16874
J. Phys. Chem. 1996, 100, 16874-16880
Structural, Electronic, and Magnetic Properties of Small Ni Clusters G. L. Estiu† and M. C. Zerner*,‡ Cequinor, Departamento de Quı´mica, Fac. Cs. Exactas, UNLP-Casilla de Correo 962, 1900 La Plata y Departamento de Ciencia y Tecnologı´a, UniVersidad Nacional de Quilmes, Roque Saenz Pen˜ a 180, 1876 Bernal, Argentina, and Quantum Theory Project, UniVersity of Florida, GainesVille, Florida 32611 ReceiVed: March 19, 1996; In Final Form: July 28, 1996X
The electronic structure and magnetic properties of small Nin clusters (n ) 4, 5, 6, 8, 13) are studied at the self-consistent-field multireference configuration interaction singles level using the intermediate neglect of differential overlap model Hamiltonian parametrized for spectroscopy. Results are in agreement with density functional theory calculations and ab initio calculations for the smaller (n ) 4-6) structures. Discrepancies develop for the larger (n ) 8, 13) structures, and these discrepancies are discussed. We examine systems and criteria for the selection of the cluster size that most accurately model heterogeneous catalysts.
1. Introduction Small transition metal clusters of various sizes have been extensively studied in recent years,1-11 mainly in relation to their magnetic properties, which are, at the same time, strongly determined by their geometry and electronic characteristics.12 Such clusters are of special interest for their particular role in catalysis. In this field, the present development of the application of quantum chemistry to catalysis research1,2,12-16 needs to answer the question of how accurately the properties and reactivity of a heterogeneous catalyst are described by the small metal particles that more closely resemble the structure of supported catalysts (despite their size, smaller than the metal crystallites) but are often the choice to model the solid side of a heterogeneous catalytic interface. There is a certain appeal in this approach, since the active metal surfaces in industrial catalysts are far from the idealized well-defined crystal surfaces that are used in the slab model, but better described as a conglomerate of small particles. Small metal particles (“real clusters”) are being used, thence, to model both supported and extended structures with catalytic activity. Real clusters allow, on the other hand, the determination of the ground state of the intermediates for each stage of a catalytic process, free from assumptions related, for example, to the spin state, assumptions that cannot be avoided when dealing with the slab model. The magnetic behavior of V, Cr, Ga, Co, Fe, Ni, and Rh clusters has been carefully measured.1,2,7,9,17-22 It has been demonstrated recently that, for the small clusters, not only paramagnetic but also diamagnetic materials can have a nonvanishing magnetic moment, which is, in both cases, larger than that in the bulk.6-11,18-22 A better understanding of magnetism, especially in small particles, is a challenging subject. Moreover, nanometric (nm) metal clusters are now synthesized and their magnetic properties measured in the laboratory, allowing an accurate comparison of the experimental and theoretical data. On the experimental side the structural characteristics are indirectly determined by chemical saturation with noninvasive gases,23 whereas the magnetic moment is measured following the deflection of the cluster under the effect of a magnetic field.10 Theoretical calculations have to deal with the treatment of †
Universidad Nacional de Quilmes. University of Florida. X Abstract published in AdVance ACS Abstracts, September 15, 1996. ‡
S0022-3654(96)00828-3 CCC: $12.00
valence d electrons and have followed different lines. While effective medium theories disregard the spin-structure,24 postHartree-Fock treatments of the magnetic properties of transition metal clusters imply difficult calculations.4,5 Most of the studies reported up to date are based on the application of density functional methodologies,1,3,25,26 which, in the local spin density approximation, allow one to calculate the electronic spin as the difference between the occupation values of the R and β spin eigenvalues. Only for small, highly symmetric clusters have configuration interaction (CI) calculations been performed.27-29 Despite their cost, they are capable of giving a detailed description of the electronic structure associated with each spin state. There are sizable problems when considering such calculations. Foremost among these is the strong energy bias Hartree-Fock (molecular orbital) models have for high spin. Overcoming this bias requires a good deal of correlation in the configuration interaction refinement. The intermediate neglect of differential overlap (INDO) model we examine here is parametrized on spectroscopy at the CI singles (CIS) level and has a high reliance on atomic spectroscopy. For this reason, some of the bias toward high spin might be expected to have been removed through the parametrization. This certainly seems to be the case in the very many organometallic compounds studied using this method. Nevertheless, one of the purposes of this work is to determine how reliable this model is when applied to metal clusters such as these. We focus this study on a self-consistent-field (SCF) multireference configuration interaction (MRCI) treatment of small Ni clusters, which opens an examination of how far we can rely on our present knowledge of the electronic and magnetic characteristics of the Ni structures for a cluster size (13 atoms) that is very small experimentally, but large from the standpoint of the size of the calculations. In this article, we present a study of the electronic and magnetic properties of small Nin clusters (n ) 4-6, 8, 13) for different geometries and lattice parameters. We examine structures that correspond to the optimized, most stable structures and those associated with the observed interatomic bulk distance. The calculated multiplicities are compared with those obtained from density functional (DF) calculations,3 ab initio calculations, and experiment, and the comparisons discussed, when possible. © 1996 American Chemical Society
Properties of Small Ni Clusters
J. Phys. Chem., Vol. 100, No. 42, 1996 16875
2. Computational Details We apply the intermediate neglect of differential overlap (INDO) model30,31 at the SCF/CI level to study the structure, electronic characteristics, and magnetic properties of small Nin (n ) 4-6, 8, 13) clusters. For each cluster size, different geometries and lattice parameters have been compared. The lattice parameters involved are either those associated with the bulk or those that result from a geometry optimization to minimum energy. Other values also have been included, in some cases, for a comparison with the results published by other authors. The calculated geometries are the result of a full optimization (interatomic distances and angles) without any constraint in their variation. Optimization is based on a minimization of the gradient, evaluated analytically, using the BFGS algorithm to update the Hessian matrix in successive geometry cycles.32,33 These calculations have been made at the RHF and ROHF levels. Special care has been taken not to destroy symmetry during the SCF cycles, an effect that can occur through a nonequivalent occupation of degenerate orbitals and which may result in spurious Jahn-Teller distortions.34 To this end, calculations are started by a configuration average HartreeFock procedure,34 with an average multiplicity (M) for the number of electrons considered. The orbitals of the CAHF calculation form the reference for a Rumer CI35 that allows us to determine the magnetic properties (M) of the different structures. The details of the MRCI calculations, regarding the number of references and symmetry imposed to the calculations, depend on the size of the geometry of the cluster under study and are discussed in the next section. For the eight-atom cluster the multiplicity (M) has also been calculated, when possible, at the UHF level, after annihilation of the next component in order to partially correct spin contamination. Two parametrizations of the INDO theory have been used in these studies: one for geometry, which utilizes two-center twoelectron integrals that are calculated ab initio (INDO/1), and one to calculate the electronic descriptors and compare different M at fixed geometries, which obtains these integrals empirically from atomic spectroscopy at the CI singles (CIS) level.31,37-39 The resonant integrals β are chosen according to formulas that take into account different electronegativities38 and reproduce the available experimental geometries and spectroscopy for the transition metal dimers. As this study involves only Ni atoms, they can easily be reproduced by setting the resonance integral βs ) βp ) -1.0 eV (the ZINDO default) for both the INDO/1 and INDO/S versions, whereas we use βd ) -90.0 eV to obtain geometry and βd ) -32.0 eV (the ZINDO default) for spectroscopy. This is a different procedure than we used in ref 12, where we used βd ) -90.0 eV for both geometry and spectroscopy. The results are very similar except where noted for Ni13. 3. Results and Discussion 3.1. Ni4 Clusters. Tetrahedral (Td) and square-planar (D4h) structures have been studied for both the interatomic bulk distance (2.49 Å) and the optimized geometries. Results in Table 1 show that, in both cases, the D4h structures are more stable than the Td, whereas high spin multiplicity (M) characterizes both symmetries. When the cluster mimics a piece of the bulk, the D4h symmetry most closely resembles a (100) surface, whereas the Td symmetry is a better model of a (111) plane. In the first case the lowest energy configuration belongs to a 9A1g state, which is 0.26 and 0.63 eV more stable than the 7A1g and the
Figure 1. Frontier orbitals of Ni4 clusters: (a) D4h symmetry, bulk interatomic distance; (b) D4h symmetry, geometry-optimized interatomic distance; (c) Td symmetry, bulk interatomic distance; (d) Td symmetry, geometry-optimized interatomic distance. 7E , respectively. States of multiplicity 5 and 3 (5B , 3E , u 2g u respectively) are almost degenerate with the 7A1g. The molecular orbitals (MOs) used for this symmetry result from CAHF calculations, which start by averaging 40 e- in 23 orbitals. Then these orbitals are used for an SCF, averaging 2 e- in two openshell orbitals. Other procedures lead to symmetry breaking. A different set of MOs can be obtained by means of restrictedHartree-Fock (RHF) calculations, but they lead to states of higher energy. The lowest energy nonet is described by the electronic configuration (Figure 1A) b1g2eu4a2u2eg4b1u2a1g2b 2a 2e 4e 4b 1a 2a 2e 2b 1a 1e 2a 1, where all the s orbitals 1g 2u g u 2g 1u 1g u 2g 1g u 2g are at least singly occupied. A Mulliken population analysis yields s0.99p0.09d8.90 on each equivalent Ni atom. Calculations have been done under D2h symmetry. The CI space includes 10 occupied and 5 virtual orbitals from the closed-shell configuration. Eleven references have been defined for M ) 1, 3, 5 and 17 references for M ) 7, 9, leading to calculations that involve 2000, 3200, 2180, 2800, and 160 configurations, respectively. In the case of Td symmetry, a 7T2 represents our most stable structure, which is 0.54, 0.76, and 0.89 eV more stable than the 5A2, 9T1, and 3T2, respectively (Table 1). There are nine electronic states within 1.0 eV of the lowest energy one. Eigenvectors obtained by means of RHF calculations lead, in this case, to the most stable configuration, associated with a a12t26e4t26e4t16t16t23a11t22a10 electronic distribution (Figure 1B). All the s orbitals but one in the HOMO are occupied in the most stable septet, leading to an orbital population of s0.85p0.12d9.03. Calculations have been done under C2V symmetry, with 1, 3, and 12 references for M ) 1, 3 and 5-9, respectively. The
16876 J. Phys. Chem., Vol. 100, No. 42, 1996
Estiu and Zerner
TABLE 1: Total Valence Energies (Hartrees) Calculated for Ni4 Structures from MRCIS Calculationsa M 1 3 5 7 9 11
energy state energy state energy state energy state energy state energy state
TABLE 2: Total Valence Energies (Hartrees) Calculated for Ni5 Structures from MRCIS Calculationsa
D4h 2.49 Å
D4h 2.30 Å
Td 2.49 Å
Td 2.30 Å
M
-162.570 1A 1g -162.578 3 Eu -162.576 5B 2g -162.590 7A 1g -162.601 9A 1g -162.414 11 A1g
-162.813 1A 1g -162.802 3A 1g -162.806 5B 2g -162.842 7A 1g -162.839 9A 1g -162.681 11A 1g
-162.485 1A 1 -162.499 3T 2 -162.510 5A 2 -162.532 7T 2 -162.505 9T 1 -162.335 11T 1
-162.538 1A 1 -162.548 3T 1 -162.533 5A 2 -162.516 7A 2 -162.551 9T 1 -162.374 11T 1
1 3 5 7 9 11
energy state energy state energy state energy state energy state energy state
C4V 2.49 Å
C4V 2.30 Å
-203.397 1A 1 -203.399 3 B2 -203.367 5A 1 -203.382 7B 1 -203.398 9E -203.343 11 E
-203.393 1B 2 -203.392 3E -203.387 5E -203.421 7A 1 -203.406 9B 1 -203.257 11 E
D3h 2.49 Å -203.247 1A′ 1
-203.257 A′′2 -203.216 5A′ 2 -203.231 7A′ 1 -203.277 9A′′ 1 -203.187 11 A′1 3
a Electronic states are also indicated. Different multplicities (calculated by ZINDO/S-MRCI) are compared for the bulk (2.49 Å) and fully optimized geometries (INDO/1) associated with D4h and Td symmetries.
a Electronic states are also indicated. Different multiplicities (calculated by means of ZINDO/S MRCI) are compared for the bulk (2.49 Å) and fully optimized geometries (INDO/1) associated with C4V and D3h symmetries.
number of configurations varies from 80 (M ) 1) to 600 (M ) 3, 7), 1200 (M ) 9), and 1372 (M ) 5), when the space of the CI involves 10 orbitals down and 9 up from the HOMO of the closed-shell structure. Optimization of the geometry has a similar structural effect on both the square-planar and the tetrahedral Ni4 clusters, decreasing the interatomic distances to 2.298 and 2.310 Å, respectively (Table 1). The multiplicity, on the other hand, changes in the opposite direction, decreasing for the first one and increasing for the latter. States of multiplicity 9 (9A1g) and 7 (7A1g) are almost degenerate for the D4h optimized structure, with the first only 0.089 eV higher in energy. A 1A1g is 0.72 eV higher in energy, whereas states of M ) 5 and 3 lie 0.98 and 1.018 eV above the septet (Table 1). There are eight electronic states within 1.0 eV of the most stable septet. The decrease of M after optimization of the geometry is the consequence of a spin pairing to give a double occupied bonding s type orbital, an effect that is favored by the stronger overlap for the smaller interatomic distances. In this way, all the s orbitals but one are occupied in the optimized structure (Figure 1C). There is no change in the orbital population, as the number of electrons in the s type orbitals remains the same. States of lower energy are achieved, in this case, after RHF calculations, by means of a CI that includes five references for M ) 1, 3, 5, and 6 for M ) 7, 9, and gives, in D2h symmetry, 270, 1200, 870, 460, and 160 configurations, respectively. The CI space involves 10 orbitals down and 5 up from the HOMO in the closed-shell structure. In the optimized Td structure, all the s orbitals are singly occupied, leading to a 9T1 state with a configuration a12t26e4t26e4t16t24t14a11t23a10 and a Mulliken population s1.04p0.19d8.76. The next state is a triplet 3T1, which is only 0.089 eV less stable. The quintet (5A2) and septet (7A2) are separated by 0.51 and 0.968 eV from the lowest nonet. There are 40 electronic states within 1.0 eV from the lowest energy one. Calculations have been also done under C2V symmetry. The same number of reference states defined for the bulk-like structure lead, in this case, to 60, 400, 1320, 480, 1000, and 60 configurations for M ) 1, 3, 5, 7, 9, and 11, respectively. On the basis of the electronic distribution of the optimized square-planar and tetrahedral structures (Figure 1), no distortions from the D4h symmetry are predicted for the 7A1g low-energy state. In the Td structure, on the other hand, the nonequivalent occupation of the 3-fold degenerate t orbitals would lead to Jahn-Teller distortions that lower the symmetry and break the degeneracy. Elongation along a 3-fold axis lowers the Td symmetry to C3V. Optimization of a slightly distorted tetrahedral
leads to an elongated structure, with three bond lengths of 2.49 Å and three of 2.52 Å, which is 0.18 hartree (1 hartree ) 627.5 kcal/mol) more stable than the 9T2 one, but still 0.11 hartree less stable than the 7A1g in D4h symmetry. A MRCI analysis of the C3V structure shows that it is not stable against distortion, as a pair of degenerate orbitals is not equally occupied in the ground state 7E. Reduction of the Td symmetry to C2V, through simultaneous elongation-compression of the tetrahedral in perpendicular directions, gives a structure that is only 0.05 hartrees less stable than the optimized D4h, characterized also by a high spin multiplicity (M ) 7), and stable against distortion. A comparative analysis shows that the distortion decreases the energy difference between the most stable structures in both symmetries. Similar results have been found by Reuse and Khanna3 in recent density functional calculations, where squareplanar and Td distorted to 2-fold symmetry compete for the definition of the ground state in Ni4 clusters. 3.2. Ni5 Clusters. Two different symmetries that resemble more closely the (111) and (100) close-packed structures (triangular bipyramid and square pyramid, respectively) have been analyzed for Ni5. As for the Ni4 case, we have compared the description associated with the interatomic bulk distances and those that result from geometry optimizations, taking into account, for the latter, symmetry-breaking distortions. For the interatomic distances frozen to the bulk value, the C4V structure is 2.9 eV more stable than the D3h one (Table 2). Singlet (1A1), triplet (3B2), and nonet (9E) electronic states are almost degenerate in the first case, with the triplet and the nonet within 0.01 eV. States of M ) 5 and 11 lie 0.80 and 2.1 eV above the ground state. There is a large number of low lying energy states of different M with very similar energies (32 within 1.0 eV), a fact that complicates the calculations, making the results dependent on the number of configurations and on the orbital space involved in the CI calculation. We have defined 6, 6, 8, 11, 8, and 11 references for M ) 1-11, respectively. When the CI space was generated with 16 orbitals down and 6 up from the HOMO of a closed-shell configuration, the calculation included 600, 840, 1300, 2100, 2200, and 1050 configurations for increasing M. While in the nonet all the s orbitals but one are occupied to give an electronic configuration a12b22e4b22e4a12b22b12e4e4a22b12a22e3b11a12b22b12e2a11e2a11a10 and a Mulliken population s0.95p0.14d8.78, the population of the d orbital increases in the triplet (s0.32p0.14d9.54) due to the coupling of the unpaired electrons in the e and b1 d orbitals (Figure 2A). Similar calculations show that the nonet is the most stable in the bipyramidal structure, separated, in this case, from the triplet,
Properties of Small Ni Clusters
J. Phys. Chem., Vol. 100, No. 42, 1996 16877
Figure 3. Frontier orbitals of Ni6 clusters of Oh symmetry.
Figure 2. Frontier orbitals of Ni5 clusters: (a) C4V symmetry, bulk interatomic distance; (b) C4V symmetry, geometry-optimized interatomic distance; (c) D3h symmetry, bulk interatomic distance.
by 0.54 eV, and characterized by an occupation of 1 electron in each s orbital but the highest, defining an electronic configuration a1′2e′4a2′′2e′′4e′4e′′4e′4a1′2e′′4a1′2a1′′1e′4e′′4e′2a2′2a2′′1a1′1a2′′1e′2a1′0 (Figure 2B) and a Mulliken population s0.95p0.16d8.84. Singlet and septet states are 0.81 and 1.22 eV higher in energy. There are 23 electronic states within 1.0 eV from the ground state. We have defined, this time, 1, 5, 1, 7, 7, and 5 starting references that lead, for the same orbital space, to 85, 920, 290, 2300, 1492, and 104 configurations involved in the calculations for M ) 1-11, respectively. Optimization of the geometry decreases the interatomic distance in the C4V structure and stabilizes the septet 7A1 (Table 2), which turns out to be 0.015 and 0.029 hartrees more stable than the nonet and the triplet. There are eight electronic states of different M within 1.0 eV from the most stable one. For the same orbital space in the CI, the definition of 1, 1, 6, 5, 3, and 3 references for M ) 1-11, respectively, generates 88, 220, 2250, 1400, 800, and 200 configurations in the calculations. According to the electronic distribution, characterized by filled or half-filled degenerate orbitals, no Jahn-Teller distortion is predicted for the optimized structure. On the other hand, the geometry of the D3h structure cannot be optimized without distortion, the latter implying an axial compression of the bipyramidal structure of 0.1 Å keeping the initial symmetry. A 7A ′′ is also the most stable state, followed by near degenerate 2 5A ′ and 3A ′ states, which are 0.41 eV higher in energy. 1 1 The results can be compared with those from Reuse and Khanna,3 who also found a compressed triangular bipyramidal structure. The multiplicity calculated by these authors, M ) 9,
is, however, higher than ours, M ) 7. Recent experimental results suggest M ) 9 for Ni5.22 We find our results, however, very sensitive to geometry, and a slight increase in bond length would also lead us to predict a nonet. A measure of this sensitivity is demonstrated in Table 2. 3.3. Ni6 Clusters. We have centered our study of Ni6 clusters on structures of Oh symmetry, which, for the interatomic bulk distances, model a piece of a (100) structure. Calculations have been done under D2h symmetry, with a CI space that includes 14 orbitals down and 6 orbitals up from the HOMO of the closed-shell configuration. Defining 1, 3, 9, 2, 8, 6, and 7 reference states for M ) 1-13, respectively, the maximum number of configurations generated is 5160 for M ) 7. The most stable state (11Eg) belongs to a t2u6eg4t2g6eg2t1u3a1g2a2g1t1u3eg1 electronic distribution in the outer orbitals (those involved in the CI). It is only 0.30, 0.55, and 0.85 eV below 9A2g, 13A2g, and 7A2g, respectively. The results for this cluster size can be compared with the ab initio CASSCF calculations by Gropen and Almlo¨ff4 (G&A), which, for the same structure that they used to model bulk Ni, found an M lower than ours (M ) 7). According to these authors and in agreement, also, with the results of our calculations (Figure 3), the electronic structure of Ni6 can be viewed in terms of two subsystems, a 3d and a 4s part. The 4s orbitals transform, under Oh symmetry, as a1g, t1u, and eg, and it is the distribution of the six electrons in the set of s orbitals that leads to different M values between our results and G&A. Whereas G&A4 found that all the low lying states are characterized by an occupation (a1g)2(eg)0(t1u)4 in the s part, coupled to a 3T1g substrate, we have found that a 5E substrate ((a1g)2(eg)1(t1u)3) defines the s part of the lowest energy eleventet. Coupling of this substrate to the 7A1u, 5Eu ((eg)2(t1u)3(a2g)1) states in the d space results in the 11Eg, 9A2g states of lowest energy. According also to the ab initio calculations,4 the promotion of an electron to the eg orbital, antibonding in character, increases the total energy 1.5 to 3.0 eV. We calculate the configuration that partially occupies the e orbital as most stable; the promotion of a second electron to the eg orbital, to give states of M ) 13, 11, decreases the stability by 0.5 and 1.0 eV, respectively. On the other hand, we predicted the 7T2u and 9T2u states, which are calculated as more stable by the ab initio CASSCF calculations, 1.7 and 2.7 higher in energy than the most stable 11E. It is
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TABLE 3: Total Valence Energies (Hartrees) Calculated for Ni6 Structures from MRCIS Calculationsa M 1 3 5 7 9 11 13
Oh 2.49 Å energy state energy state energy state energy state energy state energy state energy state
-244.054 1A 2g -244.108 3 T2u -244.122 5A 2g -244.189 7A 2g -244.213 9A 2g -244.224 11 Eg -244.204 13A 2g
TABLE 4: Total Valence Energies (Hartrees) Calculated for Ni8 Structures from RMCIS and UHF Calculationsa
Oh 2.36 Å -244.191 1A
2g
-244.239
3T 2u
-244.245
5A
2g
-244.242 7A 2g -244.284 9A 2g -244.295 11E g -244.252 13E g
a
Electronic states are also indicated. Different multiplicities (ZINDO/S MRCI) are compared for the bulk (2.49 Å) and fully optimized geometries (INDO/1) associated with Oh symmetries.
noteworthy that, for a given electronic configuration, G&A calculated the unusual result that the septet was more stable than the nonet. We get a similar description for the geometry-optimized structure, where the most stable multiplicity, M ) 11, is related to the same distribution of electrons in the s and d subsystems (Table 3). Calculations have been done, as in the previous case, defining 1, 1, 12, 12, 8, 10, and 7 references for M ) 1-13, respectively, giving a maximum number of configurations (2800) for M ) 7. According to the electronic configuration t2u6eg4t2g6eg2t1u3ag2ag1t1u3eg1, the octahedral cluster would JahnTeller distort in order to break the degeneracy of the half-filled eg orbital. This symmetry breaking is achieved by means of an axial elongation and gives a distorted structure, also characterized by a high M (11), which is only 0.13 eV more stable than the Oh one. DFT calculations by Reuse and Khanna3 also show minimal distortions from an Oh symmetry for Ni6. Although high M is also stabilized, both the optimized distance and the calculated M ()9) are somewhat smaller than ours. Recent experiments suggest that the ground state of Ni6 is a nonet.22 3.4. Ni8 Clusters. Oh symmetry has been also analyzed for this cluster size, comparing the structure built on the interatomic bulk distances and the one that results from a full geometry optimization (r ) 2.25 Å). To include the results derived from DFT calculations3 in the comparative analysis, we have also considered the structure predicted as the most stable in ref 3, associated with a 2.05 Å bond distance. The results of our MRCI calculations suggest that the multiplicity is strongly distance dependent (Table 4). At the optimized distance we find a 3T2u as most stable, lower in energy by 0.8 eV from the nearly degenerate 1T2u and 5Eg states. At the 2.49 Å bulk distance, a 1A1g is 0.49 eV more stable than the first triplet. Higher M are stabilized at the shorter (2.05 Å) interatomic distance: CAHF calculations suggest a ground state 5A 3 1g followed by a very close lying state, T2u, only 0.03 eV higher in energy. RHF calculations at this shorter geometry suggest that the 3T2u state is lower than the 5A1g by 0.16 eV. This difference is in some sense a measure of the consistency of our calculations. Supposedly higher levels of theory starting from these different approaches would yield the same result. Since both procedures are variational and the RHF level yields the lower energy, we assume that 3T2u is the ground state of this structure. The electronic distribution, as well as the number of orbitals involved in the calculations, is shown, for each case,
M r
1
3
5
7
9
2.49 Å RHF 2.25 Å RHF 2.05 Å RHF 2.05 Å CAHF 2.05 Å UHF 2.05 Å UHFA
-254.479 1A 1g -263.516 1 T2u -272.921 1 T2u -272.915 1 T2u -272.908 c -272.908 c
-254.461 3E u -263.548 3T 2u -272.951 3T 2u -272.944 3T 2u -272.925
-254.381 5A 1g -263.515 5E g -272.945 5A 1g -272.945 5A 1g b
-254.293 9A 1u -263.425 9T 2g -272.922 9A 1u -272.937 9A 1u -272.944
-272.947
b
-254.340 7A 1g -263.472 7T 1u -272.935 7A 1g -272.941 7T 1u -272.932 c -273.085 c
-273.018
a Electronic states are also indicated in the first case. Different multiplicities are compared for the bulk (2.49 Å) and fully optimized geometries in Oh symmetry. We have also considered r ) 2.05 Å, the DFT-optimized interatomic distance. b No convergency is attained for M ) 5, 11. M ) 13 is less stable than M ) 9 (272.81 hartrees) according to UHF. c Symmetry breaking in the SCF.
Figure 4. Frontier orbitals of Ni8 clusters of Oh symmetry: (a) bulk interatomic distance; (b) geometry-optimized interatomic distance; (c) interatomic distance optimized in ref 3, RHF calculations; (d) idem. c, CAHF calculations.
in Figure 4. The definition of 3, 6, 6, 11, and 4 references for M ) 1-9, respectively, leads to a maximum number of configurations (4320) for the state of M ) 7. The optimized structures (Figure 8B,C) would Jahn-Teller distort in order to break the degeneracy in the eu, t2g d type orbitals. Stabilization of the nonet, predicted by Reuse and Khanna,3 would probably lead to half-filled eu, t2g d type and t1u s type orbitals. The resulting structure would be, thence, stable against distortion, in agreement with ref 3.
Properties of Small Ni Clusters
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TABLE 5: Total Valence Energies (Hartrees) Calculated for Ni13 Structures from MRCIS Calculationsa M 1 3 5 7 9 SCF
energy state energy state energy state energy state energy state energy
Ih 2.49 Å
Oh 2.41 Å
-529.6522 1T 1g -529.6527 3 T1g -529.5585 5H u -529.4648 7H u -529.3708 9T 2u -529.6390
-529.9689 1A 1g -529.9666 3T 2g -529.8755 5A 2u -529.8206 7A 2u -529.7487 9E u -529.6692
a
Electronic states are also indicated. Different multiplicities (ZINDO/S MRCI) are compared for the Ih and Oh symmetries.
The dependence of the spin state on the characteristics of the MOs and, through them, on the calculation procedure used for their generation also applies to the results of DFT, where the smaller gap between occupied and virtual orbitals1 leads to the stabilization of higher spin multiplicities. This small gap generates quite “bulklike” densities of states, making the small clusters good models for bulk metal. According to DFT calculations,3 M ) 9 is preferred for the optimized cubic structure (2.05 Å bond lengths). To study the origin of this discrepancy, we have compared the stability of different M from UHF calculations. However, in most of the cases but M ) 3 and 9, the orbital symmetry is destroyed during the SCF cycles. Thence, we can only conclude that, in agreement with DFT, UHF favors M ) 9 over M ) 3 by approximately 1.0 eV. This result is not surprising at all, as spin-polarized DFT is, in essence, a UHF-like calculation. Thence, although spin contamination is known to be lower for DFT than for SCF calculations, one has to be aware of this effect in the analysis of the M of small transition metal clusters. The present results open a question on the accuracy of both the DFT-UHF and MRCI/SCF calculations for the study of the magnetic properties of medium size Ni clusters and the reliability of the results of different calculations when states of different M are separated by only a few tenths of an electronvolt. 3.5. Ni13 Clusters. The discrepancy between DFT3 and semiempirical MRCI calculations is most remarkable in the case of Ni13 clusters. Whereas the former strongly favors high spin M ()9), the latter suggest M ) 1 for an octahedral (Oh) structure and M ) 3 for a icosahedral (Ih) one. Triplet and singlet states are near degenerate for a given symmetry (Table 5A). We have worked on the geometry-optimized structures, as optimization leads to the interatomic bulk distance in Ih and to a value 0.07 Å smaller in Oh symmetry (Table 5). In the case of the Ih symmetry, there are 16 electronic states of M ) 1 and 3 within 0.5 eV, which result from promotion of either 1 or 2 electrons to the bonding ag s orbital (Figure 5). Further promotion of electrons to the t1u orbital increases the energy 2.7 eV. Because of the large number of references that have to be defined for systems of large degeneracy, the space of the CI (10 occupied and 4 virtuals) became rather limited in the number of virtuals included in comparison with the number of unocuppied s orbitals. However, the inclusion of nine virtuals instead of four does not change the results; the most stable state of 3T1g symmetry is associated with the promotion of 2 electrons to the bonding ag orbital. For the smaller CI space, the definition of 6, 9, 12, 7, and 36 references for M ) 1-9 leads to a maximum number of configurations of 1294 for M ) 5. In the case of Oh symmetry, there are 35 electronic states of M ) 1 and 3 that lie within 0.5 eV of the most stable 1A1g. This symmetry (studied as D2h) allows us to define a larger CI
Figure 5. Frontier orbitals of Ni13 clusters: (a) Oh symmetry; (b) Ih symmetry.
space (14 orbitals down and 9 up) and to analyze the influence of this variable on the final results. We concluded that the relative stability of the different M does not change, whereas the energy of each state decreases by 0.08 eV when the number of virtuals is increased from 4 to 9. The promotion of an electron to the t1u orbital (Figure 5B) increases the energy 2.4 eV. Working with 2, 10, 18, 39, and 35 references for M ) 1-9, the maximum number of configurations (2520) belongs to M ) 7. According to the electronic configuration of t2g6t2u6eu4t2g6eg4t2g6a1u2, the Oh Ni13 cluster would not Jahn-Teller distort. Distortions are predictable for the Ih Ni13 structure, where the hg and t1g degenerate orbitals are neither filled nor half-filled (Figure 5A). Distortions from the Ih symmetry in Ni13 structures have been discussed in a previous article.29 This distortion results in a D5d symmetry cluster and quenches the spin resulting in a 1A1g ground state. The distortion energy from Ih(3T1g) to D5d(1A1g) is estimated at 0.044 hartree ) 27.6 kcal/mol. In ref 29 we found the D5d(1A1g) structure lower in energy than Oh. This result is different from what we find here, as given in Table 5, a consequence of the choice of βd as discussed previously. The multiplicity predicted from the MRCIS INDO model is found here to be very sensitive to geometry. Recent magnetic evidence suggests that Ni13 might be an eleventet.23 4. Conclusions We have analyzed the multiplicity of small Ni clusters by means of INDO/S MRCI calculations, comparing the results
16880 J. Phys. Chem., Vol. 100, No. 42, 1996 with those derived from DFT and ab initio methodologies, when available. The results are coincident for the small Ni4, Ni5, and Ni6 clusters. However, discrepancies develop between the DFT3 and semiempirical INDO/S-MRCI calculations for Ni8 and Ni13 cluster sizes for which ab initio results are unfortunately lacking. This comparative analysis opens several questions. On the basis of the statement that the magnetic moment per atom of the small Ni clusters is larger than that in the bulk,7,9,23 one is tempted to prefer the DFT results for Ni8 and Ni13. We have demonstrated that INDO/S UHF calculations also favor large M, a fact that we associate partially with spin contamination, an effect that is not completely avoided in DFT results. We are presently working on a fully projected UHF scheme41 to attempt to separate inaccuracies that might be due to the level of theory used (MRCIS Vs, for example, larger CIs or projected UHF) and those that might be due to the model Hamiltonian itself (INDO/s). In relation to catalysis research, this analysis focuses on the size of the cluster that better models a highly dispersed or a polycrystalline material. The question is, when using a small cluster to model an Oh (100) structure, the choice between Ni6, Ni8, and Ni13 should rely on the number of atoms, or is it better to work with structures that allow a proper description of the electronic and magnetic characteristics, free from uncertainties related to the effect of spin contamination or the choice of the configuration space in the CI calculations? If too large a CI expansion is required to properly reproduce the magnetic and geometric characteristics of these small clusters, a systematic study of the catalysis that takes place might not be feasible. Acknowledgment. This work was supported in part through grants from the Consejo Nacional de Investigaciones Cientı´ficas y Te`cnicas (CONICET), Universidad Nacional de Quilmes and Fundacio´n Antorchas (Republica Argentina) and the Office of Naval Research. G.L.E. is a member of CONICET (Republica Argentina). References and Notes (1) Mele, F.; Russo, N.; Toscano, M. Surf. Sci. 1994, 307-309, 113. (2) Pacchioni, G.; Rosch, N. Surf. Sci. 1994, 306, 169. (3) Reuse, E. A.; Khanna, S. N. Chem. Phys. Lett. 1995, 234, 77. (4) Gropen, O.; Almlof, J. Chem. Phys. Lett. 1992, 191, 306. (5) Massobrio, C.; Pasquarello, A.; Car, R. Chem. Phys. Lett. 1995, 238, 215. (6) Alvarado, P.; Dorantes-Da´vila, J.; Dreysse´, H. Phys. ReV. B 1994, 50 (2), 1039. (7) Broto, J. M.; Ousset, J. C.; Rakoto, H.; Askenazy, S.; Dufour, Ch.; Brieu, M.; Mauret, P. Solid State Commun. 1993, 85, 263.
Estiu and Zerner (8) Lee, K.; Callaway, J. Phys. ReV. B 1993, 48 (20), 15358. (9) Billas, I. M. L.; Chaˆtelain, A.; de Heer, W. A. Science 1994, 265, 1682. (10) Douglas, D. C.; Cox, A. J.; Bucher, J. P.; Bloomfield, L. A. Phys. ReV. B 1993, 47 (19) 12874. (11) Miura, K.; Kimura, H.; Imanaga, S. Phys. ReV. B 1994, 50 (14), 10335. (12) Estiu, G. L.; Zerner, M. C. J. Phys. Chem. 1994, 98, 4793. (13) Bodker, F.; Morup, S.; Linderoth, S. J. Magn. Magn. Mater. 1995, 14-144, 373. (14) Maciejewski, P.; Wurth, W.; Kostlmeier, S.; Pachionni, G.; Rosch, N. Surf. Sci. 1995, 330, 156. (15) Holmgren, L.; Andersson, M.; Rose´n, A. Surf. Sci. 1994, 331333, 231. (16) Triguero, L.; Walgren, U.; Boussard, P.; Siegbahn, P. Chem. Phys. Lett. 1995, 237, 550. (17) Douglas, D. C.; Bucher, J. P.; Bloomfield, L. A. Phys. ReV. B 1992, 45, 6341. (18) de Heer, W. A.; Milan, P.; Chatelain, A. Phys. ReV. Lett. 1990, 65, 488. (19) Bucher, J. P.; Douglass, D. C.; Bloomfield, L. A. Phys. ReV. Lett. 1991, 66, 3052. (20) Reddy, B. V.; Khanna, S. N.; Dunlap, B. I. Phys. ReV. Lett. 1993, 70, 3323. (21) Cox, A. J.; Louderback, J. G.; Bloomfield, L. Phys. ReV. Lett. 1993, 71, 923. (22) Apsel, S. E.; Emmert, J. W.; Deng, J.; Bloomfield, L. A. Phys. ReV. Lett. 1996, 76, 1441. (23) Parks, E. K.; Zhu, L.; Ho, J.; Riley, S. J. J. Chem. Phys. 1994, 100, 7206. (24) Stave, M. S.; Depristo, A. E. J. Chem. Phys. 1992, 97, 3386. (25) Santamaria, R.; Kaplan, I. G.; Novaro, O. Chem. Phys. Lett. 1994, 218, 395. (26) (a) Castro, M.; Salahub, D. Phys. ReV. B 1994, 49, 11482. (b) Castro, M.; Salahub, D. Phys. ReV. B 1993, 47, 10955. (27) Bonacic-Koutecky, V.; Fantucci, P.; Koutecky, J. Chem. ReV. 1991, 91, 1035. (28) Estiu, G. L.; Zerner, M. C. Int. J. Quantum Chem. 1993, 27, 4793. (29) Estiu, G. L.; Zerner, M. C. J. Phys. Chem. 1994, 98, 4793. (30) Zerner, M. C. ZINDO; Quantum Theory Project; Williamson Hall, University of Florida: Gainesville. (31) Edwards, W. D.; Zerner, M. C. Theor. Chim. Acta 1987, 72, 347. (32) (a) Head, J. D.; Zerner, M. C. Chem. Phys. Lett. 1985, 122, 264. (b) Head, J. D.; Zerner, M. C. Chem. Phys. Lett. 1986, 131, 359. (33) Head, J. D.; Weiner, B.; Zerner, M. C. Int. J. Quantum Chem. 1988, 33, 177. (34) Jahn, H. A.; Teller, Proc. R. Soc. London Ser. A 1937, 161, 220. (35) Zerner, M. C. Int. J. Quantum Chem. 1989, 35, 567. (36) Pauncz, R. Spin Eigenfunctions; Plenum Press: New York, 1979. (37) Anderson, W. P.; Cundari, T. R.; Zerner, M. C. Int. J. Quantum Chem. 1991, 35, 31. (38) Zerner, M. C. In ReViews in Computational Chemistry; Lipkowitz, K. B., Boyd, B., Eds.; VCH Publishers: New York, 1990; Vol. 2. (39) Stewart, J. P. In ReViews in Computational Chemistry; Lipkowitz, K. B., Boyd, B., Eds.; VCH Publishers: New York, 1990; Vol. 1. (40) Cory, M.; Zerner, M. C. Work in progress.
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