Structure and Electronic Properties of Copper Clusters An Ab Iizitio LCAO-MO-SCF Study BY CHRISTIAN BACHMANN AND ALAINVEILLARD JEAN DEMUYNCK E.R. no 139 du C.N.R.S. Institut Le Bel Universitk L. Pasteur 4 rue B1. Pascal 67000 Strasbourg France Received 5th Jzcly 1979 The structure and electronic properties of copper clusters Cu (n = 2-5,8 and 13) have been studied through ab initio LCAO-MO-SCF calculations with a Gaussian basis set (12 7 5) contracted to [5,3,2]. The linear structure is more stable than the two- and three-dimensional structures for n = 3 and 4 but becomes less stable for n 3 5. Since for Cu8 the band of 3d levels and the 4s levels are well separated and for CulJ the two bands just begin to overlap the distribution of energy levels in these clusters appears to be rather different from that in the bulk metal.This is also a consequence of the fact that most of the atoms in the cluster CuI3 are surface atoms. The binding energy increases quasi-linearly with the number of atoms in the cluster up to n = 8. A knowledge of the electronic structure of metallic clusters containing up to a few tens of atoms is important for the following reasons. (i) Metal catalysts are usually found in the form of dispersed microcrystallites with a high surface/volume ratio The relationship between the electronic and structural properties of these clusters usually <10 A in size and those of the bulk metal is far from clear. A thorough understanding of the electronic and structural properties of these small metal particles is probably necessary in understanding how these metal aggregates act as catalysts.(ii) A number of experimental techniques such as the technique of con- densation within a matrix at low temperature has been developed recently for the experimental study of small metallic clusters with <10 For instance it has been possible to follow the changes in the U.V. and visible absorption spectra as a func- tion of the size of the agg~-egate.~.~ (iii) There is a strong interest in the chemistry of polynuclear organometallics with several metal-metal bonds (called molecular clusters by the chemists) as potential homogeneous catalysts. The possible relation- ship between the role of molecular clusters in homogeneous catalysis and the role of metallic clusters in heterogeneous catalysis has already been Many quantum mechanical studies deal with the electronic properties of transition metal clusters containing up to a few tens of atoms (large clusters including up to a thousand atoms have been studied in the tight-binding appro~imation).~ However nearly all the previous studies rely on semi-empirical methods.Both the extended Huckel method and the CNDO method have been used to investigate the electronic structure of transition or noble metal clusters (Ni Cu Pd Ag Au) of up to 55 atoms.10-20 Recently the SCF-Xa-scattered wave method has been used to investi- gate transition metal clusters of 8 and 13 atoms (Ni Cu Pd and Pt).21$22 The same method has been used to study the clusters of Li atoms (Li to Li13)23and of A1 atoms (up to A143).24 However the conclusions of these semi-empirical studies have been the subjects of controversy for instance regarding the rapidity with which the electronic C.BACHMANN J. DEMUYNCK AND A. VEILLARD properties of the clusters approach their bulk counterparts with increasing size. Ac-cording to Extended Hiickel and CNDO calculations the binding energy per atom increases slowly with the size of the clusters the binding energy being 4to 4of the bulk value for a 55 atom cluster. In contrast the SCF-Xcc calculations suggest that the bulk density of states (for transition metals) is largely attained for a cluster of 13 atoms.21 Ab initio calculations dealing with the electronic structure of metallic clusters are very scarce and have been restricted to clusters of lithium and beryllium atom^.^^-^^ We report here the results of an ab initio LCAO-MO-SCF study of clusters of copper atoms Cu with n between 2 and 13.In this study we have tried to answer the follow- ing questions (i) how do the electronic structure and binding energy per atom vary with cluster size and geometry; (ii) do the binding energy per atom and the density of states (DOS) structure approach those of the bulk metal for small clusters; (iii) what is the most stable structure for a cluster with a given number of atoms? A pre- liminary account of this work has been published previously.28 METHOD AND CALCULATIONS Ab initio LCAO-MO-SCF calculations have been carried out with four different basis sets of contracted Gaussian functions.Calculations for the clusters of up to eight atoms were carried out with basis set I (BSI) which is a Gaussian basis set (12 7 5) contracted to [5 3 23 (minimal basis set for the inner shells and the 4p shell double- zeta basis set for the shells 3d and 4s). A larger basis set (BSII) (13 8 5) contracted to [6,4. 21 (minimal basis set for the inner shells double-zeta for 3d and 4p triple-zeta for 4s) has also been used for Cu,. Another large basis set used for Cu and denoted BSIII is made of basis set I incremented with polarization functions namely a set of s p and d functions along the Cu-Cu axis and off the nuclei. A smaller basis set (BSIV) (12 7 4) contracted to [S 3 13 (minimal except for the 4s orbital which is split) was used for the cluster Cu, since the corresponding calculation with BSI would have exceeded the present possibilities of our open-shell SCF program.The Gaussian basis set (12 7 5) corresponds to the basis set (12 6,4) of ref. (29) incremented with one p function of exponent 0.25 and one d function of exponent 0.2. The (13 8 5) basis set has one additional s function of exponent 0.02 and one addi- tional p function of exponent 0.1. The (12 7 4) basis set is similar to the (12 7 5) set except for the exponents of the d functions which have been reoptimized. Different geometries have been considered for n = 3,4 5 and 8. For the 13-atom cluster the cubo-octahedral geometry shown in fig. I is the structure corresponding FIG.1.-Cubo-octahedral cluster containing 13 atoms.STRUCTURE OF COPPER CLUSTERS to the local arrangement ot atoms in the face-centred cubic (f.c.c.) crystalline metal. The Cu-Cu distance has been optimized for a limited number of structures corre- sponding to n = 2 3 4 and 8. Otherwise a fixed distance of 2.40 A has been used. The calculations were carried out with the system of programs A~terix.~~?~~ The open-shell SCF treatment is based on the restricted Hartree-Fock formalism proposed by Guest and Saunder~.~~ All one- and two-electron integrals were computed with single-word accuracy on the Univac 1110 (word of 36 bits). The SCF calculations were carried out with double-word accuracy for the clusters up to Cu,. For CuI3 the SCF calculation was carried out with single-word accuracy.NUMERICAL RESULTS In table 1 are reported the calculated SCF energies for Cu2 as a function of the interatomic distance and of the basis set used. The ground-state energies for the different structures and different sizes of clusters are given in table 2 together with TABLE 1.-SCF ENERGIES (IN a.u.) FOR Cu2 AS A FUNCTION OF THE BASIS SET USED AND OF THE INTERATOMIC DISTANCE (IN A) BSI BSII BSIII BSIV r E r E r E r E 2.29 -3271.0493 2.291 -3271.084 85 2.25 -3271.677 00 2.35 -3268.867 76 2.34 -3271.0496 2.341 -3271.085 14 2.30 -3271.677 81 2.40 -3269.868 02 2.40 -3271.0493 2.391 -3271.084 57 2.35 -3271.677 96 2.45 -3269.867 90 re=2.34 re=2.3 33 re=2.327 re=2.41 the optimized Cu-Cu distance whenever calculated and the binding energy per atom.The energy values and binding energies per atom of table 2 have been obtained with basis set I and a fixed Cu-Cu distance of 2.40 A except for CuI3 where they corre- spond to basis set IV. A calculation for Cu (Oh symmetry) has also been carried out FIG.2.-Orbital energies (in a.u.) for the occupied orbitals 3d and 4s of the linear clusters Cu,, n = 2-5. Open-shell orbitals are indicated by an asterisk. TABLE 2.-SCF ENERGIES OPTIMIZED Cu-Cu DISTANCES IONIZATION POTENTIALS AND BINDING ENERGIES PER ATOM FOR THE CLUSTERS Cu (WITH BASIS SET I UNLESS OTHERWISE STATED ENERGY VALUES FOR A FIXED CU-cu DISTANCE OF 2.40 A) ~ ~~ energy of optimized binding energy ionization n structure electronic the ground Cu-Cu distance per atom potential state state /a.u.IA /kcal mol-l /eV 2 linear -3 271.0493 2.343 9.7 5.7" (6.1') z 3 linear -4906.5753 2.35 10.0 5.4" 2 bent (120") -4906.5704 8.9 03h -4906.5700 2.41 8.9 4 linear -6542.1180 12.7 5.4" (5.6') square planar (&) -6 542.1128 2.43 11.9 5.2" tetrahedral (Td) -6 542.1016 10.2 IX = 150" -6542.1124 11.9 02' a = 1200 -6542.1099 11.5 5 linear -8 177.6437 12.3 trigonal bipyramid (&h) -8 177.6591 14.2 square pyramid (C4") -8 177.6560 13.8 4.6" body-centred square (&) -8 177.6384 11.6 8 cube (Oh) -13 084.321 2.43 19.5 5.9' -13 079.537" 11.5 square antiprism (D4d) -13 084.333 20.4 5.5' 13 cubo-octahedron -21 254.314" 14.6 a Calculated as the difference of the energies for the ion Cu,' and the cluster Cu,.'Calculated according to Koopmans' theorem. With basis set IV. 174 STRUCTURE OF COPPER CLUSTERS with basis set IV thus allowing for a comparison of the results obtained with basis sets I and IV. Fig. 2 is a plot of the orbital energies for the occupied orbitals 3d and 4s (closed-shells and open-shells) of the linear clusters Cu, n = 2-5. Fig. 3 is a plot of the same orbital energies for the two- and three-dimensional clusters Cu, n = 3-5 8 and 13 (Cu has also been included for the sake of comparison). Table 3 shows the depend- ence of the orbital energies of Cu on the basis set used. Fig. 4 shows the density of occupied states for the cluster CwI3 (each energy level has been represented through a Gaussian function with a broadening parameter CJ = 0.005 a.u.).TABLE 3.-oRBITAL ENERGIES OF CUz (IN a.U.) FOR THE ORBITALS 3d AND 4s AS A FUNCTION OF THE BASIS SET USED BSI BSII BSIII BSIV og (4s) UU -0.225 -0.461 -0.223 -0.459 -0.223 -0.469 -0.221 -0.471 n -0.470 -0.468 -0.478 -0.482 6 -0.481 -0.479 -0.489 -0.491 6 nu -0.487 -0.501 -0.485 -0.498 -0.494 -0.506 -0.496 -0.510 og(34 -0.511 -0.508 -0.516 -0.519 -* e" s) il 0 0 FIG.3.-Orbital energies (in a.u.) for the occupied orbitals 3dand 4s of the two- and three-dimensional clusters Cun,n = 3-5 8 and 13. Open-shell orbitals are indicated by an asterisk. DISCUSSION BOND LENGTHS AND GEOMETRIES The results of table 1 show that the calculated bond length for Cu is rather sensi- tive to the basis set used. The calculated bond length decreases (from 2.41 to 2.33 A) when the quality of the basis set increases (in the order BSIV BSI and BSIII).We C. BACHMANN J. DEMUYNCK AND A. VEILLARD & 1a.u. FIG.4.-Density of occupied states for CuI3(orbital energies in a.u.). estimate that the theoretical bond length in Cu at the Hartree-Fock limit would be close to 2.3 A. The experimental bond length is 2.22 A.33 Our calculations do not support the results of Joyes and Leleyter; these authors report an equilibrium bond length of ~2.2 A with a basis set of Slater (minimal basis set except for the d orbitals which are split) (their SCF energy of -3270.4 a.u. is higher than the SCF energies reported in table 1 for the basis sets I to 111). Calculations close to the Hartree-Fock limit usually tend to produce bond lengths which are accurate to a few hundredths of A and generally too short.35 However the discrepancy for Cu is probably larger of the order of 0.08 A with the calculated bond length too long.We tentatively ascribe this behaviour to the neglect at the SCF level of configurations of the type (4~0)~ together with the configurations arising from the atomic states 3d94s2 (including the configuration formed from the 4s orbitals34 will probably lengthen the bond). The calculated value of 1.1 mdyn A-' for the harmonic force constant in Cu is nevertheless in good agreement with the experimental value of 1.32 mdyn A-1.33 The computed bond lengths for the clusters Cu fall within two groups namely 2.34-2.35 A for the linear structures and 2.41-2.43 A for the two- and three-dimen- sional structures.This is exemplified for the Cu cluster (table 2) with the distance for the linear structure 0.06 A shorter than the corresponding value for the triangular structure. Potential energy curves for Ag, based on Extended Huckel calculations also show larger internuclear distances for the cubic and planar geometries compared to the linear geometry." A similar trend is also apparent in the calculations of Anderson.18 This increased bond length in the two- and three-dimensional structures is probably a consequence of the presence of filled antibonding levels; the anti- bonding character of these levels is comparatively larger in the non-linear structures,'O but may be reduced by increasing the internuclear distance.The internuclear separa- tions calculated for the two- and three-dimensional structures show little change with the size of the cluster. The distance of 2.43 A calculated for n = 8 differs appreciably from the value of 2.56 A for the bulk metal. Goddard et al. have optimized the bond length for a cluster of 13 Ni atoms and find a Ni-Ni distance of 2.41 A to be compared to the value of 2.49 A for the bulk For n = 3 and 4 the linear structure is slightly more stable (by z 3 kcal mol-l) than the two-dimensional structure of D3,,or D4hsymmetry. However for n = 5 the linear structure becomes less stable (by ~9 kcal mol-l) than the trigonal bipyramid. I76 STRUCTURE OF COPPER CLUSTERS We conclude that the linear structure is more stable than the two- and three-dimen- sional structures only for the smallest aggregates with n = 3 and 4.Our calculations do not support the conclusions of Extended Hiickel calculations that the linear struc- ture represents the most stable geometric form up to 30 or 50 The experimental evidence although scarce seems to be in agreement with the results of the ab initio calculations. Ag is probably linear since the corresponding Raman spectrum shows only one line.,' The e.s.r. spectra of a species Ag,' or Agj+ has been analysed by assuming two pairs of inequivalent silver atoms this being consistent with a linear ge~metry.,~ Both the anion Pbg- and the cation Bi:+ appear to have a trigonal bipyramidal struct~re.~~~~~ The Ag cluster has an octahedral structure.41 Thus the structural evidence points to a change from the linear structure to the three- dimensional structures probably between 4 and 5 atoms.Support for a change from the linear to the three-dimensional structure between Ag and Ag also comes from the observation by Schulze et al. of a decrease in the energy of the lowest ab- sorption band from Ag to Ag, followed by an increase from Ag to Ag and a subse- quent decrease from Ag to Ag6.5 The authors attribute this change in the trend of the excitation energy to a change of structure (cf. below). The greater stability of the linear chains in the extended Hiickel calculations has been assigned by Baetzold to the presence of filled antibonding levels the antibonding character of these levels being increased in the non-linear structures.1° Examination of the orbital energies for Cu show that this analysis is correct the open-shell orbital energies for Cu are respectively -0.028 a.u.for the D3hstructure and -0.108 a.u. for the linear structure. However there are two opposing factors which favour a greater stability of the two- and three-dimensional structures and which soon become predominant. First the antibonding interaction just mentioned for Cu becomes nonbonding for Cu4 (in the sense that the molecular orbital displays out-of-phase amplitude on non-adjacent atoms see for instance the open-shell orbital e of Cu4 square-planar in fig. 5). This causes a relative stabilization of the two-dimensional n -0.114 n -0.198 * ,0363 -0.282A .0.304 w-FIG.5.-Highest-occupied orbitals of the linear clusters CUZ-CUS and of the two- and three-dimen- sional clusters Cu3-Cu5.Their orbital energies are given in a.u. C. BACHMANN J. DEMUYNCK AND A. VEILLARD 177 structure as shown by the open-shell orbital energies of Cu3 DSh and CU~D~~ -0.028 and -0.096 a.u. respectively. The second factor and probably the most important one is that the number of bonding interactions for a given cluster is much larger in the two- and mostly three-dimensional structures than in the linear structures (see for instance the in-phase bonding orbitals of fig. 5). This is typified by the change in the energy of the bonding in-phase orbital in the series Cu3 Cu4 and Cu5 -0.24 -0.25 -0.26 a.u.respectively for the linear structures and -0.28 -0.30 -0.36 a.u. for the two- and three-dimensional structures. ORBITAL ENERGIES AND BINDING ENERGIES Examination of table 3 shows that the orbital energies of Cu2 are relatively insensi- tive to the basis set used. Fig. 2 and 3 show the build-up of a relatively narrow and dense band of 3d levels together with a relatively wide 4s band (with usually a small admixture of 4p orbitals). For Cu8 the 3d band and the 4s levels are well separated. For C~13 the two bands just begin to overlap with the a, level being the lowest level with s-character and just at the top of the d band Thus our results do not support the conclusion inferred from the SCF-Xa calculations that “already in the Cu8 aggregate the d band is totally overlapped by the sp band as in the solid ”.21 For C~13 the two levels t2gand e located slightly below the d band correspond to d orbitals which are bonding between the central atom and outer atoms of the cubo- octahedron but which are primarily localized on the central atom.However the splitting off of these two d orbitals from the bottom of the d-band is much smaller in our calculation (about 0.02 a.u.) than in the SCF-Xa calculation (more than 0.1 a.u.). In the SCF-Xa study by Messmer et al. it is stated that “ while the main band of d levels in C~13 corresponds closely to the d band characteristic of bulk copper the two deep-lying d levels have no close counterpart in bulk copper and are artifacts of the deeper potential energy of the central atom of C~13 compared with the surface atoms of the cluster”.We believe that the correspondence made between the main d band in C~l3 and the d band characteristic of bulk copper is erroneous. The central atom in C~13 is surrounded by twelve atoms hence the strongly bonding interactions resulting in the t2s and e levels at lower energies while each “ surface ” atom is sur- rounded by only four atoms. The central atom in Cu13 is in a situation analogous to that found for the atoms of the bulk copper. We conclude that the two levels tZgand e of Cu13 represent the origin of the d band of bulk copper (together with the corresponding antibonding levels) while the main band of d levels in Cu13 corresponds rather to surface states of bulk copper. There is some experimental evidence that the energy levels of clusters of the size considered here are different from those of the bulk metal at variance with the conclusions of the SCF-Xa calculations and in agreement with the results of the ab initio calculations.In the U.V. photoelectron spectroscopy of palladium particle arrays (probably much larger than the present clusters) the width of the Pd valence band was found to be a sensitive function of particle size.42 From a study of the absorption bands of silver clusters Schulze et al. concluded that more than ten atoms are necessary to change the electronic properties of silver clusters from a molecular type towards a metal type’ (however Ozin considers that silver clusters containing from 6 to 15 silver atoms exhibit both molecular and bulk optical characteristics).4 On the other hand the ionization potential appears rather insensitive to the size of the cluster.For the closed-shell systems these ionization potentials were calculated STRUCTURE OF COPPER CLUSTERS either according to Koopmans' theorem or as the difference of the energies of the neu- tral species and of the ion (the relaxation effects are then accounted for but they appear to be relatively unimportant). Only the latter procedure can be used for the open- shell systems. The calculated values centre around 5 eV and do not show any sig- nificant trend with the number of atoms. They appear rather close to the work func- tion of 4.7 eV for bulk copper. The binding energies per atom calculated with basis set I increase practically linearly with the number of atoms up to n = 8.A similar trend is found between the clusters of 8 and 13 atoms with basis set 11. Thus in terms of binding energy the properties of the clusters should be rather different from those of the bulk metal. The binding energy of 20 kcal mol-' calculated for Cu with basis set I is well below the experimental value of 80 kcal mol-1 for the bulk metal (however part of the difference should be traced to the SCF approximation which underestimates the binding energies). Note added in proof Since this paper was submitted we have completed the calculations (with basis set IV) for the cluster Cu13 with two different structures the cubo-octahedron and the icosahedron. The icosahedron has a lower energy (-21 254.286a.u.for 6Ag)than the cubo-octahedron (-21 254.219 a.u. for 6A,and -21 254.200 a.u. for 2T2g) (these energy values correspond to a double-word accuracy). Thus a cluster of thirteen atoms should prefer an icosahedral arrangement rather than the cubo- octahedral arrangement corresponding to the f.c.c. structure of the bulk metal. Furthermore there may be a close correspondence between the structures of the naked clusters and those of the mole- cular clusters. In this respect one may note that (i) the structure of a trigonal bipyramid is rather common for organometallics with five metal atoms; (ii) a square antiprismatic arrangement of the eight copper atoms has been found in the octameric o-anisylcopper (I); 43 (iii) an icosahedral arrange- ment of thirteen gold atoms has been reported for the cation Au13 (d~pm),.~~ This work has been supported through the A.T.P.no2454 of the C.N.R.S. Calcu-lations have been carried out at the Centre de Calcul du C.N.R.S. in Strasbourg-Cronenbourg. l J. R. Anderson Structure of Metallic Catalysts (Academic Press London 1975). M. Moskowits and J. E. Hulse J. Chem. Phys. 1977,66 3988. M. Moskowits and J. E. Hulse J. Chem. Phys. 1977 67,4271. G. A. Ozin and H. Huber Inorg. Chem. 1978 17 155. W. Schulze H. U. Becker and H. Abe Chem. Phys. 1978,35 177. B. J. Garrison N. Winograd and D. E. Harrison J. Chem. Phys. 1978,69 1440. E. L. Muetterties Bull. SOC. chim. belg. 1975,84,959. * J. M. Basset and R. 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