Statistical Mechanics and Morphology of very small Atomic Clusters BY M. R. HOARE AND J. MCINNESJ~ Department of Physics, Bedford College, Regent’s Park, London NWl 4NS Received 19th January, 1976 Mechanical and thermodynamic systems of a very few atoms (N < 100) interacting through central-forces are considered in the light of their role in an ideal model for “ precipitation ”. We discuss the stability of such systems at low energies and present detailed numerical investigations of the potential-energy surfaces of “ clusters ” of up to thirteen atoms interacting through Lennard-Jones and Morse potentials. Our main results comprise what we believe to be an almost exhaustive survey of the distinct minima available to systems of this size for the potentials used. Each such stable structure obtained is vibrationally and rotationally analysed and its symmetry is examined.The most striking feature of these results is the extreme sensitivity of the number of possible stable configurations to the range and softness of the pair potential. Thus, of no fewer than 988 minima for 13 Lennard-Jones atoms, only some 36 are supported by the (a = 3) Morse potential. The minima available are also classified geometrically and it is shown that non-crystallographic con- figurations predominate in structures of both greatest and least binding energy. A preliminary account of the statistical mechanics of cluster systems based on the rigid-rotor/har- monic oscillator approximation is given. 1. INTRODUCTION In recent years it has been possible for the first time to generate and examine very small clusters of atoms (N < 100) under effectively “ free-space ” conditions and to obtain information on their structural, mechanical and electronic properties to an accuracy sufficient to invite detailed theoretical investigation at a level hitherto reserved for stable, chemically-bonded molecules.The greatest single advance in the physics of “ microclusters ”, “ Van der Waals molecules ” or whatever else we may wish to call such aggregates, has undoubtedly been due to the development of the supersonic nozzle-beam in combination with various size and structure-discrirninat- ing detectors. Its use in conjunction with mass-spectrometry,1-6 electron diffrac- tion 7-9 and optical methods has made possible not only the determination of geometrical characteristics but also, in favourable cases, properties such as ionization potential or work f~nction.~ These experiments, which are unfortunately not represented at this Discussion, could be said to provide the most clear-cut instances of “ precipitation ” so far known and, as such, offer the most attractive possibilities for the comparison of theory and experiment. At the same time, advances in electron diffraction and microscopy have brought down the threshold for observation of small- particle deposits to the region of N < 100 atoms and supplemented the results of beam experiments under considerably more varied, if less controllable, conditions.The importance of these experiments, which we can only allude to here, ranges from catalysis to astrophysics and crosses the boundaries of subjects such as nucleation, crystal growth and surface physics in all its variety.For further background and bibliography we refer to earlier papers.10-12 Present address: Department of Physics, University of Warwick, Coventry CV4 7AL.M. R . HOARE AND J . MCINNES 13 In this Discussion paper space limitations prevent us taking up more than the first stages in an attempt to develop a mechanics of few-atom systems in what, in the absence of well-defined phase transitions, we can only call the “ solid-like ” region. By this we refer to the indefinite condition in which a cluster can be said to be at Sufficiently low energy for it to possess a definite geometrical identity on an experi- mental time-scale yet sufficiently high energy to undergo structural equilibration and perhaps sublimation.Most of the results we shall present here will be those of various explorations of the potential-energy surfaces of clusters in few-atom con- figuration space, designed to discover first the positions and multiplicity of the stable minima available and the dependence of these on the nature of the inter-atomic potential, then to construct an approximate statistical mechanics based on their local geometry. The result is an as yet primitive “ statistical morphology ” of very small particles which may provide the basis for more special treatments of electronic properties, catalytic activity, nucleation kinetics etc. 2. THE POTENTIAL-ENERGY SURFACE Consider the mechanical system of a few atoms (say for present purposes N 10) interacting in free space with a specified Hamiltonian.Let the system be classical and governed entirely by central two-body forces derivable from a realistic pair- potential ~ ( r ) . Under these conditions the total potential energy of the system can be expressed as JvN) = $2 c NI~L-rJl) (2.1) i j # i where ri is a position vector for the i’th particle and rN = (rl,r2, . . . rN} specifies a point in configuration space of 3N-6 dimensions. (Although not evident in the notation, the six degrees of freedom for rotation and centre of mass motion are assumed removed, e.g., by requiring: rl = (O,O,O), r2 = (x2,0,), r3 = (x3,y3,0).) Under these conditions V(rN) defines a potential energy surface in a space of 3N-5 dimensions inheriting certain characteristics (boundedness, differentiability, behaviour at infinity) from the pair potential ~ ( r ) , but otherwise embodying geometrical informa- tion of extraordinary complexity.We shall attempt to analyse certain aspects of this under the simplifications introduced above. Needless to say, further complica- tions would arise in a more realistic treatment involving non-central forces, many- body potentials and quantum-mechanical effects. STATICS Certain conditions are required on the pair potential function in order that it support definite stable configurations of the N atoms with finite separation. In- tuitively we expect the usual well-potentials of Lennard-Jones or Morse type to provide for this; somewhat more generally we may assert that a sufficient condition for the existence of at least one stable configuration of minimum potential energy is that ~ ( r ) should be differentiable, bounded below, convex at infinity and positive infinite at r = 0.Under these conditions, V(rN) will have a greatest lower bound at some absolute minimum configuration rN(o). The centres of the atoms will occupy points in a definite geometrical figure, the external vertices of which will form a poly- hedron in 3-space. * Physically we can refer to this object as a cluster of N atoms or, * Evidently the physical cluster at its minimum will strictly correspond to a set of N! points in configuration space differing only in the numberings of the set of atoms. We refer to these collec- tively as a minimal cluster or absolutely minimal cluster as appropriate.A further problem arises14 MECHANICS AND MORPHOLOGY OF VERY SMALL ATOMIC CLUSTERS more briefly, as an N-mer. The positive quantity V( GO)- V(rN(o,) with V( GO) the poten- tial energy when all atoms are infinitely separated, can be identified with the binding energy of the absolute minimum and this is conveniently measured in units of the well depth in the individual pair-potential. As is easily imagined, the set of configurations absolutely minimal in potential energy will be only one of a multitude of such sets with lesser binding-energy (higher potential-energy). The precise number of such local minima is a mysterious quantity, though it can be expected to be sensitive to the nature of the pair-potential v(r).Each minimum will be characterized by the vanishing of all partial derivatives (aV/ax,) and the positive-definiteness of the Hessian matrix (a2 V/ax,8x,) in any internal co- ordinate system (principle of virtual work). The eigenvalues of the Hessian matrix correspond geometrically to the principal radii of curvature at the minimum in the potential energy surface and physically to the normal-mode frequencies of vibration governing small motions. A number of invariants can be defined to test whether two given minima are in fact geometrically distinct. A sufficient condition for the geometrical distinctness of two configurations is that the inertia tensors of each corresponding cluster should differ. A less easily computed sufficient condition for their identity is that the inter- atomic distance matrices D i j = lri-vjl should be identical for some choice of number- ing.This does not, however, distinguish between enantiomorphic pairs. To test whether a structure possesses an enantiomorph we may use the fact that, if a plane of symmetry is present, two of the principal axes of the inertia ellipsoid will lie in it. To establish the absence of a plane of symmetry it therefore suffices to test the sym- metry of the three planes determined by the principal axes. 3. MINIMUM SURVEYS How many geometrically-distinct minimal structures exist for, say, ten Lennard- Jones atoms interacting through the pair-potential V(r) = r-l2--2rm6 and, of these which is absolutely minimal? If we replace this by the Morse potential V(r) = (1 -exp(3(1 -r))}2- 1 does the number change and, if so, can subsets of the minima for each potential be put into one-to-one correspondence with each other? Can the sets of minima in either case be related to the alternative question of enumerating the number of distinct packings of hard spheres in contact? Only in the cases N = 3 and N = 4 can we give unequivocal answers to these questions. (By writing the potential energy in terms of " bond-lengths " between neighbours it follows that the unique minima possible are the equilateral triangle and regular tetrahedron respectively.) Already at N = 5 the problem is highly non- trivial and the futility of trying to discover minima by systematic searching of con- figuration space becomes clear.(N.b., *N(N- 1) > 3N-6 for N > 4.Any attempt to write out the partial derivatives and solve for their stationary values is equally futile.) where two minimal configurations referred to the same co-ordinate system have identical energy by virtue of being equivalent under the operations of translation, rotation or reflection (improper rota- tion). We shall say that such configurations are geometrically equivalent and refer to all possibilities collectively as a single isomer. In the last case, however, we may wish to distinguish between enantio- morphic pairs sharing the same energy. These can occur with N 2 6.M. R. HOARE AND J . MCINNES 15 Nevertheless, the possibility exists of finding an algorithm which will discover at least a large proportion of the existing minima for N > 5 if not all of them.The key to such an algorithm is to be found in observing the way hard spheres can be composed in rigid contact. Just as is suggested by sphere packings, it is reasonable to expect that a large proportion of the minima available to N+1 atoms consists of those derivable by addition of the (N+l)’st atom to a favourable position at the surface of a previously established N-atom cluster. Although in certain cases the suggested minimum may fail to exist, and the cluster collapses to one of more or less unrelated geometry, computer experiments confirm that potentials of the general range and hardness of the Lennard-Jones function are capable of supporting the growth of successive minima by the stepwise process just described. It is too much to hope that we could find all existing minima in this way because we cannot rule out the possibility that the addition of several atoms simultaneously and their positioning in some co-operative fashion may lead to a stable structure of a different kind.As we shall see, however, it is almost inconceivable that the absolutely minimal structure for N = 10 would go undiscovered by this method. Descriptions have been given elsewhere of the methods available for the computer “ optimization ” of many variable functions, i.e., for the precise determination of local minima in the neighbourhood of a given starting configuration and with func- tions of the complexity of eqn (2.1).13 These methods are sufficiently fast and convenient, even with functions of some hundreds of variables, that we can afford to speak in physical terms of “ relaxing ” a structure to a minimum or carrying out an experiment in “soft-sphere packings ” without going into details of the numerical analysis.* To convert the above ideas into a definite algorithm we need to identify all the smallest “ seed-structures ” that seem likely to occur and “ grow ” minima upon these by making all possible positionings of additional atoms at their surface, comparing every N-atom minimum so obtained and deleting all geometrically indistinguishable structures from the list before continuing the process to size N+ 1. The choice of small seed-structures does not seem to be a serious problem. The two most important are clearly the N = 4 tetrahedron and the N = 6 octahedron. To these we may add a small number of others in the size range of N < 10 from Bernal’s list of canonical polyhedra. (The Archimedean antiprism, the capped trigonal biprism etc.15) In outline, the required algorithm goes as follows.(1) Pick a seed-structure of N atoms and relax it to its minimum in the required potential. Keeping this configuration fixed, test the structures obtained on adding an additional atom to each stable surface position in turn. Store all configurations which are geometrically distinct. Confirm that they remain distinct and delete any which collapse to another structure in the list. (3) Take each of the (N+1) atom structures discovered in this way and repeat the procedures under (1) and (2) above using it in place of the seed-structure. (4) Terminate the sequence when computing time becomes unreasonable.Pick a new seed-structure and begin again, checking each new minimum for dis- tinctness against the full list from all previous stages. We carried out this programme using the London University CDC7600 computer and investing most of our computing time in a study of the minima for the Lennard- Jones potential. Our main object was to reach the size N = 13 which we knew * Our recent work on the “relaxation” of the Bernal random close-packed structure is an example of a calculation carried out relatively easily in n. configuration space of some 3000 variables.“ (2) Relax these configurations to their true (N+l) atom minima.16 MECHANICS AND MORPHOLOGY OF VERY SMALL ATOMIC CLUSTERS would include the very important minimum configuration in which twelve atoms are grouped in an icosahedral shell about a thirteenth.This appears to be the smallest possible structure in which one atom is completely enclosed in a co-ordination shell. We were able to enumerate all the N = 13 Lennard-Jones minima generated by the algorithm, analyse the statistics of their distribution in potential energy and later carry out a vibrational analysis of each one to determine its normal-mode frequencies. The principal moments of inertia of each were also determined for application in statistical mechanics. For present purposes, however, we shall confine ourselves mainly to the first stages, which might be put under the somewhat quaint heading of ‘‘ statistical statics ”. 4. RESULTS FOR THE LENNARD-JONES POTENTIAL As we indicated earlier, one can make a reasonable guess at the outcome of the first stages of the computed growth sequence by experimenting with the packing of real spheres.Thus it is clear that, starting from the N = 4 tetrahedral seed, the first isomeric minima (3) occur at N = 7, giving rise to five distinct at N = 8 and thereafter rising extremely rapidly. * The first stages in the tree-like interrelationship of minima of the tetrahedral family are identified in fig. 1 which, though very limited, nevertheless FIG. 1 .-Interrelationships for tetrahedral minima of up to eight Lennard-Jones atoms. The letters in each circle identify the structures, the small numbers identify numbers of equivalent surface facets for the particular growth step.The following have recognizable geometry: 6A,7A,8A (Boerdijk spiral) ; 7B (Pentagonal bipyamid) ; 8E (Stellated tetrahedron). The other structures have various planes and axes of symmetry some of which may be recognized in the pictorial representations in fig. 2. The Boerdijk spiral configurations alone have no plane of symmetry and thus possess enantiomorphs. illustrates several of the features rspeated in far greater complexity for the larger isomers. When the computation was stopped at N = 13 it was found that no less than 988 distinct minima had been verified and listed, of which 131 possessed enantiomorphs. MULTIPLICITY OF ISOMERS The final tally of isomers for N = 6 to N = 13 is set out in table 1 with sub-totals of the three main types: tetrahedral, octahedral and others.It is relatively easy to * Enumeration of structures by hand model-building is somewhat unreliable. In an earlier paper we listed only four of the five N = 8 isomers, a mistake later corrected by Bonnissent and Mutafschiev.16M. R. HOARE AND J . MCINNES 17 make computer-generated drawings of the actual minimum configurations in sphere- packing, ball-and-spoke and other conventional renderings. We illustrate a few of these in fig. 2 and 3. Fig. 2 shows the tetrahedral subset of minima for N = 7 to N = 10. In fig. 3 we have listed separately parts of the lists of minima for N = 13 TABLE 1 .-MULTIPLICITY OF L-J ISOMERS size 6 7 8 9 10 11 12 13 tetrahedral 1 3 5 11 25 69 171 483 octahedral 1 1 3 6 29 60 143 338 others 0 0 0 1 3 16 52 167 total 2 (1) 4 (1) 8 (1) 18 (3) 57 (11) 145 (19) 366 (47) 988 (131) with tetrahedral and octahedral types respectively.In each case the most stable structure appears top left with isomers in descending order of binding energy along the rows. Thus the 13-atom icosahedron-beyond reasonable doubt the absolute minimum-appears first in fig. 3a and one notices immediately the trend towards more extended structures as the binding energy decreases. With the single exception of the N = 6 octahedron (a " seed-structure ") the N = 7 (3) N = 8 (51 N = 9 (11 I I N = 10 (251 FIG. 2.-Tetrahedral isomers for N = 7 to N = 10. Automatically printed projections of the computed minimum configurations for the Lennard-Jones potential. Binding energies decrease from left to right and in successive rows.The Boerdijk spiral configurations are marked with a cross.18 MECHANICS AND MORPHOLOGY OF VERY SMALL ATOMIC CLUSTERS FIG. 3a.-Tetrahedral isomers for thirteen Lennard-Jones atoms. The figure shows only the first and last 43 configurations of the total of 483 arranged in order of decreasing binding energy. The structure of greatest binding energy, the 13-atom icosahedron, is seen top left. The Boerdijk spiral is again marked with a cross. minima of greatest binding energy for the L-J potential are all of " polytetrahedral " type and follow the Werfelmeier sequence-the progression of structures obtained on adding atoms around the five-fold axis of a 7-atom pentagonal bipyramid to give the icosahedron.ll9 l7 A few of the configurations seen have been mentioned in sphere- packing studies, for example the Boerdijk spiral configuration resembling a chain of tetrahedra face to face.18 These minima (marked with a cross in the figures) are, some- what surprisingly, not the least stable available but are followed in the lists by numer- ous others of less systematic extended structure.It is interesting that in the course of generating the sequence of tetrahedral minima, no major structural rearrangements were found to occur on carrying out stage (2) of the algorithm with the Lennard-Jones potential. An example of mechanical collapse was, however, found in the octahedral sequence. The eight-atom structure consisting of an octahedron with two extra atoms in a skewed configuration (one of four possi- bilities) proves unstable and moves to a structure of tetrahedral type already accounted for.Another of the four N = 8 octahedral structures is also anomalous-it distortsM. R . HOARE AND J . MCINNES 19 FIG. 36.-Octahedral isomers for thirteen Lennard-Jones atoms. The first and last 38 structures of the total of 338 are shown in order of decreasing binding energy. Note how in many structures, though not invariably, the 6-atom octahedral sub-unit has distorted to part of an 8-atom " dodeca- deltahedron " at the minimum. spontaneously with a fundamental change of symmetry to form the " dodecadelta- hedron " (" DD "), a minimum envisaged by Werfelmeier in his 1937 study of alpha-particle structures and later listed by Bernal as a canonical polyhedron.15* l9 These special features of octahedral minima not only affect the number of larger isomers which can form, but may point to a mechanical disability shared by very small nuclei containing vestiges of cubic close-packed structure.This could be of importance in crystal growth-theory. (See also discussion in ref. (1 l).) Numerous " DD " structures can be distinguished on close examination of fig. 3b. STATISTICAL DISTRIBUTION OF POTENTIAL ENERGY OVER MINIMA The N = 13 isomers are sufficiently numerous to provide an adequate sample for statistical analysis of the distribution of binding energies over the whole set. Fig. 4 shows a histogram of the full set of N = 13 minima without inclusion of enantio- morphs. The curve obtained is noticeable asymmetric with a pronounced " foot " at high energies leading to the detached single point for the icosahedral structure.The latter is well separated from its closest competitor in energy (E = 44.33 and E = 41.47 respectively in pair units). The width of the distribution is narrower than20 MECHANICS AND MORPHOLOGY OF VERY SMALL ATOMIC CLUSTERS might be expected considering the variations of shape and compactness which occur. This, however, reflects the fact that, so long as clusters are too small to contain an appreciable number of " interior " atoms, changes of shape can only alter potential energy appreciably through variation in the distribution of second nearest neighbour distances. The asymmetry of the peak indicates that it is easier to form alternative structures by variation of the less stable, more elongated structures than the more compact units tending to the icosahedral configuration.35.0 45.0 binding energy FIG. 4.Distribution of number of isomers by binding energy for thirteen Lennard-Jones atoms. Solid line: tetrahedral isomers. Dashed line: octahedral isomers. Energy is given in units of the L-J pair-energy and the isolated point is the icosahedral structure. The points shown make up 521 of the 988 minima found. ANALYSIS BY TYPES Although we have previously studied the relative stability of crystalline and non- crystalline units by means of isolated examples l1 it was not previously possible to make a systematic survey of the symmetry properties of minima over a list as exhaus- tive as that generated here. The main trend of the results is that octahedral structures in the size-range N 5 13 are comparable in numbers to the tetrahedral types but possess binding energies more narrowly peaked about the most probable.For N = 13 the first fourteen minima in order of binding energy are tetrahedral and thus seem likely, in the absence of special entropic effects, to dominate the equilibrium population of isomers in a condensing system. Fig. 5 is an attempt to present the statistics of structural types graphically. Each of the three symbols marks the structural type of a particular minimum in the complete N = 13 Lennard-Jones set when these are arranged in decreasing order of binding energy as in the pictorial plots of fig. 2 and 3. The predominance of tetrahedral minima at both extremes of binding energy and the relatively uniform distribution of other types nearer the peak of the distribution is clearly distinguishable. 5.RESULTS FOR THE MORSE POTENTIAL We earlier posed the question of whether the numbers of distinguishable minima for a few atoms under central two-body forces would be markedly sensitive to the shape of the well in the pair-potential. Previous studies on particular structures suchM. R. HOARE AND J . MCINNES 21 FIG. 5.-Distribution of structural types for 988 Lennard-Jones minima of 13 Lennard-Jones atoms. Each structure is coded by a symbol according to type and these are printed as in fig. 3. in order of decreasing binding energy, illustrating the " interleaving " of different structural types not apparent in fig. 4. The lower figure shows the tetrahedral minima alone and illustrates the bunching of these at lower and higher energy ranges.Q = tetrahedral, 0 = octahedral, El = other. as the icosahedron have indicated that the range and softness of the potential can have considerable effect on the compactness of the minimum found 2o and in certain cases be a deciding factor in whether a particular minimum has perfect or broken symmetry.21 It did not seem justifiable to repeat the extensive computing of the growth algorithm described in Section 3 with a change of potential. Instead we used a short cut which seemed likely to generate a comparable set of Morse minima from the Lennard-Jones22 MECHANICS AND MORPHOLOGY OF VERY SMALL ATOMIC CLUSTERS set already obtained without the costly repetitiveness of the original scheme.We accordingly took each distinct N-atom Lennard-Jones minimum and applied a Morse potential to the same configuration. The resulting points on the new potential-energy surface were no longer minimal, but from this starting point a new minimum could be sought by a relaxation programme as before. While there could be no certainty that the programme would not " miss " a very shallow minimum in this neighbourhood and lead the configuration point over a saddle to an alternative one, it seemed reason- able to assume that the method would discover those isomeric minima which, while altering somewhat their position in configuration space, nevertheless retained their nearest neighbour topology and thus could be said to be " supported " by the Morse potential.In the event a surprisingly small fraction of the Lennard-Jones minima survived in the Morse potential energy surface-no more than 36 of the original 988! (table 2). This is less surprising at second sight when we consider the tree-like interrelationship of the minima on potential energy surfaces of progressively higher dimension. Thus, if the minima corresponding to some particular geometrical motif is " squeezed out " on softening or lengthening the pair-potential, this can only eliminate progressively more minima from each generation. We are still investigating the geometrical characteristics of the minima which are interconverted on replacing the Lennard-Jones potential with the Morse. It should not be difficult to identify precisely which structural features lose their stability on softening the potential, and to verify that the disappearance of only one or two types of sub-structure can suffice to cut the number of distinct minima as drastically as found.Our preliminary finding is that certain tetrahedral configurations (including the single N = 6 structure) '' open-up '' to give octahedral-type minima on softening the potential, the longer-range potential thus showing a greater tendency for form crystallographic structures. Whether this type of rearrangement is alone responsible for the very considerable smoothing-out of the topography of the P-E surface remains to be seen. A particularly striking result is the reduction in the number of stable Morse isomers on going from N = 11 to N = 12 (24 reducing to 22).This result need not be paradoxical if we visualize what is involved. With N = 11 the apparently absolute minimum is the icosahedron with two adjacent gaps. It may well be that, for the Morse potential a number of variants of this structure have a precarious existence close to low saddle-points. Addition of the twelfth atom may have sufficient shrink- ing effect to tip the balance and destroy a number of arrangements with stable 11- atom sub-units. The role of the form of the pair potential in determining the stability of alternative structures is evidently a complex one, though not altogether impossible to imagine- TABLE 2.-MULTlPLICITY OF MORSE, L-J AND HARD-SPHERE ISOMERS size 6 7 8 9 10 1 1 12 13 morse 1 3 5 8 16 24 22 36 L-J 2 4 8 18 57 145 366 988 hard-sphere 2 4 >10 >32 >113 >473 * * * The numbers quoted for hard-sphere isomers refer only to tetrahedral-type configurations.M.R. HOARE AND J . MCINNES 23 in somewhat picturesque terms we might say that the packing of Morse spheres is like packing frog-spawn; packing Lennard-Jones spheres is nearer to the packing of caviar. These peculiarities are entirely a function of the extreme smallness of the clusters considered here and can only be weakly reflected in the properties of the bulk phase or the infinite surface. Nevertheless, they could prove to be one of the more important on the properties of special systems such as nucleating films or ultra-finely dispersed metal catalysts. 6. HARD-SPHERE ISOMERS We also commented on the relationship between the multiplicity of minima for realistic potentials and the problem of enumerating configurations of a few hard spheres in contact.As a number of authors have pointed out,22 this problem is not well-defined inasmuch as the tetrahedral space-filling deficit leads to small gaps or " clearance regions " which, in clusters of more than seven atoms, can themselves be permuted around a structure, leading to a considerable increase in stable configura- tions only marginally different from each other. On imposing a realistic potential and " relaxing " the structure, the clearance regions predictably " heal-up ", usually to give almost perfectly symmetrical pentagonal motifs. * The numbers of sphere- packing isomers for a few atoms are thus of limited interest.For the sake of com- parison, however, we determined the multiplicities up to N = 11 and have included the figures in table 1. We have also investigated the possibility of enumerating packing isomers by graph theoretic methods with only limited success.23 The analogous, though simpler, problem of finding the numbers of " animals " of N adjacent points on a lattice is well known to be extraordinarily diffi~ult.'~ 7. CLUSTER EQUILIBRIA The extension of our results on the statics of cluster isomers to a statistical mech- anics of equilibria and growth-kinetics is possible, though subject to a number of drastic assumptions. If we take it that all the configurational states available to an " N-mer " at a given temperature are solid-like and characterized by small motions in the neighbourhood of stable minima, then the problem of describing the equilibrium populations of each is analogous to that of describing the state of a molecule with an unusually large number of tautomers interconvertible via energy-barriers.Provided we accept the harmonic-oscillator/rigid-rotor approximation, a partition function can be written for each distinct isomer and these in turn can be summed to give the partition function for all states, both configurational and internal, of the N-atom cluster. By further combination of these a grand partition function may also be constructed as a function of N. To implement this calculation the normal vibration frequencies and principal moments of inertia of every single isomer are required- these we have obtained for the full set listed in table 1 up to N = 13.It is less easy to obtain rotational symmetry factors in every case and we have so far been unable to apply these systematically. (N.b., a-factors can be as high as 60 for the N = 13 icosahedron and have a pronounced effect on the free-energy and equilibrium con- centrations.) It will be noted that a tacit assumption in the above is that, for tem- * Much larger soft-packed clusters may show a breaking of their symmetry at certain minima if the well in the pair-potential is sufficiently narrowed. This effect, analogous to the formation of clearance regions in hard packings, seems only to occur for the Lennard-Jones potential in the size- range N > 40, too large to be studied by the present mefhod.'O* *'24 MECHANICS AND MORPHOLOGY OF VERY SMALL ATOMIC CLUSTERS peratures of interest, the whole range of accessible isomers remain solid-like and harmonic even though the lowest in potential energy may tend to melt before the highest isomers can be “ excited ”.The results of these calculations, which we have only in preliminary form, may be applied in various ways. It is relatively easy to compute the “ mix ” of isomers of particular size as a function of temperature as the higher-energy structures become “excited”, and to study the effect of the potential on this. By computing free- energies of formation it is also possible to obtain equilibrium constants for growth and decay of clusters and to evaluate the “ single-configuration ” approximation used by previous authors 16- 25 who assumed that the structure of lowest potential energy is minimal in free-energy also and effectively determines equilibrium concentrations and growth rates.In all, the surprising features of these results lie more in the static properties dis- cussed earlier than in the statistical calculations. Within the approximations used, there appears to be a general tendency for vibrational frequency patterns to be some- what insensitive to the energy and character of the respective minima and thus for equilibria to be determined largely by energetic and configurational factors rather than vibrational and rotational entropy. We hope to expand on these remarks in the Discussion and subsequent publications. 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