A modern chemist has access to good computational methods that generate numerically useful information on molecules, e.g., energy, geometry and vibrational frequencies. But we also have a collection of models based on orbital ideas incorporating concepts of symmetry, overlap and electronegativity. In this text we focus on the latter as these ideas have been a huge aid in understanding the connections between stoichiometry, geometry and electronic structure. The connections can be as simple as an electron count yielding user-friendly “rules.” Our problem here, the electronic structure of a cluster or a more extended structure of the type encountered in solid-state chemistry, requires the application of models beyond those reviewed in the Appendix. Models are like tools – they permit us to disassemble and assemble the electronic structure of molecules. For each problem we choose a model that will accomplish the task with minimum effort and maximum understanding. Just as one would not use a screwdriver to remove a hex nut, so too we cannot use highly localized models to usefully describe the electronic structures of many clusters and extended bonding systems. We must use a method that is capable of producing a sensible solution as well as one that is sufficiently versatile to treat both the bonding in small clusters and bulk materials.

The proven method we will use is one that generates solutions based on the orbitals and electrons that the atoms or molecular fragments bring to the problem.

The theme of this text, clusters as a bridge to solid-state chemistry, requires that we now consider the geometric and electronic aspects of substances that are solids. In doing so we will focus our attention initially on the nature of the atomic structures inside a bulk material; that is, we will completely ignore the surfaces. Towards the end of this chapter we will reincorporate surfaces into the problem and, in doing so, complete the bridge. The electronic-structure problem presented by periodic structures exhibiting extended bonding has been effectively dealt with in several earlier texts some of which are listed at the end of this chapter. These works go beyond what we need to establish our theme; however, the reader interested in more depth and breadth is referred to them.

Cluster molecules with extended bonding networks

As usual, let us begin with a discussion of geometric ideas relevant to a transition from molecular clusters to the solid state.

Surface vs. core atoms

In the structure of [Al69R18]3− (Figure 2.32) the number of nearest-neighbor Al atoms and bonding parameters changes in going from the outer shell made up of Al–R fragments deeper into the inner shells constructed from Al atoms alone. The internal cluster atoms display coordination numbers and inter-atomic distances more closely associated with bulk elemental Al than single-shell clusters. Is this reasonable? For the single-shell clusters discussed in preceding chapters the requirement for external ligands dominates the cluster stoichiometry/shape relationship.

1. (a) Approach: Cr possesses six valence electrons, thus ligands must be chosen to supply 12 more to satisfy the 18-electron rule, e.g., six CO ligands; Mn possesses seven valence electrons, thus ligands need to supply 11, e.g., 5 CO + 1 H. (b) This problem requires recognition of the types of bonds in the molecule and which ones, if any, are unusual. The answer to part (a) suggests the Cr–CO bonds can be adequately described as two-center donor–acceptor bonds. The Cr–H–Cr with a two-coordinate H atom is clearly the unusual situation; hence, a fragmentation into two Cr(CO)5 fragments and an H–anion is appropriate. A Cr(CO)5 fragment is a 16-electron species with an empty orbital available to accept an electron pair. The H– anion possesses one filled orbital; hence, the three orbitals (two from the metal fragments and one from H) can be used to form one bonding, one non-bonding and one antibonding three-center orbital with the bonding combination containing the two available electrons.

2. S contributes four AOs and the six H atoms contribute six AOs for a total of ten leading to ten MOs. The central atom has functions of symmetry a1g (3s) and t1u (3p), whereas the six ligand functions have symmetry-adapted combinations of a1g (3s) and t1u, and eg just like an octahedral transition-metal complex. As shown below, interaction between the central atom and the ligands generates four bonding MOs plus their antibonding partners and two non-bonding orbitals having ligand character only for atotal of ten MOs.

Of the millions of different chemical systems discovered since chemistry began, many are solids at room temperature. From the early days these solids have been classified in the four families, molecular, ionic, covalent and metallic solids, based on the nature of the forces which bind the atoms. Molecular solids are composed of groups of covalently bound atoms, i.e., molecules, held by weak charge-polarization (van der Waals) forces. In ionic solids, electrostatic attraction is the primary force binding cations and anions. Bonding in covalent solids is similar to that within molecules but extends over the whole crystallite. Metallic solids also exhibit extended bonding but, in addition, possess weakly bound, highly delocalized electrons easily moved by applied fields. Of course, this classification is somewhat artificial and many solids exhibit complex bonding in which more than one type of bonding is displayed. Molecular clusters in the solid state are naturally described nowadays with molecular-orbital models. Intermolecular interactions are weak. Although this is not true for solids with extended bonding networks, the solid-state machinery we developed in Chapter 6 shows that MO ideas smoothly transfer to crystalline solids. Hence, we have an analogous language for treating these more complex structures.

This is not a text of solid-state chemistry and the purpose of this chapter is to illustrate the use of the theoretical model of Chapter 6 with experimental examples. In doing so, we firmly establish the other foundation of our cluster bridge.

Definitions

Symmetry operations leave a set of objects in indistinguishable configurations which are said to be equivalent. A set of symmetry operators always contains at least one element, the identity operator E. When operating with E the final configuration is not only indistinguishable from the initial one, it is identical to it. A proper rotation, or simply rotation, is effected by the operator R(φn), which means “carry out a rotation of configuration space with respect to fixed axes through an angle φ about an axis along some unit vector n.” The range of φ is − π < φ ≤ π. Configuration space is the three-dimensional (3-D) space ℛ3 of real vectors in which physical objects such as atoms, molecules, and crystals may be represented. Points in configuration space are described with respect to a system of three space-fixed right-handed orthonormal axes x, y, z, which are collinear with OX, OY, OZ (Figure 2.1(a)). (A right-handed system of axes means that a right-handed screw advancing from the origin along OX would rotate OY into OZ, or advancing along OY would rotate OZ into OX, or along OZ would rotate OX into OY.) The convention in which the axes x, y, z remain fixed, while the whole of configuration space is rotated with respect to fixed axes, is called the active representation.

Crystallographic magnetic point groups

Because the neutron has a magnetic moment, neutron diffraction can reveal not only the spatial distribution of the atoms in a crystal but also the orientation of the spin magnetic moments. Three main kinds of magnetic order can be distinguished. In ferromagnetic crystals (e.g. Fe, Ni, Co) the spin magnetic moments are aligned parallel to a particular direction. In antiferromagnetically ordered crystals, such as MnO, the spins on adjacent Mn atoms are antiparallel, so there is no net magnetic moment. In ferrimagnetic crystals (ferrites, garnets) the antiparallel spins on two sublattices are of unequal magnitude so that there is a net magnetic moment. In classical electromagnetism a magnetic moment is associated with a current, and consequently time reversal results in a reversal of magnetic moments. Therefore the point groups G of magnetic crystals include complementary operators ΘR, where Θ is the time-reversal operator introduced in Chapter 13. The thirty-two crystallographic point groups, which were derived in Chapter 2, do not involve any complementary operators. In such crystals (designated as type I) the orientation of all spins is invariant under all R ∈ G. In Shubnikov's (1964) description of the point groups, in which a positive spin is referred to as “black” and a negative spin as “white,” so that the time-reversal operator Θ induces a “color change,” these groups would be singly colored, either black or white.