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electron affinities and gain energy by accepting an electron in a chemical reaction. Thus, when an alkali atom and a halogen atom approach each other, electron transfer from the alkali to the halogen atom satisfies the octet rule of both the atoms, resulting in the formation of a salt and an ionic bond between the cations and anions. In other cases, such as H2, O2, and N2, electron shell closure is achieved not by transferring electrons from one atom to the other but rather by sharing electrons. This leads to a covalent bond, which, in general, is stronger than an ionic bond. A fourth kind of bonding appears as atoms from groups 2–13 and some elements in the higher groups upto 16 come together to form a crystal. Here, each atom contributes its outer valence electrons to a common pool. These electrons move “freely” and collectively, forming a metal. Note that, the ionic cores of the metal also have electronic shell closure.
The periodic table (see Figure 2.1) currently consists of 118 elements among which 94 occur in nature. All materials are created by combining atoms from one or more groups. Among the 94 naturally occurring elements, some are expensive while some others occur in trace quantities. Some of these elements are even toxic. Is it possible to replace the expensive elements by earth abundant materials and the toxic elements by nontoxic ones? This has been the dream of alchemists for centuries. With the birth of cluster science, we have arrived at a stage where this dream may not be as farfetched as it once seemed.
Atomic clusters are groups of atoms whose size and composition can be varied by design, one atom at a time. More than half a century of research has made it clear that the properties of clusters are very different from any other form of matter [1]. Because their properties are size‐, composition‐, and shape‐specific, clusters can be tailored with atomic precision. In 1992, Khanna and Jena [2] coined the word “superatom” to describe a cluster that has the same chemistry as an atom in the periodic table and suggested that these superatoms can be used as the building blocks of a new three‐dimensional periodic table, with superatoms forming the third dimension [3]. If these superatoms can retain their geometry and properties when assembled, a new class of cluster‐assembled materials with tailored properties can be formed. A classic example of such a crystal is based on C60 fullerene, which was discovered by Smalley and coworkers in the gas phase in 1985 [4] and later synthesized in bulk quantities by Kratschmer et al. [5]. Once assembled, C60 fullerenes retain their shape, but the property of the fulleride crystal is very different from that of graphite and diamond. The former is the ground state of carbon while the latter is metastable but protected by a very large energy barrier. Note that the discovery of C60 was not the result of a rational design approach. The question is: can other clusters like C60 be rationally designed by using some prescribed rules?
Figure 2.1 The periodic table of elements.
In this chapter we examine if the electron‐counting rules, known to explain the stability of atoms and compounds, can be used to rationally design stable clusters. The first glimpse of such a possibility came from the experiment of Knight and collaborators in 1984 [6]. The authors observed conspicuous peaks in the mass spectra of Na clusters containing 2, 8, 20, 40, . . . atoms. Realizing that similar observation was made in nuclear physics where nuclei with 2, 8, 20, 40, . . . nucleons were found to be very stable, Knight et al. suggested an electronic shell model, analogous to the nuclear shell model [7], to explain the magic numbers in Na clusters. They used the jellium model where free electrons move in a uniform distribution of positive ion charge. Assuming that a Na cluster has a spherical geometry and the charges on the positive ion cores are distributed uniformly, they showed that the stability of the magic Na clusters is due to shell closures of their electronic orbitals such as 1S2, 1S2 1P6, 1S2 1P6 1D10 2S2, 1S2 1P6 1D10 2S2 1F14 2P6, . . . .
In the following, we first study clusters of simple metals whose stability can be well explained by the jellium model and see if magic clusters can be assembled to make a bulk material. Next, we explore a number of other electron‐counting rules such as the octet rule for sp elements [8–10], 18‐electron rule for transition metal elements [11], 32‐electron rule for rare earth elements, Hückel's aromaticity rule for organic molecules [12, 13], and Wade‐Mingos rule [14–17] for boron‐based clusters and Zintl ions [18, 19]. We focus not only on neutral but also on charged clusters that can be stabilized by using any one of the above rules and combinations thereof.
2.2 Electron‐Counting Rules
2.2.1 Jellium Rule
One of the early works using the jellium rule to study clusters as “giant atoms” was due to Saito and Ohnishi [20]. The authors studied if a closed shell Na cluster will interact weakly as noble gas atoms do and if an open shell Na cluster will be reactive. They showed that two Na8 clusters interact weakly just as two noble gas atoms do, thus implying that Na8 clusters with 1S2 1P6 closed electronic shells are chemically inert. In a similar fashion, they showed that a Na19 cluster with electronic configuration of 1S2 1P6 1D10 2S1 can be viewed as an alkali atom as both need one extra electron to close the s‐shell. In Figure 2.2 we show the binding energy of two Na19 clusters as a function of distance computed by these authors. Note that there is an initial attraction leading to the formation of a Na19 dimer with the centers of the Na19 clusters 17 a. u. apart. As the distance between the two Na19 jellium clusters is further reduced, the clusters face a significant energy barrier and eventually coalesce to form a Na38 jellium cluster that is magnetic with two unpaired spins. But, does Na19 cluster mimic the chemistry of a Na atom? To understand this, we plot in Figure 2.3 the binding energy as function of distance between two Na atoms using the Gaussian 16 code [21] and the density functional theory with the Perdew, Burke, Ernzerhof (PBE) form for the generalized gradient approximation [22] and 6‐31 + G* basis function. As can be seen, the energy profile in Figure 2.3 is very different from that in Figure 2.2. Clearly, Na19 cannot be regarded as a superatom mimicking the chemistry of a Na atom.
Figure 2.2 Binding‐energy curves of (Na19)2 for two different electronic configurations, (N↑, N↓ = 20, 18) (filled circles) and (N↑, N↓ = 19,19) (open circles). For several inter‐jellium distances, schematic pictures for positive background are shown.
Source: Saito and Ohnishi [20]. © American Physical Society.
Figure 2.3 Binding energy as a function of distance between two Na atoms. The computed bond length (3.0 Å) of the Na2 dimer agrees well with the experimental value of 3.08 Å.
To understand the effect of geometry of a cluster on its electronic structure, we focus on Na20, which is a closed shell cluster in the jellium model. In Figure