Inhomogeneous contraction of interatomic distances in metallic clusters: Calculations for Cs_{n} and OCs_{n}

The equilibrium geometrical structures of Cs„and OCs„clusters have been obtained by a method that, in the framework of density-functional theory, describes the ion-electron interaction by means of a pseudopotential that is spherically averaged about the cluster center. In the size range studied (up to n =78 and 70, respectively) the clusters present well-separated atomic layers for which the distances to the two closest neighbors of each atom have been analyzed, showing that there is an inhomogeneous shrinking of the geometrical structure, in the sense that the distances obtained for atoms in the inner layers are lower than those for the outer ones. The analysis of the growing of the clusters suggests that the equilibrium geometries are those for which any atom of the aggregate has its closest neighbor at a distance lower than the nearest-neighbor distance of the pure bulk metal. For small (n ~ 20) pure clusters, geometries of high symmetry have been found, which are completely modified by the presence of the oxygen impurity. A regular octahedron is formed around the oxygen atom in most cases.


I. INTRODUCTION
Metallic clusters are remarkable examples of finite quantum systems.The size dependence of some of their properties, for example, the abundance spectra and the ionization potentials (IP), is well understood, for the case of simple metallic elements, in the framework of the jellium model, ' which is an extension of the free-electron model of bulk metals.In the case of clusters, the quan- tum states of the valence electrons are determined by the self-consistent mean field produced by the finite positive ionic distribution, considered as continuous (jellium mod- el), and the electron-electron interaction.For clusters with number of atoms n smaller than 100, the assumption of a homogeneous spherical jellium distribution allows the reproduction of experimentally found local maxima in the abundances, magic numbers, which coincide with those clusters with completely filled electronic shells and with the next empty shell well separated in energy.'  The corresponding ionization potentials present a step- like behavior when we pass from the magic cluster n to its neighbor n+1, in agreement with experiment, al- though the calculated drops are, in general, greater than the measured ones.
Several refinements of this self- consistent spherical model have been proposed, such as the inclusion of spheroidal distortions and the use of diffuse positive distributions.
Recently, experimental results on the abundances have been obtained by Martin and co-workers for large clus- ters of OCs", with n ~600 (Ref. 8) and Na", with n ~1500 (Refs.9 and 10), which have been explained by means of an inhomogeneous jellium model.A positive density that is higher than the bulk average value is need- ed at the center of the cluster in order to produce the bunching of the electronic levels necessary to give the right sequence of magic numbers.Martin and co- workers found it impossible to predict the experimental values of the most abundant clusters, for n + 100, with a homogeneous distribution, although both distributions give the same sequence for the light clusters, n ~100.
The calculated ionization potentials for the centrally compressed spherical background present also steps at those magic numbers observed experimentally for large clusters and a reduction of step size in the IP values for light clusters, n ~100.'  In this contribution we present an analysis of the equi- librium geometries obtained for Cs"with 3 ~n ~78 and OCs", 3 ~n ~70, using a model that, in the framework of density functional theory, includes also the ionic structure of the aggregate, transcending the jellium model but retaining the spherical assumption for the external poten- tial.The ion-electron interaction for the valence elec- trons is thus approximated by the spherical average of the pseudopotential (SAPS method").Some results concerning the cohesive energies, electronic structure, ion- ization potentials, embedding energy of the oxygen atom, and general geometrical trends have already been published.' ' We have compared the IP's for OCs"with the experimental values, showing that the SAP S scheme reproduces the general trend and the drop after a magic number, those drops greater than the experimental ones.' The inadequacy of the homogeneous jellium mod- el to describe the expected sequence of magic numbers of Cs"was discussed in Ref. 13, where we have shown that the inclusion of the ionic structure gives the expected se- quence for pure clusters (expected from trends in the measured ionization potential' ) and the experimental one for suboxide QCs"clusters.' In the present work we analyze the ion-ion distances in the clusters and the de- tailed geometrical structure showing that an inhomogene- ous shrinking is obtained from our calculations in agree- ment with the conclusions of T. P. Martin's group.'   Also, a recent SAPS calculation' for aluminum clusters Al", 3~n ~2S, explains the deviations of the predicted values of the polarizability within the homogeneous jelli- um model from the experimental results as due to the ac- tual contraction of the small Al, clusters.Finally, we point out that lattice contractions have been measured in Au and Pt clusters, ' and also that a size-dependent con- traction of the nearest-neighbor distances for clusters iso- lated in solid argon has been measured for Ag and Cu clusters, ' ' thus indicating that the effect could be a general property of any metallic cluster.

II. THK MABEL
The model used to calculate the electronic and geometrical structures of the clusters has been presented in de-  tail elsewhere" ' ' and we stress here only the main points.For an initial geometry of the cluster, the Kohn- Sharn equations of density-functional theory ' are self- consistently solved, using the local-density approximation for the exchange-correlation energy.Afterward the ionic coordinates are moved, with a steepest-descent method and preserving the self-consistency between the ionic positions and the electronic density, to obtain the equilibri- um geometry and the corresponding ground-state total energy E. To avoid trapping in a local minimum several initial random geometries are considered (40 for pure clusters and 70 for the OCs").An empty-core pseudopo- tential is used to describe the ion-electron interaction with a core radius, r, (Cs) =2.74 a.u., which leads to the experimental ionization potential in the free-atom limit for the same density functional method.The total ionic potential of the cluster is spherically averaged around the center of mass of the cluster.This is equivalent to using the spherically averaged electronic density in the calcula- tions and it is the most drastic approximation of our model which must be more accurate the bigger the clus- ter becomes.However, the exact ion-ion repuslion is in- cluded in the calculations, which makes the total energy a function of the precise location of the atoms.For QCs" all the electrons of the oxygen atom are considered in the calculation and the positive nuclear charge is always lo- cated at the center of mass of the cluster reinforcing the spherical symmetry assumed for the aggregate.
The results of our calculations' ' reproduce the elec- tronic shell effects associated with the local maxima in the abundance spectrum of' QCs".n =10,20, 36, and 60 due to electronic effects arises from the SAPS calculations, in agreement with the experimen- tal results.' For pure Cs clusters the sequence obtained ( ls, 1 p, 1d, 2s, 1 f,2p, 1 g, 2d, 3s, 1 h ) presents three shells nearly degenerate in energy (2d, 3s, and lh), so the fol- lowing sequence of magic numbers was predicted: n =8, 18, 20, 34, 40, and 58, in accordance with the results for other alkali-metal clusters.

III. THE GRQ%ING QF THK CLUSTERS
The main trends of the geometrical structures of Cs" and QCs"are summarized in Figs. 1 and 2, respectively.FICx. 1. Mean radius and standard deviation as functions of n ' ' for the atomic layers of Cs"clusters.The straight lines, R;, given by Eq. {1),separate the three regions defined in the clus- ters.R& corresponds to the radius of the homogeneous spheri- cal jellium.scribed such that new shells appear in the innermost re- gion of the cluster when certain conditions are met.This can be appreciated in Fig. 1, where we have indicated with two vertical arrows the biggest clusters with one central atom and one and two additional atomic layers, Cs, 8 and Cs63, respectively.For Cs, 8 the distance be- tween the central atom and any atom in the surface layer becomes slight1y larger than the nearest-neighbor distance in bulk Cs, dNN =9.893 a.u.Our calculation indi- cates that the cluster Cs» reconstructs its geometry in such a way as to avoid neighbor interatomic distances larger than dNN.This is achieved by locating the added atom in the interior of the cluster (see the geometrical structures obtained for Cs, 8 and Cs, 9 in Fig. 7) instead of placing it on the surface, a situation which would have increased the cluster radius (and distances between neigh- bors) further.For Cs63 the situation is analogous to that of Csis.Now the central atom is surrounded by an atom- ic shell of 19 atoms at a mean radius nearly identical to that of Cs&8 and again clearly larger than dNN.The next cluster, Cs64, has two atoms in the inner region, like Cs», forming a new shell that is populated as the cluster grows in size.This behavior is again an indication that neighbor interatomic distances greater than dN& are avoided in the equilibrium geometries of the clusters.
In the case of OCs"a stable inner structure, typically of six atoms arranged as a regular octahedron, is formed around the oxygen impurity for any cluster size.' This means that, in general, the geometrical structure of Cs" is strongly perturbed by the presence of the oxygen, as a comparison of Cs6o and OCs6O in Fig. 3 clearly shows.For these suboxide clusters the growth pattern divers from that in pure Cs.An intermediate Cs shell appears at OCs47 midway between the surface shell and the inner core.The appearance of this new atomic shell can be ex- plained by the same argument as in the case of pure Cs: From Fig. and 22 with a more stable value for the radius also.

IV. THK INHQMQGKNKQUS CQNTRACTIQN
In order to show the inhomogeneous shrinking of the interatomic distances of the clusters we have divided each For each atom within one of those regions we look for the two closest ions at any point in the cluster and not only inside its own atomic shell.Finally, the mean value of the dis- tances to the nearest atom D, and of the distances to the two nearest atoms D2 are obtained for the collection of atoms within the chosen regions.The regions considered are separated by the radii given by the following relationship: R;(n)=a;+b;n'~, i =1, 2, 3, where, for Cs", b, =b2=b3=r, (r, is the Wigner-Seitz radius for bulk cesium, r, =S.67 a.u.), a, =0, a2= -8.0 a.u., a3= -16.0 a.u., and only the positive values of R;(n) are considered.These radii clearly separate the atomic shells that actually exist in the Cs aggregates.Actually, only two radii are necessary to separate the three regions considered.
The most external radius, R, = r, n ', which is the radius of a cluster of n atoms in the spherical jellium model, is included only to illustrate the global contraction of the cluster volume with respect to a piece of the macroscopic metal with the same num- ber of atoms.For OCs" the three regions considered cor- respond to the following values of the coeKcients in (1): b& =r" b2 =63 =0, and a& =0, a2 =8.5 a.u., a3 = 15.0 a.u. ; in this case only two regions are considered for 9 ~n ~45, and three regions are analyzed for 46 ~n ~70.
The results for the mean values and the standard deviations for the distances to the closest atom D, and to the two closest atoms D2 are given in Figs. 5 and 6 for Cs" and OCs", respectively.Also the experimental value of the nearest-neighbor distance for bulk cesium, d», is given for comparison.
At this point it is important to notice that the two closest atoms (of a given one) mentioned here have not the same meaning as the nearest-neighbor and secondnearest-neighbor for a perfect crystal in solid state phys- ics.As we show in the following, the interatomic dis- tances for a Cs cluster are not as well defined as in a per- fect crystal (that is, there is a larger dispersion).By means of D i and D2 we are trying to obtain the average nearest-neighbor distance for atoms in the different re- gions defined above.Actually, an average over a larger number of closest atoms, D3, D4, etc. , may be better.
Ideally one should average over the first coordination shell, but the problem is that its population around an inner atom is different from that for a surface atom.To avoid these additional complications we stop with D2, since we consider that it is enough for our purposes.atomic shell plus one atom at the center, and Fig. S(a)   shows that for 7 ~n ~13 the closest atom to any one in the external shell is the central atom and vice versa, so the lines for Di corresponding to the outer shell (crosses)   and to the central atom (stars) are practically identical.However, from Cs&4 to Cs, 8 the atom closest to one in the surface is also on the surface at a distance which is lower than the mean radius of the atomic layer.Now the line corresponding to the central atom is clearly over the mean value D, for the external shell.The sudden drop in the value of D i at Csi9 is due to the formation of the new internal shell already described.For Cs, 8 the value of D, corresponding to the central atom is slightly larger than dNN (see arrow in the figure), so the transition reflects the fact that the preferred equilibrium geometry is one in which the values of D, for any region of the cluster are always lower than dNN, as is the case for Cs&9.
A similar behavior is obtained for D& in the range 40 ~n ~63 in which there is also only one atom (the cen- tral one) in the innermost region defined by the radius R3.For this region, D, increases with the cluster size, associated with the corresponding increment of the ra- dius of the intermediate shell, and a sudden decrease occurs when a new inner shell of two atoms is formed at Cs64.However, in this size range the values of D& for the intermediate shell are lower (only in three cases are slightly larger) than those for the central atom.This is due to the contraction of this shell with respect to the case of single layered clusters, as can be shown when its population is considered.The number of atoms in this intermediate shell is 10 (for n =40, 41, and 44) and 12 (for n =42, 43, 45, 46, and 50).The values of D, for the central atom are, in both cases, clearly lower than those corresponding to the central atom of Cs&& and Csi3, re- spectively.The contraction is about 0.8 a.u.for most of the clusters considered.Also the values of Di corre- sponding to the intermediate layer are lower than those corresponding to the outer part of the single layered clus- ters, Cs&& and Cs&3, indicating that these intermediate lay- ers are more closely packed.When the population in this intermediate shell is greater than 12, the corresponding values of D, for the two regions defined by R2 and R 3 are clearly different, showing that for these clusters the closest atom to one in the intermediate shell is never the central one.A maximum value of 19 atoms surrounding the central one is reached at Cs63, with a radius for the spherical shell of the order of dNN like in the case of Cs, 8, thus promoting the formation of a new internal shell for the larger clusters.
The values of D"and also of D2 in Fig. 5(b), are fairly constant in the range studied except at the first steps of the formation of a new shell, for example, the most inter- nal shell, for 64+ n ~78, which only has a number of atoms between 2 and 6.The oscillation in D, for the external shell of the light clusters, when n ~13, is smoothed by the inclusion of the distance to the second closest atom in D2, Fig. 5(b), with the corresponding in- crement in the standard deviations.The reason is that for 7 n ~13 the closest distance between atoms in the outer shell is, as mentioned above, bigger than the mean radius of the shell, being Cs&3, a centered icosahedron, the first cluster for which both distances are similar.&n Fig. 5(b) the values of D2 also show the same effect of inhomogeneous contraction of the clusters.For those aggregates with a central atom, D2 coincides with D, for the region that only contains the central atom due to the spherical arrangement of the layer surrounding the center of the cluster.It is important to point out that if the cen- tral atom had been included as part of its closest shell for the calculation of D, and D2, the values obtained should not be very different from those quoted because of the small relative influence of one distance (with the value given in the figures) in the mean values of 7 -19 (the pop- ulations of the shell closest to the central atom) distances.
Actually, we feel that defining a region with only one atom is not satisfactory, as shown by the peculiar behav- ior of this region in Fig. S. The reason for showing ex- plicitly this one atom region is that it helps to understand the process of cluster growth.
Comparison of Figs.5(a) and 5(b) shows that, for the atoms in the outer shell, the distance to the second closest atom is, in average, 0.2S a.u.greater than the dis- tance to the first one.The same qualitative behavior is present in the intermediate shell; however, the effect is in this case weaker than before and the values for Di and The results for OCs"are given in Figs.6(a) and 6(b) for the two distances D, and Dz corresponding to the three regions in which the aggregates have been divided (see Fig. 2).In this case the strong ionic bonding due to the oxygen impurity gives as a result the very stable and compressed structure of the inner atoms of the clusters, the layer associated with the radius R2.From Figs. 5(a) and 5(b) we can conclude that, in spite of the overlapping between the standard deviations that is present mainly for the larger aggregates, Cs atoms in finite clusters are closer than in the bulk material and that the atoms in the inner layers are more closely packed than the atoms of the surface.Of course, when cluster size increases it is expected that at least the distances in Figs.5(a) and 5(b) corresponding to the internal layers should converge to the nearest-neighbor distance in bulk cesium, but what our results show is that this conver- gence is rather slow and that for n ~100 we are still far away from the bulk limit, because even for Cs78 more than half of the atoms of the cluster are located at its sur- face.It can be expected that, as the number of atoms in the internal part grows, an expansion toward the bulk distances should occur in the internal regions of the clus- ters.

B. OCs"clusters
the actual geometrical structures of these regions in the next section.For these clusters only the distances be- tween Cs atoms have been analyzed to obtain D, and D2.
All the information about the oxygen-cesium distances was already given in Fig. 2, in particular, the very stable radii of the innermost shells, which are around 5.3 a.u. in the case of octahedral structure, giving a value around 7.5 a.u.for the edge of the octahedron and, consequently, for the values of D, and D2 corresponding to this inter- nal Cs layer.The drop in the values of D, and Dz be- tween OCs34 and OCs35 is due to the formation of a pentagonal bipyramid, with seven atoms, as the inner structure of the cluster, instead of the octahedron.The former is also the inner shell for OCs36 and OCs37 whereas eight atoms forming a square antiprism is the geometry ob- tained for n =38, 39, 40, and 42.The spike in the values of DI and D2 for OCs4& is due to the new structure of the internal layer, now formed by nine atoms arranged as a tricapped trigonal prism.Finally, for OCs&5 a bicapped square antiprism (10 Cs atoms) is the geometry for the inner part before a transition, again to the octahedron, occurs at OCs46.These structures will be also comment- ed in the next section.
The mean values of D, and D2 for the external shell are very similar to those obtained for Cs"; however, the intermediate shell, for 46~n ~70, shows a contraction which is smaller than the one obtained for the pure case.
A very significant overlapping of the standard deviations is present here, indicating the strong dispersion on the values of the distances used to calculate D, and D2 for these two atomic shells.The number of atoms in the in- termediate shell and its corresponding mean radii have values that are more stable (independent of n) than those of pure clusters.
The inhomogeneous contraction of the clusters is clear- ly shown by Figs.6(a) and 6(b) just by comparing the values of D, and Dz for the outer shells with those of the internal one, the latter being determined by the presence of the oxygen impurity.This inhuence of the oxygen on the compression of the internal shell is very clear even for those clusters for which the populations of the two atom- ic layers are identical to those of the pure clusters, n =26 -32, except n =29, as can be seen by comparing Figs. 5 and 6: The values of D, and D2 for the internal shell of these clusters are nearly one atomic unit lower in the case of OCs" than for the pure clusters.However, the values for the external shell are not very much influenced by the impurity due to the strong screening produced by the inner layer.When D& and D2 for this internal shell are compared with those for the most inter- nal one in pure Cs", 64~n ~78, the conclusion can be drawn that the outer shells of the pure cluster produce an effect of compression that is analogous to the one due to the presence of the oxygen atom in the suboxide clusters.
Finally, we point again that when the cluster size in- creases the distances D, and D2 for the internal layers should converge to the bulk distance d NN, except perhaps in the first coordination shell around the oxygen impuri- ty.Our results suggest that the inner structure, in most cases an octahedron, will probably remain stable even in the very large clusters.

V. THE EQUILIBRIUM GEOMETRICAL STRUCTURES
In a previous work' we have pointed out that the well-known difticulty of determining the ground-state geometry of an atomic cluster, due to the large number of local minima in the energy hypersurface, is also present in the SAPS method.In fact, from our results we con- clude there that the determination of the cluster geometry for a not too small cluster (e.g. , Cs5~) is a nearly impossible task because of the huge number of local mini- ma.However, many of these minima can be considered as small stable distortions of the ground-state geometry, because of the softness of the aggregates, ' and in that case we expect that the geometry associated with the en- ergy minimum obtained in the SAPS method, starting from a relatively small number of random initial geometries, should be quite close to the true ground-state geometry within our method.This should be especially true for the smaller clusters and for the internal structure of the larger ones for which we present in this section some examples of their geometries.Two effects can be shown from the actual geometries obtained: The com- plete reconstruction of the pure clusters when they grow in size and also when the oxygen impurity is included in the cluster, and the formation of a compressed inner lay- er for large clusters.
Our results can be compared with other calculations that avoid the spherical average for the total ionic poten- tial but are restricted to small clusters of light alkaline metals (Na, Li) (see the references quoted in Ref. 22).
Unfortunately, direct experimental evidence on the geometrical structures of small clusters is very scarce and dificult to obtain for unsupported aggregates, although information about the actual geometries of the clusters can be extracted from the comparison of the experimental and calculated static electric polarizabilities, which is a measurable property strongly sensitive to the cluster shape.These polarizabilities can be calculated in the SAPS scheme' once the geometry is obtained and then compared with the experimental data.
The geometries for Cs"with 4 ~n ~13 are identical to those of Na"calculated by the same method and de- scribed in Ref. 22, with the only exception of Cs8 for which a centered pentagonal bipyramid is obtained in- stead of the square antiprism of Na8.Of course, the in- teratomic distances are different, and we have found that, at least for n ~13, they are 29 -31%%uo greater than those of Na", approximately the same relative variation of the corresponding nearest-neighbor distances of the bulk ma- terials [dNN(Na) =6.917 a.u. and dNN(Cs) =9.896 a.u.j.
Summarizing, the calculated geometries of Cs"are the following: tetrahedron, n =4; trigonal bipyramid, n =5; octahedron, n =6; centered octahedron (Oh), n =7; cen- tered square antiprism (C4"), n =9; centered tricapped trigonal prism (D3&), n =10; centered square antiprism bicapped at the square faces ( C~, ), n = 11; distorted and centered icosahedron with one vertex lost, n =12, and a perfect centered icosahedron for n = 13 with a value for its edge of 9.9+0.that present relevant properties of symmetry.Also, the corresporlding geometries for QCs"are given for compar- ison.Cs&5 is a bicapped hexagonal antiprism with a cen- r 1 atom Csi6 has a D3a symmetry and can be described as a centered trigonal prism with two atoms over each rectangular face and one more centered over each of the long edges of these faces, Cs» is a centered pentagonal prism capped in all its faces preserving a fivefold symme- tI y aIld Cs ] 9 is the first cluster with two atoms in the inner part, its external structure being identical to the oIle of Cs i 8. Finally CS20 is given in Fig. 8 and its re- Inalkahle geollietly (D3$ ) is completely different from the one obtained for Na20, in which only two atoms are lo- cated inside the structure.A visualization of the efFect of shrinking can be obtained from this last cluster by comparing the mean values of the edges of the inner and the outer polyhedra.An internal tetrahedral structure is also obtained for Cs"with n =21 -2S, and an octahedral one is formed in the inner region for n =26 -32 with the only exception of n =29.Cs cover are completely modified by the presence of the Q atom, except in those light clusters for which the cen- tral site is empty (n =4, 5, 6).The presence of the oxygen atom produces a general contraction of the clusters, for example, the edge of the octahedron changes from 9.S2 a.u. in the pure case to 7.96 a.u.for QCs6, indicating again the strong ionic bonding between the impurity and the Cs atoms.This contraction is also present for those Cs"clusters with their internal layer formed by six atoms, as we have mentioned in Sec.III and was shown in Fig. 4. For the other clusters we obtain pentagonal bi- pyramid, n =7; square antiprism, n =8; tricapped trigo- nal prism, n =9 (this is the first cluster with two separat- ed atomic layers, of six and three atoms, respectively, and its geometry is given in Fig. 9).The geometries for OCsIO and QCs» are given in Fig. 7, where it is seen that the inner Cs layer has five atoms for both clusters but the corresponding structures are completely difFerent (a square pyramid and a trigonal bipyramid, respectively) and the outer atoms can be considered as coordinated to each of the faces of those internal structures.QCs&2 has also a trigonal bipyramid as internal geometry, but for 13 & n + 34 and 46 + n ~70 the internal structure is al- ways an octahedron with the only exception being OCs6" for which a trigonal bipyramid is obtained.The structure of QCs&4 was already given in Ref.   (right).The values of (d) are as in Fig. 9. faces (0& ), and those for n = 15, 16, 18, and 19 are plot- ted in Fig. 7. Particularly notorious is the distorted cu- boctahedron for OCs&8 which surrounds the inner octahedron, preserving the global 0& symmetry.Some of the structures for the internal layers with a low number of atoms are given in Figs. 9 -11 to show that our method is able to describe the inhuence of the external shells on the inner structure producing geometries different from those of the single layered clus- ters.In Fig. 9, OCs9 and the nine Cs atoms of the inner shell of OCs44 are compared, showing a completely different geometry.Also, the internal shell of OCs4& is formed by nine atoms surrounding the oxygen, with a structure identical to that of OCS9 but with different values for the distances quoted in Fig. 9(a).These are now the following: a =7.8 a.u., b =a, and b'=8.6 a.u., indicating that the presence of the outer shell of Cs atoms produces a contraction of the radial distances of the atoms in the inner region of the cluster.Figure 10 gives the case of Cs4o and OCs45, both with 10 atoms surround- ing the central one, which can be compared with Cs&& and OCs, o in Fig. 7; it is clear that the equilibrium geometries are determined by the actual environment of the clusters and that there is a complete reconstruction of the aggre- gate when the total number of atoms is modified.Two more examples of centered internal structures are given in Fig. 11 for Cs46 and Cs53 because of its symmetry prop- erties.For n =46, 12 atoms are arranged around the central one in a form which is different from the icosahedron of Cs, 3 or from the cuboctahedron.The structure of 15 atoms corresponding to Cs53 is a distorted cuboctahedron bicapped on its square faces and with a central atom, a structure which is different from the one obtained for pure Cs» given in Fig. 7, showing again the inhuence of the external Cs atoms in the equilibrium geometries of the internal layer.

VI. SUMMARY
The geometrical structures found in our calculations indicate that atoms in the clusters are arranged in spheri- d =9.1   FIG.11.The structures of the internal layer for Cs46 (upper) and Cs53 (lower).The values of (d) are as in Fig. 9. cal layers around the cluster center and that those geometries for which no atom has its closest neighbor at a distance clearly greater than dNN (bulk nearest neigh- bor) are preferred as equilibrium geometries.The ionic bonding of the oxygen impurity strongly modi6es the structural arrangement of Cs"clusters and, on the other side, it produces a stable inner structure (mostly an octahedron) around which the rest of the Cs atoms are lo- cated, also in spherical layers.In the growing of this external part the formation of a new intermediate shell seems to be ruled by the same condition as for pure clus- ters: no closest neighbor at a distance greater than d NN.
The geometrical arrangement of the aggregates in atomic spherical layers allows for the analysis of the in- teratomic distances for atoms located at different regions of the clusters.When the distances from one atom to its two closest atoms are considered, the values obtained show that the atoms in the inner parts of the clusters are more densely packed than those on the outer parts.This effect should be related to the fact that, for the size range studied, more than half of the atoms of the aggregates are located at the surface and then the strong surface tension produces the shrinking of the cluster, and what is re- markable is that, according to our calculations, it is an inhomogeneous one.It should be interesting to check these conclusions for other simple metal clusters.
FIG. 2. The same quantities as inFig. 1 now for the Cs atoms in OCs"clusters.The oxygen atom is in the cluster center.The vertical arrow indicates the biggest cluster with only two atomic shells.

For
FIG. 3. R ' FIG. 4.Radial distribution of the atoms and electronic densities for Cs26, OCs26, Cs32, and QCs32.Symbols and units are as in Fig.
FIG. 5. (a)  Mean distance to the nearest Cs atom and stan- dard deviation, for the three regions defined in Fig.1for Cs" clusters, as a function of n.(b) Mean values of the distances to the two nearest Cs atoms for each of the regions.In both cases the horizontal arrow corresponds to the nearest-neighbor dis- tance in pure bulk cesium, dNN =9.893 a.u.
FIG. 6.(a) and (b).The same quantities as in Figs.5(a) and 5(b), now for the case of OCs"clusters.
1 a.u.The structures for Cs, o and Cs» are given in Fig. 7, together with those of larger clusters a=

For
FIG. 8. Equilibrium geometry for Cs20.The distances are in atomic units.
FIG. 9.The structure of OCs9 (left) and of the internal region of QCs«, both formed by nine Cs atoms surrounding the O atom.The symbols and distances are as in Fig. 7.(d) is the mean value of the plotted edges.
The vertical arrows indicate the largest clusters with a central atom and one and two atomic layers, respectively.
2 the values of the radii of the two atomic shells of OCs46 are 6.5 a.u. and 15+1.5 a.u., respectively, and this means that the minimum distance between a Cs atom in the inner part and another in the outer part is of the order of, or greater than, dNN, because two atoms in difFerent shells are never in the same radial direction (in this cluster); if it is assumed that no Cs atom in any finite aggregate can be located at a distance from a11 the other atoms that is clearly greater than dNN, as the results for pure Cs suggest, a new intermediate she11 should form in order to fulfill this condition.This intermediate layer bridges the two parts of the OCs"clusters and makes the radius of the surface layer increase faster (see Fig.2).The number of atoms in the intermediate layer increases from 13 to 22 between OCs47 and OCs7o, although with an oscillatory behavior.However, the corresponding shell radius is rather stable.In fact, for 60~n ~70 the number of atoms in this layer oscillates only between 19 13: an octahedron with one Cs atom coordinated to each one of its eight Equilibrium geometries for Cs"(left-hand column) and QCs"(right-hand column) for n = 10,11, 15, 16, 18, and 19.