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Properties and structure of small cobalt clusters.
Vacuum 54 (1999) 143 — 149
Properties and structure of small cobalt clusters. Nonlocal density functional calculations P. M"ynarski , M. Iglesias, M. Pereiro, D. Baldomir *, L. Wojtczak Department of Theoretical Chemistry, University of %o& dz& , Poland Facultade de Fı& sica, Universidad Santiago de Compostela, 15706 Santiago de Compostella, Spain Solid State Physics Department, University of %o& dz&, Poland
Abstract The results of self-consistent nonlocal (GGA) density functional calculations are reported for small cobalt clusters (n)6;13). An emphasis is made on a proper treatment of correlation effects. The magnetic as well as the bonding properties of these clusters are discussed in terms of a delicate balance between intra- and interatomic exchange and correlation. The enhancement of magnetic moments is analysed in terms of cluster symmetry. Finally, the present results are compared with other theoretical calculations obtained for these systems. 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction The rapid development of nanofabrication techniques followed by the discovery of some new and interesting effects such as the giant magnetoresistance as a primary example, provides another impact for the theoretical study of transition metal clusters. A serious challenge confronts those systems for any first principle calculation, due to the heavy demand of a proper description of both exchange and correlation effects. Usually, the size of the cluster as well as the high level of correlation treatment needed causes a rapid failure of traditional ab initio quantum chemistry methods based either on configuration interaction (CI) expansion of Slater determinants or Moller—Plesset perturbation techniques. Fortunately, the density functional theory (DFT) [1, 2] offers a computational alternative yet retaining the ab initio level. Among transition atom systems the cobalt clusters have been much less studied than others from the first transition row of elements. Recent papers devoted to this subject are works of Castro et al. [3], Jamorski et al. [4] and Li and Gu [5]. While the work of Li and Gu is based on local density aproximation (LDA), in the other papers a generalized gradient aproximation (GGA) is used where all electrons are treated explicitly. Here we present a systematic study of both electronic and magnetic
* Corresponding author.
properties of cobalt clusters Co (2)n)6; 13) using L a fully selfconsistent spin polarized nonlocal (GGA) Gausian density functional program [6], describing core electrons of cobalt atoms with a model potential (MP). The main purpose of the present work is to show how the electronic as well as magnetic properties evolve with the increase of cluster size. The geometry of all clusters, except Co , has been optimized. Also, for n"4 differ ent less symmetrical structures have been also fully optimized for the sake of possible Jahn—Teller effects. The GGA offers the state-of-the-art treatment of both exchange and correlation effects and, at the same time provides a computationally efficient and more precise way of evaluating molecular integrals by means of using a linear combination of Gaussian-type orbitals (LCGTO).
2. Computational details The present results have been obtained using a computational scheme of Sambe and Felton [7] later improved by Dunlap [8]. Despite the use of orbital basis sets this method introduces two auxiliary basis sets which are used in the self-consistent scheme to fit the charge density and exchange—correlation potential. However, after the self-consistent field (SCF) convergency is reached the exchange—correlation energy is recalculated in an analytical manner. The core electrons of cobalt atom are
0042-207X/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 2 - 2 0 7 X ( 9 8 ) 0 0 4 5 0 - 3
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P. M!ynarski et al. / Vacuum 54 (1999) 143—149
described here by the use of a model potential (MP) in concert with 15 electron orbital basis sets. This MP has been optimized for a specific use in the DFT scheme [9] while the inner 3p shell has been moved into a basis set thus allowing the explicit treatment of important 3p—3d interactions in transition metal systems. Also, the use of MP minimizes the basis set superposition (BSSE) errors which should be taken into account when analysing close lying states on an all electron basis set level and this allows to keep the same computational treatment of cobalt clusters for all nuclearities studied. Having in mind the latter purpose the grid used in the fitting procedures consisted of 12 angular grids along with a 32 point radial mesh per atom. The GGA treatment of exchange—correlation was done through Perdew and Wang’s [10] nonlocal functional for exchange and Perdew’s [11] nonlocal functional for correlation along with the Vosko—Wilk—Nusair (VWN) parametrization of the correlation energy of a homogenous electron gas [12]. Details of the implementation of these density gradient nonlocal functionals as well as the corresponding nonlocal exchange correlation (XC) potentials are described in [6]. The orbital valence basis set with contraction pattern (2111/211>/311*) with (5,3,3;5,3,3) auxiliary set was chosen to be appropriate for all clusters. The present calculations upon cobalt atom predict the ds electronic configuration as the ground state placing its ds state 1.04 eV higher while the experimental s—d interconfigurational energy is 0.43 eV [13]. Although this result gives an improvement over its LDA value of !0.8 eV [13] it is still not satisfactory. The remaining error can be accounted for by either the need of a separate basis set optimization for both atomic states which can give a more appropriate value as it was shown in [14] for the Nickel atom or one has to keep in mind that the ground state of an atom with an open shell is described by a single Slater determinant with M "S and M "¸ 1 * where S, ¸ describe a given multiplet term with highest spin S and highest orbital angular momentum ¸. The latter deficiency can be partially removed by increasing the polarization space in an orbital basis set. Due to the use of model potential the relativistic corrections should be of somewhat lesser importance. In atomic calculations the electron density has been spherically averaged and its total energy ds state has been used as a reference for calculating binding energies of the clusters.
3. The electronic structure, geometry and magnetic properties of cobalt clusters We start the molecular calculations with cobalt dimer. Very few studies have been performed for this system. An excellent review for the Co as well as for other transition metal dimers has been given by Salahub [15]. From that reference a limited CI calculation cited gives the unbound
state with respect to the ds state of the Co atom with bond length of 2.56 As while LSD calculations give 2.07 As . In the present study we use a spin-polarized version of DFT. This means that for every possible difference between the number of spin-up and spin-down electrons (from now on denoted as N) we look for an electronic configuration giving the lowest energy. Also, in all molecular systems we use symmetry rules for reducing the number of molecular calculations. In the case of a dimer, a D symmetry has been adopted though we assign the F electronic configuration in terms of the infinite symmetry group. The ground state has been found for N"4 with the following electron assignment 1p1n2p1d1n1p E S E S E S giving the bond distance of 3.71 bohr and binding energy of 0.865 eV (or 0.433 eV per atom — in latter calculations the binding energy per atom will be given as being more appropriate for viewing its dependence on cluster nuclearity). By making the following promotion: 1di into S 1pi we also optimized this excited state which gave the S same optimized bond distance of 3.71 bohr. Also, for the equilibrium separation of the ground state another excited state has been calculated by promoting electron 1pi S into 1ni. The adiabatical and vertical energy differences S in these cases were 0.213 an 0.531 eV, respectively. The potential energy curve of the ground state near equilibrium has been fitted using a third-degree polynomial in order to obtain vibrational frequency as well as the anharmonicity term. The calculated values are 452.9 cm\ for u and 1.90 for u x . This value of vibraC C C tional frequency seems to be too large but this can be accounted for by a ‘‘cooperative’’ effect of moderate grid size and MP use. The value of binding energy can be compared to the estimated experimental value [16] (1.2 eV. The highest occupied orbital for the Co ground state is the 1b of minority spin with its eigenS value !5.97 eV which in the view of DFT [2] has its physical interpretation as being the ionization potential (IP) of a system (taken with the reversed sign). It is worthwhile to mention here that although the highest occupied Kohn—Sham orbital is the only one having a direct physical meaning while all the remaining ones are just pure mathematical constructions their numerical similarity to real orbitals is used to make estimations of the degree of (sp;sd) hybridizations by means of Mulliken population analysis as well as plots of orbitals which, in turn, provides help in the analysis of the bonding mechanisms in molecular structures. The atomic state of Co in a dimer has been evaluated as s p d thus closely resembling the ds state. In Co case only one structure has been studied: the equilateral triangle; aditional symmetries are being calculated and will be published elsewhere. Using the C symmetry the ground state has been found for T electron spin up/spin down difference N"7 with Co—Co equilibrium distance 3.86 bohr. The binding energy per atom is 0.56 eV and the Co atomic state is estimated as
P. M!ynarski et al. / Vacuum 54 (1999) 143—149
s p d . The bonding is mainly due to the s electrons though as it can be seen in Fig. 1 there is some admixture of d-type electrons in the bonding molecular orbitals. The cobalt tetramer has been inspected in a more detailed way than other cobalt clusters where only the most symmetric structures have been extensively examined. At first we investigated the tetrahedral structure. Even with the imposed T symmetry constraint there is B a huge number of closely lying states and care has to be taken in order to properly assign the ground state. The one we have found is for N"10 with optimized Co—Co
Fig. 1. Plots of the 2ai (up) and 1aj (down) molecular orbitals of Co cluster.
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distance of 4.33 bohr and binding energy per atom equal to 0.98 eV. The highest occupied Kohn—Sham orbital is of T symmetry and of majority spin hav ing 2 electrons, thus being a candidate for the Jahn— Teller distorsion. The atomic state electron configuration in Co is s p d . Next, we have assumed a square geometry for the cobalt tetramer imposing the D F constraint. The optimized Co—Co distance was also equal to 4.33 bohr with N"10. The adjacent spin up/spin down differences N"8 and N"12 produced less stable states which were placed 0.56 and 1.85 eV above the ground state, respectively (vertical values). The optimized T structure is more stable than the optimized D one by B F
Fig. 2. Plots of the 3ei orbital of a Co (D ) cluster. S F
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P. M!ynarski et al. / Vacuum 54 (1999) 143—149
0.96 eV. The analysis of eigenvalue spectra of Co in the square geometry shows that the highest occupied orbital containing 1 electron has E symmetry (Fig. 2) thus the Jahn-Teller effect is possible here so the symmetry has been lowered from D to D and the rhombus structure F F has been optimized. Ground state has been found for Co—Co distance equal to 4.14 bohr along with 72° angle for N"10. Indeed, the optimized D structure is more F stable than the square geometry by 0.29 eV. Clearly, by choosing the square geometry, other distortions are possible though due to the rather small energy differences between different Jahn—Teller structures the quality of both the grid and basis sets must be enhanced possibly with BSSE corrections. On the other hand the change of geometry may imply the change in N therefore changing the magnetic moments. That, in turn, is directly connected with a rather big rearrangement of exchange and correlation energies so the Jahn—Teller effects in transition metal clusters must be discussed along with the level of exchange—correlation treatment. The geometry of Co has been assumed as an equilat eral trigonal bipyramid, so the C symmetry has been T adopted in order to reduce the time of computations during geometry optimizations. The lowest state has been obtained for N"13 and a Co—Co distance of 4.45 bohr although another state with N"11 has also been optimized leading to the same bond distance as for N"13. This state lies only 0.34 eV higher than the ground state and possible Jahn—Teller distortions may lead to a new state with, after geometry optimization process, an increased width of a d ‘‘band’’, leading to the partial collapse of magnetic moment, thus yielding the ground state with a smaller local magnetic moment. The Mulliken population analysis gives an average atomic hybridization of Cobalt as s p d . The Co cluster has been assumed to have a square bipyramid geometry within O symmetry constraint. The F lowest state has been obtained for a Co—Co distance of 4.33 bohr and N"14. This time the highest occupied orbitals are of T and T symmetry for majority and S E minority spin, respectively, with its full occupancy of 3 electrons. This state is strongly bound giving the binding energy per atom as 1.74 eV. Such bond strength can be explained considering the directional nature of d orbitals. The present structure makes the d orbitals point to each other directly thus giving the strongest overlap among all clusters studied. This is also confirmed by auxiliary analysis of Mulliken population giving an average atomic hybridization as s p d . The last cluster studied was Co within O symmetry. F Its geometry has not been optimized and the assumed Co—Co distance between the central atom and its 12 neighbours corresponds to the Co bulk value of 2.44 As . However, the spin up/spin down electron difference N has been optimized in order to get a proper account for the average magnetic moment. The lowest energy was
obtained for N"27 while the states with N"25 and N"29 were placed higher by 1.14 and 0.13 eV, respectively. One has to notice the small difference between the states with N"27 and N"29. If the geometry had been optimized, the state with N"27 would be more favoured and though there would be a tendency of the magnetic moment to collapse due to the increase of the band width, it is very likely that N"27 would be preferred for optimal geometry due to the over 1 eV separation with respect to the N"25 state. The obtained average magnetic moment 2.08 k is in complete agreement with the extrapolated experimental value [17]. However, there is a big difference between local magnetic moments of the central atom and the neighbouring ones. The central Co atom is the least bulk-like. It has an excess of charge and a very small magnetic moment, 0.56 k with large negative sp contribution almost!1 k while for the ‘‘surface’’ atoms their magnetic moment is enhanced: 2.204 k where the d contribution to this value is essential and is 1.92k . Using the Mulliken population analysis of the eigenvalue spectra the estimated d ‘‘band’’ width is 3.65 eV for spin up and 3.12 eV for spin down. The lowest molecular level is an s-like with p-type contributions and eigenvalues for different spins yield an sp exchange splitting of 0.17 eV. The ‘bottom’ of d-type molecular levels is split by 0.33 eV.
4. Model potential and all electron basis set calculations In this section a comparison between our results and all electron basis set calculations [4, 3] will be made. Our work extends to larger cluster nuclearities up to Co . Therefore, the common ground for comparison is limited only to Co in the most extensive way and with some limitations to Co and Co due to the choice made in our study where only the most symmetric structures have been chosen. Our data obtained by the use of a model potential will be confronted with the Jamorski et al. and Castro et al. results obtained through the all electron basis sets along with the use of very extensive grid sizes. While the latter treatment is appropriate for small transition metal clusters, the former method used in our work to study greater cluster nuclearities is the only one computationally manageable. The nonlocal GGA level has been kept the same in both approaches. In the case of Co both calculations agree with spin up and spin down electron difference N"4 for the ground state as well as for the molecular diagrams. Our calculated bond length of 1.96 As along with u "453 cm\ C are superior towards a ‘‘default’’ option with respect to the grid size used in that work. (See option ‘‘a’’ in Table I, Ref. [4]) though less accurate with respect to the very extended grid size used exclusively for the dimer (options ‘‘d’’ and ‘‘e’’ in Table I, Ref. [17]). The calculated
P. M!ynarski et al. / Vacuum 54 (1999) 143—149
GGA binding energy of Co in Ref. [17], 2.26 eV, is much greater than our result of 0.865 eV. In the light of experimental assumptions for Co binding energy being less than 1.2 eV [16] the result of these papers seem to be too high. Besides the known difficulties of the DFT approach to the multiplet atomic structure the spherical densities used in our calculations correctly predict the ds structure as the atomic ground state, so we strongly believe that our GGA value is the best theoretical estimate for Co binding energy reported so far. The use of very extensive grid sizes along with all electron basis set may produce a big BSSE error not calculated in Ref. [4]. The use of MP greatly reduces the BSSE corrections. In the case of Co we calculated only one structure, the equilateral triangle. Due to the use of spin densities along with odd number of electrons the symmetry has been lowered to C . The minimum in our case has been T obtained for N"7 with a Co—Co distance of 2.04 As . This result should be compared to the three GGA best structures obtained in works of Jamorski and Castro: D (N"5), C (N"7) and C (N"5) all lying within F T T 0.08 eV range with C (N"5) assigned for the ground T state. In the light of the discussion given in Ref. [4], on the Co ground state though our result N"7 agrees well with all electron 0.08 eV uncertainity. The last common ground for comparisons provides cobalt tetramer. The Co system shows an abundance of low-lying states and because we calculated only three of the most symmetrical structures so the comparison will be given to our chosen geometries. Our GGA D F (rhomb) structure with optimized geometry (2.19, 2.57 As ) and N"10 corresponds to the D (rhomb) structure of F paper 17 also with optimal N"10 and geometry (2.13, 2.73 As ). This cluster is the most stable one within all D structures calculated (see ‘‘GGA/GGA’’ column in Table VII of Ref. 17). Both remaining structures D and F T were optimized only on LSDA level in Ref. [17] B therefore our work provides the unique data for these two geometries within nonlocal GGA treatment. The Jahn—Teller effect has been very extensively studied on the LSDA level in work [17] while ours in a limited way exclusively via GGA nonlocal corrections in the case of D PD (rhomb) transition. Both Jamorski and F F Castro LSDA and ours GGA treatments correctly predict the D (rhomb) stability over the D structure by F F 0.12 and 0.29 eV, respectively. The present comparison confirms a good quality of the model potential used in our calculation along with the moderate grid size. Both factors are the only choice when greater transition metal clusters are investigated. Also, the prediction with respect to cluster magnetic moments, stability and cluster optimized geometries within several hundreds of an angstrom are valid. In our opinion, the present results and those taken from Ref. [4] when viewed together provide both an
147
extensive treatment of cobalt cluster with high level of exchange and correlation descriptions and also valuable experience within density functional methodology.
5. Analysis of the obtained results and conclusions The electronic, structural and magnetic properties of small cobalt clusters have been studied within the framework of the density functional method. The nonlocal (GGA) treatment of exchange and correlation effects within DFT scheme offers the most accurate description of a ‘‘many-body’’ problem in present day ab initio calculations upon transition metal systems. All clusters show an abudance of close-lying states. Therefore, for each cluster nuclearity, its ground state has to be assigned with care. The main purpose of this paper was to show how the basic properties of Co clusters evolve with the cluster size. We imposed some constrains in order to achieve a reasonably short computing time. The key among them was the assumed high symmetry of each cluster studied. The highly symmetric structures exhibit degeneracy of the highest occupied Kohn—Sham orbitals leading therefore to JT distortions. This was investigated partially in the case of a planar tetramer, obtaining results that throw an affirmative answer. Nevertheless, its energetics, as well as the changes of cluster magnetic moments, are small and our general conclusions are still valid. The main results are collected in Table 1. As it can be seen, the magnetic moment of Co clusters is enhanced when compared to its bulk value of 1.72k . This enhancement can be accounted for by an increased exchange splitting due to the decrease of the coordination number. On the other hand, the splitting of the majority and minority ‘‘bands’’ strongly depends on both cluster symmetry and Co—Co distances, thus the ‘‘size
Table 1 Electronic and magnetic properties of cobalt clusters as a function of cluster size Property Binding energy per atom (eV) Magnetic moment per atom (k ) ‘‘Kohn—Sham’’ IP (eV) Optimized Co—Co distance (As ) Population of d orbital
Co
Co
Co
Co
Co
Co
0.433
0.596
0.980
0.856
1.743
0.942
2.0
2.33
2.50
2.60
2.33
2.08
5.97
5.65!
5.91!
5.99!
6.01!
6.21!
1.96
2.04
2.29
2.35
2.28
2.44
8.06
7.87
7.58
7.56
7.51
8.05
The spin up/spin down electron difference N has been optimized. Eigenvalue of highest occupied Kohn—Sham orbital with opposite sign (!/ denote whether it is of majority/minority spin). Bulk Co—Co distance.
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P. M!ynarski et al. / Vacuum 54 (1999) 143—149
effect’’ with respect to the local magnetic moments has an oscillatory character. The change of bond length directly affects the bandwidths [15] thereby, in turn affecting the local magnetic moments. Therefore, the cluster geometry has to be optimized if the calculated magnetic moment is to be confronted with cluster experimental data. As the cluster size increases so does the ‘‘band’’ width. This can be clearly seen in Fig. 3 where the density-of-states (DOS) curves are sketched. The DOS has been obtained by a Gaussian broadening (0.2 eV) of each eigenenergy level within s—d—p manifolds. The cluster Fermi level presented in Fig. 3 as a dashed vertical line obtained by the integration of a DOS, lies between partially bonding and mainly antibonding spd orbitals specially in the minority spin. This gives a clear indication of the huge number of closely lying excited states for each cluster studied. Also, well below the E there is mainly an s-like peak, common $ to all clusters while the peaks closer to E have pre$ dominantly d character, and increasingly, antibonding properties. Another property, which is commonly studied in metal clusters, is the IP, which, as the cluster size increases, should converge to the bulk Fermi level. The IP is usually calculated as a difference between total energies of neutral and cationic systems. In the present
study, the IP is viewed as the eigenvalue of the highest occupied Kohn—Sham orbital and, except for the dimer, it has a weak monotonic rather than an oscillatory character with respect to cluster size while the binding energies are oscillatory along with it. The major factor here is the assumed symmetry of the cluster. It is cleary seen in Co , where the overlap of d orbitals is the strongest one due to their ‘‘head-to-head’’ positioning, thus resulting in the highest binding energy. This is also confirmed by the degree of the d orbitals localization obtained by Mulliken analysis. Our results for Co , when compared to other theoretical calculations [5] are markedly different with respect to the magnetic nature of the Co central atom. In our view, it is the least bulk-like one, with negative sp contributions to the net spin moment — a feature which has been recently observed for small Fe L clusters [18]. Also, the Mulliken population of the molecular levels shows a decreasing contribution of the central atom as their eigenvalues approach the cluster Fermi level. The eigenstates near the Fermi energy are mainly responsible for the creation of the magnetic moment, so its value for the central atom should be small. In the light of these arguments, the sp contibutions via sp—d hybridizations indirectly influence local magnetic moments though, undoubtedly, the role of 3d electrons is
Fig. 3. Density of states (DOS) curves for 3-D cobalt clusters. The vertical dashed lines indicate the DOS integrated Fermi level.
P. M!ynarski et al. / Vacuum 54 (1999) 143—149
dominant. As if the d orbitals were considered alone they would give the net spin moment of central atom 1.53k . Clearly, in small transition metal clusters, the sp as well as the spd hybridizations are very important and their presence has to be reflected in an appropriate basis sets. Finally, due to the big sensitivity of the local magnetic moments with respect to cluster environment, the subtle balance between mutually opposite XC rearrangements has to be fully taken into account via the high-level accuracy of the XC potentials used in self-consistent calculations.
References [1] Hohenberg P, Kohn W. Phys Rev B 1964; 136:864. [2] Kohn W, Sham LJ. Phys Rev A 1965;140:1133. [3] Castro M, Jamorski C, Salahub D. Chem Phys Lett 1997;271:133.
149
[4] Jamorski C, Martı´ nez, A, Castro M, Salahub D. Phys Rev B 1997;55:10 905. [5] Li Z, Gu B. Phys Rev B 1993;47:13 611. [6] M"ynarski P, Salahub DR. Phys Rev B 1991;43:1399. [7] Sambe H, Felton RH. J Chem Phys 1975;62:1122. [8] Dunlap BI, Connolly JW, Sabin JR. J Chem Phys 1979;71: 3396. [9] Andzelm J, Radzio E, Salahub DR. J Chem Phys 1985;83:4573. [10] Perdew JP, Wang Y. Phys Rev B 1986;33:8800. [11] Perdew JP. Phys Rev B 1986; 33:8822; Phys Rev B 1986;34:7406E. [12] Vosko SH, Wilk L, Nusair M. Can J Phys 1980;58:1200. [13] Gritsenko OV, Cordero NA, Rubio A, Balbas LC, Alonso JA. Phys Rev A 1993;48:4197. [14] M"ynarski P, Salahub DR. J Chem Phys 1991;95:6050. [15] Salahub DR. In: Lawley KP, editor. Ab-initio Methods in Quantum Chemistry — II, New York: Wiley, 1987. [16] Hales DA, Su C-X, Lian L, Armentrout PB. J Chem Phys 1994; 100:1049. [17] Bucher JP, Douglas DC, Bloomfield LA. Phys Rev Lett 1991; 66:3052. [18] Vega A, Dorantes-Da´vila J, Balba´s LC. Phys Rev B 1993;47:4742.
Properties and structure of small cobalt clusters. Nonlocal density functional calculations P. M"ynarski , M. Iglesias, M. Pereiro, D. Baldomir *, L. Wojtczak Department of Theoretical Chemistry, University of %o& dz& , Poland Facultade de Fı& sica, Universidad Santiago de Compostela, 15706 Santiago de Compostella, Spain Solid State Physics Department, University of %o& dz&, Poland
Abstract The results of self-consistent nonlocal (GGA) density functional calculations are reported for small cobalt clusters (n)6;13). An emphasis is made on a proper treatment of correlation effects. The magnetic as well as the bonding properties of these clusters are discussed in terms of a delicate balance between intra- and interatomic exchange and correlation. The enhancement of magnetic moments is analysed in terms of cluster symmetry. Finally, the present results are compared with other theoretical calculations obtained for these systems. 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction The rapid development of nanofabrication techniques followed by the discovery of some new and interesting effects such as the giant magnetoresistance as a primary example, provides another impact for the theoretical study of transition metal clusters. A serious challenge confronts those systems for any first principle calculation, due to the heavy demand of a proper description of both exchange and correlation effects. Usually, the size of the cluster as well as the high level of correlation treatment needed causes a rapid failure of traditional ab initio quantum chemistry methods based either on configuration interaction (CI) expansion of Slater determinants or Moller—Plesset perturbation techniques. Fortunately, the density functional theory (DFT) [1, 2] offers a computational alternative yet retaining the ab initio level. Among transition atom systems the cobalt clusters have been much less studied than others from the first transition row of elements. Recent papers devoted to this subject are works of Castro et al. [3], Jamorski et al. [4] and Li and Gu [5]. While the work of Li and Gu is based on local density aproximation (LDA), in the other papers a generalized gradient aproximation (GGA) is used where all electrons are treated explicitly. Here we present a systematic study of both electronic and magnetic
* Corresponding author.
properties of cobalt clusters Co (2)n)6; 13) using L a fully selfconsistent spin polarized nonlocal (GGA) Gausian density functional program [6], describing core electrons of cobalt atoms with a model potential (MP). The main purpose of the present work is to show how the electronic as well as magnetic properties evolve with the increase of cluster size. The geometry of all clusters, except Co , has been optimized. Also, for n"4 differ ent less symmetrical structures have been also fully optimized for the sake of possible Jahn—Teller effects. The GGA offers the state-of-the-art treatment of both exchange and correlation effects and, at the same time provides a computationally efficient and more precise way of evaluating molecular integrals by means of using a linear combination of Gaussian-type orbitals (LCGTO).
2. Computational details The present results have been obtained using a computational scheme of Sambe and Felton [7] later improved by Dunlap [8]. Despite the use of orbital basis sets this method introduces two auxiliary basis sets which are used in the self-consistent scheme to fit the charge density and exchange—correlation potential. However, after the self-consistent field (SCF) convergency is reached the exchange—correlation energy is recalculated in an analytical manner. The core electrons of cobalt atom are
0042-207X/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 2 - 2 0 7 X ( 9 8 ) 0 0 4 5 0 - 3
144
P. M!ynarski et al. / Vacuum 54 (1999) 143—149
described here by the use of a model potential (MP) in concert with 15 electron orbital basis sets. This MP has been optimized for a specific use in the DFT scheme [9] while the inner 3p shell has been moved into a basis set thus allowing the explicit treatment of important 3p—3d interactions in transition metal systems. Also, the use of MP minimizes the basis set superposition (BSSE) errors which should be taken into account when analysing close lying states on an all electron basis set level and this allows to keep the same computational treatment of cobalt clusters for all nuclearities studied. Having in mind the latter purpose the grid used in the fitting procedures consisted of 12 angular grids along with a 32 point radial mesh per atom. The GGA treatment of exchange—correlation was done through Perdew and Wang’s [10] nonlocal functional for exchange and Perdew’s [11] nonlocal functional for correlation along with the Vosko—Wilk—Nusair (VWN) parametrization of the correlation energy of a homogenous electron gas [12]. Details of the implementation of these density gradient nonlocal functionals as well as the corresponding nonlocal exchange correlation (XC) potentials are described in [6]. The orbital valence basis set with contraction pattern (2111/211>/311*) with (5,3,3;5,3,3) auxiliary set was chosen to be appropriate for all clusters. The present calculations upon cobalt atom predict the ds electronic configuration as the ground state placing its ds state 1.04 eV higher while the experimental s—d interconfigurational energy is 0.43 eV [13]. Although this result gives an improvement over its LDA value of !0.8 eV [13] it is still not satisfactory. The remaining error can be accounted for by either the need of a separate basis set optimization for both atomic states which can give a more appropriate value as it was shown in [14] for the Nickel atom or one has to keep in mind that the ground state of an atom with an open shell is described by a single Slater determinant with M "S and M "¸ 1 * where S, ¸ describe a given multiplet term with highest spin S and highest orbital angular momentum ¸. The latter deficiency can be partially removed by increasing the polarization space in an orbital basis set. Due to the use of model potential the relativistic corrections should be of somewhat lesser importance. In atomic calculations the electron density has been spherically averaged and its total energy ds state has been used as a reference for calculating binding energies of the clusters.
3. The electronic structure, geometry and magnetic properties of cobalt clusters We start the molecular calculations with cobalt dimer. Very few studies have been performed for this system. An excellent review for the Co as well as for other transition metal dimers has been given by Salahub [15]. From that reference a limited CI calculation cited gives the unbound
state with respect to the ds state of the Co atom with bond length of 2.56 As while LSD calculations give 2.07 As . In the present study we use a spin-polarized version of DFT. This means that for every possible difference between the number of spin-up and spin-down electrons (from now on denoted as N) we look for an electronic configuration giving the lowest energy. Also, in all molecular systems we use symmetry rules for reducing the number of molecular calculations. In the case of a dimer, a D symmetry has been adopted though we assign the F electronic configuration in terms of the infinite symmetry group. The ground state has been found for N"4 with the following electron assignment 1p1n2p1d1n1p E S E S E S giving the bond distance of 3.71 bohr and binding energy of 0.865 eV (or 0.433 eV per atom — in latter calculations the binding energy per atom will be given as being more appropriate for viewing its dependence on cluster nuclearity). By making the following promotion: 1di into S 1pi we also optimized this excited state which gave the S same optimized bond distance of 3.71 bohr. Also, for the equilibrium separation of the ground state another excited state has been calculated by promoting electron 1pi S into 1ni. The adiabatical and vertical energy differences S in these cases were 0.213 an 0.531 eV, respectively. The potential energy curve of the ground state near equilibrium has been fitted using a third-degree polynomial in order to obtain vibrational frequency as well as the anharmonicity term. The calculated values are 452.9 cm\ for u and 1.90 for u x . This value of vibraC C C tional frequency seems to be too large but this can be accounted for by a ‘‘cooperative’’ effect of moderate grid size and MP use. The value of binding energy can be compared to the estimated experimental value [16] (1.2 eV. The highest occupied orbital for the Co ground state is the 1b of minority spin with its eigenS value !5.97 eV which in the view of DFT [2] has its physical interpretation as being the ionization potential (IP) of a system (taken with the reversed sign). It is worthwhile to mention here that although the highest occupied Kohn—Sham orbital is the only one having a direct physical meaning while all the remaining ones are just pure mathematical constructions their numerical similarity to real orbitals is used to make estimations of the degree of (sp;sd) hybridizations by means of Mulliken population analysis as well as plots of orbitals which, in turn, provides help in the analysis of the bonding mechanisms in molecular structures. The atomic state of Co in a dimer has been evaluated as s p d thus closely resembling the ds state. In Co case only one structure has been studied: the equilateral triangle; aditional symmetries are being calculated and will be published elsewhere. Using the C symmetry the ground state has been found for T electron spin up/spin down difference N"7 with Co—Co equilibrium distance 3.86 bohr. The binding energy per atom is 0.56 eV and the Co atomic state is estimated as
P. M!ynarski et al. / Vacuum 54 (1999) 143—149
s p d . The bonding is mainly due to the s electrons though as it can be seen in Fig. 1 there is some admixture of d-type electrons in the bonding molecular orbitals. The cobalt tetramer has been inspected in a more detailed way than other cobalt clusters where only the most symmetric structures have been extensively examined. At first we investigated the tetrahedral structure. Even with the imposed T symmetry constraint there is B a huge number of closely lying states and care has to be taken in order to properly assign the ground state. The one we have found is for N"10 with optimized Co—Co
Fig. 1. Plots of the 2ai (up) and 1aj (down) molecular orbitals of Co cluster.
145
distance of 4.33 bohr and binding energy per atom equal to 0.98 eV. The highest occupied Kohn—Sham orbital is of T symmetry and of majority spin hav ing 2 electrons, thus being a candidate for the Jahn— Teller distorsion. The atomic state electron configuration in Co is s p d . Next, we have assumed a square geometry for the cobalt tetramer imposing the D F constraint. The optimized Co—Co distance was also equal to 4.33 bohr with N"10. The adjacent spin up/spin down differences N"8 and N"12 produced less stable states which were placed 0.56 and 1.85 eV above the ground state, respectively (vertical values). The optimized T structure is more stable than the optimized D one by B F
Fig. 2. Plots of the 3ei orbital of a Co (D ) cluster. S F
146
P. M!ynarski et al. / Vacuum 54 (1999) 143—149
0.96 eV. The analysis of eigenvalue spectra of Co in the square geometry shows that the highest occupied orbital containing 1 electron has E symmetry (Fig. 2) thus the Jahn-Teller effect is possible here so the symmetry has been lowered from D to D and the rhombus structure F F has been optimized. Ground state has been found for Co—Co distance equal to 4.14 bohr along with 72° angle for N"10. Indeed, the optimized D structure is more F stable than the square geometry by 0.29 eV. Clearly, by choosing the square geometry, other distortions are possible though due to the rather small energy differences between different Jahn—Teller structures the quality of both the grid and basis sets must be enhanced possibly with BSSE corrections. On the other hand the change of geometry may imply the change in N therefore changing the magnetic moments. That, in turn, is directly connected with a rather big rearrangement of exchange and correlation energies so the Jahn—Teller effects in transition metal clusters must be discussed along with the level of exchange—correlation treatment. The geometry of Co has been assumed as an equilat eral trigonal bipyramid, so the C symmetry has been T adopted in order to reduce the time of computations during geometry optimizations. The lowest state has been obtained for N"13 and a Co—Co distance of 4.45 bohr although another state with N"11 has also been optimized leading to the same bond distance as for N"13. This state lies only 0.34 eV higher than the ground state and possible Jahn—Teller distortions may lead to a new state with, after geometry optimization process, an increased width of a d ‘‘band’’, leading to the partial collapse of magnetic moment, thus yielding the ground state with a smaller local magnetic moment. The Mulliken population analysis gives an average atomic hybridization of Cobalt as s p d . The Co cluster has been assumed to have a square bipyramid geometry within O symmetry constraint. The F lowest state has been obtained for a Co—Co distance of 4.33 bohr and N"14. This time the highest occupied orbitals are of T and T symmetry for majority and S E minority spin, respectively, with its full occupancy of 3 electrons. This state is strongly bound giving the binding energy per atom as 1.74 eV. Such bond strength can be explained considering the directional nature of d orbitals. The present structure makes the d orbitals point to each other directly thus giving the strongest overlap among all clusters studied. This is also confirmed by auxiliary analysis of Mulliken population giving an average atomic hybridization as s p d . The last cluster studied was Co within O symmetry. F Its geometry has not been optimized and the assumed Co—Co distance between the central atom and its 12 neighbours corresponds to the Co bulk value of 2.44 As . However, the spin up/spin down electron difference N has been optimized in order to get a proper account for the average magnetic moment. The lowest energy was
obtained for N"27 while the states with N"25 and N"29 were placed higher by 1.14 and 0.13 eV, respectively. One has to notice the small difference between the states with N"27 and N"29. If the geometry had been optimized, the state with N"27 would be more favoured and though there would be a tendency of the magnetic moment to collapse due to the increase of the band width, it is very likely that N"27 would be preferred for optimal geometry due to the over 1 eV separation with respect to the N"25 state. The obtained average magnetic moment 2.08 k is in complete agreement with the extrapolated experimental value [17]. However, there is a big difference between local magnetic moments of the central atom and the neighbouring ones. The central Co atom is the least bulk-like. It has an excess of charge and a very small magnetic moment, 0.56 k with large negative sp contribution almost!1 k while for the ‘‘surface’’ atoms their magnetic moment is enhanced: 2.204 k where the d contribution to this value is essential and is 1.92k . Using the Mulliken population analysis of the eigenvalue spectra the estimated d ‘‘band’’ width is 3.65 eV for spin up and 3.12 eV for spin down. The lowest molecular level is an s-like with p-type contributions and eigenvalues for different spins yield an sp exchange splitting of 0.17 eV. The ‘bottom’ of d-type molecular levels is split by 0.33 eV.
4. Model potential and all electron basis set calculations In this section a comparison between our results and all electron basis set calculations [4, 3] will be made. Our work extends to larger cluster nuclearities up to Co . Therefore, the common ground for comparison is limited only to Co in the most extensive way and with some limitations to Co and Co due to the choice made in our study where only the most symmetric structures have been chosen. Our data obtained by the use of a model potential will be confronted with the Jamorski et al. and Castro et al. results obtained through the all electron basis sets along with the use of very extensive grid sizes. While the latter treatment is appropriate for small transition metal clusters, the former method used in our work to study greater cluster nuclearities is the only one computationally manageable. The nonlocal GGA level has been kept the same in both approaches. In the case of Co both calculations agree with spin up and spin down electron difference N"4 for the ground state as well as for the molecular diagrams. Our calculated bond length of 1.96 As along with u "453 cm\ C are superior towards a ‘‘default’’ option with respect to the grid size used in that work. (See option ‘‘a’’ in Table I, Ref. [4]) though less accurate with respect to the very extended grid size used exclusively for the dimer (options ‘‘d’’ and ‘‘e’’ in Table I, Ref. [17]). The calculated
P. M!ynarski et al. / Vacuum 54 (1999) 143—149
GGA binding energy of Co in Ref. [17], 2.26 eV, is much greater than our result of 0.865 eV. In the light of experimental assumptions for Co binding energy being less than 1.2 eV [16] the result of these papers seem to be too high. Besides the known difficulties of the DFT approach to the multiplet atomic structure the spherical densities used in our calculations correctly predict the ds structure as the atomic ground state, so we strongly believe that our GGA value is the best theoretical estimate for Co binding energy reported so far. The use of very extensive grid sizes along with all electron basis set may produce a big BSSE error not calculated in Ref. [4]. The use of MP greatly reduces the BSSE corrections. In the case of Co we calculated only one structure, the equilateral triangle. Due to the use of spin densities along with odd number of electrons the symmetry has been lowered to C . The minimum in our case has been T obtained for N"7 with a Co—Co distance of 2.04 As . This result should be compared to the three GGA best structures obtained in works of Jamorski and Castro: D (N"5), C (N"7) and C (N"5) all lying within F T T 0.08 eV range with C (N"5) assigned for the ground T state. In the light of the discussion given in Ref. [4], on the Co ground state though our result N"7 agrees well with all electron 0.08 eV uncertainity. The last common ground for comparisons provides cobalt tetramer. The Co system shows an abundance of low-lying states and because we calculated only three of the most symmetrical structures so the comparison will be given to our chosen geometries. Our GGA D F (rhomb) structure with optimized geometry (2.19, 2.57 As ) and N"10 corresponds to the D (rhomb) structure of F paper 17 also with optimal N"10 and geometry (2.13, 2.73 As ). This cluster is the most stable one within all D structures calculated (see ‘‘GGA/GGA’’ column in Table VII of Ref. 17). Both remaining structures D and F T were optimized only on LSDA level in Ref. [17] B therefore our work provides the unique data for these two geometries within nonlocal GGA treatment. The Jahn—Teller effect has been very extensively studied on the LSDA level in work [17] while ours in a limited way exclusively via GGA nonlocal corrections in the case of D PD (rhomb) transition. Both Jamorski and F F Castro LSDA and ours GGA treatments correctly predict the D (rhomb) stability over the D structure by F F 0.12 and 0.29 eV, respectively. The present comparison confirms a good quality of the model potential used in our calculation along with the moderate grid size. Both factors are the only choice when greater transition metal clusters are investigated. Also, the prediction with respect to cluster magnetic moments, stability and cluster optimized geometries within several hundreds of an angstrom are valid. In our opinion, the present results and those taken from Ref. [4] when viewed together provide both an
147
extensive treatment of cobalt cluster with high level of exchange and correlation descriptions and also valuable experience within density functional methodology.
5. Analysis of the obtained results and conclusions The electronic, structural and magnetic properties of small cobalt clusters have been studied within the framework of the density functional method. The nonlocal (GGA) treatment of exchange and correlation effects within DFT scheme offers the most accurate description of a ‘‘many-body’’ problem in present day ab initio calculations upon transition metal systems. All clusters show an abudance of close-lying states. Therefore, for each cluster nuclearity, its ground state has to be assigned with care. The main purpose of this paper was to show how the basic properties of Co clusters evolve with the cluster size. We imposed some constrains in order to achieve a reasonably short computing time. The key among them was the assumed high symmetry of each cluster studied. The highly symmetric structures exhibit degeneracy of the highest occupied Kohn—Sham orbitals leading therefore to JT distortions. This was investigated partially in the case of a planar tetramer, obtaining results that throw an affirmative answer. Nevertheless, its energetics, as well as the changes of cluster magnetic moments, are small and our general conclusions are still valid. The main results are collected in Table 1. As it can be seen, the magnetic moment of Co clusters is enhanced when compared to its bulk value of 1.72k . This enhancement can be accounted for by an increased exchange splitting due to the decrease of the coordination number. On the other hand, the splitting of the majority and minority ‘‘bands’’ strongly depends on both cluster symmetry and Co—Co distances, thus the ‘‘size
Table 1 Electronic and magnetic properties of cobalt clusters as a function of cluster size Property Binding energy per atom (eV) Magnetic moment per atom (k ) ‘‘Kohn—Sham’’ IP (eV) Optimized Co—Co distance (As ) Population of d orbital
Co
Co
Co
Co
Co
Co
0.433
0.596
0.980
0.856
1.743
0.942
2.0
2.33
2.50
2.60
2.33
2.08
5.97
5.65!
5.91!
5.99!
6.01!
6.21!
1.96
2.04
2.29
2.35
2.28
2.44
8.06
7.87
7.58
7.56
7.51
8.05
The spin up/spin down electron difference N has been optimized. Eigenvalue of highest occupied Kohn—Sham orbital with opposite sign (!/ denote whether it is of majority/minority spin). Bulk Co—Co distance.
148
P. M!ynarski et al. / Vacuum 54 (1999) 143—149
effect’’ with respect to the local magnetic moments has an oscillatory character. The change of bond length directly affects the bandwidths [15] thereby, in turn affecting the local magnetic moments. Therefore, the cluster geometry has to be optimized if the calculated magnetic moment is to be confronted with cluster experimental data. As the cluster size increases so does the ‘‘band’’ width. This can be clearly seen in Fig. 3 where the density-of-states (DOS) curves are sketched. The DOS has been obtained by a Gaussian broadening (0.2 eV) of each eigenenergy level within s—d—p manifolds. The cluster Fermi level presented in Fig. 3 as a dashed vertical line obtained by the integration of a DOS, lies between partially bonding and mainly antibonding spd orbitals specially in the minority spin. This gives a clear indication of the huge number of closely lying excited states for each cluster studied. Also, well below the E there is mainly an s-like peak, common $ to all clusters while the peaks closer to E have pre$ dominantly d character, and increasingly, antibonding properties. Another property, which is commonly studied in metal clusters, is the IP, which, as the cluster size increases, should converge to the bulk Fermi level. The IP is usually calculated as a difference between total energies of neutral and cationic systems. In the present
study, the IP is viewed as the eigenvalue of the highest occupied Kohn—Sham orbital and, except for the dimer, it has a weak monotonic rather than an oscillatory character with respect to cluster size while the binding energies are oscillatory along with it. The major factor here is the assumed symmetry of the cluster. It is cleary seen in Co , where the overlap of d orbitals is the strongest one due to their ‘‘head-to-head’’ positioning, thus resulting in the highest binding energy. This is also confirmed by the degree of the d orbitals localization obtained by Mulliken analysis. Our results for Co , when compared to other theoretical calculations [5] are markedly different with respect to the magnetic nature of the Co central atom. In our view, it is the least bulk-like one, with negative sp contributions to the net spin moment — a feature which has been recently observed for small Fe L clusters [18]. Also, the Mulliken population of the molecular levels shows a decreasing contribution of the central atom as their eigenvalues approach the cluster Fermi level. The eigenstates near the Fermi energy are mainly responsible for the creation of the magnetic moment, so its value for the central atom should be small. In the light of these arguments, the sp contibutions via sp—d hybridizations indirectly influence local magnetic moments though, undoubtedly, the role of 3d electrons is
Fig. 3. Density of states (DOS) curves for 3-D cobalt clusters. The vertical dashed lines indicate the DOS integrated Fermi level.
P. M!ynarski et al. / Vacuum 54 (1999) 143—149
dominant. As if the d orbitals were considered alone they would give the net spin moment of central atom 1.53k . Clearly, in small transition metal clusters, the sp as well as the spd hybridizations are very important and their presence has to be reflected in an appropriate basis sets. Finally, due to the big sensitivity of the local magnetic moments with respect to cluster environment, the subtle balance between mutually opposite XC rearrangements has to be fully taken into account via the high-level accuracy of the XC potentials used in self-consistent calculations.
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