Bond topology and structure-generating functions: graph-theoretic prediction of chemical composition and structure in polysomatic T–O–T (biopyribole) and H–O–H structures

Abstract Aspects of the bond topology and chemical composition of a mineral may be incorporated into a general formula by writing the local topological details of each cation and anion, along with their chemical identity, as a general expression called a structure-generating function. Here, this procedure is described for polysomatic T–O–T and H–O–H structures. We may write tetrahedrally coordinated cations and their associated anions as {T 2n Θ m }. For {T 2n Θ m } to be a chain or ribbon, 5n < m ≤ 6n, and we may write m as 5n + N, where N is an integer. Within the {T 2n Θ (5n+N)} unit, we may recognize three types of anion vertices: (1) bridging anions, Θ br , that are bonded to two T cations; (2) apical anions, Θ ap , that are involved in linkage to other cations out of the plane of the bridging anions; and (3) linking anions, Θ l , that link to non-T cations in the plane of the bridging anions. We may incorporate the connectivity of the cations in our algebraic representation of the chain as follows: {T 2n Θ br a Θ ap b Θ l c } where a + b + c = 5n + N. The apical anions of the T- or H-sheets provide some anions of the layer of octahedra. We may use the handshaking di-lemma of graph theory to examine the interaction between the two types of layers, and write a Structure-Generating Function , S (N;n) , that gives both the stoichiometry and aspects of the bond topology of the structures. Where N = 1, the T-sheet consists of ribbons of the form {T 2n Θ (5n+1) } = {T 2n Θ br (3n–1) Θ ap 2n Θ l 2 }. Each T–Θbr–T linkage spans an octahedron, and hence there are (3n – 1) octahedrally coordinated cations between opposing {T 2n Θ br (3n–1) Θ ap 2n Θ l 2 } ribbons. There are an additional (n–1) vertices, Ψ, required to complete the coordination of the M cations on one side of the O-sheet, and we may write the structure-generating function for biopyriboles as follows: S( 1;n) = Xi[M (3n–1) Ψ 2(n–1) {T 2n Θ br (3n–1) Θ ap 2n Θ l 2 } 2 ] = [M (3n–1) Ψ 2(n–1) {T 2n Θ (5n+1) } 2 ]. Where N = 2, the general form of the T-ribbon is {T 2n Θ (5n+2) }, a component of the H-sheet in the polysomatic H–O–H minerals in which the T-ribbons are linked laterally by [5]- or [6]-coordinated high-valence cations, D, which have the coordination (Dφ 4 1 φ ap φ t ), where f t may or may not be present depending on the coordination number, [6] or [5], of the D cation. The general formula for an H-sheet is [Dφ ap {T 2n Θ br (3n–2) Θ ap 2n Θ l 4 }φ t 0–1 ], where φ t (written after the T-sheet) occurs on the outside of the H-sheet and may be involved in linkage between adjacent H–O–H blocks. The H-sheet links via its apical anions to the O-sheet, giving the general formula of an H–O–H block as [M (3n+1) (Dφ ap Ψ n {T 2n Θ (5n+2) }φ t 0–1 ) 2 ]. These H–O–H blocks may link directly or indirectly through the φt anions of the (DΘ l 4 φ ap φ t ) octahedra, giving S (2;n) = Xi[M (3n+1) Ψ 2n (D 2 φ ap 2 {T 2n Θ br (3n–2) Θ ap 2n Θ l 4 } 2 )φ t 0–2 ]. Combining the expressions for the structure-generating functions gives a single function for T–O–T and H–O–H structures: S (N;n) = X i [M (3n+2N–3) ? 2(n+N–2) (D 2(N–1) f 2 ap (N–1) {T 2n T (3n–N) br T 2n ap T 2N 1 } 2 )f 0–2(N–1) t ] This expression also generates mixed-ribbon polysomatic structures. Thus S (1;2+3) gives the chemical composition and structure of the mixed-chain pyribole chesterite, and S (2;1+4) gives the chemical composition and structure of the mixed-chain H–O–H mineral, veblenite.

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