The goals of theoretical crystallography may be summarized as follows: (1) predict the stoichiometry of the stable compounds; (2) predict the bond topology (i.e. the approximate atomic arrangement) of the stable compounds; (3) given the bond topology, calculate accurate bond lengths and angles (i.e. accurate atomic coordinates and cell dimensions); (4) given accurate atomic coordinates, calculate accurate static and dynamic properties of a crystal. For oxides and oxysalts, we are now quite successful at (3) and (4), but fail miserably at (1) and (2). The current situation in the first two areas is briefly reviewed, prior to discussing in some detail an approach to topological aspects of structure in oxide and oxysalt crystals. The structure of a molecule or crystal may be represented by a graph, in which the vertices represent orbitals, atoms or groups of atoms, and the edges represent orbital interactions or chemical bonds. The topological characteristics of the bond network are contained in the (weighted) adjacency matrix of the graph and the corresponding eigenvalues constitute the spectrum of the graph. Simple graph theory arguments show that molecular (fundamental) building blocks are actually orbital (or energetic) building blocks, showing that there is an energetic basis for the use of fundamental building blocks in the representation and hierarchical analysis of complex structures. The electronic energy density of states may be derived by inverting the collection of moments of the energy, which may be evaluated directly from the topology of the bond network. Of particular importance in infinite structures is the observation that the energy difference between two structures is primarily dependent on the first few disparate moments of their respective electron energy density of states. Putting this in structural terms, the important energetic differences between structures involve differences in coordination number and local polyhedral connectivity. This supports the general idea that structures may be hierarchically ordered according to the polymerization of coordination polyhedra with higher bond valences. It is shown that Pauling's rules may be intuitively related to bond topology and its effect on the lower-order moments of the electronic energy density of states. It is also concluded that arguments of ionicity and/or covalency are secondary to the overriding influence of bond topology on the stability and energetics of structure. Bond-valence theory is reviewed in some detail. It may be considered as a simple form of molecular-orbital theory, parameterized via interatomic distance rather than electronegativity or ionization potential, and arbitrarily scaled via the valence-sum rule. Combination of bond-valence theory with bond topology/energetic considerations leads to a very simple way of expressing complex structures. The structural unit is a strongly bonded, usually anionic, polyhedral array whose charge is balanced by large low-valence interstitial cations. This gives a simple binary representation of even the most complicated structure; moreover, we can calculate the Lewis basicity and acidity of the two components and examine their interaction via the valence-matching rule. This enables us to examine several aspects of structural chemistry that have hitherto been intractable. The principal idea behind this work is to develop a coherent approach that is reasonably transparent to chemical and physical intuition, and that can be simply applied to complex crystals.
