A Paper Presented at the 2008 ISNSCE Foundations of Nanoscience (FNANO) Conference: A Formal Crystal Description System

We outline primary components of an algorithm based on geometric first principles for general crystal net prediction. The algorithm, and its computer implementation as a MAPLE program, is under development, and we present some examples of its operation. This algorithm arose out of an outline for nanostructure design heuristic proposed in FNANO 2007. † Department of Mathematics, University of South Florida, Tampa, FL 33620; [email protected]. ‡ Department of Chemistry, University of South Florida, Tampa, FL 33620; [email protected]. § Department of Mathematics, University of South Florida, Tampa, FL 33620; [email protected]. As the demand increases for more complex nanostructures, the demand will increase for mathematical and computational tools for structure design. As structures grow more complex, more systematic methods of design would be helpful. In [13] and [14], a theoretical rationale for such a computer based design method is proposed. The motivation was purely geometric, and presented a theoretical method for designing molecular building blocks (or tiles) that could be assembled in a given form. Here, we follow the initial impetus of that proposal applied to crystal design, and this presentation is a status report on a computer project for designing crystals originating from this approach. There are at least two good reasons for starting with crystals. – First, despite considerable extant activity in crystal design (as we shall see in a moment), there is is a feeling that more mathematical support and collaboration would be helpful, a point expressed in O. Yaghi’s plenary presentation here last year (abstract at [26]). There is a thread of scientific and engineering op-ed pieces – e.g., [17], [10], and [6] – expressing a great distance travelled but an even greater distance to go. Recently, [12] proposed an essentially geometric and combinatorial approach. This is an important goal even in DNA computing, for Seeman (e.g., [21]) has presented as a major goal the construction of a DNA crystal able to serve as a skeleton or scaffold. – In addition, crystals are fairly well understood, with a tradition of mathematical analysis going back to the seventeenth century. Indeed, several approaches for crystal prediction and design were proposed in the papers mentioned above, most notably the minimization of entropy or a similar parameter, although the subject retains its initial geometric impetus. The initial proposal being based on a purely geometric analysis, we began a programme for the geometric synthesis of crystalline structures. The very first investigation of crystals now regarded as both modern and theoretical – namely Kepler’s analysis of the snowflake (see [22]) was geometric. And underlying our approach is the fundamentally geometric perspective of A. F. Wells ([25]), who described crystals as periodic (graphical) nets embedded in twoor three-dimensional space. Nevertheless, several extant systems incorporate energy or entropy optimization considerations; here are some examples of projects which design crystals out of pre-assembled building blocks. The design process was dubbed reticular synthesis in [27], and there is a preference for designing porous crystals composed of large and rigid blocks like zeolites or metal-organic frameworks. – For example, C. Mellot-Draznieks and G. Ferey and their group are working on an Automated Assembly of Secondary Building Units (AASBU) method, which takes large and complex “secondary” units and using (energy) minimization and stochastic search techniques develop candidate designs for crystals composed of these units (see, e.g., [18] and [19]). – Meanwhile, M. D. Foster and M. M. J. Treacy maintain a database [9], which relies on stochastic minimalization and group symmetries to search for candidate nets In fact, Treacy and colleagues have developed a program to exhaustively enumerate the 4-connected (essentially the tetrahedral) uninodal nets using the symmetries and cost (energy) function minimalization ([24]). – Two threads met when theoretical and computational work of J. Conway, D. H. Huson and W. Thurston ([2]), O. Delgado Friedrichs, A. Dress, and others (see [3]; see also [4]) met O. Yaghi and M. O’Keefe (see [20]) in developing an a cluster of projects as part of a Reticular Chemistry Structure Resource (RCSR), in the most overtly geometric approach we’ve seen, but uses barycentric placement of nodes within cells, which is a (local) minimization approach (see [11]). This is only a sample. Here we will, perhaps näıvely, eschew optimization in order to concentrate on the geometry. We describe on of our projects, and this one follows a bottom up approach inspired by [13] and [14] (and developed further in [15] and [16]). There, a completed structure was viewed as a complex of molecular building blocks, and one imagined that a bug placed on a distinguished block could walk from block to block to block, etc., following a set of instructions of how to step from one block to the next. The list of instructions for just one walk would be a string of substrings, each substring incorporating a particular step; if each substring could be interpreted as a particular rigid motion (translation and rotation), then the composition of rigid motions would itself be a rigid motion which would fix the relative placement of the last block with respect to the first. The set of all such instructions would thus fix the entire structure with respect to the initial block. This bottom up approach will motivate a MAPLE program one of us (W. E. Clark) has composed to exhaustively enumerate prospective crystals (as defined within tight criteria, thus eliminating paracrystals and similar variants). This particular project is one of an array of interconnected programs, which we envision will eventually make up a functioning ensemble that will assist crystallographers and others in designing such structures.