MSA ROEBLING MEDAL LECTURE: TOWARD A BASIS FOR THEORETICAL MINERALOGY

We may represent a bond network as a weighted chromatic digraph, the adjacency matrix of which gives us a quantitative representation of the bond-topological characteristics of the structure. The moments approach to the electronic-energy density-of-states (EDOS) has a topological interpretation in terms of closed paths in the graph of the structure. The energy difference between two structures is dependent mainly on the first few disparate moments of their corresponding energy density-of-states. What does this mean: (1) zero-order moments define chemical composition; (2) second-order moments define coordination number; (3) fourth- and sixth-order moments define local connectivity of coordination polyhedra. We may recognize two types of closed-system mineral reactions: (1) reactions where bond topology is conserved; (2) reactions where bond topology is not conserved. Conservation of Bond Topology: This produces variation in patterns of Short-Range Order. Non-conservation of Bond Topology: In a closed system, zero moments are fixed and changes in second-order moments are energetically of greatest importance, i.e., changes in coordination number. We may identify such changes in mineral reactions, and variations in coordination number correlate with enthalpies of reaction. By partitioning complicated minerals into two parts, a strongly bonded structural unit and a weakly bonded interstitial complex, the correspondence principle of Lewis acidity – Lewis basicity (a mean-field analogue of the valence-sum rule) may be used to explain the structural and chemical complexity of many minerals. The ranges of observed and unobserved chemical compositions of structural units of minerals can be understood from this perspective, as can variations in chemical composition and structure of the interstitial complex. Where data are available, species in aqueous solution follow the valence-sum rule, and their Lewis basicities scale with the pH values of the solution at maximum abundance of the species in solution, suggesting that complex species in aqueous solution form building blocks of the crystallizing minerals. This approach focuses on the topological arrangements of chemical bonds in minerals and allows consideration of many problems that are difficult to approach from a solely atomistic perspective.