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Theoretical treatment of the electronic structure of small metallic particles
Surface Science 106 (1981) 225-238 North-Holland Publishing Company
THEORJiYl’ICAL TREATMENT METALLIC PARTICLES
OF THE ELECTRONIC
STRUCTURE
OF SMALL
R.P. MESSMER General Electric Company, CorporateResearch and Development, Schenectady, New York 12301, USA Received 8 September 1980
A brief review is given of various theoretical methods which have been applied to small metal particles to elucidate their electronic structure, This survey includes discussions of simple models and semi-empirical methods as well as first principles methods. With a few exceptions, these theoretical methods are carried over from molecular physics, where they have been used to study the electronic structure of molecules. From a theoretical perspective, metal clusters are of great interest for two reasons. The first reason is that the intrinsic electronic properties of such clusters are of fundamental importance. The second reason is that such clusters might serve as models for understanding localized electronic phenomena in bulk metals. Theoretical work which considers clusters from both of these viewpoints will be presented.
1. Introduction The most obvious characteristic of small metal particles is that a large percentage of the atoms in the particle are surface atoms. For example, if we assume a particle has a simple cubic geometry, then such a particle which contains 1000 atoms will have 49% surface atoms. Thus, any particle containing fewer than roughly 1000 atoms will contain over 50% surface atoms. Most of the small metal particles (hereafter referred to as “clusters”) to be discussed below contain far fewer than 1000 atoms, and hence the question of surface effects and their contribution to observed properties is a very important one. This leads one to ask how many atoms a cluster should contain in order to exhibit bulk-like properties. If one uses the fraction of atoms on the surface as a criterion for convergence to the bulk, then (using the simple cubic cluster geometry again) even with a million atoms (corresponding to a particle diameter of a few hundred Angstroms) 6% of the atoms are on the surface. Thus one might suppose that clusters of this size are necessary to attain reasonable convergence to the bulk. However, it should be clear that one cannot discuss the convergence of cluster properties to the bulk values in such simple terms. There will be some properties, which arise from local electronic structure effects, which may require relatively few atoms to describe. On the other hand to obtain the properties of the bulk Fermi surface may require very many atoms. Thus in general, no simple statement regarding the convergence of cluster properties can be made. The rate of convergence to the bulk values of a property will depend upon the property. 0039-6028/81/0000-0000/$02.50
@ North-Holland
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R.P. Messmer I Theoretical treatment of electronic structure
A few general comments about the electronic structure of metal clusters and theoretical methods would be appropriate at this point. As many of the atoms in a cluster are surface atoms and these surface atoms may have a significantly different potential than those in the interior of the cluster, one expects the inclusion of the effects of a self-consistent field (SCF) in the calculation to be essential for this type of problem. Likewise, the effects of a substrate or support on the electronic structure of a cluster can be appreciable if the contact is over a reasonably large fraction of the cluster’s surface area. Thus one must be cautious in comparing even SCF calculations for isolated clusters with experimental results on supported metal particles, unless the effects of the support are explicitly known. There have been numerous theoretical investigations of metal clusters using many theoretical techniques. In general, the more sophisticated the theoretical method is, the fewer the atoms which can be treated. In comparison to the understanding of electronic structure in molecules and solids, very little is known about clusters. Thus there is a need for both simple and sophisticated models. With simple models one can investigate the qualitative behavior of relatively large clusters; with sophisticated models one hopes to obtain quantitative information on rather small clusters. However, the information obtained from the latter can be used to refine or re-parameterize the simple models. In table 1, a summary of some theoretical methods and their application to metal clusters is given. It includes the various methods discussed by authors at this conference. Most of these methods have the molecular orbital approach as their starting point. Due to time limitations, the present author’s own research interests and the convenience of discussing a number of methods from a common framework, the discussion will be restricted in the following to those methods which derive from the molecular orbital viewpoint.
Table 1 Theoretical techniques applied to the investigation of metal clusters Theoretical technique
Investigators
Simple models : Cubium/simple Hiickel theory [l, 2) Modified jellium [6,7] Parameterized LCAO (9-111 Model Hamiltonians (Anderson [ 161, Hubbard [17j, etc.)
Messmer [3]; Bilek and Scala [4,5] Martins, Car and Buttet [8] Baetzold [ 12-141; Khanna et al. [ 151 Joyes [ 181
First principles mod&: Pseudopotentials Xcr method [23,24] Hartree-Fock [28] Configuration interaction [32]
Stall et al. [21]; Car and Martins [22] Messmer et al. [25,26]; Salahub and Messmer [27] Schwartz and Quinn [29]; Basch and Newton [30]; Veillard [31] Noel1 et al. [33]
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In the next section some specific examples of intrinsic properties of clusters will be discussed, following which a few cases of calculations which have employed clusters as models of bulk or surface properties will be presented.
2. Intrlnslc
properties of clusters
A primary problem which plagues all electronic structure calculations of clusters is the lack of information on the geometrical structure (i.e. the positions of the atoms) of the clusters. In most situations in molecular and solid state physics, this structural information is known at the outset. However, for the case of clusters one is forced to assume “reasonable structures” in order to proceed. The usual wisdom is to assume highly symmetrical clusters (e.g. spherical) reflecting the bulk structure and to assume internuclear distances close to the bulk values. Neither of these assumptions is likely to be true for extremely small clusters, that is, on the order of 10 atoms and less. Ideally it would be desirable to have the electronic structure calculations determine the geometry of the clusters. However, at present, this appears to be an almost impossible task. The simple models which can treat reasonably large clusters (of the order of 100 atoms) are not accurate enough to reliably determine the small energy differences between various geometric configurations. On the other hand, the sophisticated models are so computationally complex that only a few atoms can be treated reliably. 2.1. Cluster density of states Keeping in mind the discussion above, one must proceed on reasonable assumptions to investigate the intrinsic electronic properties of clusters. One of the most readily accessible theoretical characteristics of a metal cluster is its density of states (DOS). A number of studies on metal clusters using at least three different theoretical methods have discussed the calculated DOS. Results from three different methods, the extended Hiickel [12], X&-scattered wave [25] and Hartree-Fock [31] approaches, will be discussed. First, it would be useful to briefly characterize these methods. The extended Hiickel method is a semi-empirical LCAO molecular orbital approach which is defined by its Hamiltonian matrix elements. The diagonal elements are set equal to -4, where 4 is an atomic orbital ionization energy deduced from experiment and the off-diagonal ionization energy deduced from experiment and the off-diagonal elements are proportional to an average of two such terms multiplied by an overlap integral between the appropriate atomic orbitals. Thus the specification of the atomic orbital functions is the critical input to the calculation. The method has been very valuable as an interpolation and extrapolation aid in understanding the electronic structures of series of molecules particularly in organic chemistry. The XcY-scattered wave (Xcu-SW) method is a first-principles self-consistent-field molecular orbital approach which employs Slater’s Xa exchange-correlation approximation and the muffin-tin approximation to the potential. Except for the change of boundary conditions the method is the same as the KKR method of band theory. The latter has given
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R.P. Messmer I E3eoreticaltreatmentof electronicstructure
very accurate descriptions of the band structure of metals. The Xc&W method has been widely applied to transition metal-containing molecules and treatments of metal-metal bonding in inorganic chemistry with considerable success [34]. The Hartree-Fock method is well known in the context of molecules, because of the numerous LCAO molecular orbital calculations which have employed it. Less familiar to those engaged in molecular physics are the problems of Hartree-Fock theory as applied to the electron gas [35] or in fact any metal [36]. A prime failure of the Hartree-Fock method as applied to metals is that it predicts a vanishing density of states at the Fermi level. Furthermore, applications to inorganic chemistry have shown problems with the method’s ability to describe metal-metal bonding between transition metal atoms. There has also been considerable recent discussion on the vital matter of the determination of adequate basis sets needed for the description of d-orbitals of first row transition metals [33,37]. The only metal for which calculations have been carried out by all three methods is copper, and thus we are restricted to a comparison of results for Cu clusters. The Hartree-Fock [31] and Xa-SW [2.5] calculations employed Cu13 clusters, the extended Hiickel calculations [12] employed a Cu 19 cluster. The calculated distribution of orbital energies or “density of states” obtained from these different methods on Cu clusters are shown in fig. 1. The disagreement among the methods is rather profound. For each method the orbitals which are predominantly Cu 4s are labelled “s”; and likewise, the d-band is shown as a rectangular area. For the case of the Hartree-Fock calculation, the d-band 1
I CUl3
HARTREE-FOCK
Gl3
%9
X,-SW
EXTENDEDHUCKEL
-5
q
S,d
-S
-s
-d -s
_/e9 -S
-s
-‘ep
-S
Fig. I. Comparison of orbital energies for copper clusters as determined by the Hartree-Fock, Hiickel methods.
Xa-SW and extended
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which is -3.0 eV wide is entirely below the s-band. The Xa-SW result, which has a d-band width of -2.5 eV is found in the midst of the s-band and the s character in this region is strongly hybridized with Cu 3d character. This situation is analogous to that found in the bulk metal. The extended Hiickel result for Cu19shown in the third column of fig. 1, also has the d-band completely overlapped by the s-band, but the d-band width is less than 0.2 eV. This latter value is an order of magnitude less than that obtained in the other two calculations. The narrow d-band is probably an artefact of the basis functions chosen to describe the d-orbitals, because similar calculations for Pd [12] gave a similarly narrow d-band. However, when different basis functions were used for Pd the d-band width increased dramatically [13]. If more realistic atomic functions for Cu were employed in the extended Huckel method, the results could possibly be brought into qualitative agreement with those of the Xcr-SW results. There are certainly significant discrepancies between the Xa-SW and Hartree-Fock results for Cui3. One point of agreement is that an e, and a t2, level, which are largely d-orbitals localized on the central atom, are split off from the rest of the d-band. It clearly would be extremely valuable to perform experiments on beams of clusters (of the size being discussed here), in order to determine which, if either, of the distributions in the first two columns of fig. 1 are valid. One particularly useful experiment in this regard, if it could be done, would be photoelectron spectroscopy. 2.2. Cluster magnetism One of the fascinating aspects of investigating transition metal clusters is the prospect of finding unusual magnetic effects - again due to the large surface to volume ratio of such clusters. Of the cluster calculations carried out to date using molecular-based methods, only the X&W method has been used to investigate the magnetic properties of various metal clusters. The first Xcr-SW work on transition metal clusters [12] was concerned with Nis, Ni13, Pd13 and Pt13. It was found that Nis and Nil3 clusters were magnetic while Pd13 and Pti3 were not. For the Nis cluster the calculated average spin magneton number per atom was 0.25, whereas for the Nil3 cluster the value was 0.46. These may be compared to the bulk value of 0.54. It should be pointed out that the spin density on the central atom is quite different from that found on the 12 surface atoms. Further discussion of such effects in a variety of metal clusters are presented in another contribution to this conference [27]. It is rather remarkable that the 13-atom clusters of Ni, Pd and Pt which all have the same number of valence electrons turn out to be magnetic for Ni and non-magnetic for Pd and Pt, and thus mimic the behaviour found in the bulk metals. One cannot expect that this will be a general phenomenon however. In fact, recent calculations on a 15atom Cr cluster have shown that such clusters carry a net magnetic moment. Bulk chromium is known to be antiferromagnetic, and the cluster results try to mimic this behavior to a large degree but due to surface effects there is not an exact cancellation of moments of opposite spin orientation. A more complete discussion of the interesting behavior of this cluster is given elsewhere [27].
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@earntent
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structwe
Calculations on Fe clusters have also been carried out [38,39]. For this case it has been observed that clusters of four, nine and fifteen atoms have average spin magneton numbers per atom which exceed the bulk value. However, the values decrease as a function of cluster size.
3. Clusters as models In this section we consider the theoretical use of clusters as models for focalized surface or bulk phenomena by discussing a few examples. Perhaps the most common application of this view point is in the study of chemisorption of atoms and molecules on metals. A review of such applications and of this view point has been given recently by the author [26]. Here, we can discuss only rather briefly three such studies related to chemisorption.
3.1. chemisotption A recurring theme through the three examples, is the question of how many atoms the cluster must contain in order to give a reasonably converged description of the electronic structure of the actual surface or the actual su~ace-admolecule interaction. Let us first consider a simple theoretical method which has been used to discuss the local density of states (LDOS) on the surface and in the interior of both a semi-infinite solid and a cluster. In the context of the semi-infinite solid the theoretical method has been referred to as cubium [l, 401, whereas in the cluster context [3-51 it is simple Hiickel theory. One assumes a simple cubic array of atoms with one s-orbital per site. The Hamiltonian consists of equal diagonal elements which may be taken as zero for convenience, and an offdiagonal element or “hopping integral” between nearest neighbors as familiar in tight binding calculations. In fig. 2a, the LDOS of the semi-infinite cubium model for atoms in three different planes are shown. The curves labelled (i, j, l), (i, j, 2) and (i, j, 3) correspond to the LDOS of atoms in the top, second and third (100) planes of the crystal, respectively. The same model can be solved analytically for finite clusters of arbitrary size [3]. However for such finite clusters, discrete energy levels are obtained. Thus in order to compare LDOS results with the semi-infinite case, each level is replaced by a gaussian of width 0.15 (in units of 2t, where t is the nearest neighbor matrix element). The resultant LDOS curves for a 9 X 9 X 9 cube of atoms is shown in fig. 2b. The curves labelled (5,5,1), (5,5,2) and (5,5,3) correspond to the LDOS of the central atom in each of the top, second and third (100) planes of the cubic cluster, respectively. Comparison of the two sets of curves shows that the cluster results reproduce the main features of the semi-infinite results, although “fine structure” features can be discerned in some of the cluster curves, showing that the discreteness of the eigenvalue spectrum continues to manifest itself. Nonetheless, the agreement is quite striking and for many purposes the cluster results would be more than adequate. Thus, for this particular property, one could conclude that the ?19-atom cluster constitutes a “usefully converged”
R.P. Messmer 1 Theoretical treatment of electronic
(0)
structure
231
l=(i,j,31
-9259 I:(i,i,Z
1
L_
r’_
”
fzCi,j,l)
2
/# Y-L_
g
‘I/: -3 -2
-I
0
I
2
3
ENERGY iIN UNITS OF 21)
Y
I
I
I
I
I
-3 -2 -I 0 I 2 3 ENEfW(IN UNITSOF21)
Fig. 2. Comparison of local density of states (LDOS) for three different sites as determined by: (a) the semi-infinite cubium modeland @)the simpleHiickelcluster model with a cluster of 729 atoms.
of the semi-infinite result; this, in spite of the fact that in an absolute sense this cluster is still quite far from the semi-infinite system -having only 129 discrete energy levels and 95% of the proper band width. The fact that such a large cluster is needed to achieve this degree of agreement with the cubium result, is largely due to the simplicity of the model, i.e. only one orbital per site and a high degeneracy of orbitals due to a symmetrical cluster. This problem of high degeneracy of cluster orbitals is also brought out in the work of Martins, Car and Buttet [S] using their variational spherical model for clusters. The second example deals with the chemisorption of 0 atoms on an Al(100) surface, as modelled by clusters of Al atoms [41]. It was assumed that an oxygen atom would chemisorb in a four-fold hollow site on the surface and clusters of 5, 9 and 25 Al atoms were used to model the (100) surface. X&W calculations were performed for the oxygen atom at 0.0, 2.0, and 4.0 bohr (1 bohr = 0.5292 A) above the surface plane of aluminum atoms. The calculations at 0.0 bohr were used to simulate the incorporation of the oxygen atom into the Al lattice. Various DOS and LDOS curves were generated and compared with photoelectron spectra available at the time. It was found that only the 25 atom Al cluster provided a reasonably converged description of the metal-adsorbate interaction. This conclusion is consistent with a study of convergence of the DOS for pure Al clusters containing up to 43 atoms [42].
representation
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As the XCI-SWmethod, in its presently employed form, calculates only an approximate total energy, it cannot be used reliably to compute the equilibrium metal-adsorbate distance. Thus, such info~ation must be arrived at by indirect means. That is, one must calculate some properties of the system as a function of metal-adsorbate separation and compare these to the experimentally observed properties in order to deduce the most likely separation. For oxygen atoms chemisorbed on aluminum, two types of information were available with which to make a comparison with theory. These were photoelectron spectra of the valence region (at three different photon energies) and the Al 2p level shifts due to chemiso~tion of oxygen as determined by phot~lectron s~ctroscopy. From a comparison of the theoretical results with the experimental information, it was concluded that the oxygen atoms did not sit on the Al surface but were incorporated into the Al lattice, i.e. the best agreement with experiment was obtained at the metal adsorbate distance d = 0.0 bohr. Fig. 3 shows the calculated local DOS on the oxygen atom for the AlzsO cluster, where the discrete levels have been replaced by a gaussian as in the previous case discussed above. Fig. 4 shows the photoelectron spectra for clean aluminum and for two rather low coverages of oxygen [43]. A comparison of the calculated and observed positions of structure in the spectra shows rather good agreement. By contrast the calculated result for the Al50 cluster (also at d = 0.0) shows only two peaks in the LDOS - one at 2.5 eV, the other at 7.0 eV below EF. The A12p level shifts which take place on interaction with an oxygen atom were also calculated using the Alz50 cluster. The results are shown in table 2. Again the results for lattice incorporation of oxygen (i.e. d = 0.0) are more consistent with the experimental findings [44] than are those for oxygen above the surface (i.e. d = 2.0). /
---
Fig. 3. Xa-SW cluster results for the local density of states on the oxygen atom in an Al=0 cluster in which the oxygen atom is coplanar with the (100) surface layer of aluminum atoms (i.e. d = 0.0).
R.P. Messmer I llzeoretical treatment of electronic structure
233
1
Fig. 4. Photoelectron spectra for oxygen chemisorption exposures of 2 and 10 L of oxygen; see ref. [43].
on Al(100) surface. Spectra are for clean Al surface and for
These calculations also were able to describe the nature of the bonding between oxygen and aluminum and to provide a detailed account of the origin of the structure in the photoelectron spectrum. They also demonstrated how such results can change with cluster size and the importance therefore of investigating the convergence of cluster results. Subsequent work has also investigated oxygen chemisorption on the (111) surface and the (110) surface [46]. As the third and final example of the use of metal clusters in elucidating surfaces and their interaction with adsorbates, we consider the chemisorption of the CO molecule on the Cu(100) surface. Experimentally, one observes in the X-ray photoelectron spectrum in both the C 1s and 0 1s core regions two rather intense peaks displaced to higher binding energy from the main peak. In the isolated CO molecule these intense satellites do not occur. Hence, the question: what in the chemisorption process leads to these satellite peaks? The explanation deduced from Xcr-SW cluster calculations will be briefly described. As the Xa-SW method treats all electrons of the system it is thus very convenient to discuss core level spectroscopy within this framework. Calculations were performed for Table 2 Calculated Al 2p level shifts due to chemisorption Atom ”
d = 2.0
d = 0.0
Expt. b,
AI(l) Al(4)
0.30 0.65
1.21 1.05
1.30 1.30
of oxygen (in eV)
a) Al(l), atom just beneath the oxygen atom; A](4), one of the four atoms on the surface plane closest to the oxygen atom, b, See ref. [44].
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CL&O and Cu&O clusters with the CO molecule in l-fold and 4-fold sites on the Cu cluster. Again, in order to investigate how the calculated properties depended on metal cluster size, 5atom and 9-atom clusters were investigated [47,48]. In this instance, it was found that the CusCO cluster was sufficient to obtain all the basic physics of the problem, although the Cu&O cluster does provide a more complete description. Further, although the l-fold site using the Cu&O cluster provides the best quantitative agreement with experiment, the qualitative aspects are very much the same as for the 4-fold Cu5C0 results. As a consequence, the latter cluster results [47] will be used to illustrate the basic physical conclusions of these studies. Fig. 5 shows the X&W orbital energies for the Cu&O cluster. The orbitals primarily derived from the isolated CO molecular orbitals are labelled by the CO orbital designations in parentheses. Note that the 27r level, which is unoccupied in the isolated CO molecule, splits (due to mixing in of Cu orbital character) into bonding (27rb) and antibonding (27rJ orbitals with respect to the Cu atoms on interaction with the Cu cluster. The former orbital is partially occupied and the latter is empty. When absorption of an X-ray photon by a core electron (say the C 1s) occurs, a core hole is created and the electrons rearrange themselves to screen this core hole. The most effective orbital in the screening process is the 1~ orbital which is almost exclusively on the CO molecule. However due to the electronic relaxation which takes place due to the presence of the core hole, the 27rborbital which is partially occupied also changes its character and increases its CO content at the expense of Cu content. This latter effect contributes to the extramolecular screening of the core hole, i.e. it is an effect which cannot occur in the isolated CO molecule. The extra, rather intense, satellites which occur in the photoelectron spectrum can be Cu$O
0 -0.2
0 s -0.6 c Lz
lOa, 5b, 9a, -
=-IOe -19eMn,)
4b,
L
-EF
& -0.8 6 z -1.0 -1.2
Fig. 5. Xa-SW results for the ground state orbital energies of a CusCO cluster. The CUSCO cluster is used as a model to investigate the chemisorption on CO on Cu(100).
R.P. Messmer / ‘Iheoretial treahent of electronic
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attributed to “shake-up” processes. That is, due to the strong perturbation which occurs on removal of a core electron, the system can be left in an excited state of the ion. As a consequence the outgoing electron has less kinetic energy (appears at higher binding energy). In the sudden approximation the various possible excited states are determined by monopole selection rules, which means that the initial and final orbitals involved in the shake-up process must have the same symmetry. Thus, for example, returning to fig. 5 and assuming a core electron has been removed from the C 1s level, possible shake-up transitions would include an electronic excitation from the 8e+ 9e or the 8e+ lOe, etc. Energetically there are many possible shake-up transitions in the range of O-10eV above the main peak and thus the intensities have to be calculated [47,48] in order to make definite assignments. It turns out that there are only two types of transitions which have any appreciable intensity and these occur roughly where the satellites in the photoelectron spectrum occur. The two types of transitions are: (1) 8e(2n,,)+ 9e(2ra) and (2) le(lrr)+ 8e(27rJ and le(lP)+ 9e(2rJ. The former is responsible for the lower energy satellite and the latter for the higher energy satellite. The lower energy satellite cannot exist in the isolated CO molecule because the 2~ orbital is not split into two components (8e and 9e) as occurs for the chemisorption case. Furthermore the 8e(2mJ analog in the isolated CO case (i.e. the 27r orbital) is unoccupied. The analog of the le+ 8e, 9e(lP+ 27r) transitions does exist in the isolated CO molecule, but has a rather small intensity. The increased intensity in the chemisorbed case arises from the additional electronic relaxation which is permitted via the CO interaction with the metal. 3.2. Magnetic impurity problem As an example of a situation where clusters have been used to gain some insight into a non-surface phenomenon, the case of a magnetic impurity in a Cu host will be discussed. It turns out that the picture which emerges [50] by treating an impurity, such as V, Fe or Mn in a cluster of Cu atoms containing the first two shells of neighbors to the impurity, is rather different than the theoretical descriptions previously proposed. The two previous models are the virtual impurity state approach of Friedel [51] and Anderson [16] and the impurity-ion crystal field model of Hirst [52]. These latter two models take very different points of view. In the former model one assumes that the d-orbitals of the magnetic impurity are so hybridized with the conduction electrons of the host that they lose such atomic-like characteristics as the manifestation of ligand field effects, multiplet effects and spin-orbit coupling. In the latter model, the opposite extreme is assumed, that the atomic effects are very important, i.e. multiplet effects, crystal field effects and spin-orbit coupling must all be considered as a primary part of the problem, and the interaction with the conduction electrons is a perturbation. A great deal of work has been devoted to the Anderson type model of magnetic impurities but little has been done by way of theoretical development of the Hirst model because of its inherent difficulties.
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The Xa-SW cluster model calculations have provided some realistic appraisals of the magnitude of certain physical effects in this problem. The results of the calculation, although they do not support the assumptions of either model completely, are much more in accord with the characterization of the problem which has been given by Hirst [52]. The main conclusions of the calculations in terms of the important physical effects are given here. The chemical bonding effects of the impurity to the host are of the utmost importance. This aspect of the problem has been entirely neglected in previous models. The impurity-host bonding indicates that the Hirst ionic “crystal field” model must be modified to a “ligand-field” model so as to properly account for covalent bonding effects between the impurity and the host. The calculations show that localized effects are important. For example, they predict a ligand field splitting (ELF) of the d-orbitals of the impurity. The sign of the splitting is opposite to that given by the Hirst model, but recent experiments [53] which have conclusively demonstrated the importance of ligand field effects in these systems, have determined the sign of ELF to be the same as that predicted by the Xa-SW calculations. The calculations suggest that a better approximation to the magnetic impurity problem is to consider the impurity and a shell or two of neighbors such that the local aspects of the problem are taken into account properly. Then this larger entity, not just a single site, could be considered as interacting with the conduction electrons of the host.
4. Conclusions There have been many other applications of metal cluster calculations to various problems. Two recent studies by the author have been directed at amorphous metals [54] and grain boundaries [55]. In fact the number of investigations in which metal clusters have been used as models, far exceeds the number in which their intrinsic properties have been addressed. This is likely to change as the experimental techniques for determining geometrical structure and electronic structure of metal clusters are further developed. Clearly experimental information on the electronic structure of clusters is badly needed in order to resolve the question of the appropriate theoretical description, as the present Hartree-Fock and XO descriptions are so very different. The question of cluster convergence is a very important one and must be constantly kept in mind in all theoretical studies which employ clusters as models. We have seen that the size of cluster needed to describe a given property can vary quite significantly with the property which is being considered. There are still many questions to be resolved, but this merely demonstrates the fact that the physics of clusters is a very new field of investigation. The field is certainly a dynamic one, and we can certainly look forward to many exciting findings and some important surprises which will force us all to re-think some of our cherished concepts regarding small aggregates of metal atoms.
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Note added in proof In addition to the extended Hiickel results 1121shown in fig. I, another c~culation using the extended Hiickel method, but with a different parameterization, has been pubhshed by the same author [%I. In the latter calculation, the d-band width was found to be an order of magnitude larger than in the former work. Also in the latter work, the top of the d-band is found to be about 4eV below the Fermi level, for both the cluster calculation and the corresponding bulk band calculation. Experimentally, it is well known that the top of the d-band is about 2eV below the Fermi level. This demonstrates the problems with parameterization of the extended Hiickel method for metal clusters.
References [1] D. Kaikstein and P. Soven, Surface Sci. 26 (1971) 85. [Z] E. Huckel, 2. Physik 70 (1931) 204. [3] R.P. Messmer, Phys. Rev. B15 (1977) 1811. [S] 0. Bilek and L. Skala, Czech. J. Phys. 288 (1978) 1003. [S] 0. Bilek and P. Kadura, Phys. Status Solidi (b) 85 (1978) 225. [6] J. Bardeen, Phys. Rev. 49 (1936) 653. [7] N.D. Lang and A.R. Williams, Phys. Rev. Letters 34 (1975) 531. [S] J.L. Martins, R. Car and J. Buttet, Surface Sci. 106 (1981) 265. [9] J.C. Slater and G.F. Koster, Phys. Rev. 94 (1954) 1488. [lo] R. Hoffmann, J. Chem. Phys. 39 (1963) 1397. (111 J.A. Pople and G.A. Segal, J. Chem. Phys. 43 (1965) S136. [12] R.C. Baetzold and R.E. Mack, J. Chem. Phys. 62 (1975) 1513. [13] R.C. Baetzold, M.G. Mason and J.F. Hamilton, J. Chem. Phys. 72 (1980) 366. [14] R.C. Baetzold, Surface Sci. 106 (1981) 243. [15] S.N. Khanna, F. Cyst-~ckmann, Y. Boudevihe and J. Rous~au-Voilet, Surface Sci. 106 (1981) 287. [16J P.W. Anderson, Phys. Rev. 124 (1961) 41. [17] J. Hubbard, Proc. Roy. Sot. (London) A276 (1%3) 238. [18] P. Joyes, Surface Sci. 106 (1981) 272. (191 H. Hellmann, Acta Physicochim. URSS 1 (1935) 913. [ZO] J.C. Phillips and L. Kleinman, Phys. Rev. 116 (1959) 287. [21] H. Stall, J. FIad, E. Golka and T. Kruger, Surface Sci. 106 (1981) 2.51. [22] R. Car and J.L. Martins, Surface Sci. 106 (1981) 280. (231 J.C. Slater and K.H. Johnson, Phys. Rev. B5 (1972) 844. [24] K.H. Johnson and F.C. Smith, Jr., Phys. Rev. B5 (1972) 831. [25] R.P. Messmer, SK. Knudson, K.H. Johnson, J.B. Diamond and C.Y. Yang, Phys. Rev. 813 (1976) 1396. [26] R.P. Messmer, in: The Nature of the Surface Chemical Bond, Eds. T.N. Rhodin and G. Ertl (North Holland, Amsterdam, 1979) p, 53. [27] D.R. SaIahub and R.P. Messmer, Surface Sci. 1% (1981) 414. [28] C.C.J. Roothaan, Rev. Mod. Phys. 23 (1951) 69. [29] ME. Schwartz and C.M. Quinn, Surface Sci. 106 (1981) 258. [30] H. Basch and M.D. Newton, J. Chem. Phys., in press. [31] C. Bachmann, J. Demuynck and A. Veillard, Faraday Symposium 14, in press. (32) See, e.g., H.F. Schaefer, III, Ed. Methods of Electronic Structure Theory, (Plenum, New York, 1977). [33J J.O. Noell, M.D. Newton, P.J. Hay, R.L. Martin and F.W. Bobrowicz, J. Chem. Phys., in press. 1341 F.A. Cotton, Ace. Chem. Res. 11 (1978) 225.
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lreahnenl
of electronic slrucrure
[3S] C. Kittel, Quantum Theory of Solids (Wiley, New York, 1963). [36] H.J. Monkhorst, Phys. Rev. B20 (1979) 1504. [37] T.H. Dunning, Jr., B.H. Botch and J.F. Harrison, J. Chem. Phys. 72 (~980) 3419. (38) C.Y. Yang, Ph.D. Thesis, M~sachusetts Institute of Technology (1977). f39] C.Y. Yang and K.H. Johnson, to be published. [40] J.R. Schrieffer and P. Soven, Phys. Today 28 (1975) 24. [41] R.P. Messmer and D.R. Salahub, Phys. Rev. B16 (1977) 3415. [42] D.R. Salahub and R.P. Messmer, Phys. Rev. B16 (1977) 2526. [43] W. Eberhardt and C. Kunz, Surface Sci. 75 (1978) 709. (441 S.A. Flodstom. R.Z. Bachrach, R.S. Bauer and S.B.M. Hagstrom, Phys. Rev. Letters 37 (1976) I282 [45] D.R. Satahub, M. Roche and R.P. Messmet, Phys. Rev. B18 (1978) 6495. [46] D.R. Salahub, L.-C. Niem and M. Roche, Surface Sci. 100 (1980) 199. [47] R.P. Messmer, S.H. Lamson and D.R. Salahub, Solid State Commun. 36 (1980) 265. [48] R.P. Messmer, S.H. Lamson and D.R. Salahub, to be published. [49] G. Loubrief, Phys. Rev. B20 (1979) 5339. [SO] K.H. Johnson, D.D. Vvedensky and R.P. Messmer, Phys. Rev. Bl9 (1979) 1519. [Sl] J. Friedel, Can. J. Phys. 34 (1956) 11%. ]S2) L.L. Hirst, Physik Kondens. Materie 11 (1970) 255. 153) D.C. Abbas. T.J. Aton and C.P. Slichter, Phys. Rev. Letters 41 (1978) 719. 1541 R.P. Messmer, Phys. Rev. B23 (1981) 1616. [SS] C.L. Briant and R.P. Messmer, Phil. Mag. B42 (1980) 569. 1561 R.C. Baetzold, J. Phys. Chem. 82 (1978) 738.
THEORJiYl’ICAL TREATMENT METALLIC PARTICLES
OF THE ELECTRONIC
STRUCTURE
OF SMALL
R.P. MESSMER General Electric Company, CorporateResearch and Development, Schenectady, New York 12301, USA Received 8 September 1980
A brief review is given of various theoretical methods which have been applied to small metal particles to elucidate their electronic structure, This survey includes discussions of simple models and semi-empirical methods as well as first principles methods. With a few exceptions, these theoretical methods are carried over from molecular physics, where they have been used to study the electronic structure of molecules. From a theoretical perspective, metal clusters are of great interest for two reasons. The first reason is that the intrinsic electronic properties of such clusters are of fundamental importance. The second reason is that such clusters might serve as models for understanding localized electronic phenomena in bulk metals. Theoretical work which considers clusters from both of these viewpoints will be presented.
1. Introduction The most obvious characteristic of small metal particles is that a large percentage of the atoms in the particle are surface atoms. For example, if we assume a particle has a simple cubic geometry, then such a particle which contains 1000 atoms will have 49% surface atoms. Thus, any particle containing fewer than roughly 1000 atoms will contain over 50% surface atoms. Most of the small metal particles (hereafter referred to as “clusters”) to be discussed below contain far fewer than 1000 atoms, and hence the question of surface effects and their contribution to observed properties is a very important one. This leads one to ask how many atoms a cluster should contain in order to exhibit bulk-like properties. If one uses the fraction of atoms on the surface as a criterion for convergence to the bulk, then (using the simple cubic cluster geometry again) even with a million atoms (corresponding to a particle diameter of a few hundred Angstroms) 6% of the atoms are on the surface. Thus one might suppose that clusters of this size are necessary to attain reasonable convergence to the bulk. However, it should be clear that one cannot discuss the convergence of cluster properties to the bulk values in such simple terms. There will be some properties, which arise from local electronic structure effects, which may require relatively few atoms to describe. On the other hand to obtain the properties of the bulk Fermi surface may require very many atoms. Thus in general, no simple statement regarding the convergence of cluster properties can be made. The rate of convergence to the bulk values of a property will depend upon the property. 0039-6028/81/0000-0000/$02.50
@ North-Holland
Publishing Company
226
R.P. Messmer I Theoretical treatment of electronic structure
A few general comments about the electronic structure of metal clusters and theoretical methods would be appropriate at this point. As many of the atoms in a cluster are surface atoms and these surface atoms may have a significantly different potential than those in the interior of the cluster, one expects the inclusion of the effects of a self-consistent field (SCF) in the calculation to be essential for this type of problem. Likewise, the effects of a substrate or support on the electronic structure of a cluster can be appreciable if the contact is over a reasonably large fraction of the cluster’s surface area. Thus one must be cautious in comparing even SCF calculations for isolated clusters with experimental results on supported metal particles, unless the effects of the support are explicitly known. There have been numerous theoretical investigations of metal clusters using many theoretical techniques. In general, the more sophisticated the theoretical method is, the fewer the atoms which can be treated. In comparison to the understanding of electronic structure in molecules and solids, very little is known about clusters. Thus there is a need for both simple and sophisticated models. With simple models one can investigate the qualitative behavior of relatively large clusters; with sophisticated models one hopes to obtain quantitative information on rather small clusters. However, the information obtained from the latter can be used to refine or re-parameterize the simple models. In table 1, a summary of some theoretical methods and their application to metal clusters is given. It includes the various methods discussed by authors at this conference. Most of these methods have the molecular orbital approach as their starting point. Due to time limitations, the present author’s own research interests and the convenience of discussing a number of methods from a common framework, the discussion will be restricted in the following to those methods which derive from the molecular orbital viewpoint.
Table 1 Theoretical techniques applied to the investigation of metal clusters Theoretical technique
Investigators
Simple models : Cubium/simple Hiickel theory [l, 2) Modified jellium [6,7] Parameterized LCAO (9-111 Model Hamiltonians (Anderson [ 161, Hubbard [17j, etc.)
Messmer [3]; Bilek and Scala [4,5] Martins, Car and Buttet [8] Baetzold [ 12-141; Khanna et al. [ 151 Joyes [ 181
First principles mod&: Pseudopotentials Xcr method [23,24] Hartree-Fock [28] Configuration interaction [32]
Stall et al. [21]; Car and Martins [22] Messmer et al. [25,26]; Salahub and Messmer [27] Schwartz and Quinn [29]; Basch and Newton [30]; Veillard [31] Noel1 et al. [33]
R.P. Messmer / Theoreticaltreatmentof electronicstruchue
227
In the next section some specific examples of intrinsic properties of clusters will be discussed, following which a few cases of calculations which have employed clusters as models of bulk or surface properties will be presented.
2. Intrlnslc
properties of clusters
A primary problem which plagues all electronic structure calculations of clusters is the lack of information on the geometrical structure (i.e. the positions of the atoms) of the clusters. In most situations in molecular and solid state physics, this structural information is known at the outset. However, for the case of clusters one is forced to assume “reasonable structures” in order to proceed. The usual wisdom is to assume highly symmetrical clusters (e.g. spherical) reflecting the bulk structure and to assume internuclear distances close to the bulk values. Neither of these assumptions is likely to be true for extremely small clusters, that is, on the order of 10 atoms and less. Ideally it would be desirable to have the electronic structure calculations determine the geometry of the clusters. However, at present, this appears to be an almost impossible task. The simple models which can treat reasonably large clusters (of the order of 100 atoms) are not accurate enough to reliably determine the small energy differences between various geometric configurations. On the other hand, the sophisticated models are so computationally complex that only a few atoms can be treated reliably. 2.1. Cluster density of states Keeping in mind the discussion above, one must proceed on reasonable assumptions to investigate the intrinsic electronic properties of clusters. One of the most readily accessible theoretical characteristics of a metal cluster is its density of states (DOS). A number of studies on metal clusters using at least three different theoretical methods have discussed the calculated DOS. Results from three different methods, the extended Hiickel [12], X&-scattered wave [25] and Hartree-Fock [31] approaches, will be discussed. First, it would be useful to briefly characterize these methods. The extended Hiickel method is a semi-empirical LCAO molecular orbital approach which is defined by its Hamiltonian matrix elements. The diagonal elements are set equal to -4, where 4 is an atomic orbital ionization energy deduced from experiment and the off-diagonal ionization energy deduced from experiment and the off-diagonal elements are proportional to an average of two such terms multiplied by an overlap integral between the appropriate atomic orbitals. Thus the specification of the atomic orbital functions is the critical input to the calculation. The method has been very valuable as an interpolation and extrapolation aid in understanding the electronic structures of series of molecules particularly in organic chemistry. The XcY-scattered wave (Xcu-SW) method is a first-principles self-consistent-field molecular orbital approach which employs Slater’s Xa exchange-correlation approximation and the muffin-tin approximation to the potential. Except for the change of boundary conditions the method is the same as the KKR method of band theory. The latter has given
228
R.P. Messmer I E3eoreticaltreatmentof electronicstructure
very accurate descriptions of the band structure of metals. The Xc&W method has been widely applied to transition metal-containing molecules and treatments of metal-metal bonding in inorganic chemistry with considerable success [34]. The Hartree-Fock method is well known in the context of molecules, because of the numerous LCAO molecular orbital calculations which have employed it. Less familiar to those engaged in molecular physics are the problems of Hartree-Fock theory as applied to the electron gas [35] or in fact any metal [36]. A prime failure of the Hartree-Fock method as applied to metals is that it predicts a vanishing density of states at the Fermi level. Furthermore, applications to inorganic chemistry have shown problems with the method’s ability to describe metal-metal bonding between transition metal atoms. There has also been considerable recent discussion on the vital matter of the determination of adequate basis sets needed for the description of d-orbitals of first row transition metals [33,37]. The only metal for which calculations have been carried out by all three methods is copper, and thus we are restricted to a comparison of results for Cu clusters. The Hartree-Fock [31] and Xa-SW [2.5] calculations employed Cu13 clusters, the extended Hiickel calculations [12] employed a Cu 19 cluster. The calculated distribution of orbital energies or “density of states” obtained from these different methods on Cu clusters are shown in fig. 1. The disagreement among the methods is rather profound. For each method the orbitals which are predominantly Cu 4s are labelled “s”; and likewise, the d-band is shown as a rectangular area. For the case of the Hartree-Fock calculation, the d-band 1
I CUl3
HARTREE-FOCK
Gl3
%9
X,-SW
EXTENDEDHUCKEL
-5
q
S,d
-S
-s
-d -s
_/e9 -S
-s
-‘ep
-S
Fig. I. Comparison of orbital energies for copper clusters as determined by the Hartree-Fock, Hiickel methods.
Xa-SW and extended
R.P. Messmer / lleoretical
treatment of electronic structure
229
which is -3.0 eV wide is entirely below the s-band. The Xa-SW result, which has a d-band width of -2.5 eV is found in the midst of the s-band and the s character in this region is strongly hybridized with Cu 3d character. This situation is analogous to that found in the bulk metal. The extended Hiickel result for Cu19shown in the third column of fig. 1, also has the d-band completely overlapped by the s-band, but the d-band width is less than 0.2 eV. This latter value is an order of magnitude less than that obtained in the other two calculations. The narrow d-band is probably an artefact of the basis functions chosen to describe the d-orbitals, because similar calculations for Pd [12] gave a similarly narrow d-band. However, when different basis functions were used for Pd the d-band width increased dramatically [13]. If more realistic atomic functions for Cu were employed in the extended Huckel method, the results could possibly be brought into qualitative agreement with those of the Xcr-SW results. There are certainly significant discrepancies between the Xa-SW and Hartree-Fock results for Cui3. One point of agreement is that an e, and a t2, level, which are largely d-orbitals localized on the central atom, are split off from the rest of the d-band. It clearly would be extremely valuable to perform experiments on beams of clusters (of the size being discussed here), in order to determine which, if either, of the distributions in the first two columns of fig. 1 are valid. One particularly useful experiment in this regard, if it could be done, would be photoelectron spectroscopy. 2.2. Cluster magnetism One of the fascinating aspects of investigating transition metal clusters is the prospect of finding unusual magnetic effects - again due to the large surface to volume ratio of such clusters. Of the cluster calculations carried out to date using molecular-based methods, only the X&W method has been used to investigate the magnetic properties of various metal clusters. The first Xcr-SW work on transition metal clusters [12] was concerned with Nis, Ni13, Pd13 and Pt13. It was found that Nis and Nil3 clusters were magnetic while Pd13 and Pti3 were not. For the Nis cluster the calculated average spin magneton number per atom was 0.25, whereas for the Nil3 cluster the value was 0.46. These may be compared to the bulk value of 0.54. It should be pointed out that the spin density on the central atom is quite different from that found on the 12 surface atoms. Further discussion of such effects in a variety of metal clusters are presented in another contribution to this conference [27]. It is rather remarkable that the 13-atom clusters of Ni, Pd and Pt which all have the same number of valence electrons turn out to be magnetic for Ni and non-magnetic for Pd and Pt, and thus mimic the behaviour found in the bulk metals. One cannot expect that this will be a general phenomenon however. In fact, recent calculations on a 15atom Cr cluster have shown that such clusters carry a net magnetic moment. Bulk chromium is known to be antiferromagnetic, and the cluster results try to mimic this behavior to a large degree but due to surface effects there is not an exact cancellation of moments of opposite spin orientation. A more complete discussion of the interesting behavior of this cluster is given elsewhere [27].
230
R.P. Messmer I Theoretical
@earntent
of electronic
structwe
Calculations on Fe clusters have also been carried out [38,39]. For this case it has been observed that clusters of four, nine and fifteen atoms have average spin magneton numbers per atom which exceed the bulk value. However, the values decrease as a function of cluster size.
3. Clusters as models In this section we consider the theoretical use of clusters as models for focalized surface or bulk phenomena by discussing a few examples. Perhaps the most common application of this view point is in the study of chemisorption of atoms and molecules on metals. A review of such applications and of this view point has been given recently by the author [26]. Here, we can discuss only rather briefly three such studies related to chemisorption.
3.1. chemisotption A recurring theme through the three examples, is the question of how many atoms the cluster must contain in order to give a reasonably converged description of the electronic structure of the actual surface or the actual su~ace-admolecule interaction. Let us first consider a simple theoretical method which has been used to discuss the local density of states (LDOS) on the surface and in the interior of both a semi-infinite solid and a cluster. In the context of the semi-infinite solid the theoretical method has been referred to as cubium [l, 401, whereas in the cluster context [3-51 it is simple Hiickel theory. One assumes a simple cubic array of atoms with one s-orbital per site. The Hamiltonian consists of equal diagonal elements which may be taken as zero for convenience, and an offdiagonal element or “hopping integral” between nearest neighbors as familiar in tight binding calculations. In fig. 2a, the LDOS of the semi-infinite cubium model for atoms in three different planes are shown. The curves labelled (i, j, l), (i, j, 2) and (i, j, 3) correspond to the LDOS of atoms in the top, second and third (100) planes of the crystal, respectively. The same model can be solved analytically for finite clusters of arbitrary size [3]. However for such finite clusters, discrete energy levels are obtained. Thus in order to compare LDOS results with the semi-infinite case, each level is replaced by a gaussian of width 0.15 (in units of 2t, where t is the nearest neighbor matrix element). The resultant LDOS curves for a 9 X 9 X 9 cube of atoms is shown in fig. 2b. The curves labelled (5,5,1), (5,5,2) and (5,5,3) correspond to the LDOS of the central atom in each of the top, second and third (100) planes of the cubic cluster, respectively. Comparison of the two sets of curves shows that the cluster results reproduce the main features of the semi-infinite results, although “fine structure” features can be discerned in some of the cluster curves, showing that the discreteness of the eigenvalue spectrum continues to manifest itself. Nonetheless, the agreement is quite striking and for many purposes the cluster results would be more than adequate. Thus, for this particular property, one could conclude that the ?19-atom cluster constitutes a “usefully converged”
R.P. Messmer 1 Theoretical treatment of electronic
(0)
structure
231
l=(i,j,31
-9259 I:(i,i,Z
1
L_
r’_
”
fzCi,j,l)
2
/# Y-L_
g
‘I/: -3 -2
-I
0
I
2
3
ENERGY iIN UNITS OF 21)
Y
I
I
I
I
I
-3 -2 -I 0 I 2 3 ENEfW(IN UNITSOF21)
Fig. 2. Comparison of local density of states (LDOS) for three different sites as determined by: (a) the semi-infinite cubium modeland @)the simpleHiickelcluster model with a cluster of 729 atoms.
of the semi-infinite result; this, in spite of the fact that in an absolute sense this cluster is still quite far from the semi-infinite system -having only 129 discrete energy levels and 95% of the proper band width. The fact that such a large cluster is needed to achieve this degree of agreement with the cubium result, is largely due to the simplicity of the model, i.e. only one orbital per site and a high degeneracy of orbitals due to a symmetrical cluster. This problem of high degeneracy of cluster orbitals is also brought out in the work of Martins, Car and Buttet [S] using their variational spherical model for clusters. The second example deals with the chemisorption of 0 atoms on an Al(100) surface, as modelled by clusters of Al atoms [41]. It was assumed that an oxygen atom would chemisorb in a four-fold hollow site on the surface and clusters of 5, 9 and 25 Al atoms were used to model the (100) surface. X&W calculations were performed for the oxygen atom at 0.0, 2.0, and 4.0 bohr (1 bohr = 0.5292 A) above the surface plane of aluminum atoms. The calculations at 0.0 bohr were used to simulate the incorporation of the oxygen atom into the Al lattice. Various DOS and LDOS curves were generated and compared with photoelectron spectra available at the time. It was found that only the 25 atom Al cluster provided a reasonably converged description of the metal-adsorbate interaction. This conclusion is consistent with a study of convergence of the DOS for pure Al clusters containing up to 43 atoms [42].
representation
R.P. Messmer / Theoreticaltreatmentof electronicstructure
232
As the XCI-SWmethod, in its presently employed form, calculates only an approximate total energy, it cannot be used reliably to compute the equilibrium metal-adsorbate distance. Thus, such info~ation must be arrived at by indirect means. That is, one must calculate some properties of the system as a function of metal-adsorbate separation and compare these to the experimentally observed properties in order to deduce the most likely separation. For oxygen atoms chemisorbed on aluminum, two types of information were available with which to make a comparison with theory. These were photoelectron spectra of the valence region (at three different photon energies) and the Al 2p level shifts due to chemiso~tion of oxygen as determined by phot~lectron s~ctroscopy. From a comparison of the theoretical results with the experimental information, it was concluded that the oxygen atoms did not sit on the Al surface but were incorporated into the Al lattice, i.e. the best agreement with experiment was obtained at the metal adsorbate distance d = 0.0 bohr. Fig. 3 shows the calculated local DOS on the oxygen atom for the AlzsO cluster, where the discrete levels have been replaced by a gaussian as in the previous case discussed above. Fig. 4 shows the photoelectron spectra for clean aluminum and for two rather low coverages of oxygen [43]. A comparison of the calculated and observed positions of structure in the spectra shows rather good agreement. By contrast the calculated result for the Al50 cluster (also at d = 0.0) shows only two peaks in the LDOS - one at 2.5 eV, the other at 7.0 eV below EF. The A12p level shifts which take place on interaction with an oxygen atom were also calculated using the Alz50 cluster. The results are shown in table 2. Again the results for lattice incorporation of oxygen (i.e. d = 0.0) are more consistent with the experimental findings [44] than are those for oxygen above the surface (i.e. d = 2.0). /
---
Fig. 3. Xa-SW cluster results for the local density of states on the oxygen atom in an Al=0 cluster in which the oxygen atom is coplanar with the (100) surface layer of aluminum atoms (i.e. d = 0.0).
R.P. Messmer I llzeoretical treatment of electronic structure
233
1
Fig. 4. Photoelectron spectra for oxygen chemisorption exposures of 2 and 10 L of oxygen; see ref. [43].
on Al(100) surface. Spectra are for clean Al surface and for
These calculations also were able to describe the nature of the bonding between oxygen and aluminum and to provide a detailed account of the origin of the structure in the photoelectron spectrum. They also demonstrated how such results can change with cluster size and the importance therefore of investigating the convergence of cluster results. Subsequent work has also investigated oxygen chemisorption on the (111) surface and the (110) surface [46]. As the third and final example of the use of metal clusters in elucidating surfaces and their interaction with adsorbates, we consider the chemisorption of the CO molecule on the Cu(100) surface. Experimentally, one observes in the X-ray photoelectron spectrum in both the C 1s and 0 1s core regions two rather intense peaks displaced to higher binding energy from the main peak. In the isolated CO molecule these intense satellites do not occur. Hence, the question: what in the chemisorption process leads to these satellite peaks? The explanation deduced from Xcr-SW cluster calculations will be briefly described. As the Xa-SW method treats all electrons of the system it is thus very convenient to discuss core level spectroscopy within this framework. Calculations were performed for Table 2 Calculated Al 2p level shifts due to chemisorption Atom ”
d = 2.0
d = 0.0
Expt. b,
AI(l) Al(4)
0.30 0.65
1.21 1.05
1.30 1.30
of oxygen (in eV)
a) Al(l), atom just beneath the oxygen atom; A](4), one of the four atoms on the surface plane closest to the oxygen atom, b, See ref. [44].
234
R.P. Messmer I Theoretical
treatment
of electronic structure
CL&O and Cu&O clusters with the CO molecule in l-fold and 4-fold sites on the Cu cluster. Again, in order to investigate how the calculated properties depended on metal cluster size, 5atom and 9-atom clusters were investigated [47,48]. In this instance, it was found that the CusCO cluster was sufficient to obtain all the basic physics of the problem, although the Cu&O cluster does provide a more complete description. Further, although the l-fold site using the Cu&O cluster provides the best quantitative agreement with experiment, the qualitative aspects are very much the same as for the 4-fold Cu5C0 results. As a consequence, the latter cluster results [47] will be used to illustrate the basic physical conclusions of these studies. Fig. 5 shows the X&W orbital energies for the Cu&O cluster. The orbitals primarily derived from the isolated CO molecular orbitals are labelled by the CO orbital designations in parentheses. Note that the 27r level, which is unoccupied in the isolated CO molecule, splits (due to mixing in of Cu orbital character) into bonding (27rb) and antibonding (27rJ orbitals with respect to the Cu atoms on interaction with the Cu cluster. The former orbital is partially occupied and the latter is empty. When absorption of an X-ray photon by a core electron (say the C 1s) occurs, a core hole is created and the electrons rearrange themselves to screen this core hole. The most effective orbital in the screening process is the 1~ orbital which is almost exclusively on the CO molecule. However due to the electronic relaxation which takes place due to the presence of the core hole, the 27rborbital which is partially occupied also changes its character and increases its CO content at the expense of Cu content. This latter effect contributes to the extramolecular screening of the core hole, i.e. it is an effect which cannot occur in the isolated CO molecule. The extra, rather intense, satellites which occur in the photoelectron spectrum can be Cu$O
0 -0.2
0 s -0.6 c Lz
lOa, 5b, 9a, -
=-IOe -19eMn,)
4b,
L
-EF
& -0.8 6 z -1.0 -1.2
Fig. 5. Xa-SW results for the ground state orbital energies of a CusCO cluster. The CUSCO cluster is used as a model to investigate the chemisorption on CO on Cu(100).
R.P. Messmer / ‘Iheoretial treahent of electronic
structrrre
235
attributed to “shake-up” processes. That is, due to the strong perturbation which occurs on removal of a core electron, the system can be left in an excited state of the ion. As a consequence the outgoing electron has less kinetic energy (appears at higher binding energy). In the sudden approximation the various possible excited states are determined by monopole selection rules, which means that the initial and final orbitals involved in the shake-up process must have the same symmetry. Thus, for example, returning to fig. 5 and assuming a core electron has been removed from the C 1s level, possible shake-up transitions would include an electronic excitation from the 8e+ 9e or the 8e+ lOe, etc. Energetically there are many possible shake-up transitions in the range of O-10eV above the main peak and thus the intensities have to be calculated [47,48] in order to make definite assignments. It turns out that there are only two types of transitions which have any appreciable intensity and these occur roughly where the satellites in the photoelectron spectrum occur. The two types of transitions are: (1) 8e(2n,,)+ 9e(2ra) and (2) le(lrr)+ 8e(27rJ and le(lP)+ 9e(2rJ. The former is responsible for the lower energy satellite and the latter for the higher energy satellite. The lower energy satellite cannot exist in the isolated CO molecule because the 2~ orbital is not split into two components (8e and 9e) as occurs for the chemisorption case. Furthermore the 8e(2mJ analog in the isolated CO case (i.e. the 27r orbital) is unoccupied. The analog of the le+ 8e, 9e(lP+ 27r) transitions does exist in the isolated CO molecule, but has a rather small intensity. The increased intensity in the chemisorbed case arises from the additional electronic relaxation which is permitted via the CO interaction with the metal. 3.2. Magnetic impurity problem As an example of a situation where clusters have been used to gain some insight into a non-surface phenomenon, the case of a magnetic impurity in a Cu host will be discussed. It turns out that the picture which emerges [50] by treating an impurity, such as V, Fe or Mn in a cluster of Cu atoms containing the first two shells of neighbors to the impurity, is rather different than the theoretical descriptions previously proposed. The two previous models are the virtual impurity state approach of Friedel [51] and Anderson [16] and the impurity-ion crystal field model of Hirst [52]. These latter two models take very different points of view. In the former model one assumes that the d-orbitals of the magnetic impurity are so hybridized with the conduction electrons of the host that they lose such atomic-like characteristics as the manifestation of ligand field effects, multiplet effects and spin-orbit coupling. In the latter model, the opposite extreme is assumed, that the atomic effects are very important, i.e. multiplet effects, crystal field effects and spin-orbit coupling must all be considered as a primary part of the problem, and the interaction with the conduction electrons is a perturbation. A great deal of work has been devoted to the Anderson type model of magnetic impurities but little has been done by way of theoretical development of the Hirst model because of its inherent difficulties.
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The Xa-SW cluster model calculations have provided some realistic appraisals of the magnitude of certain physical effects in this problem. The results of the calculation, although they do not support the assumptions of either model completely, are much more in accord with the characterization of the problem which has been given by Hirst [52]. The main conclusions of the calculations in terms of the important physical effects are given here. The chemical bonding effects of the impurity to the host are of the utmost importance. This aspect of the problem has been entirely neglected in previous models. The impurity-host bonding indicates that the Hirst ionic “crystal field” model must be modified to a “ligand-field” model so as to properly account for covalent bonding effects between the impurity and the host. The calculations show that localized effects are important. For example, they predict a ligand field splitting (ELF) of the d-orbitals of the impurity. The sign of the splitting is opposite to that given by the Hirst model, but recent experiments [53] which have conclusively demonstrated the importance of ligand field effects in these systems, have determined the sign of ELF to be the same as that predicted by the Xa-SW calculations. The calculations suggest that a better approximation to the magnetic impurity problem is to consider the impurity and a shell or two of neighbors such that the local aspects of the problem are taken into account properly. Then this larger entity, not just a single site, could be considered as interacting with the conduction electrons of the host.
4. Conclusions There have been many other applications of metal cluster calculations to various problems. Two recent studies by the author have been directed at amorphous metals [54] and grain boundaries [55]. In fact the number of investigations in which metal clusters have been used as models, far exceeds the number in which their intrinsic properties have been addressed. This is likely to change as the experimental techniques for determining geometrical structure and electronic structure of metal clusters are further developed. Clearly experimental information on the electronic structure of clusters is badly needed in order to resolve the question of the appropriate theoretical description, as the present Hartree-Fock and XO descriptions are so very different. The question of cluster convergence is a very important one and must be constantly kept in mind in all theoretical studies which employ clusters as models. We have seen that the size of cluster needed to describe a given property can vary quite significantly with the property which is being considered. There are still many questions to be resolved, but this merely demonstrates the fact that the physics of clusters is a very new field of investigation. The field is certainly a dynamic one, and we can certainly look forward to many exciting findings and some important surprises which will force us all to re-think some of our cherished concepts regarding small aggregates of metal atoms.
R.P. Messmer I lheoreticai treatment of electronicstructure
237
Note added in proof In addition to the extended Hiickel results 1121shown in fig. I, another c~culation using the extended Hiickel method, but with a different parameterization, has been pubhshed by the same author [%I. In the latter calculation, the d-band width was found to be an order of magnitude larger than in the former work. Also in the latter work, the top of the d-band is found to be about 4eV below the Fermi level, for both the cluster calculation and the corresponding bulk band calculation. Experimentally, it is well known that the top of the d-band is about 2eV below the Fermi level. This demonstrates the problems with parameterization of the extended Hiickel method for metal clusters.
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