On the shapes of weakly adsorbed two-dimensional clusters

Surface Science 97 (1980) 537-552 0 North-Holland Publishing Company

ON THE SHAPES OF WEAKLY ADSORBED TWO-DIMENSIONAL

CLUSTERS

Giuseppe DEL RE, Vincenzo BARONE and Nicola MONTELLA Cattadra di Chimica Teorica, Universitd di Napoli, Via Mezzocannone

4, I-80134 Napoli, Italy

and Andre’ JULG Laboratoire de Chimie Thkorique,

Universite’ de Provence, F-13331

Marseille CPdex 3, France

Received 5 October 1979; accepted for publication 24 March 1980

A simple iterative method previously applied to study distortions induced by boundaries in finite clusters is applied to the study of the geometrical relaxation of model two dimensional islands of univalent metals, weakly adsorbed on a substrate. Particular attention is devoted to the formation of one or two vacancies by partial desorption. The rearrangement results in the formation of “ghost-vacancies” surrounding the site where an atom was removed, and/or pieces of a distorted triangular lattice. In some cases the islands are practically divided into subunits. The general picture obtained for the adsorbed phase is a system of non-interacting distorted islands, whose shapes result from competition between “memory” of the shape prior to formation of vacancy, tendency to form a triangular lattice, and tendency of each atom to form a well defined number of bonds.

1. Introduction Two-dimensional clusters of atoms are interesting both as limiting cases of certain three-dimensional systems [ 1,2], and as models for phases adsorbed on surfaces [3-61, especially in the case of weak adsorption. In that case, the adsorbed phase can be viewed as a two-dimensional solid [7] which, in the low-coverage limit, must be studied in terms of the formation, the properties, and the evolution of islands [8,9]. Of course, interactions between the substrate and the adsorbed phase play an important role even in physisorption; however, by analogy with molecules, the major features of adsorbed islands can be expected to be determined by interactions of the chemical-bond type, and should be quite insensitive to the substrate; whereas finer details, which correspond to the so called “non-bonded” interactions [lo], will depend on the substrate. In other words, properties associated with strongly interacting pairs of atoms, like their equilibrium distances (“bond lengths”), should be largely independent on the existence of the substrate; whereas properties, like geometrical characteristics compatible with lengths and angles associated with existing chemical bonds, must be expected to vary with the nature 537

538

G. De1 Re et al. /

~~~p~~

of weakly adsorbed 2D clusters

of the substrate. In particular, the substrate could play a role by keeping physisorbed islands planar or quasi-planar and by imposing a given symmetry, without actually affecting the distances of neighbouring atoms. Consider, for example, four identical strongly interacting atoms forming a physisorbed island on a weakly attractive substrate (we avoid the term “surface molecule” because it is already used for another purpose, see e.g. ref. [I I]). If the atoms under consideration tend to form bonds with one another, such an island will take one of two limiting shapes, a tetrahedron or a losange, for it is characteristic of chemical valence forces to produce situations where there are either strong bonds (with the involved atoms at relatively short distances) or very weak bonds (with the involved atoms at comparatively large distances), and the tetrahedron and the losange correspond precisely to situations where each atom forms three or two strong bonds, respectively. The actual shape is largely independent on the substrate, the losange being favoured if it has an energy of the same order or lower than that of the tetrahedron. The substrate will play a role in determining how close to planar and how elongated the losange will be. Under the assumption that the interaction with the substrate is governed by a Van der Waals potential energy having periodic minima along the surface [12], a planar or quasi-pl~ar situation can be expected for the losange arrangement; the ratio of the axes of the losange, the deviation from planarity and slight adjustments in bond distances, will be determined by the symmetry and the nature of the substrate, and by the residual “non-bonded” interactions between the adatoms (a useful macroscopic analogy is a floating parallelogram with hinged corners). To summarize, we can say that chemical forces determine what chemists would call the “configuration” of an island, whereas complete specification of the geometrical situation, called by chemists the “conformation”, also requires knowledge of the effect of the substrate. With the above picture in mind, we now discuss the following problem: consider the low-coverage limit of an adorbate-substrate system where, at full coverage, a physisorbed phase forms a square-lattice monolayer. At low-coverage, the adsorbed phase may be thought to consist of islands which retain the regularity of the infinite lattice as far as it is compatible with boundary effects. Thus, more or less distorted square-lattice islands should be expected. When one of those islands loses one or two atoms by desorption, further more dramatic changes in geometry are expected: will the islands collapse to a more compact form or will they break down into several sub-islands? What will be the geometry of the resulting system? A complete treatment of a specific concrete instance of island formation and evolution is one of the ways to gain insight into these questions. It has seemed to us, however, that a model study laying down the general qualitative features of the phenomenon, without the admixture of less pertinent information, is a necessary first step. Therefore, we have carried out an investigation of the geometries of “fr.ee” islands and of their changes upon formation of vacancies, using a Hiickel scheme coupled with a recipe for determining optimum geometries. The results of

G. Del Re et al. /Shapes

of weakly adsorbed 20 clusters

539

that investigation are presented in this article, which is based on previous work on boundary effects in two- and three-dimensional clusters [ 13,141.

2. The method A previously described. iterative TB scheme 113,141 has been used. By its structure, it simulates an ideal relaxation process taking place upon formation of the edges and of the defects, for it assumes that the boundaries and the defects initially coexist with the geometry of the perfectly regular lattice. From such an unstable structure, a stable structure is determined (by iteration), which is characterized by the property that the interatomic distances correspond to bond orders which are also obtained from the effective one-electron Hamiltonian matrix associated with these distances. In the computation reported here, the iteration process has been applied to ah the near nei~bours of a given atom, without any distinction between directly bonded and non-bonded atoms, and self-consistency with respect to charges has also been required. Such extensions with respect to our earlier work do not affect previously obtained results, but eliminate certain implicit assumptions (like equality of atomic Mulliken populations) which do not hold for ail types of systems. We have started from a Roothaan (LCAO-SCF) one-electron scheme defined in the general framework of the Born-Oppenheimer and of the frozen-core approximation. The A0 (Atomic Orbital) basis has been taken to consist just of one s-like orbital 1~~)per atom. Such a small (minimal) basis is reasonable in model studies con~er~ng atoms with just one valence electron. Its elements are not exactly standard s orbitals of the given atomic species (say, 2s orbitals of Li) if, as is currently done, the basis is assumed to be an orthogonal one. However, their precise functional form is not important in a parameterized procedure such as the present one: indeed, it can be assumed that the basis orbitals actually satisfy the more restrictive zero-differential overlap (ZDO) condition [ 1.51, a condition widely used in the ND0 semiempirical methods (cf., e.g., ref. 1161). Let us the basis be the row vector

where N is the number of centres of the given cluster; the molecular orbitals (MO’s) can then be represented by the row-vector iJI>= lx) U )

ta

U being the unitary matrix defined by the eigenvalue equation LU= UE

(UflJ=I)

.

(3)

Here L is the matrix which represents the Hamiltonian operator of the electronic system under consideration in the basis (1), E is the diagonal matrix whose non-

G. Del Re et ai. / Shapes of weakly adsorbed 20 clusters

540

van~s~ng elements are the orbital energies el, e2, .... eiv. The matrix L is given by L = LCO’C t G)

(4)

where G is the matrix representing electron-electron ZDO appro~mation we have [ 17-201

c Ps,(vIss)

G,, =

;P&F IFF)t

G,., =

-$P& is) ,

sfr

interactions. Because of the

,

Sal

where l’,, and P,, are electron populations and bond orders, respectively. For molecular orbitals whose occupation numbers are n,, n2, .,,, nN:

Pys = Re c nju&j i

.

(6b)

In the ground states under consideration the ni values are 0 or 2 depending on whether the jth orbital is empty or occupied (in our case, there are doubly-degenerate singly-occupied orbitals, but as far as energy is concerned, it is sufficient to count one of them). The diagonal elements of L are evaluated as L,,‘-W,

f;P,,(Vl~)

- t=*,(rrlss), .s

where IV,is the ionization potential, and f&=iV-P

(8)

s.s

is the net charge associated with the orbital I%) which shares N, electrons. If the net charges are close to zero, as is the case in a system of practically equivalent atoms, eq. (7) reduces to L,, ‘V-W, + @7(w).

(9)

The off~iagon~ elements of matrix& are expressed in terms of the off-diagonal Hamiltonian element &, of the molecule formed of two atoms of the species under study (the value of pa being chosen so as to fit the experimental binding energy), and of prs =&, -Rk

>

where RF’ is the equilibrium bond length of the diatomic molecule: L,, = Ml + aPrs + bk)

exp(-kp,,)

,

(10)

G. Del Re et al. / Shapes of weakly adsorbed 20 clusters

541

and all distances are given by the general formula R,, = R( 1 + 3.4Pr,)/( 1 + 5.9P,,) ,

(11)

where R is the only parameter characteristic of the particular species under study. The above expressions allow inclusion in the computations of all neighbours of the various atoms (instead of distinguishing between nearest and next nearest neighbours). However, as has been recently shown [21], for long distances L,, falls off more rapidly than when it is computed by exponential functions or by formulas involving overlap [22]. Hence, we assume that L,,vanish at the distance corresponding to zero bond order (i.e. E):

L,.,=O if

R,,S=R.

(12)

In such cases it is understood that there is no chemical binding between r and s: therefore, R,,should be considered undefined, except for purely geometrical or symmetry considerations, unless “non-bonded” interactions are introduced. If core repulsions (Rcore) are evaluated from the approximation (valid for univalent atoms)

then the total energy is given by

Etot = c niq- ;c P,,G,, +RCo'e r,s i =

2

+f

c

j,occ

Ej

-$

[

cPgrlrr) r

- r$s2F~s(Frlss)]

rG(1 - PrJss,,)(rr Iss).

As has been mentioned, our systems often have doubly degenerate levels. In such circumstances, the bond orders have been computed on the basis of Hund’s rule one electron per orbital. The computational procedure is as follows. From a starting set of distances and of charges, the matrix L is computed according to eqs. (7) and (10) and used to determine the eigenvector matrix U which corresponds to the solution for the given geometry: then the distances are changed according to eq. (ll), so that from the Pr,'s and Prs's obtained, new elements of L can be evaluated. Diagonalization is carried out again, and the whole procedure is repeated until two successive matrices L coincide within the considered limit (i.e. to self-consistency). As regards connection between method used here and sophisticated methods, it may be noted that our procedure simulates an iterative SCF scheme by which, instead of reaching self-consistency for various geometries and then determining the best geometry: (i) a given set of charges and bond orders produce a geometry between the standard

542

G. Del Re et

ad. f

Shapes of weakly adsorbed 20 cbtsiers

steps (consisting in determination of the 15matrices for various geometries, determination of energy, and choice of the geometry for which energy reaches a minimum); (ii) a new set of charges and bond orders is produced from the L matrix associated with the geometry thus determined, and so on to convergency. In our scheme the double variation with respect to the orbital coefficients and to the distances is replaced by the assumptions that the distances which minimize the energy are always related to the coefficients (through the bond orders) by eq. (I 1). Thus, the procedure used here may be seen as a semi-empirical (parameterized) version of a min~al-basis SCF determination of optimal geometries. The justification of use of a simplified method is well known: more sophisticated computational schemes would be more reliable only if very large basis sets were used, they would require and produce a large amount of information specific of the particular case under consideration, and would be feasible only for very small systems. Furthermore it has recently been shown [23] that a procedure similar to that used here and in previous work [13,14] is able to give very accurate geometries when coupled with Htickel type methods.

3. Pararneterization and choice of the model systems The coefficients of eq. (11) have been obtained by fitting the experimental distances [24] to the bond orders corresponding to the nearest neighbour approximation, i.e. P,, = 1 for the diatomic molecule (M,) and P,, = 0.19 and 0.26 respectively for the regular fee and bee lattices 125,261. Table 1 shows that eq. (11) leads to very accurate results for all kinds of univalent atoms. Therefore it can be expected to obtain geometries realistic enough to suggest general trends. In the present paper we have used parameters determined for lithium, i.e. W,.=5AeV,

(rr]vp)=3.3eV,

/3,=-l.l3eV,

RFs=2.67A.

To obtain the coefficients of eq. (10) we have reproduced at best ab-initio results for Li4 [27] and have obtained a = -0.662,

b = -0.048,

k = 1 .lO .

The two-centre two-electron integrals (n’[ss) have been computed according to Ohno’s formula [28]. Single and double vacancies have been introduced in the position shown in the figures for two-dimensional islands with up to 64 centers. Also a few cases representing boundary defects have been studied. The actual drawing of the figure representing the distorted clusters obtained from the computer has involved a number of difficulties, due to the fact that certain distances were not defined (I?,, 2 4.10 a) or were compatible with both planar and distorted structures. We have used symmetry conditions in the former case, and have drawn planar projections of the

G. Del Re et al. /Shapes Table 1 Interatomic distances given the experimental

for different univalent data (from ref. [ 241)

Species

E

M2

Li Na K CU Ag

4.10 5.00 6.20 3.32 3.73

2.62 3.19 3.95 2.12 2.38

metals

computed

bee (2.67) (3.08) (3.92) (2.12) (2.50)

543

of weakly adsorbed 20 clusters

3.05 3.12 4.61 2.47 2.77

by eq. (11); in parentheses

are

fee (3.04) (3.72) (4.62) (2.48) (2.79)

3.16 3.86 4.78 2.56 2.88

(3.16) (3.85) (4.76) (2.56) (2.88)

actual clusters in the latter case, for the reasons discussed in the introduction. A classification that may help in the search for trends, and that has guided the choice of the systems to be studied is as follows: (a) systems with no vacancy; (b) systems with one vacancy : (b 1) one inner vacancy, (b2) one missing edge center, (b3) one missing corner; (c) systems with two vacancies: (cl) no vacancy on the boundary, (~2) one vacancy on the boundary, (~3) both vacancies on the boundary. The systems of type a were already studied [ 13,141 and only a brief summary of the results will be given in the next section. Over thirty model clusters of the types b and c have been studied, most of which are presented in the figures and in table 2.

4. Results and discussion The results are presented in figs. l-8. A discussion must start from the effect of boundaries. In our previous work [13,14] we found that square clusters tend to swell so as to form rounded sides (fig. 3A) and rectangular clusters tend to form “drops”. Therefore, we can say that, once weakly adsorbed islands are formed, they tend to round-off slightly if they are close to squares, and to approach sets of smaller, nearly-square islands when they originate from rectangles (cf. fig. 2 of ref. [13]). This tendency persists when an atom leaves the island and a vacancy is formed, as is suggested e.g. by the 7 X 7 cluster of fig. IC. There, as well as in the clusters of figs. 6 and 7, the rounding-off at the edges smoothes the defect out. The most impressive example’ of relaxation tending to suppress boundary defects is the 3 X 3 cluster of fig. 6B: as no far neighbours are present to counter relaxation, that

544

G. Del Re et al. /Shapes

of weakly adsorbed 20 clusters

i

l

,$’ c

_IL~I

i

ll ll

l

Fig. 1. Equilibrium geometries of some clusters with a vacancy in the central position. In this figure, as well as in all the following ones, the broken lines indicate the atom positions corresponding to a section of the regular bee lattice of lithium.

is very similar to the one with a central vacancy (fig. 1A). In general the loss of a single atom, even from the interior of an island, results in shifts of the surrounding atoms tending to eliminate the vacancy, a phenomenon quite analogous to the rounding-off of boundary defects. An example is given by the 5 X 5 cluster with cluster

A

Fig. 2. Equilibrium geometries of some 4

X

4 clusters with a single vacancy.

G. Del Re et al. /Shapes

. :.:*:. I-I .

l

‘.

*

.

l

9.

:

.

*

-.-.-‘p---

.

.

:*

:

.

of weakly adsorbed 20 clusters

.

.’

.

l

D

Fig. 3. Equilibrium geometries of some 5 X 5 clusters (B) and with two vacancies (C-F).

without

vacancies

(A), with one vacancy

a vacancy in the 4.4 position (fig. 3B). This cluster, as well as the 4 X 4 cluster of fig. 2A, also shows a new remarkable effect: whereas the atoms immediately neighbouring the vacancy move in to fill it, farther neighbours act as they were repelled, so that a “ghost-vacancy” appears as a reaction to those inward shifts. The most impressive case of ghost vacancies is provided by the 5 X 5 cluster with one central vacancy (fig. lB), where the formation of the ghost vacancies gives a square which

Fig. 4. Equilibrium diagonal.

geometries

of the 6 X 6 (A) and 8 X 8 (B) clusters

with two vacancies

on a

G. Del Re et al. /Shapes

546

Fig. 5. Equilibrium

geometries

of weakly adsorbed 20 clusters

of ‘some 7 X 7 clusters

with two vacancies.

is rotated by 90” with respect to the initial one. One may wonder whether the external atoms of the cluster 1B (and 3D, although there are no ghost vacancies there) could no really move in to fii the holes. In fact, the iteration procedure used to obtain equilibrium geometries is such that if a bond order becomes zero at the nth step, it stays zero or becomes negative. Therefore, once an internuclear distance has become long enough to produce a zero bond order, it cannot revert to a shorter value. We suggest that this limitation of our method has a physical counterpart in that the formation of gaps may well give stable “sub-islands” by some sort of surface-tension effect: only a strong perturbation could force the system to change

A

Fig. 6:Equilibrium corner vacancy.

r

geometries

of two-dimensional

clusters

of increasing

dimensions

with

a

G. Del Re et al. ! Shapes of weak@ adsorbed 20 clusters

0

.......

l

541

i r_______~. ......

i

:

.........

~~

li B *

:o I I ~ ..._~___ ...... ........i........ * 0 . ; j ......_A&. ...... ~

e

0 e

Fig. ‘7.Equilibrium geometries of two-dimensional atoms missing on a corner.

;

*

l l

l

clusters of increasing dimensions with two

back to a compact cluster. Thus, a structure with the outermost atoms drifted to fiu. the vacancies is stable, but cannot result of a gradual rearrangement of a frag mented structure. The 5 X 5 clusters illustrate another interesting trend: the tendency of the square lattice to become triangular as far as possible - but with some constr~nt due to “memory” of the initial square arrangement. The mechanism of that “memorization” depends on the fact that atoms initially equivalent by symmetry tend to stay so. It can be understood by considering the case of a 5 X 5 cluster with two vacancies (figs. 3C and 3E): the 23 atoms can form almost perfect triangular lattices and

a Fig. 8. Equilibrium table 2).

b

geometry of the most stable 5 X 5 cluster with two vacancies (case q of

548

G. Del lie et al. / Shapes of weakly adsorbed ZD clusters

yet retain the initial symmetry. On the other hand, persistance of one or more vacancies (as such or as “ghosts”) takes place when either 24 (rather than 23) atoms must be accomodated in the initial square, and/or the initial symmetry (after creation of vacancies) is not compatible with a regular triangular pattern. The conservation of symmetry is characteristic of our method and we have used it to overcome a number of difficulties. By imposing a given initial symmetry, and by using that symmetry also to fix geometrical parameters that were not determined by the iteration procedure, completely determined results can be obtained, and most structures keep their planarity. The consistency of the symmet~ constraint with the general trends is proven by cases where no symmetry exists (e.g. fig. 7). On the other hand, small symmetry-destroying perturbations on highly symmetric cases (e.g. rectangles; cf. refs. [13,14]) do produce non-symmetric results. Then only non-planar final clusters are consistent with the results of the iteration procedure. This is actually the reason why we have focused our attention on symmetric cases; the more so as there are no differences in general trends. When two vacancies are formed with a time interval far shorter than the lifetime of unreconstructed single-vacancy systems either coalescence or repulsion are expected. Actually we find a different effect: each vacancy tends to be surrounded by a round (llexagon~~ “wall” of atoms. Some tendency to coalescence of vacancies which are not nearest neighbours is found only in large clusters, e.g. in passing from the 6 X 6 to the 8 X 8 cluster with two vacancies on the diagonal (fig. 4): that tendency is associated with the tendency of the bigger cluster to retain its square shape, Further insight can be obtained by considering energies computed according to eq. (15) and reported in table 2 for the 5 X 5 case. The factors playing a role in those energies are the positions and numbers of vacancies, but also the ensuing distortion. As regards systems with a single vacancy, it appears that the greatest work is necessary to break a side in highly symmetric positions (case b): this is related to the above remark that symmetry is a factor opposing formation of triangular cells. Inner vacancies are associated with a smaller increase in energy, except the central vacancy, where, as has been seen, very important distortions take place. Systems with two vacancies present a more complicated picture. The highest work of formation is found where the vacancies are on diagonals; a possible explanation is again that this favours symmetry agains formation of triangles. The competition between “memory” of the shape prior the formation of vacancies and the tendency to form a triangular lattice is best evidenced by the fact the square q of table 2, where the best compromise between the two competitive effects is reached (fig. 8), is that with energy closest to the full square: the central hole of that “atoll” island has a stabilizing effect because the average number of bonds formed’by each atom is smaller and is closer to the number for which the destabilizing effect of core repulsion to the energy reaches a minimum. On the other hand, systems like s of table 2 (fig. 3D), where the vacancies are preserved or like i, m

G. Del Re et al. /Shapes

of weakly adsorbed 20 clusters

549

Table 2 Cohesion

energies

(eV) of 5 X 5 square

clusters

with one and two vacancies,

P

k

a

after relaxation

FE! 37.606

39.236

b

EEI 33.356

34.965

36.663

m

h

El3

El!@

36.395

d

36.566

r

EEI

RI 35.376

35.570

n

i

34.612

S

E!a BEiEi3 E#

EEB

35.959

e

34.309

1

9

M

C

35.260

lzl!

EEI

j

35.943

35.591

3 5.420

I -

t

0

35.403

34.618

35.756

I I 35.160

, _

550

G. Del Re et al. / Shapes of weakly adsorbed 20 clusters

(figs. 3C and 3E), where the triangular lattice is obtained at the expense of great distortion of the shape prior to formation of vacancies, have a lower cohesion energy. That distortion plays a major role is also supported by comparison of cases like q and d of table 2: it appears that removal of one corner eases the tension due to retention of a high symmetry in d (or in c), so that the formation of the second vacancy is, so to speak, an energy releasing process. The case of corner vacancies can be interpreted along similar lines: examples are the systems h and r of table 2. To complete the present discussion we consider the question whether the present results could be obtained by other schemes, like the rigid-sphere model, or other pairwise potentials [ lo,1 21. Use of such models is not justified because in the systems studied here, the electrons are completely delocalized so that the energy of a single island must be obtained by quantum-chemical computations involving at least all the mobile electrons: empirical potentials could rather be of use, in the present context, in connection with the effect of the substrate on island conformation. A similar hybrid procedure has already given very good results for the similar problem of molecules containing delocalized 71systems [29,30] .

5. Conclusion We have shown that a simple quantum-chemical method can provide the framework of tentative rules and trends to be used in the study of monolayer formation, properties and evolution in the low-coverage limit. The practical implications of our results can be illustrated by what they suggest about the life of a specific island, e.g. the 5 X 5 square. We can imagine that a rectangular island (say containing 5.X n atoms) is formed from a continuous adsorbed monolayer having a square-lattice structure by desorption of a sufficient number of atoms. According to our method, such a rectangular island tends to divide into “drops” (cf. fig. 2 of ref. [13]), so that we expect that, sooner or later, it will divide into nearly square islands. One such island is distorted with respect to the original geometry, because the atoms of its boundary are no longer subject to the attraction of outer atoms: therefore, there is a sort of “surface-tension” effect resulting in a rounding-off of the sides. As the desorption process goes on, the island will lose one atom from any of its sites. The factors causing a given structure to be more stable than another in the framework of our model are essentially the number of local transitions to triangles and the rounding-off of the sides, possibly by forming “atoll” type structures. The greater stability of structures with certain positions of the vacancy may mean that a vacancy will migrate to those positions if the potential energy barriers are not too high and if the lifetime of the initial arrangement with respect to other processes is long enough. The most significant process that can take place in competition with migration of the vacancy is further “division” of the island, as is sug-

G. Del Re et al. /Shapes

of weakly adsorbed 20 clusters

551

gested by the geometry found in certain cases (fig. 3B). Further division can also take place if another atom is lost at a suitable site before the structure has had time to rearrange to a more stable situation (fig. 3D). When two vacancies are formed simultaneously, then the final geometry is affected by both; whereas a second vacancy formed on a structure already defective, where there is not a certain bond, cannot restore that bond (in so far as our model simulates reality) and hence leads to a different final structure. Early simultaneous formation of two vacancies is expected from our model to be followed either by further subdivision into new types of smaller islands (fig. 3D) or by formation of an “atoll” island (fig. 8) - the shape to which all islands should tend if they lived enough after formation of one or two vacancies. Thus, the properties of the low coverage limit of weak adsorption should be dominated by those of a system of non-interacting “atolls” of a variety of sizes. The theoretical results presented above appear to be quite reliable (see also table 1) and this means that their practical implications can be taken as a basis for the interpretation of experimental evidence. In fact, earlier experience with ‘II systems [31] and recent work on saturated molecules [23] confirm the above statement. Furthermore, more sophisticated (ab-initio) computations carried out for the smaller islands have given results in full agreement with the simpler method reported here [27,32]. Nevertheless improvements, especially in connection with the distance-bond order relationship and the evaluation of core repulsion, are desirable extensions of the present study, let alone explicit consideration of the role of the substrate. On the other hand, experimental work on island formation and shape is only at its beginning, but the importance and the accessibility to experiment of island formation is already well established. For instance, adsorption has been shown to proceed in certain cases via the formation of two-dimensional islands (StranskiKrastanov growth mode) not much larger than those considered in the present study (e.g., refs. [33-361). The practical importance of the detailed shape of the islands in the early stage of adsorption (or the last stages of desorption) is obvious if one thinks of the interpretation of crystal growth data and of the reactivity of supported catalysts (e.g., refs. [26,37,38]). In both cases the experimental evidence is indirect, and requires some preliminary notion about the shape and number of reactive centers, i.e. of individual islands.

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