744 -43.173 950 -8886.404 897 -8886.430 264 -8886.441 253 -8886.347 138 -31.733 503 -31.690 270

Rh

Ru

a Gaussian ) all-electron Gaussian basis set; ECP ) relativistic effective core potential. b Pd atom has a closed-shell 1S0 ground state arising from 4d10 electronic configuration. c Rh atom has a open-shell 4F 8 1 d 9/2 ground state arising from 4d 5s electronic configuration. Ru atom has a open-shell 5F5 ground state arising from 4d75s1 electronic configuration. e See: Moore, C. E. Atomic Energy LeVels As DeriVed From the Analyses of Optical Spectra; Natl. Bur. Stand. (U.S.) Circular 467; U.S. GPO: Washington, DC, 1958; Vol. III.

Computational Methods The density functional theory22-25 calculations were carried out by the use of Dgauss program developed by Andzelm and Wimmer.26 The all-electron Gaussian-type basis set27,28 was used at the double-zeta split-valence plus polarization (DZVP) level: (633321/53211/531) for Pd, Rh, and Ru atoms, with the auxiliary basis set of (10/5/5). A norm-conserving pseudopotential (NCPP),29 which was generated by the method of Troullier and Martins,30 was also used for all the transition metals we considered including Au and Pt atoms. The valence basis set for Pd, Rh, Ru, Au, and Pt atoms was chosen as (4,3/ 4/3,2) with a fitting basis set of (5/5/5). The calculations were, at the first stage, implemented at the self-consistent local spin density (LSD) functional level with the local potential of Vosko, Wilk, and Nusair (VWN).31 Subsequently, the calculations were performed at the self-consistent gradient-corrected nonlocal spin density (NLSD) with the nonlocal exchange potential of Becke32-34 together with the nonlocal correlation functional of Perdew (BP).35 Geometries were optimized by using analytical gradients,26,36 and no symmetrical restrictions were used during the optimizations of a series of dimers. Vibrational analysis was done for the energy-optimized structures. Second derivatives were calculated by numerical differentiation of the analytic first derivatives. A two-point method with a finite difference of 0.01 au was used. Additionally, the standard Mulliken population analysis was carried out to characterize the interaction between pairs of atoms. The calculations were performed either using a CRAY C-94 (or C-98) in single processor mode or a Silicon Graphics Indigo2 IRIX 4.0.5H workstation. Results and Discussion Pd2, Rh2, and Ru2 Dimers. Pd2, Rh2, and Ru2 dimers are good systems to evaluate the results obtained by means of the Dgauss program as far as the quality of the basis set or the accuracy of the initial parametrization is concerned. The total energies of the optimized Pd2, Rh2, and Ru2 structures, calculated at the nonlocal level at various spin multiplicities, are shown in Table 1. In the case of the Pd2 dimer, the detailed electronic and geometrical structures have been previously calculated by

Electronic Structure of Transition Metal Clusters

J. Phys. Chem., Vol. 100, No. 2, 1996 567

TABLE 2: Calculated Properties of Pd2, Rh2, and Ru2 Dimers dimer

basisa

spin

re (Å)

ωe (cm-1)

De (kcal/mol)

Pd

Gaussian Gaussian ECP ECP Gaussian Gaussian Gaussian ECP Gaussian Gaussian Gaussian ECP

1 3 1 3 1 3 5 5 1 5 7 7

2.805 2.547 2.766 2.531 2.348 2.524 2.564 2.589 2.111 2.184 2.325 2.528

122 185 137 195 245 186 191 204 406 348 274 203

13.04 12.83 13.95 24.22 5.93 43.02 48.08 30.61 36.23 52.14 59.04 27.12

Rh

Ru

a Gaussian ) all-electron Gaussian basis set; ECP ) relativistic effective core potential.

the Dgauss program at the local and nonlocal levels.37 Thus, the reproducibility has been first tested in our present calculations, and our results for the optimized Pd2 dimer shown in Table 1 have similar tendencies to the previously obtained values.37 By means of the same procedure as used for Pd2 dimer, the total energies of the optimized Rh2 and Ru2 dimers have also been calculated at various spin multiplicities. A more detailed description of the comparison between DFT results with and without nonlocal corrections, such as total energies, optimized geometries, and vibrational frequencies of the monometallic Pd2, Rh2, and Ru2 dimers, is given elsewhere.38 In the present calculations, we have paid attention to the choice of the numerical grid of points, which is essential to yield the accuracy of numerical integration for exchange-correlation contributions, as reported by Goursot et al.39 Table 2 shows the calculated molecular properties, i.e., interatomic bond distances (re), vibrational frequencies (ωe), and binding energies (De) of optimized Pd2, Rh2, and Ru2 dimers, at a variety of spin multiplicities. In the case of the Pd2 dimer, the bond distance of the singlet is longer than that of the triplet, since the singlet is generally formed from the interaction between two 4d10 (1S0) ground state of Pd atoms. On the other hand, there is a much smaller dependence on the use of relativistic ECPs. The ECPs tend to shorten by 0.04 and 0.01 Å the bond distances for the singlet and the triplet, respectively. The vibrational frequencies indicate the same dependence on the spin multiplicities as the bond distances do. For the binding energies (De) obtained using the all-electron Gaussian basis set, the singlet is slightly more stable (0.2 kcal/mol) than the triplet. When a relativistic ECP is used, the triplet is much more stable than the singlet. At the nonlocal level, the triplet is 10.2 kcal/ mol more stable than the singlet, and the dissociation energy (binding energy) of the triplet is 24.2 kcal/mol, a result which agrees with the experimental one40 (16.1 kcal/mol) within 25% error bar. Dixon et al.37 finally concluded that the ground state is predicted to be a triplet formed from a σ*(4dz2)1σ(5s)1 occupancy with a bond distance of 2.54 Å and a vibrational frequency of 185 cm-1. This conclusion has been reobtained by our present calculations, although our corresponding values are slightly different, 2.53 Å and 195 cm-1, respectively, since the basis and auxiliary basis sets for Pd atom in our calculations might be different from those they used in their previous calculations. Zerner et al.19b also obtained a triplet (3Σu+) ground state for Pd2 dimer with a configuration (σg2πu4δg4δu4πg4σu1σg1) and an interatomic bond distance of 2.46 Å, using INDO calculations. Our results for the Pd2 dimer are in agreement with INDO as well as CAS-MCSCF/MRSDCI/RCI [(complete active space) multiconfiguration self-consistent field/multireference singles + doubles configuration interaction/relativistic configuration interaction] calculations.41

For the Rh2 dimer, the ground state was considered as 5Σg+ from Shim’s configuration interaction (CI) calculations.42 The calculated properties are re ) 2.86 Å, ωe ) 118 cm-1, and De ) 0.85 eV (19.6 kcal/mol), the details being described in the literature.22,43 Balasubramanian et al.44 also reported that the ground state is 5∆g with a bond length (re) of 2.26 Å and a vibrational frequency (ωe) of 305 cm-1 by using CAS-MCSCF/ MRSDCI/RCI methods. More recently, the local and nonlocal calculations for the two lowest states of Rh2 dimer, 5Σu (ground state) and 5∆g, have been performed using the deMon program within LCGTO-MCP-DF formalism.39 Based on these backgrounds, the spin multiplicity for Rh2 dimer has been predominantly chosen as 5 in our nonlocal calculations together with ECPs, as shown in Tables 1 and 2. With the all-electron Gaussian basis set, the total energy of spin multiplicity 5 is obviously the smallest one, comparing with those of the singlet and the triplet. This is consistent with the results obtained from another quantum calculations.39,42,44 The calculated properties, using ECPs with nonlocal corrections, are re ) 2.589 Å, ωe ) 204 cm-1, and De ) 30.6 kcal/mol at the ground-state (spin multiplicity ) 5). Contrary to the Pd2 dimer, the bond distance of the singlet is the smallest one among a series of multiplicities. The use of ECPs gives little effects for bond distance in the case of multiplicity 5, as it tends only to lengthen the bond distance by 0.02 Å. For the binding energies, the inclusion of ECPs apparently makes the binding energy reduced to 30.6 kcal/mol. Here we should note that there are Rh(4d85s1) and Rh(4d9) mixing of configurations in the case of the Rh2 dimer, so that the assignment of highly open-shell electronic states is more difficult in contrast to Pd2 dimer. Chen et al.29 previously published the results of bond distance and vibrational frequency of Rh2 dimer by means of the Dgauss program. The bond distances and frequencies at the ground state (spin multiplicity ) 5) are re ) 2.268 Å and ωe ) 301 cm-1 in the case of using all-electron Gaussian basis set and are re ) 2.291 Å and ωe ) 331 cm-1 in the case of using relativistic ECPs, respectively. They had calculated these bond distances as well as frequencies only at the local level with both all-electron Gaussian basis set and relativistic ECPs.38 Taking into account these considerations, our calculated results for bond distances and frequencies are in good agreement with their results. In the case of the Ru2 dimer, Andzelm et al.45 have combined the local spin density method with a model potential for the inner-shell electrons. With the use of the model potential method, the Ru2 dimer was found to possess a 7Σ ground state, with re ) 2.42 Å, ωe ) 330 cm-1, and De ) 3.0 eV (69.2 kcal/ mol). These values were compared with the values, which are re ) 2.41 Å, ωe ) 380 cm-1, and De ) 2.7 eV (62.3 kcal/mol), obtained by means of all-electron LCGTO-LSD(VWN) calculations using a compact basis set, resulting in a 7Σ ground state of Ru2 dimer. On the contrary, Balasubramanian et al.46 obtained the molecular properties of re ) 2.36 Å, ωe ) 273 cm-1, and De ) 2.0 eV (46.1 kcal/mol) at the 7∆u ground state of the Ru2 dimer by using MRSDCI treatments. Furthermore, Goursot et al.6b recently reported that the calculated ground state was found to be 7∆u, corresponding to the configuration (1σg21πu41δg32σg21δu21πg21σu), with re ) 2.22 Å and ωe ) 380 cm-1. Therefore, we mainly intend to choose the spin multiplicity for Ru2 dimer as 7 at nonlocal calculations in the case of using relativistic ECPs, as shown in Tables 1 and 2. It is clearly demonstrated that the total energy for the spin multiplicity 7 is the smallest using the all-electron Gaussian basis set. The bond distance of the singlet is the shortest one, as it was the case for the Rh2 dimer. In particular, the calculated bond distance depends very

568 J. Phys. Chem., Vol. 100, No. 2, 1996

Harada and Dexpert

TABLE 3: Total Energies (au) for Au2 and Pt2 Dimers dimer

basisa

spin

energy (au)

Au

ECP ECP ECP (∞) ECP ECP ECP ECP (∞)

1 3 2b,d 1 3 5 3c,d

-66.734 150 -66.672 980 -66.666 566 -52.667 634 -52.687 917 -52.673 497 -52.610 920

Pt

a ECP ) relativistic effective core potential. b Au atom has a openshell 2S1/2 ground state arising from 5d96s2 electronic configuration. c Pt atom has a open-shell 3D3 ground state arising from 5d96s1 electronic configuration. d See: Moore, C. E. Atomic Energy LeVels As DeriVed From the Analyses of Optical Spectra; Natl. Bur. Stand. (U.S.) Circular 467; U.S. GPO: Washington, DC, 1958; Vol. III.

TABLE 4: Calculated Properties of Au2 and Pt2 Dimers dimer

basisa

spin

re (Å)

ωe (cm-1)

De (kcal/mol)

Au

ECP ECP ECP ECP ECP

1 3 1 3 5

2.630 2.952 2.406 2.579 2.653

154 477 297 225 212

42.39 4.02 35.58 48.30 39.25

Pt

a

ECP ) relativistic effective core potential.

much on the use of ECPs, which tends to lengthen by 0.2 Å the bond distance. The ground state properties, using relativistic ECPs, are re ) 2.528 Å, ωe ) 203 cm-1, and De ) 27.12 kcal/ mol. Related to our calculations, Chen et al.29 also previously reported the results of bond distances and vibrational frequencies of Ru2 dimer using the Dgauss program. Their bond distances and frequencies at the ground state (spin multiplicity 7) are re ) 2.231 Å and ωe ) 333 cm-1 in the case of using the allelectron Gaussian basis set and are re ) 2.288 Å and ωe ) 335 cm-1 in the case of using ECPs, respectively. Based on our calculations, it is known that they had calculated these bond distances and frequencies only under the condition of local levels both with the all-electron Gaussian basis set and with relativistic ECPs.38 Au2 and Pt2 Dimers. Au2 and Pt2 dimers have often been chosen to develop the accurate relativistic effective core potentials. The relativistic effective core potential MCSCF/CI calculation was applied to the ground (1Σg+) and excited electronic states of Au2.47,48 Lee et al. reported that, within the SCF level, the bond of Au2 dimer contracts by 0.3 Å and increases in vibrational frequency by about 50% when relativistic effects are included. The relativistic effects,49 therefore, are very important for the appropriate description of electronic properties of Au2 dimer. In the case of the Pt2 dimer, Basch et al.50 reported that the calculated properties are re ) 2.58 Å, ωe ) 267 cm-1, and De ) 0.93 eV (21.4 kcal/mol) for the 1Γg(δδ) state, using relativistic effective core potential in low-level MCSCF calculations. Balasubramanian49,51 has also performed CAS-MCSCF/FOCI/RCI calculations and concluded that the ground state of Pt2 dimer is 3Σg- with re ) 2.456 Å and ωe ) 189 cm-1. The total energies and calculated molecular properties (interatomic bond distances, vibrational frequencies, and binding energies) of Au2 and Pt2 dimers at various spin multiplicities in our calculations are shown in Tables 3 and 4, respectively. For the Au2 dimer, the comparison of the total energy for the singlet with that for the triplet, using ECPs with nonlocal corrections, indicates that the singlet is much more stable than the triplet. Similarly, as for the total energies of the Pt2 dimer at various spin multiplicities, the triplet is the most stable among the other spin multiplicities, as shown in Table 3. Therefore, the spin multiplicities of ground state of Au2 and Pt2 dimers can be estimated, from the comparison of total energies, as a

singlet and a triplet, respectively. A more detailed description of the comparison between DFT results with and without nonlocal corrections for the monometallic Au2 and Pt2 dimers is given elsewhere.38 Concerning the calculated interatomic bond distances and vibrational frequencies of Au2 dimer, the bond distance of the singlet is 0.32 Å shorter than that of the triplet. The nonlocal frequency of the singlet is much lower than that of the triplet. The molecular properties at the ground state of singlet for the Au2 dimer are re ) 2.630 Å, ωe ) 154 cm-1, and De ) 42.39 kcal/mol, which are in agreement with experimental results (re ) 2.472 Å and ωe ) 191 cm-1).40 On the other hand, in the case of the Pt2 dimer, the bond distance of the triplet is 0.17 Å longer than that of the singlet. The molecular properties at the ground state of triplet are re ) 2.579 Å, ωe ) 225 cm-1, and De ) 48.30 kcal/mol. Chen et al.29 previously also published the results of bond distance and vibrational frequency of Pt2 dimer using the Dgauss program. The bond distance and frequency at the ground state (triplet) in their calculations is re ) 2.404 Å and ωe ) 305 cm-1, respectively, in the case of using ECPs. The bond distance (2.405 Å) and vibrational frequency (307 cm-1) of the triplet in our local calculation, results which are described elsewhere,38 are in excellent agreement with their obtained values. PdPt, RhPt, RuPt, and PdAu Dimers. PdPt, RhPt, RuPt, and PdAu dimers are one of the smallest systems in order to consider the general features of electronic structure of PdPt,5a RhPt,10b RuPt,10c and PdAu5b bimetallic clusters with and without supporting materials. The electronic states and the nature of the chemical bonding of PtAu dimer as well as Pt3Au clusters have recently been elucidated by means of CAS-MCSCF/MRSDCI/RCI calculations.52 This investigation successfully gives us a lot of useful results of several low-lying electronic states for the mixed clusters such as PtAu and Pt3Au with the corresponding electronic configurations, because this kind of information is very hard to obtain from experiments. Here it should be noted that the Dgauss program allows us to calculate only the lowest electronic configuration for a given multiplicity so that the electronic configuration of an excited state cannot be calculated with the Dgauss program. The total energies and calculated molecular properties of PdPt, RhPt, RuPt, and PdAu dimers at various spin multiplicities are shown in Tables 5 and 6, respectively. In the case of the PdPt dimer, the triplet is the most stable state at the nonlocal level; hence, the ground state of the PdPt dimer is a triplet. The bond distance of the triplet is 0.12 Å longer than that of the singlet. The frequency of the triplet is 31 cm-1 lower than that of the singlet. The ground state of the PdPt dimer is, therefore, estimated to be a triplet with the calculated properties of re ) 2.524 Å, ωe ) 221 cm-1, and De ) 46.4 kcal/mol. For the RhPt dimer, the spin multiplicity 4 is a slightly more stable state than the other spin multiplicities at the nonlocal level, as shown in Table 5. The bond distance of the spin multiplicity 4 is 0.02 Å longer than that of the doublet. The nonlocal frequency of the spin multiplicity 4 is only 7 cm-1 higher than that of the doublet. The ground state of the RhPt dimer has a spin multiplicity 4 with the calculated properties of re ) 2.576 Å, ωe ) 225 cm-1, and De ) 43.6 kcal/mol. Similarly, for the RuPt dimer, the spin multiplicity 5 is slightly more stable than the other spin multiplicities at the local level, as shown in Table 5. The bond distance of the spin multiplicity 5 is 0.087 Å shorter than that of the triplet and only 0.007 Å shorter than that of the spin multiplicity 7. The local frequency of the spin multiplicity 5 is 92 cm-1 higher than that of the

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TABLE 5: Total Energies (au) for PdPt, RhPt, RuPt, and PdAu Dimers dimer

basisa

spin

energy (au)

PdPt

ECP ECP ECP ECP (∞) ECP ECP ECP ECP (∞) ECP ECP ECP ECP ECP (∞) ECP ECP ECP ECP (∞)

1 3 5 Pd(1),c,h Pt(3)d,h 2 4 6 Rh(4),e,h Pt(3)d,h 1 3 5 7 Ru(5),f,h Pt(3)d,h 2 4 6 Pd(1),c,h Au(2)g,h

-55.451 450 -55.481 131 -55.417 036 -55.407 154 -47.940 328 -47.961 991 -47.937 220 -47.892 435 -41.933 907 -41.962 663 -41.998 223 -41.988 012 -41.863 840 -62.494 729 -62.428 221 -62.232 670 -62.434 977

RhPt

RuPtb

PdAu

a ECP ) relativistic effective core potential. b Local level calculation was carried out in the case of RuPt dimer. c Pd atom has a closed-shell 1 S0 ground state arising from 4d10 electronic configuration. d Pt atom has a open-shell 3D3 ground state arising from 5d96s1 electronic configuration. e Rh atom has a open-shell 4F9/2 ground state arising from 4d85s1 electronic configuration. f Ru atom has a open-shell 5F5 ground state arising from 4d75s1 electronic configuration. g Au atom has a openshell 2S1/2 ground state arising from 5d96s2 electronic configuration. h See: Moore, C. E. Atomic Energy LeVels As DeriVed From the Analyses of Optical Spectra; Natl. Bur. Stand. (U.S.) Circular 467; U.S. GPO: Washington, DC, 1958; Vol. III.

TABLE 6: Calculated Properties of PdPt, RhPt, RuPt, and PdAu Dimers dimer

basisa

spin

re (Å)

ωe (cm-1)

De (kcal/mol)

PdPt

ECP ECP ECP ECP ECP ECP ECP ECP ECP

1 3 2 4 3 5 7 2 4

2.400 2.524 2.558 2.576 2.436 2.349 2.356 2.864 2.935

252 221 218 225 228 320 328 232 314

27.79 46.40 30.04 43.63 61.99 84.30 77.89 37.48 -4.24

RhPt RuPtb PdAu

a ECP ) relativistic effective core potential. b Local level calculation was carried out in the case of RuPt dimer.

triplet. The ground state of the RuPt dimer has a spin multiplicity 5 with the calculated properties of re ) 2.349 Å, ωe ) 320 cm-1, and De ) 84.3 kcal/mol. For the PdAu dimer, the doublet is the most stable state at the nonlocal level, as shown in Table 5. The bond distance of the doublet is 0.07 Å shorter than that of the spin multiplicity 4, and the frequency of the doublet is lower than that of the spin multiplicity 4. The ground state of the PdAu dimer is, therefore, a doublet with the calculated properties of re ) 2.864 Å, ωe ) 232 cm-1, and De ) 37.5 kcal/mol. This binding energy of the PdAu dimer in our calculation agrees well with the experimental result (32.3 kcal/mol).40 Moreover, a comparison of the binding (dissociation) energies53 for the bimetallic dimers with those for the monometallic dimers has been often carried out in recent years to establish the general trends which underlie the bonding. From comparing the De values for the PdPt and RhPt dimers with those for the homonuclear dimers such as Pd2, Rh2, and Pt2, the De value (46.40 kcal/mol) of the PdPt dimer as well as that (43.63 kcal/ mol) of the RhPt dimer tends to be more close to that (48.30 kcal/mol) of the Pt2 dimer, although De values for Pd2 and Rh2 dimer are 24.22 and 30.61 kcal/mol, respectively. The differences of De value between Pd-Pt and Pt-Pt as well as between Rh-Pt and Pt-Pt are 1.90 and 4.67 kcal/mol, respectively. In

TABLE 7: Mulliken Net Charges (Electrons) and Bond Orders Obtained for the Ground State of PdPt, RhPt, RuPt, and PdAu Dimers after Geometry Optimizations at Nonlocal Level dimer

metal

charge

net charge

bond orders

PdPt

Pd Pt Rh Pt Ru Pt Pd Au

10.036 9.964 8.941 10.059 7.963 10.037 9.824 11.176

-0.036 0.036 0.059 -0.059 0.037 -0.037 0.176 -0.176

0.71

RhPt RuPta PdAu a

0.64 0.84 0.60

Local level calculation was carried out in the case of RuPt dimer.

the case of the PdAu dimer, the De value (37.48 kcal/mol) of the PdAu dimer also becomes close to that (42.39 kcal/mol) of the Au2 dimer, resulting in the difference of De value equal to 4.91 kcal/mol. Mulliken charge analyses can provide some qualitative insight into the nature of metal atoms in molecular electronic states although they cannot be used in an absolute way, due to their basis set dependence. Also, it is well-known that the d orbitals polarize to one way and the s orbital polarizes in the opposite way in bimetallic dimers, so the net Mulliken population is not the best measure of the bonding. However, the comparison of the overlap populations and the trends of electronic states in a series of bimetallic dimers is very meaningful in our study. The values of the Mulliken charge analysis and Mulliken bond orders for the ground state of PdPt, RhPt, RuPt, and PdAu dimers at the nonlocal level are shown in Table 7. Note that the bond orders,54 closely related to Mulliken overlap populations, are very useful indices to link between the details of density matrices obtained from quantum chemical calculations and the qualitative chemical characters in a desired molecule. In the case of the PdPt dimer, Pd and Pt atoms have a net charge (electrons) of -0.036 and 0.036, respectively, with a bond order equal to 0.71. On the other hand, for the RhPt dimer, Rh and Pt atoms have a net charge of 0.059 and -0.059, respectively, and the bond order is 0.64. For the RuPt dimer, Ru and Pt atom have a net charge of 0.037 and -0.037, respectively, with its bond order of 0.84 at the local level. For the PdAu dimer, the Pd and Au atom have a net charge of 0.176 and -0.176, respectively, and its bond order is 0.60. These results suggest that the Pd atom of the PdAu dimer, the Rh atom of the RhPt dimer, and the Ru atom of the RuPt dimer are expected to be more electronically populated than the Pd atom of the PdPt dimer. These values indicate that there will be a possibility of the charge transfers between different metal elements in the PdPt, RhPt, RuPt, and PdAu bimetallic cluster systems, for example, in the case of the polymer-protected PdPt bimetallic cluster consisting of 55 atoms.5a The distribution of electrons on the surface of metal cluster highly influences the reactivity of hydrogenation of olefins and dienes.5a Thus, in general, the charge populations might be one of the important factors to improve the reactivities of monometallic as well as bimetallic transition metal cluster systems. In the case of the PdPt dimer, the Mulliken populations of the Pd atom (5s0.385p0.124d9.55) and of the Pt atom (6s0.626p0.035d9.31) are calculated at the ground state of triplet. Comparing these results with the Mulliken population of Pd atom (5s0.2615p0.0564d9.683) of the Pd2 dimer as well as that of Pt atom (6s0.4146p0.0545d9.533) of the Pt2 dimer, the contributions of Pd(5s) and Pt(6s) orbitals to Pd-Pt bond substantially increase from 0.261 to 0.38 and from 0.414 to 0.62, respectively, in the PdPt dimer. On the contrary, the contributions of Pd(4d) and Pt(5d) orbitals to the Pd-Pt bond decrease from 9.683 to 9.55 and from 9.533

570 J. Phys. Chem., Vol. 100, No. 2, 1996 to 9.31, respectively. The populations of Pd(5p) and Pt(6p), however, do not change so much, comparing them in the case of PdPt, Pd2, and Pt2 dimers. For the Pd-Pt bond, the Pd-Pt bond in PdPt dimer is more strongly attributed to the interaction of Pd(5s) and Pt(6s) orbitals, when comparing with the interaction of Pd(5s) orbitals in the Pd-Pd bond as well as that of Pt(6s) orbitals in the Pt-Pt bond. The nature of Pd(5s) and Pt(6s) orbitals for bond of Pd-Pt is in accord with the result that the bond order (0.71) of the Pd-Pt bond in the PdPt dimer is larger than that (0.59) of the Pd-Pd bond in the Pd2 dimer as well as that (0.64) of the Pt-Pt bond in the Pt2 dimer. This result also corresponds well to the fact that the Pd-Pt bond distance (2.524 Å) of the PdPt dimer is shorter than the Pd-Pd bond distance (2.531 Å) of the Pd2 dimer as well as the Pt-Pt bond distance (2.579 Å) of the Pt2 dimer. Similarly, in the case of the RhPt dimer, the Mulliken populations of Rh atom (5s0.285p0.094d8.57) and of Pt atom (6s0.476p0.035d9.56) are also calculated at the ground state of spin multiplicity 4. Comparing these results of RhPt dimer with the Mulliken population of Rh atom (5s0.345p0.054d8.62) of the Rh2 dimer as well as that of the Pt atom (6s0.4146p0.0545d9.533) of the Pt2 dimer, the contributions of Rh(5s) and Rh(4d) orbitals to the Rh-Pt bond slightly decrease. On the contrary, the contributions of Pt(6s) and Pt(5d) orbitals to its bond slightly increase. The contribution of the Pt(6p) orbital to the Rh-Pt bond, however, remains nearly constant, comparing the Rh-Pt bond of the RhPt dimer with the Pt-Pt bond of the Pt2 dimer. In contrast to the Pd2 dimer, it is well-known that the electronic states in the Rh2 dimer have a mixture of electronic configurations. To resolve the extent of s and d orbitals in the Rh-Pt bond is, therefore, more difficult than in the Pd-Pt bond. The bond order (0.64) of the Rh-Pt bond in the RhPt dimer is nearly the same value as that (0.65) of the Rh-Rh bond in the Rh2 dimer as well as that (0.64) of the Pt-Pt bond in the Pt2 dimer. This trend corresponds to the fact that the Rh-Pt bond distance (2.576 Å) of the RhPt dimer is nearly the same value as the Rh-Rh bond distance (2.589 Å) of the Rh2 dimer as well as the Pt-Pt bond distance (2.579 Å) of the Pt2 dimer. In the case of the RuPt dimer, the Mulliken populations of Ru atom (5s0.5655p0.0904d7.308) and of Pt atom (6s0.3836p0.005d9.654) are also obtained with the local level at the ground state of spin multiplicity 5. The comparison of these populations with the population of the Ru atom (5s0.5315p0.0584d7.412) of the Ru2 dimer as well as that of the Pt atom (6s0.3916p0.0365d9.573) of the Pt2 dimer at the local level can provide us with the fact that the contribution of the Ru(4d) orbital to the Ru-Pt bond apparently decreases and the contribution of the Pt(5d) orbital to its bond increases. The trend of both decreasing contribution of Ru(4d) and increasing contribution of Pt(5d) orbitals in the Ru-Pt bond is more remarkably observed than that in the case of Rh(4d) and Pt(5d) in the Rh-Pt bond, not in the Pd-Pt bond. In the case of the Pd-Pt bond, the decreasing contribution of Pt(5d) as well as the decreasing contribution of Pd(4d) orbitals is observed. The trend of both increasing contribution of Ru(5s) and Pt(6s) orbitals is not obtained, whereas the case of the PdPt bond has the trend of increasing Pd(5s) and Pt(6s) orbital contributions as mentioned above. The electronic states in the Ru2 dimer are composed of highly open-shell states of Ru atom and have a mixture of electronic configurations. Thus, in our calculations, the extent of s and d orbitals in the Ru-Pt bond is also very difficult to solve. The bond order (0.84) of the Ru-Pt bond in the RuPt dimer is much lower than that (1.11) of the Ru-Ru bond in the Ru2 dimer, but the former is higher than that (0.76) of the Pt-Pt bond in the Pt2 dimer. This trend has indicated that, at the local level, the Ru-Pt bond distance (2.349 Å) of the RuPt dimer is consistent with the Ru-Ru bond

Harada and Dexpert distance (2.288 Å) of the Ru2 dimer as well as the Pt-Pt bond distance (2.405 Å) of the Pt2 dimer. Moreover, in the case of the PdAu dimer, the Mulliken populations of Pd atom (5s0.295p0.064d9.48) and of Au atom (6s1.186p0.025d9.97) are obtained at the ground state of the doublet, too. Comparing these results of the PdAu dimer with the Mulliken population of the Pd atom (5s0.2615p0.0564d9.683) of the Pd2 dimer as well as that of the Au atom (6s0.9516p0.0355d10.013) of the Au2 dimer, the contribution of Pd(5s) and Au(6s) orbitals to the PdAu bond substantially increases in the PdAu dimer. On the contrary, the contribution of Pd(4d) and Au(5d) orbitals to the Pd-Au bond decreases. This tendency of s and d orbitals in the Pd-Au bond is very similar to that in the Pd-Pt bond, if the Au atom in the Pd-Au bond is replaced by a Pt atom. The bond order (0.60) of the Pd-Au bond in the PdAu dimer is the same as that (0.59) of the Pd-Pd bond in the Pd2 dimer as well as that (0.60) of the Au-Au bond in the Au2 dimer. This seems to be inconsistent with the fact that the Pd-Au bond distance (2.864 Å) of the PdAu dimer is longer than both the Pd-Pd bond distance (2.531 Å) of the Pd2 dimer and the AuAu bond distance (2.630 Å) of the Au2 dimer. Comparing with the Pd-Pt bond distance (2.524 Å) in the PdPt dimer, this trend for the longer Pd-Au bond distance in the PdAu dimer is a unique character due to the electronic nature of the Au atom. Pd4, Rh4, Au4, and Pt4 Clusters. Pd4, Rh4, Au4, and Pt4 clusters, chosen to apply density functional theory (DFT) with relativistic ECPs at nonlocal level in order to investigate the electronic structure of small metal clusters, are fundamental units for modeling the cuboctahedral fcc cluster structures. The geometry optimizations of Pd4, Rh4, Au4, and Pt4 clusters have been carried out starting from an exact Td symmetry without fixing the Pd-Pd, Rh-Rh, Au-Au, and Pt-Pt bond distance to the bulk Pd bond length (2.73 Å), to the bulk Rh bond length (2.67 Å), to the bulk Au bond length (2.85 Å), and to the bulk Pt bond length (2.76 Å), respectively. This is because, for example, the Pd-Pd bond distance (2.76 Å) of a polymerprotected monometallic Pd cluster, which consists of 13-55 Pd atoms (average coordination number ) 6.2 obtained from EXAFS measurements5a,10a), is slightly longer than that (2.73 Å) of the bulk Pd foil. Similarly, in the case of the Pt-Pt bond, the Pt-Pt bond distance (2.75 Å) of the polymer-protected monometallic Pt cluster, which consists of 55 Pt atoms (average coordination number ) 8.0 obtained also from EXAFS measurements5a,10b), is slightly different from that of the bulk Pt foil. Concerning the spin multiplicities of small transition metal clusters with Td symmetry, several higher spin multiplicities in addition to the singlet state should be taken into account in some cases. Especially, the cases of small Cu, Ni, and Co clusters are typical examples because they obviously have magnetic properties. Furthermore, in the case of Pd, Rh, and Ru clusters, they have magnetic moments which appear to occur in the region around 13 atoms. From the experimental and theoretical investigations, Pd4 clusters are paramagnetic19b and Rhn (n ) 12-32) clusters have a giant internal magnetic moment55 which is associated with the reduced coordination and high symmetry. In the present DFT calculations, we have calculated the Pd4, Rh4, Au4, and Pt4 clusters, assuming that these clusters have singlet multiplicities because nonmagnetic Pd, Rh, Au, and Pt solids exist. For this concern, the geometry optimizations of these clusters at the higher spin multiplicities should be carried out in our future investigations. Table 8 shows total valence energies, atomization energies, Mulliken population analyses, HOMO energies, and bond distances of Pd4 and Rh4 clusters at the nonlocal levels using relativistic ECPs. In the case of the Pd4 cluster, the relative

Electronic Structure of Transition Metal Clusters TABLE 8: Total Valence Energies (au), Atomization Energies (kcal/mol), Mulliken Populations (Electrons), HOMO Energies (eV), and Bond Distances (Å) Calculated for Pd4 and Rh4 Clusters with Singlet Multiplicities at the Nonlocal Levels Using Relativistic ECPs

J. Phys. Chem., Vol. 100, No. 2, 1996 571 TABLE 9: Total Valence Energies (au), Atomization Energies (kcal/mol), Mulliken Populations (Electrons), HOMO Energies (eV), and Bond Distances (Å) Calculated for Au4 and Pt4 Clusters with Singlet Multiplicities at the Nonlocal Levels Using Relativistic ECPs

cluster

properties

values

cluster

properties

values

Pd4

total energy atomization energy MP HOMO energy bond distances Pd1-Pd2 Pd1-Pd3 Pd1-Pd4 Pd2-Pd3 Pd2-Pd4 Pd3-Pd4 total energy atomization energy MP HOMO energy bond distance

-116.584678 111.6 5s0.2705p0.1324d9.598 -4.63

Au4

total energy atomization energy MP HOMO energy bond distances Au1-Au2 Au1-Au3 Au1-Au4 Au2-Au3 Au2-Au4 Au3-Au4 total energy atomization energy MP HOMO energy bond distances Pt1-Pt2 Pt1-Pt3 Pt1-Pt4 Pt2-Pt3 Pt2-Pt4 Pt3-Pt4

-133.473500 88.1 6s0.9246p0.1155d9.961 -5.66

Rh4

2.81 2.59 2.59 2.59 2.59 2.81 -86.476039 80.4 5s0.0915p0.1234d8.786 -5.10 2.54

values of valence energy and the atomization energy are -116.6 au and 111.6 kcal/mol, respectively. The Mulliken population analysis indicates that the Pd4 structure has 9.6 electrons in the d orbitals and 0.4 electron in the (s + p) orbitals. The HOMO energy of the Pd4 structure is -4.63 eV. Furthermore, the two kinds of Pd-Pd bond distances are obtained as 2.81 and 2.59 Å (average ) 2.66 Å). This optimized geometry of Pd4 cluster has D2d symmetry. The distortion from an exact Td symmetry is observed owing to the Jahn-Teller first-order effects.19b,c,56 Therefore, the highly symmetric structure of the Pd4 cluster with Td symmetry is not very stable. On the other hand, in the case of the Rh4 cluster, the relative values of valence energy and the atomization energy are -86.5 au and 80.4 kcal/mol, respectively. The Mulliken population analysis demonstrates that the Rh4 structure has 8.78 electrons in the d orbitals as well as 0.22 electron in the (s + p) orbitals. The HOMO energy of the Rh4 structure is -5.10 eV, and the Rh-Rh bond distance is obtained as 2.54 Å. This bond distance is, obviously, shorter than both that (2.67 Å) of the bulk Rh foil and that (2.66 Å) of the polymer-protected monometallic Rh cluster consisting of 13 Rh atoms (average coordination number ) 5.7 obtained from EXAFS measurements10b). This optimized geometry of the Rh4 cluster still has an exact Td symmetry, in the contrast to the optimized Pd4 cluster. Table 9 shows total valence energies, atomization energies, Mulliken population analyses, HOMO energies, and bond distances of Au4 and Pt4 clusters at the nonlocal level using relativistic ECPs. For the total valence energies, the relative values of energies for Au4 and Pt4 clusters are -133.5 and -105.5 au, respectively. The atomization energies for Au4 and Pt4 clusters are also 88.1 and 144.2 kcal/mol, respectively. The Mulliken population analysis reveals that the Au4 structure has 1.04 electrons in the (6s + 6p) orbitals while the Pt4 structure has 0.45 electron in the (6s + 6p) orbitals. The HOMO energies of Au4 and Pt4 structures are -5.66 and -4.47 eV, respectively. In addition, two kinds of Au-Au and Pt-Pt bond distances have been obtained, i.e., 2.62 and 3.27 Å (average Au-Au bond distance ) 3.05 Å) in the case of the Au4 structure and 2.76 and 2.64 Å (average Pt-Pt bond distance ) 2.68 Å) in the case of the Pt4 structure, respectively. The optimized geometries of Au4 and Pt4 clusters have D2d symmetries the same in the case of the optimized Pd4 cluster. In this case, the distortion from an exact Td symmetry is also due to the Jahn-Teller first-order effects. The calculated structure of Au4 cluster has bond lengths of 3.27 and 2.62 Å, as shown in Table 9. These values might be

Pt4

2.62 3.27 3.27 3.27 3.27 2.62 -105.451789 144.2 6s0.3156p0.1335d9.553 -4.47 2.76 2.64 2.64 2.64 2.64 2.76

compared with the 2.63 Å which we obtained for the Au2 dimer, assuming a singlet state with the use of relativistic ECPs. The existence of a longer Au-Au bond distance in the Au4 cluster might suggest that the electronic properties, especially bond distances of naked metal clusters, are strongly attributed to their cluster size, that is, the number of atoms which compose the naked metal cluster. As for the atomization energies of the Pd4, Rh4, Au4, and Pt4 clusters, the ordering for these values appears as Pt4 (144.2 kcal/ mol) > Pd4 (111.6 kcal/mol) > Au4 (88.1 kcal/mol) > Rh4 (80.4 kcal/mol). This ordering of the atomization energies indicates that the optimized geometry with D2d symmetry (Pt4, Pd4, and Au4) tends to be more stable than that with exact Td symmetry (Rh4) in the present calculations. The Pt4 cluster apparently has the most stable structure, comparing with the other clusters we considered here. Balasubramanian et al.57 reported the results for geometries and energy separations of the low-lying electronic states of Au4 cluster by using CAS-MCSCF/MRSDCI/RCI calculations. The tetrahedral geometry of 1A1 symmetry has an Au-Au bond length of 2.86 Å. This bond length is an intermediate value between our obtained values, i.e., 2.62 and 3.27 Å, using the Dgauss program, as shown in Table 9. On the basis of the electronic distribution for the tetrahedral symmetry structure (Td), the distortions should not be predicted for the Pd4, Au4, and Pt4 clusters from the geometrical point of view. In fact, in the case of the Rh4 clusters, the distortion of the Rh-Rh bond distance has not been observed at the nonlocal level, as shown in Table 8. Therefore, the improvements of the valence basis set and the fitting basis set for the Pd, Au, and Pt atoms in the ECP methods might be necessary for DFT calculations of large transition metal clusters when using Dgauss. Conclusion The total energies and molecular properties of Pd2, Rh2, Ru2, Au2, Pt2, PdPt, RhPt, RuPt, and PdAu dimers as well as Pd4, Rh4, Au4, and Pt4 clusters have been calculated using the Dgauss program based on density functional theory (DFT) at the nonlocal level. The Mulliken charge analysis shows that partial charge transfer occurs much more between Pd and Au atoms and between Rh and Pt atoms than it does between Pd and Pt

572 J. Phys. Chem., Vol. 100, No. 2, 1996 atoms and between Ru and Pt atoms in each of the corresponding bimetallic dimers. The application of DFT is of great interest and importance to small metal clusters possessing magic numbers of atoms in order to predict the electronic structures of their surface since the optimized geometries and the electronic properties of the bimetallic clusters are strongly related both to the size of metal clusters and to the nature of the second metal element. This prediction of surface electronic structures can also offer a comprehensive comparison of molecular properties (bond distance, charge population, and so on) obtained by DFT calculations with those obtained by EXAFS spectroscopy. These DFT applications to large monometallic and/or bimetallic clusters will be a promising work to better understand cluster science as well as quantum chemistry. Acknowledgment. We are grateful to I.D.R.I.S. (Orsay, France) for providing a grant of computer time of the CRAY C-94 (or C-98). We gratefully acknowledge the assistance of Dr. Jean-Marie Teuler at I.D.R.I.S. for the connection to CRAY computers and of Cray Research, Inc., for the useful support. M.H. is also thankful to L.U.R.E.-C.N.R.S. for a postdoctoral fellowship as well as Japan Society for the Promotion of Science for the JSPS Fellowship for Japanese Junior Scientists. References and Notes (1) (a) Schmid, G. Chem. ReV. 1992, 92, 1709. (b) Vogel, W.; Rosner, B.; Tesche, B. J. Phys. Chem. 1993, 97, 11611. (2) (a) Teo, B. K. Polyhedron 1988, 7, 2317. (b) Teo, B. K.; Zhang, H.; Shi, X. J. Am. Chem. Soc. 1990, 112, 8552. (c) Fackler, Jr., J. P.; McNeal, C. J.; Winpenny, R. E. P.; Pignolet, L. H. J. Am. Chem. Soc. 1989, 111, 6434. (3) (a) Schmid, G.; Klein, N. Angew. Chem., Int. Ed. Engl. 1986, 25, 922. (b) Schmid, G. Polyhedron 1988, 7, 2321. (c) Feld, H.; Leute, A.; Rading, D.; Benninghoven, A.; Schmid, G. J. Am. Chem. Soc. 1990, 112, 8166. (d) Schmid, G. Clusters and Colloids: From Theory to Applications; VCH: Weinheim, 1994. (4) Coq, B.; Kumbhar, P. S.; Moreau, C.; Moreau, P.; Figueras, F. J. Phys. Chem. 1994, 98, 10180. (5) (a) Harada, M.; Asakura, K.; Ueki, Y.; Toshima, N. J. Phys. Chem. 1992, 96, 9730. (b) Harada, M.; Asakura, K.; Toshima, N. J. Phys. Chem. 1993, 97, 5103. (c) Harada, M.; Asakura, K.; Toshima, N. Jpn. J. Appl. Phys. 1993, 32 (Suppl. 32-2), 451. (6) (a) Coq, B.; Goursot, A.; Tazi, T.; Figue´ras, F.; Salahub, D. R. J. Am. Chem. Soc. 1991, 113, 1485. (b) Goursot, A.; Pedocchi, L.; Coq, B. J. Phys. Chem. 1994, 98, 8747. (7) (a) Meitzner, G.; Via, G. H.; Lytle, F. W.; Fung, S. C.; Sinfelt, J. H. J. Chem. Phys. 1988, 92, 2925. (b) Meitzner, G.; Via, G. H.; Lytle, F. W.; Sinfelt, J. H. J. Phys. Chem. 1992, 96, 4960. (8) Caballero, A.; Dexpert, H.; Didillon, B.; LePeltier, F.; Clause, O.; Lynch, J. J. Phys. Chem. 1993, 97, 11283. (9) (a) Sinfelt, J. H. Acc. Chem. Res. 1987, 20, 134. (b) Sinfelt, J. H.; Meitzner, G. Acc. Chem. Res. 1993, 26, 1. (10) (a) Harada, M.; Asakura, K.; Ueki, Y.; Toshima, N. J. Phys. Chem. 1993, 97, 10742. (b) Harada, M.; Asakura, K.; Toshima, N. J. Phys. Chem. 1994, 98, 2653. (c) Harada, M.; Toshima, N. Unpublished results. (11) Kohn, W.; Sham, L. J. Phys. ReV. A 1965, 140, 1133. (12) Becke, A. D. J. Chem. Phys. 1986, 84, 4524. (13) Pitzer, K. S. Acc. Chem. Res. 1979, 12, 271. (14) Pyykko¨, P.; Desclaux, J. P. Acc. Chem. Res. 1979, 12, 276. (15) (a) Radzio, E.; Andzelm, J.; Salahub, D. R. J. Comput. Chem. 1985, 6, 533. (b) Raatz, F.; Salahub, D. R. Surf. Sci. 1986, 176, 219. (16) Go¨rling, A.; Ro¨sch, N.; Ellis, D. E.; Schmidbaur, H. Inorg. Chem. 1991, 30, 3986. (17) Baba, M. F.; Mijoule, C.; Godbout, N.; Salahub, D. R. Surf. Sci. 1994, 316, 349. (18) Pacchioni, G.; Chung, S. C.; Kru¨ger, S.; Ro¨sch, N. Chem. Phys. 1994, 184, 125.

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