Magic numbers and growth sequences of small face-centered-cubic and decahedral clusters

29 December 1995

CHEMICAL PHYSICS LETTERS ELSEVIER

Chemical Physics Letters 247 (1995) 339-347

Magic numbers and growth sequences of small face-centred-cubic and decahedral clusters Jonathan EK. Doye, David

J.

Wales

University Chemical Laboratories, Lensfield Road, Cambridge CB2 I EW, UK

Received 5 October 1995; in final form 25 October 1995

Abstract

Growth sequences for small decahedral and face-centred-cubic (fcc) clusters are predicted by maximisation of the number of nearest-neighbour contacts. These sequences allow us to deduce likely magic numbers for fcc and decahedral clusters which may be appropriate to neutral clusters of C60 molecules.

1. Introduction

Clusters provide a bridge between isolated atoms and molecules and bulk material. As a consequence, one of the main themes in cluster science is the evolution of properties to their bulk limit [ 1 ]. However, one can also find new and interesting properties that are specific to this intermediate size regime, and structure is perhaps the best known example. For many types of small clusters the predominant morphologies exhibit fivefold symmetry which is incompatible with the translational periodicity of bulk crystals. Such species were first discovered in investigations o f clusters bound by the Lennard-Jones potential [ 2]. Although early theoretical studies considered structures that were fragments of a crystalline lattice, it soon became clear that the lowest energy structures were based on Mackay icosahedra [3]. The LennardJones potential provides a reasonable description of the interaction between rare gas atoms, and it has been confirmed by electron diffraction [ 4,5] and mass spectoscopy [6,7] that rare gas clusters are icosahedral when they contain up to at least 750 atoms. Fur-

thermore, the intensity variations in the mass spectra show very good agreement with those expected from the icosahedral growth sequence of LennardJones clusters [8]. Icosahedral structures have since been found for many systems, including metals [ 9 1 1 ], and molecular clusters [ 12]. The energetics underlying the favourability of icosahedral structures at small sizes can be easily understood by partitioning the potential energy into its different contributions. For a pair potential there are three terms, V = -n..

+ E~tr.i. + E . . . ,

(1)

where the unit of energy is the pair well depth. The number of nearest neighbour contacts, nnn, is given by l,

nnn ~

(2)

i
where ro is a nearest neighbour criterion. The strain Estrain , which measures the energetic penalty for deviation of nearest-neighbour contacts from the equilibrium pair distance, is given by energy,

0009-2614/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 0009-26 I 4(95) 01 223-0

J.P.K. Doye, D.J. Wales/Chemical Physics Letters 247 (1995) 339-347

340

Estrain =

v(rij ) - 1,

Z

(3)

i
where u(rij) is the pair potential. The contribution to the energy from non-nearest neighbours, Ennn, is Ennn=

Z

~,(r/j).

(4)

i/r.

Mackay icosahedra have a larger value of ttnn than the optimal decahedral or face-centred-cubic clusters because they have entirely {111 } faces and are nearly spherical in shape. However they are also the most strained of the three morphologies. At small sizes the number of nearest neighbours is the most important factor, and hence icosahedra are the lowest energy structures for the Lennard-Jones potential. As the size of the clusters increases the strain energy grows rapidly (it is proportional to the volume of the cluster), and so destabilises first the icosahedral, and then the decahedral structures [ 13,14]. The predominance of icosahedral structures has caused the study of small fcc and decahedral clusters to be neglected. However, there is now a growing body of evidence that some small clusters may be decahedral or fcc. For example, icosahedral structures are not so universal for small Lennard-Jones clusters as was first thought: it has been discovered recently that the lowest energy structure for a cluster with 38 atoms is fcc [ 14,15], and that the lowest energy structures for clusters with 75, 76, 77, 102, 103 and 104 atoms are probably decahedral [14]. (The I02-, 103- and 104-atom clusters are based on the 101atom Marks' decahedron [ 16], and have energies of -569.364, -575.766 and -582.087e, respectively.) This is because nnn is a sensitive function of the size of the cluster and the most favourable members of the different growth sequences do not occur at the same sizes. However, further calculations suggest that the 75-atom decahedral structure is the global free energy minimum only at very low temperatures [14], and the other cases are likely to be similar. In reduced units the Lennard-Jones potential has no free parameters. The Morse potential, though, has a single variable parameter which determines the range of the potential. This parameter gives Morse clusters a much richer variety of structural possibilities than Lennard-Jones clusters [ 14]. For a given strained structure, E.~train (Eq. (1)) increases rapidly as the

range of the Morse potential decreases because the potential wells become narrower, thus destabilising the strained structures. Consequently, the lowest energy structures of short-ranged Morse clusters are not icosahedral (except at very small sizes) but decahedral or fcc. Similarly, the inclusion of many-body forces, which are essential to accurately model metals, may alter the relative energies of the { 11 1} and { 100} faces, and consequently affect the energetic competition between the structural types [ 17,18 ]. Girifalco's intermolecular potential for C60 is very short-ranged with respect to the equilibrium pair separation [ 19]. Therefore, we anticipate that clusters of neutral C60 molecules may not have icosahedral structures. However, mass spectroscopic studies have only been performed for C6o clusters grown as positive ions. The results showed intensity peaks typical of icosahedral structures [20]. We do not think this finding contradicts the above theory because the positive charge introduces significant long-range character into the intermolecular interactions. It would therefore be very interesting if an experiment were performed to probe the size distribution of neutral C6o clusters. This would actually test whether the Girifalco potential, which has been used in several studies [ 21,22 ] of bulk C60, gives a good account of the C60 intermolecular potential. What magic numbers would we expect to be observed for neutral C60 clusters, if the Girifalco potential is appropriate? The morphology of optimal fcc clusters is given by the Wulff construction [23,24]. If only nearest-neighbours are considered in the calculation of the surface energies (a good approximation for a short-ranged potential), then the Wulff polyhedron is the truncated octahedron with regular hexagonal faces. Complete regular truncated octahedra occur at N = 38,201,586 . . . . . Similarly, a modified Wulff construction gives the optimal decahedral morphology, which is a Marks' decahedron, and complete symmetrical structures occur for N = 75,192,389 . . . . . We expect these two sequences to represent magic numbers for fcc and decahedral clusters, respectively. To make detailed predictions of the intensity pattern between the above sizes one needs complete growth sequences. Surprisingly, growth sequences have not, to the best of our knowledge, been deduced for decahedral and fcc clusters. These sequences should also be very useful to experimentalists studying the structure of clusters using methods such as chemical probes.

J.P.K. Doye, D.J. Wales/Chemical Physics Letters 247 (1995) 339-347

341

Table 1 The number of nearest neighbours for growth sequences of decahedral and fcc clusters. The number of atoms along the (pseudo-fivefold) axis of the decahedral structures, h, is also given. For some sizes there are isomers with different h but the same nnn ~c 3 4 5 6 7 8 9 20 21 22 23 24 25 26 27 28 29 3O 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

37 41 45 49 53 57 61 65 69 73 77 80 85 89 94 98 103 107 112 116 121 125 130 134 139 143 148 152 157 161 166 170 175 179 184 188 193 197 201 206 211 217 221 226 231 235 240

3 3 3 3 3 3 3 3 3 3 3 3/4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4/5 4/5 5 5 5 5 5 5 5 5

36 40 44 48 52 56 60 64 68 72 76 81 85 90 94 98 102 106 111 115 119 124 129 133 138 144 148 152 156 160 165 169 174 178 183 187 191 196 201 207 211 216 220 225 229 234 240

~c 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106

245 249 254 259 265 269 274 279 283 288 293 299 303 308 313 319 323 328 332 337 341 346 351 355 360 364 369 374 378 383 387 392 397 403 408 414 418 423 428 434 439 445 449 454 459 463 468

5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5/6 5 5 5/6 5/6 5/6 5/6 5/6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

244 249 253 258 262 267 271 276 280 285 291 296 300 305 310 316 320 325 330 336 340 344 348 353 358 363 369 373 378 382 387 391 396 402 407 411 416 421 427 431 436 441 447 451 456 460 465

ffc 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152

473 479 483 488 493 499 504 508 513 518 524 528 533 538 544 549 553 558 563 569 573 578 583 589 594 598 603 608 614 618 623 628 634 639 643 648 653 659 664 670 674 679 683 688 693 697

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6/7 6 6/ 7

469 474 480 485 490 496 500 505 510 516 520 525 530 536 540 545 550 556 560 565 570 576 580 585 590 596 600 605 610 616 620 625 630 636 640 644 648 653 658 663 669 674 679 685 690 696

342

J.P.K. Doye, D.J, Wales/Chemical Physics Letters 247 (1995) 339-347

The structural information obtained in such experiments is indirect, and requires model candidate structures. In their studies of nickel clusters [ 2 5 - 2 7 ] , Parks et al. have shown that icosahedral structures dominate for N ~< 28 and 49 ~< N ~< 71. However, preliminary results indicate that for some sizes between these ranges the structure may be fcc [26]. There has also been some experimental evidence that small gold clusters may adopt fcc structures. EXAFS (extended Xray absorption fine structure) spectra have been interpreted as providing evidence for truncated octahedral clusters, and particularly for the 38-atom truncated octahedron [ 28 ].

simply by comparing nnn. Fig. 1 shows the second finite differences of the energy, A 2E = E ( N + 1) + E( N - 1 ) - 2 E ( N ) for both growth sequences assuming E = -nnn. Peaks in A 2 E correspond to clusters which are stable compared to adjacent sizes and have been found to correlate with magic numbers in mass spectra of sodium clusters [29]. The decahedral sequence has eleven peaks in A 2E and the fcc sequence sixteen peaks. In Fig. 2 we compare the values of nnn to those for the optimal fcc and decahedral morphologies interpolated between the predicted 'magic' sizes. The interpolated values for the truncated octahedron with hexagonal faces are obtained from

2. Results

Nto = 16n 3 + 15n 2 + 6n + l

Here we calculate growth sequences for decahedral and fcc clusters with 13 ~< N ~< 152 by finding the structures which maximise the number of nearest neighbours. For a given structural type E~tr,i, and En,,~ (Eq. ( 1 ) ) do not vary much between possible structures, and so the main determinant of the lowest energy structures is n,~, assuming that the forces can be reasonably described by a pair potential. This method has been used successfully in studies of icosahedral clusters [7] and has been employed to generate candidate structures for the global minima of LennardJones [8] and Morse [14] clusters. In fact, comparison of Northby's approach [8], based on counting nearest neighbours, and a recent study employing hypersurface deformation to find global minima [15], reveals that nineteen of Northby's minima are lower in energy, one is higher and the remainder are the same. The values of n~n for the growth sequences are given in Table 1. For many sizes there are a number of isomers with the same maximum value of nn,. As usual for problems of global optimisation, there is no guarantee that there are not structures with even larger nnn- The present values were generated by considering structures that maximise sphericity and minimise the number of low coordinate faces and surface atoms, and by making use of the experience gained from previous studies [ 14]. This makes us fairly confident that the values are at least good lower bounds to the maximal nnn •

AS Estrain and Ennn are relatively smooth functions of N we can compare the stability of different sizes

and

to = 96n 3 + 4 2 n 2 + 6n, nnn

(5)

where n is the number of nearest-neighbour contacts along an edge. The first equation is solved for n as a function of N and the result is substituted into the second equation. As expected, no fcc clusters have a value of nnn larger than the interpolated values (Fig. 2). A similar procedure is possible for the optimal Marks' decahedron using Nmd = 71 0 n 3 + 20n2 +

ma = 20n 3 _[_ ~2 0-5 n nnn

2

101 t /

3

+ 18

+ 72 7-9n _ + 57,

and (6)

where n is the number of nearest-neighbour contacts along an edge of a {100} face. A Marks' decahedron is formed from a pentagonal bipyramid (hence the name decahedral) by truncation parallel to the fivefold axis and the equatorial edges to reveal five { 100} faces, thus making the shape more spherical. Re-entrant {111} faces are also introduced between adjacent { 100} faces. The optimal Marks' decahedron has square {100} faces, and the depth of the reentrant faces is dependent on the size. For small clusters, this depth is one atomic layer. An example of such a Marks' decahedron is the 75-atom cluster shown in Fig. 3. If the interpolated values of nnn for the Marks' decahedra with re-entrant faces of depth one and two atomic layers are compared, the latter decahedron has a larger nnn for N > 371. N and nnn for this Marks' decahedron are given by

J.P.K. Doye, D.J. Wales/Chemical Physics Letters 247 (1995) 339-347

54

a

64 71 75

343

95 101 108 117 126 135

146

2-

1

0

A2E

-1 -2

2'0

4'0

8'0

6'0

1()0

1½0

1'~0

N

b

A2E

01

38

52 59

75 79 86 98 102 112

140

~i~

-1 2;

40

6;

8'0 1;0 li0 140 N Fig. I. Plots of A2E as a function of N for (a) the deeahedral growth sequence, (b) the ~ c growth sequence.

4-

3

.... 75

2

...............r

-1

,,

,

_ I~+"',,~/Y+ ,'7 b., fi>i'!~W~,+. -Z I ~, I 4~t

,' ~.'

-

146

,0,

~ ~. 140

,02

$: i~

~7

....---

...............

~ ~,, !4,:,fi,,k>I ~,' tl ~ ~,~ II A I I I II

4<

+"

{

2, II I~ II

ti

-4

i

i

7

~

5 20

40

i

i

i

60

80

100

|

120

i

,7

140

N Fig. 2. Plot of ntntn c - ntn° (dashed line with crosses) and -'nn"deca_ nnnto (solid line with diamonds). The horizontal dotted line is the interpolation for the truncated octahedron with regular hexagonal faces and the dotted line with positive slope is the interpolation for the optimal Marks' decahedron.

344

J.P,K. Doye, D.J. Wales/Chemical Physics Letters 247 (1995) 339-347

54

64

95

71

101

75

117

108

126

135

146

F i g . 3. D c c a h c d r a l s t r u c t u r e s w i t h ,5. 2 E = 2.

Nmd2 = ran3 3 q- 35n2 + 2sJ 3 n + 80

rod2 tlnn

=

20n 3 + ~ '385 -n

2

+ - 5909 - n+339,

and

(7)

where n is the number of nearest-neighbour contacts along an edge of a { 100} face. Fig. 3 illustrates the structures in the decahedral growth sequence for which A zE = 2, which are the best candidates for additional magic numbers. The sequence from N = 13 - 23 is illustrated in Fig. 4 of Ref. [ 14]. It is based on the 13-atom truncated decahedron, and proceeds by capping of the square faces and then by further addition of atoms until the 23-atom pentagonal bipyramid is complete. Further growth is not based upon this structure because decahedra with more atoms along the fivefold axis have larger nnn

above this size. Hence, a significant structural change must occur as the cluster grows if it is to retain the lowest energy geometry. Similar structural changes must also occur at N ~ 50, 90 and 150 when the decahedral axis again increases in length. For N > 24 the structures are based on the 19-atom truncated decahedron with (2 × 1 ) { 100} faces. The growth sequence proceeds by capping of the {100} faces, and addition of atoms to the grooves created, thus producing new {100} faces. The cycle then begins again by capping of these { 100} faces. Symmetric structures with Dsh symmetry are possible at N = 29, 39 and 49, although there are also asymmetric structures for the two smaller sizes that have the same tlnn. The sequence would lead to the 54-atom pentagonal

J.P.K. Doye, D.J. Wales/Chemical Physics Letters 247 (1995) 339-347

bipyramid, but before this size is reached decahedral structures with a longer axis become lower in energy. The growth sequence for N = 24-50 is fairly typical. As atoms are added to a decahedral cluster with a particular number of atoms along the decahedral axis, h, the shape changes from prolate to approximately spherical to oblate. This is the reason for the four broad maxima, corresponding to h = 3-6, for the decahedral growth sequence in Fig. 2. (The maximum is fairly rugged for h = 5.) The maxima correspond to approximately spherical clusters, which minimise the surface area and hence maximise n~o. The growth sequence for the decahedral structures with h = 5 and 6 proceeds in a similar manner to h = 4 except that the growth is asymmetrical (the decahedral axis does not pass through the centre of mass of the cluster) except when the Marks' decahedra are completed at N = 75, 101 and 146 (Fig. 3). This asymmetry reduces the fraction of the surface with low coordinate { 100} faces. For example, the 54-atom structure in Fig. 3 is a fragment of the 75-atom Marks' decahedron that has a major part of the decahedron missing from one side. Also, the growth sequence between the 101- and 146-atom Marks' decahedra proceeds by generation of the surface structure of the 146atom decahedron on one side which then propagates around the cluster. This leads to geometries, such as the 126-atom structure in Fig. 3, which have a surface that is a mixture of the two Marks' decahedra. As a consequence of these favourable asymmetric geometries, structures like the 55-atom truncated decahedron, which has (2 x 2) {100} faces and Dsh point group symmetry, are not part of the growth sequence. The fcc structures with A 2E = 2 are shown in Fig. 4. Unlike the decahedral growth sequence, a more continuous sequence of structures is possible. Growth can proceed from one truncated octahedron to the next by addition of overlayers to the four {111} faces surrounding a point of the original truncated octahedron, and by capping of any new (2 × 2) {100} faces created. Such sequences are possible from the 38-atom to the 79-atom truncated octahedron, and from the 79atom to the 140-atom truncated octahedron. For example, the 86-atom cluster in Fig. 4 is formed by the addition of a single overlayer, the 52-, 98- and 102atom clusters by the addition of two overlayers, and the 112- and 116-atom clusters by the addition of three overlayers.

345

However, the growth sequence with largest nnn is not quite that simple. First, between the 38- and 79atom truncated octahedra, another set of structures is competitive. These structures are based on a 31-atom truncated tetrahedron with additional hexagonal overlayers on the { 111 } faces occupying twin sites. The 52- and 59-atom structures shown in Fig. 4 are examples with three and four complete overlayers. These geometries have larger nn, than the octahedral-based structures for N = 58-60 and the same nnn for N = 45, 46, 49-53 and 57. Secondly, for N > 116 the growth sequence with largest nnn differs from that outlined in the preceding paragraph: it is produced by capping the six { 100} faces of the 116-atom truncated octahedron (Fig. 4). This gives rise to the regular peaks in A 2E every four atoms from N = 116-140 as every face is capped. For small ( N < 38) fcc clusters, the growth sequence is more complicated and includes structures that are either fragments of the 38-atom truncated octahedron, that are based on the 3 l-atom tetrahedron, or that involve twin planes. It is noteworthy that the 55- and 147-atom cuboctahedra are not part of this growth sequence. Cuboctahedra are isomeric to Mackay icosahedra and these two structures have often been compared in studies of the energetic competition between fcc and icosahedral structures [ 17,30,31 ]. However, this is not such a good comparison since cuboctahedra are not the optimal fcc structures for these sizes.

3.

Discussion

We have found growth sequences for fcc and decahedral clusters and illustrated the probable magic numbers for a cluster which is exclusively fcc or decahedral. However, for a real cluster the energetic competition between structural types is also likely to be important in determining the morphology. To examine this competition requires that the growth sequences for the structural types - icosahedral, decahedral and fcc - be compared for a specific potential, rather than just counting nearest neighbours, as we have done. Our results, however, allow us to make some comments on the relative stabilities of the most likely morphologies for clusters with short-ranged interactions, which may be of particular relevance to clusters of C6o molecules. Comparison of sequences of truncated octahedra

346

J.P.K. Doye, D.J. Wales / Chemical Physics Letters 247 (1995) 339-347

38

52

00 0 lib 0 0 0 0 79

86

98

102

112

116

128

132

136

Fig. 4. Fcc structures with A 2 E - 2.

and Marks' decahedra for the Morse potential with a value of the range parameter appropriate to C60 indicate [32] that the truncated octahedra become lower in energy for N > 270. However, this figure does not indicate a sudden crossover between structural types, but rather that for a particular N above this size the lowest energy structure is more likely to be fcc; indeed the two structural types may 'coexist' for a range of size. This is indicated by our previous studies of Morse clusters [ 14], where we found that the lowest energy clusters with 24, 25, 38 and 79 atoms are fcc and those with 14-23, 46, 55, 70 and 75 atoms are decahedral at a value of the range parameter appropri-

140

0

ate to C60. For the Morse potential the fcc structure is fcc deca always lower in energy if nnn ~> n.. ", because Estrain is always larger for the decahedral structures, and E..n is unimportant for a short-ranged potential. Therefore, we can deduce from our results that the lowest energy structures will be fcc for N = 24-28, 38-40, 51-53, 86, 88 and 90. The majority of these occur near to the sizes at which the number of atoms along the decahedral axis changes, where the decahedral structures are far from spherical. The growth sequences outlined here allow comparison with experiments on small clusters that may exhibit fcc or decahedral structures. However, it must be

J.P.K. Doye, D.J. Wales/Chemical Physics Letters 247 (1995) 339-347 remembered that the structures exhibited by clusters in e x p e r i m e n t s d e p e n d o n s u c h f a c t o r s as t h e m e t h o d o f p r o d u c t i o n , the t i m e scale o f the e x p e r i m e n t a n d t h e g r o w t h k i n e t i c s , a n d so t h e r m o d y n a m i c e q u i l i b r i u m is n o t a l w a y s o b t a i n e d . F u r t h e r m o r e , a m e t h o d s u c h as o u r s w h i c h m i n i m i s e s the energy, n o t t h e free energy, o n l y g i v e s the z e r o t e m p e r a t u r e e q u i l i b r i u m structure. H a v i n g n o t e d t h e s e caveats, t h o u g h , the g r o w t h sequences presented here provide a good starting point f o r u n d e r s t a n d i n g t h e s t r u c t u r e s o f s m a l l fcc a n d deca h e d r a l clusters.

Acknowledgement W e are g r a t e f u l to the the E P S R C ( J P K D ) a n d the R o y a l S o c i e t y ( D J W ) for financial s u p p o r t .

References [1] J. Jortner, Z. Physik D 24 (1992) 247. [2] M.R. Hoare and P. Pal, Nature (Physical Science) 236 (1972) 35. [3] A.L. Mackay, Acta Cryst. 15 (1962) 916. [4] J. Farges, M.E de Feraudy, B. Raoult and G. Torchet, J. Chem. Phys. 78 (1983) 5067; J. Chem. Phys. 84 (1986) 3491; Advan. Chem. Phys. 70 (1988) 45. [5] J.W. Lee and G.D. Stein J. Phys. Chem. 91 (1987) 2450. [6] O. Echt, K. Sattler and E. Recknagel, Phys. Rev. Letters 47 (1981) 1121. [7] I.A. Harris, R.S. Kidwell and J.A. Northby, Phys. Rev. Letters 53 (1984) 2390; I.A. Harris, K.A. Norman, R.V. Mulkem and J.A. Northby, Chem. Phys. Letters 130 (1986) 316. [8] J.A. Northby, J. Chem. Phys. 87 (1987) 6166. [9] T.D. Klots, B.J. Winter, E.K. Parks and S.J. Riley, J. Chem. Phys. 92 (1990) 2110; J. Chem. Phys. 95 (1991) 8919.

347

[10] T.E Martin, U. N~iher, T. Bergmann, H. G6hlich and T. Lange, Chem. Phys. Letters 183 (1991) ll9. [ 1 l ] M. Pellarin, B. Baguenard, J.L. Vialle, J. Lerm6, M. Broyer, J. Miller and A. Perez, Chem. Phys. Letters 217 (1994) 349. [ 12] O. Echt, O. Kandler, T. Leisner, W. Miehle and E. Recknagel, J. Chem. Soc. Faraday Trans. 86 (1990) 2411. [13] B. Raoult, J. Farges, M.E de Feraudy and G. Torchet, Z. Physik D 12 (1989) 85; Phil. Mag. B 60 (1989) 881. [ 14] J.P.K. Doye, D.J. Wales and R.S. Berry, J. Chem. Phys. 103 (1995) 4234. [15] J. Pillardy and L. Piela, J. Phys. Chem. 99 (1995) 11805. [16] L.D. Marks, Phil. Mag. A 49 (1984) 81. [17] J. Uppenbrink and D.J. Wales, J. Chem. Phys. 96 (1992) 8520. [ 18] H.S. Lim, C.K. Ong and E Ercolessi, Surface Sci. 269/270 (1992) 1109. [19] L.A. Girifalco, J. Phys. Chem. 96 (1992) 858. [20] T.E Martin, U. N~iher,H. Schaber and U. Zimmerman, Phys. Rev. Letters 70 (1993) 3079. [21] M.H.J. Hagen, E.J. Meijer, G.C.A. Mooij, D. Frenkel and H.N.W. Lekkerkerker, Nature 365 (1993) 425. [22] A. Cheng, M.L. Klein and C. Caccamo, Phys. Rev. Letters 71 (1993) 1200. [23] G. Wulff, Z. Krist 34 (1901) 449. [24] C.L. Cleveland and U. Landman, J. Chem. Phys. 94 ( 1991 ) 7376. [25] E.K. Parks, L. Zhu, J. Ho and S.J. Riley, J. Chem. Phys. 100 (1994) 7206. [26] E.K. Parks, L. Zhu, J. Ho and S.J. Riley, J. Chem. Phys. 102 (1995) 7377. [27] E.K. Parks and S.J. Riley, Z. Physik D 33 (1995) 59. [28] A. Pinto, A.R. Pennisi, G. Faraci, G. D'Agostino, S. Mobilo and E Boscherini, Phys. Rev. B 51 (1995) 5315. [29] K. Clemenger, Phys. Rev. B 32 (1985) 1359. [30] J. Xie, J.A. Northby, D.L. Freeman and J.D. Doll, J. Chem. Phys. 91 (1989) 612. [31] Q. Wang, M.D. Glossman, M.P. Iniguez and J.A. Alonso, Phil. Mag. B 69 (1994) 1045. [32] J.P.K. Doye and D.J. Wales, in preparation.