Magic Numbers in Clusters

67

A deltahedron is a polyhedron with triangular faces only. Important examples are also shown in Chart 2 (left side of the double arrows). These are commonly called closo clusters. Boron hydrides of general formula [BnHn]2 are prime examples of this class of clusters. Other representative main-group and transition-metal clusters can be found in Refs. 70â&#x20AC;&#x201C;73. Skeletal electron pair (SEP) theory (58) predicts that the number of skeletal electron pairs for this class of clusters (except tetrahedron) is given by BV1

for closo-deltahedral clusters

(3)

where V is the number of vertices. The B values for the common deltahedra are given in Chart 2. For example, an icosahedron has V  12 vertices and a B value of 13, thereby satisfying the SEP theory. It turns out that the SEP theory applies not only to closo-deltahedral clusters (excluding tetrahedron) but also to polyhedra, which can be visualized as derivable from closo-deltahedra with one, two, or three missing vertices, commonly called nido, arachno, or hypho clusters, respectively. Equation (3) is appropriately modified to B  V  2, V  3, V  4 for these latter polyhedra. For example, a square pyramid (V  5) can be considered as an octahedron (V  6) with a missing vertex (a nido cluster); hence B  V  2  5  2  7. This may be called the debor principle. Another way of stating the debor principle is that removal of a few atoms from a closo-deltahedral cluster does not alter the number of skeletal electron pairs (B). The opposite of the debor principle is the capping principle (74,75), which states that capping does not alter the number of skeletal electron pairs of the parent polyhedron. For example, a tetracapped tetrahedron is a deltahedron with eight vertices, yet there are only six skeletal electron pairs (B  6) within the central tetrahedron. 2. 3-Connected Clusters Generating function Gn  2n

Magic numbers 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, . . .

3-Connected clusters are polyhedral clusters in which each vertex is connected to three other vertices. Some examples of 3-connected polyhedra are shown in Chart 2 (right side of the double arrows). Polyhedral CnHn belongs to this class of clusters. Other examples of main-group or metal clusters can be found in Refs. 70â&#x20AC;&#x201C;73. Since each vertex is connected to three other vertices via three cluster orbitals, these clusters (with a total of V vertices) can be considered as electron precise in that one-half of the 3V orbitals will be bonding: B  3V 2

for 3-connected polyhedral clusters

(4)

Nanoparticles

Chemistry of metal nanoparticles

Nanoparticles

Chemistry of metal nanoparticles