Cluster Arithmetic Made Simpler

NEWS AND VIEWS Cluster arithmetic made simpler Clusters of small numbers of identical atoms are all the rage, but the calculation of their properties taxes even large computers. That is why a simpler method deserves a welcome, but it also has heuristic value. THE cluster business, the study of small required for comprehensive calculations. trans are centred. These are what they numbers of identical atoms held together For each cluster is intrinsically more diffi call their glue-bonds. by a combination of chemical bonds and cult than a molecule. In principle, the The outcome of the calculations is im electrostatic forces, is booming away, and configuration and even its symmetry may pressive. First, the stability ( or the nega for a variety of reasons. As always, new not be known, while telling what is the tive energy) of lithium clusters is accu techniques have had a powerful influence, optimum configuration depends on a com rately reproduced up to the limit of strict notably those allowing for the self parison between that and all the others ab initio calculations based on self assembly of clusters from the ingredients possible. The calculation of clusters with consistent field optimization. Li,, for ex of well-cooled atomic beams. A further large numbers of atoms is, for practical ample, emerges (as it should) as one of incentive is the prospect that atomic clus purposes and at least for the time being, the most stable clusters. It is a curious ters may have important practical advan only exceptionally feasible. structure, which may be considered as two tages, as catalysts for example, or poss So is there a simpler way? Sugino and Li, rhombi, one above the other, which ibly as sub-components of electronic de Kamimura go back to an earlier attempt at are puckered into boat shapes in opposite vices. Evidently there are prizes to be generalization by Mark H. McAdon and directions with one turned through 90°. won. William A. Goddard (Phys. Rev. Lett 55, There are four glue-bonds, each occupied But the cluster business has also turned 2,563; 1985), also based on calculations by a pair of electrons, which are them up surprises, not least that the configura of lithium clusters. Their objective was selves arranged tetrahedrally (whence, tions of small clusters are often strikingly to construct configurational rules on say the authors, the stability). different from what would be expected the basis of accurate calculations of a few The authors carry their calculations from the interatomic configurations in the simple lithium clusters in a few simple beyond Li 14 , the most complicated cluster bulk solids. Both the four-atom and six configurations. to yield to self-consistent field treatment, atom clusters of lithium are planar struc One of their more striking findings was as far as Li36 • They conclude that there are tures; in Li,, the atoms are at the vertices that the distribution of electron density in, "magic numbers" of stability at clusters of of a rhombus, in Li6 , three atoms are at the say, a square Li, configuration is concen 8, 14, 18, 20, 26, 30 and 34 atoms, vertices of an equilateral triangle and the trated about the sides of the square, or although, with almost half the even num others at the mid-points of the three sides. symmetrically about the mid-points of the bers between 8 and 36 in the list , there Why should these unexpected structures interatomic bonds to be precise, and not at may be too much magic for some tastes. be stable? And, more generally, what can the atoms themselves. Those who choose The numbers 8, 20 and 30 might be a more be said about the transition from small to regard this as a hint of the electron prudent choice, to judge from the results clusters to the interatomic configuration delocalization found in the conduction of the calculations. of bulk solids? band of an electronic solid, a metal for What does this mean for the transition These are questions from which Osamu example, will not be mistaken. from cluster to bulk solid? The optimum Sugino and Hiroshi Kamimura of the McAdon and Goddard also found that configuration for any cluster is evidently a University of Tokyo embark on a sim each of the electron orbitals in Li, is occu trade-off between the electrostatic energy plified way of calculating the stability of pied by a single electron only, with the and the electron exchange or Pauli small clusters (Phys. Rev. Lett 65 2,696; spins of neighbouring electrons paired (in exclusion energy. The case of Li, illus 1990). Their starting-point is a set of data the sense of being antiparallel) in pairs. trates that an increase of the former on for the stability of small lithium-atom clus The distortion of the square into a rhom distortion of a symmetrical structure may ters. In the process, they describe a novel bus is advantageous because of the be more than offset by a decrease of the kind of bond between atoms, a "glue physical repulsion between parallel ( or latter involving the physical separation of bond" as they call it. Pauli-excluded) electrons. Another way glue-bonds from each other. That the cluster business needs a sim of putting this is to say that the acute angle What Sugino and Kamimura argue is plified method of calculation is not dis of the rhombus (measured at about 60°) that there comes a point, as clusters be puted. The techniques of computational includes two paired electrons in a roughly come larger, at which it is no longer chemistry may be used to calculate the spherical orbital filling that triangular half energetically favourable to place all the properties of an arbitrary atom cluster, of the rhombus. atoms on the outside surface of an open which for these purposes is simply another What Sugino and Kamimura have now network. Then, there is less scope for kind of molecule. For lithium clusters, for done, in the hope of simplying accurate geometrical distortion of the kind re example, in which each atom is a singly calculations of lithium clusters, is to quired if exchange energy is to be mini charged helium-like core with a single ex embark on quantum chemistry with elec mized, so that magic numbers melt more ternal electron, it would be reasonable to tron orbitals which more accurately rep completely into the background. begin with the assumption that there is resent this kind of electron distribution. The calculations show that this transi one potentially bonding electron on each They represent them as ellipsoids centred tion to bulk constitution must be under atom. For Li,, it would then suffice to on moveable points and, allowing for the way at Li", the least energetic form of calculate the energy of the four-electron electrostatic repulsion between the which has glue-bonds in a pattern with system for all possible configurations of helium-like cores of the lithium atoms, icosahedral symmetry, which in turn im the four nuclear cores. carry through a calculation to determine plies that one of them is enclosed within The snag, as Sugino and Kamimura not only the parameters defining the ellip a shell of all the others. The cluster is not note, is that even those with access to soids but the positions of the nuclei and yet a metal, but nearly so. supercomputers are lucky to win the time the points around which the different elec- John Maddox NATURE · VOL 348 · 29 NOVEMBER 1990 387 .

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