How the hypothetical frameworks are generated
SCIBS - A Symmetry-Constrained Intersite Bonding Search
Site identification
For each space group, the topological properties of the fundamental region are examined [1-4]. The fundamental region is the smallest asymmetric volume of the crystalline unit cell that is repeated by the symmetry operations of the space group to fill space. For space group P6/mmm for example, the fundamental region is a 5-faced wedge, as shown in Figure 1. The basis atoms for the unit cell reside in the volume, or on the surfaces, edges or vertices of the basis fundamental region. We confine our attention to these basis atoms, and how they interconnect to images of each other. The symmetry of the space group generates the rest of the crystal.
Figure 1. Basis fundamental region for space group P6/mmm
The symmetry operators of the space group that generate neighboring images of the basis fundamental region are first identified. This is not the same set of operators that generate the unit cell of the crystal. The neighboring fundamental regions have a face, an edge or a vertex that is in contact with the basis fundamental region. The 50 neighboring fundamental region images for P6/mmm, plus the basis fundamental region (shaded), are outlined in Figure 2.
Figure 2. The basis fundamental region (shaded) and the 50 symmetry equivalent images of the fundamental region that are in contact with it.
The crystallographically "special sites" (also known as Wyckoff positions) in and on the basis fundamental region are the invariant subspaces of the set of space group operators that generate these neighboring images of the basis fundamental region. In the case of P6/mmm, this set contains 51 operators (50 neighbors plus the identity operator). Because of the strong mirror symmetry in P6/mmm, every face edge and vertex is a special site. The interior volume of the fundamental region is the general site. For P6/mmm, the fundamental region has 5 faces (mirror sites), 9 edges and 6 vertices, plus 1 interior volume, to give 21 topologically distinct crystallographic sites. In general, in other space groups not every face, edge or vertex of the fundamental region is necessarily a special site.
The basis atoms of all crystals can be thought of as residing on these sites of the basis fundamental region. They can be bonded to each other directly, and can also be bonded to images of each other (or images of themselves) in the neighboring fundamental regions. To ensure strict adherence to connectivity rules for the crystal (i.e. for zeolites, tetrahedral atoms are 4-connected) it is necessary to establish intersite connection tables for each space group. When an atom sitting on site i connects to another atom on site j, the site symmetries will commit a certain number of bonds to each atom. For example, an atom sitting on a mirror plane commits two bonds to itself when it connects to an atom in the general position, but only one bond to the atom on the general site. Two distinct tables are needed; one for connections between images of the same atom (homoatomic bonding) and another for connections between different atoms (heteroatomic bonding).
Not all sites can support regular tetrahedral, or even four-connected atoms. For example, the edge and vertices along the six fold axis of P6/mmm cannot support a four-connected atom.
Graph generation
In our algorithm, N (where N ≥1) basis tetrahedral atoms (T-atoms) are distributed systematically over all the sites. For each distribution of atoms, all permutations of the interatomic bonding are examined. All those intersite bonding configurations that are consistent with 4-connected atoms are retained. At this stage, each viable configuration is just a "colored" graph. Atoms are "vertices", bonds are "edges". No unit cell dimensions or atom locations are known. Only the connectivity is known. Colored graphs are explained schematically in Figure 3. Each color represents a different symmetry operator (which generates an adjoining image of the basis fundamental region).
Figure 3. Schematic representation of uninodal (N=1) and binodal (N=2) colored 4-connected graphs representing potential zeolite frameworks. The colors of the edges (bonds) represent different possible symmetry operators. The operators are not necessarily independent of each other. For example, every edge operator Op in the uninodal graphs, implies that the inverse operator Op-1 also generates an edge. Frequently, Op = Op-1. The symmetry-constrained intersite bonding search (SCIBS) iterates over all possible combinations of sites, vertices and edges (i.e. atom locations and permutations of bonds) to generate 4-connected graphs, each representing a potential zeolite framework.
The equivalent colored graph for zeolite L (framework code LTL, space group P6/mmm, N=2) is explicitly listed in Figure 4. It lists atom bonds in the format <i Op j>, where i (in the basis fundamental region) is bonded to the image of atom j that is generated by the symmetry operator Op. If Op is the identity operator, then atom j is also in the basis fundamental region.
Figure 4. representation of the graph for LTL. (x, y, z) is the identity operator, for example. The shaded lines contain redundant information.
The final structural configuration of the LTL framework that this graph ultimately corresponds to is shown in Figure 5. However, the actual atom locations on their sites, or the extent of the sites (related to unit cell dimensions) are not known at this stage, only the bonding is known.
Figure 5. Representation of the graph of LTL. The bold atoms are the basis atoms inside, or on, the basis fundamental region. The dimmed atoms are symmetry-equivalent images of the basis atoms, located in adjoining images of the fundamental region. These images are generated by the operators listed in the graph. The labels refer to the entries in the colored graph shown in Figure 4.
EMBEDDING GRAPHS - Does the graph correspond to a plausible zeolite?
Once the graph is determined, we need to see if the bonding configuration can be realized as a regular tetrahedral framework - i.e. is it a plausible zeolite?
Step 1: Geometric centering.
First, the T-atoms are placed at the geometric centers of their bonded neighbors (in relative coordinates) with each T-atom constrained to sit on its site. This procedure generates self-affine candidate starting locations for the T-atoms, but does not yet define the unit cell dimensions.
Step 2: Simulated annealing of the unit cell.
Keeping the relative atom locations fixed, candidate unit cell dimensions are found by adjusting the cell dimensions (and angles if the space group permits) so that reasonable T-T distances and T-T-T angles are obtained. This is done by simulated annealing using a simple cost function [2,4].
Step 3: Simulated annealing of the T-atoms and unit cell combined.
Next, the T-atoms and cell dimensions are annealed together to achieve T-T distances of approximately 3.05Å and T-T-T angles of about 109.47°, the tetrahedral angle. If a low cost is achieved, signaling a promising tetrahedral framework, we proceed to a more elaborate (and computationally costly) refinement step. In real zeolites, the bridging oxygen atoms modify the ideal T-T-T angle to be closer to 145°, and this needs to be simulated more accurately [3].
Step 4: Simulated annealing of the T-atoms, bridging O-atoms and unit cell combined, using the Boisen-Gibbs-Bukowinsky cost function.
Oxygen atoms are placed at the midpoints of the bonded T-atoms. Then the whole cell is annealed (cell dimensions, plus T-atom locations, plus O-atom locations) using a modified version of a cost function devised by Boisen, Gibbs and Bukowinsky (BGB) [5] for a pure SiO2 composition. This cost function (in its modified form) is ideal for our problem because our graphs strictly specify the bonding. Bond breakage is not allowed for a given graph, and this requirement is easily imposed in the BGB formalism. The important structural parameters contributing to the BGB cost terms are depicted graphically in Figure 6.
Figure 6. Graphical representation of the structural parameters contributing to the cost terms in the BGB cost function.
The database
If a low BGB cost is achieved, the framework is stored in the "Bronze" portion of our database. The framework is then re-optimized using the General Utility Lattice Program (GULP, written by Julian Gale [6]) using atomic potentials due to Sanders, Leslie and Catlow [7]. If a low GULP energy is achieved as a pure SiO2 composition, the framework is stored in the "Silver" portion of the database. All entries in the "Silver" portion of our databse have a counterpart in the "Bronze" portion, but not necessarily vice-versa.
The database is searchable. Frameworks with certain unit cell dimensions can be listed, or frameworks can be sorted by energy (eV per SiO2). The BGB energies are systematically lower than the GULP energies by about a factor of ten, but correlate reasonably well.
Presently, the database contains many duplicate frameworks. For example quartz occurs as a uninodal graph in many space groups. It does not always refine to a low energy because the space group may not allow the topology to relax appropriately. For now, both the "Bronze" and "Silver" databases contain these duplicates. It is intended to develop a "Gold" database that will contain the GULP-refined graphs in the maximum symmetry that is compatible with tetrahedral atoms, with all duplicates removed. The tools to accomplish this are under development.
Where is this headed?
This database is a work in progress, with its roots going back to some unwieldy seed ideas that were sown in 1982, and which eventually germinated in 1991. Initially, and inevitably, the database is going to resemble a somewhat untidy (but vast) stamp collection. There is already a huge amount of information contained here, but it is only the beginning. As each annealing run finishes, we will upload the results as CIF files to the database. By adding more powerful tools, it is our intention to turn this database into a useful and powerful tool for zeolite scientists. For example, we plan to add (automated) information about pore sizes and pore volume. We already have an interactive molecular viewer as well as a powder pattern simulation tool. We are not going to stop there.
Presently, we have a 64-processor computer cluster grinding away non-stop at these once-unwieldy ideas, generating graphs and annealing them
We hope that the database will be useful to zeolite researchers, perhaps for identifying interesting synthetic targets, or to assist with structure determination of unknown materials, or simply to help guide towards the next research directions.
Mike Treacy and Martin Foster
Department of Physics and Astronomy
Arizona State University
P.O. Box 871504
Tempe AZ 85287-1504
Phone: (480) 965-5359 (Treacy)
Fax: (480) 965-7954
Email: treacy@asu.edu and martin@foster.nu