Construction and Characterization of Structure Models of Crystalline

Here, we demonstrate applications of this tool in crystal structure modeling tasks for both MOFs and COFs. Our algorithm is made available within our ...
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Construction and Characterization of Structure Models of Crystalline Porous Polymers Richard Luis Martin and Maciej Haranczyk* Computational Research Division, Lawrence Berkeley National Laboratory, One Cyclotron Road, Berkeley, CA 94720-8139, United States ABSTRACT: Metal−organic frameworks (MOFs) and covalent organic frameworks (COFs) are examples of advanced porous polymeric materials that have emerged in recent years. Their crystalline structure and modular synthesis offer unmatched versatility in their design. By exchanging chemical building blocks, one can both explore the unlimited space of possible structural chemistry within an isoreticular (same crystal topology) series and achieve a wide range of alternative topologies. This reticular paradigm potentially enables the design of structures with any desired porosity and internal surface chemistry. Reliable structure models are typically required in order to predict material properties using a broad spectrum of molecular modeling techniques. In this work, we introduce an algorithm for the assembly of crystalline porous polymer structure models which permits precise control over the underlying topologies of the generated models. This tool has been applied to high-throughput combinatorial structure enumeration and optimization-based automated design. Here, we demonstrate applications of this tool in crystal structure modeling tasks for both MOFs and COFs. Our algorithm is made available within our open source Zeo++ software suite.

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INTRODUCTION Many classes of crystalline porous polymeric materials have emerged in recent years. For example, metal−organic frameworks (MOFs),1 covalent organic frameworks (COFs),2 and zeolitic imidazolate frameworks (ZIFs)3 have been widely investigated in the context of applications such as separations,4 including CO2 capture,5 hydrocarbon and other liquid separations;6,7 gas storage;8,9 catalysis;10 sensing;11,12 and drug delivery.13 The unprecedented tunable porosity and surface chemistry of these materials provides an opportunity for atom-scale design of advanced materials to meet the specific technological and economic requirements of many diverse applications: from energy, to security, to human health technologies. The unmatched versatility of these advanced crystalline polymers has its origin in their modular structure. They are formed by connecting distinct chemical building blocks (secondary building units, SBUs) into periodic networks: the basis of reticular chemistry.14 For example, MOFs comprise metal or metal oxide clusters bridged by organic linkers to form two- or three-dimensional (2D or 3D), and often porous, frameworks. In recent years, significant progress has been made in identifying, classifying, and enumerating the underlying topologies of crystal structures such as MOFs15,16 in terms of nets for 3D and layers for 2D frameworks. The searchable Reticular Chemistry Structural Resource (RCSR)17 is the comprehensive online database of catalogued nets and layers (for brevity, “nets” will be used henceforth). The RCSR currently contains approximately 2100 nets, the majority of which are relatively simple, e.g., nets with one kind of edge and © 2014 American Chemical Society

one or two kinds of vertex (which are generally the most likely to form experimentally18,19), and there is an ongoing effort to extend the RCSR to include more complex nets.20,21 The International Union of Pure and Applied Chemistry (IUPAC) recently recommended the use of RCSR net symbols22 in describing the topology of MOFs and related materials. The RCSR is therefore of great importance for the classification of synthesized materials as well as the design of new structures, where the nets serve as blueprints. Each net within the RCSR typically comprises the positions of vertices and edges of a graph that are unique by symmetry along with the symmetry operations required to complete a periodic unit cell; as such, a net is not equivalent to the crystal structure of any one material, rather, it describes the connectivity of SBUs within materials of that particular topology. In many applications of computational materials science, reliable structure models are required in order to investigate material properties. Structures from either crystallographic measurement or computational predictions can be used to analyze and characterize materials.9,23,24 Computationally derived models of crystal structures can be analyzed to identify experimentally observed crystalline phases, for example, by comparing simulated X-ray powder diffraction (XRD) patterns of structure models with the experimental XRD pattern obtained for an investigated sample. In principle, approximate 3D structure models for a wide variety of material classes and Received: January 28, 2014 Revised: March 5, 2014 Published: March 6, 2014 2431

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Structure models constructed by our algorithm have multiple applications relevant to crystal structure prediction and design. Large data sets of material models constructed through such an approach can be (a) analyzed and sampled in terms of structural diversity,29 (b) prescreened, for example, using molecular simulation30 to identify promising candidate materials for a particular application, and (c) mined to elucidate relationships between the structure, chemistry and properties of materials and their underlying building blocks.31 Our algorithms can also be used to build simplified structure models, which can be utilized in materials design.32−34 Our structure building algorithms extend our Zeo++ code.35 Zeo++ is an open source software package for analysis, comparison, categorization, and manipulation of porous structures. This article provides a description of our structure modeling algorithms as well as illustrative examples of their applications in a number of crystal analysis and design tasks.

chemical species can be designed computationally. Indeed, a variety of approaches for computational design of such material models have been explored, including chemical substitution or functionalization based on modifying known experimental structures,25 and recursive, geometry-based assembly of molecular building units in their various combinations.26 Naturally, each of these approaches has advantages and disadvantages. Modification of known structures allows for the design of new, topologically analogous (i.e., isoreticular) materials based on a reliable experimental baseline but does not permit easy exploration of alternative underlying topologies. Recursive geometric assembly allows for expansive, combinatorial design of large data sets of material models; however, recursive algorithms have an intrinsic tendency to be computationally expensive and, therefore, may require additional termination criteria or limits to the depth of recursion to be selected and implemented. Furthermore, the coordination of pairs of building blocks based on prespecified connection rules26 provides some confidence in the viability of the proposed coordination phases explored in a recursive approach; however, by the same token this strategy provides little direct control over the resulting topology, symmetry, etc. of material models, instead requiring precise tuning of nontrivial building block coordination parameters in order to achieve particular topologies. Clearly, the choice of material design strategy depends on many application-specific factors. In this article, we present an alternative approach for construction of crystalline porous structures which allows for both high-throughput enumeration and screening of large data sets of materials and precise control over their underlying crystal topologies. In this approach, information on the geometry/topology of molecular building blocks is combined with reference data describing a net; by utilizing topologyspecific connectivity information and symmetry operations, building units can be positioned in accordance with a particular underlying net (e.g., those provided by the RCSR). The assembly of porous crystal structure models based on a specific topology has previously been attempted in the literature, typically coupled with computationally demanding analysis techniques (such as electronic structure calculations) for ascertaining the feasibility of specific topological phases.27,28 However, these efforts have been limited to studying a small number of materials or topologies in great detail; here, we introduce a general algorithm appropriate for easily assembling structure models for a wide range of topologies and types of material, including the enumeration of large material databases. Of course, this approach too has advantages and disadvantages. For example, our approach makes it easy to model the various distinct topological phases achievable with a given combination of SBUs, whereas the recursive approach described above typically requires specific modification of the rules governing SBU connections (angles, etc.) in order to model each specific topology. While our approach allows the easy exploration of various structural phases, it does not provide the same degree of confidence that the explored phase is the most likely. Our approach can also be used to rapidly enumerate large data sets of isoreticular material models. However, our approach requires information on the desired topology to be provided. This information is not required in the recursive approach; however, with the advent and continued growth of resources such as the RCSR, it is becoming significantly easier to access and utilize topological information.

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METHODS

Assembly of Structure Models. Our algorithm is designed to assemble structure models from a given set of building blocks and according to a specified material topology, described by a net. The number and connectivity of the building blocks determines a set of possible nets that may be achieved. There is an ongoing effort to enumerate possible nets, with the RCSR database storing nets that have been predicted mathematically, together with those of experimental crystal structures. As part of this endeavor, the frequency of occurrence of each type of net experimentally has also been examined, leading to an important result in the field of reticular chemistry, namely, that the highest symmetry,18 minimal transitivity21 nets are generally most likely to form. Accordingly, assembling structure models based on the most prevalent net(s) for a given set of building block shapes ensures that the generated models are likely to be similar to materials that can be, or have been, achieved experimentally. The specific information of interest for structure assembly is the representation of a desired topology as a periodic graph (net), i.e., the set of vertices and edges between vertices that are unique by symmetry and the symmetry operations which must be performed in order to complete a periodic unit cell. The RCSR database provides this information in the Systre file format;36 expanding the basic vertices and edges using the symmetry space group provided results in the complete periodic graph illustrating the net (see Figure 1, left). The vertices and edges of this graph encode the relative positions and connectivity of the material’s building blocks; our algorithm constructs

Figure 1. Example using rhr net. The rhr net, showing the unit cell (black box), vertices (green spheres), and edges (red lines) (left). A framework assembled to exhibit the rhr net, comprising the building blocks illustrated in Figure 2: Cu2 paddlewheels as the metal SBU (brown atoms; replacing the green spheres) and benzene 1,4dicarboxylic acid as the organic linker (gray C atoms, red O atoms, and white H atoms; replacing the red edges) (right). 2432

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Figure 2. Building blocks for an example MOF. A copper paddlewheel (top) and a linear dicarboxylic linker (bottom) are illustrated. These building blocks coordinate to one another by shared COO groups; these are shown in both building blocks only for clarity (left). The markers identifying the connection sites in each SBU are shown in black. During framework construction, these markers are used to align the building blocks, to produce a framework wherein a single COO group is seen to connect both building blocks (see Figure 1, right). Note that the COO group is preserved on the metal SBU only. This allows for the linker to be treated as a rigid entity, while implicitly allowing for torsion or other rotation about this bond, which may be required for modeling certain topologies (center). These markers are removed from the final framework model. The geometric shape representing each SBU’s arrangement of connection sites is also shown (right). Brown represents Cu atoms, gray C atoms, red O atoms, and white H atoms. corresponding framework models by positioning molecular building units according to this periodic graph. In addition to topology information, our algorithm requires molecular building blocks to be provided as input. The algorithm identifies a molecule’s connection sites (for instance, carboxylic functional groups by which organic components coordinate to metal/ metal−organic components) and designates these positions with temporary markers, which provides a convenient connectivity-based abstraction such that the framework assembly algorithm need only consider arranging building blocks based on these markers (e.g., a copper paddlewheel is essentially abstracted into its arrangement of sites, i.e., a square, see Figure 2). By overlaying SBUs described in this manner with the specific vertices they correspond to, a particular topology can be guaranteed in the resulting framework model. Specifically, this process involves the following steps: (1). Interpreting the Net. The net information provided (vertices, edges, and symmetry space group) is used to construct the complete periodic graph of the net by performing the required symmetry operations on the symmetrically distinct vertices/edges provided (see Figure 1, left). (2). Aligning the Building Blocks. Having parsed the net information, the algorithm now parses the building block information. The connectivity of the building blocks (i.e., the number of connecting positions they exhibit, and their arrangement) is analyzed, and each building block is allocated to the appropriate symmetrically distinct vertices of the net. Furthermore, each building block is oriented in 3D space such that its site markers are aligned to the edges of the corresponding vertex, and symmetry operations are performed. This procedure results in a set of SBUs, each aligned to a vertex; in this manner, a framework can be constructed by simply translating these SBUs in 3D space such that they are positioned in accordance with the specified net (step 3). (3). Assembling the Framework. The complete set of aligned building blocks must now be positioned relative to one another in order to achieve the connectivity described by the net. Two approaches to framework assembly are explored, as described below. Resulting frameworks are evaluated based on the maximum distance between pairs of connectivity markers in adjacent building blocks (the

ideal case being zero distance, i.e., all connections are made perfectly); the framework exhibiting the lowest deviation is returned as output. Method 1, Net-Based Assembly. In this assembly method, the unit cell of the net is scaled to accommodate the building blocks. Each aligned building block can then be simply translated such that the centroid of its connection sites lies at the corresponding vertex (see Figure 1, right). This elegant method is most appropriate for building blocks whose connection sites are arranged very similarly to the edges of the corresponding vertices (e.g., square vs square). Method 2, Connection-Based Assembly. In this assembly method, the aligned building blocks are connected together one by one, according to the connections between the corresponding vertices. Once all building blocks are connected, the resulting unit cell is determined. This method allows the unit cell to scale, stretch, or shear freely based on the SBU shapes provided and is more appropriate for building blocks whose connection sites are arranged distinctly to the edges of the corresponding vertices (e.g., the irregular hexagonal arrangement of sites in hexacarboxylic acids such as those in ntt-net MOFs37,38). Depending on the net in question, one assembly approach may provide more accurate models. As an example, if an oblong SBU is assigned to a perfectly square vertex, the connection-based method may result in a more accurate model since it allows for the unit cell parameters to adapt to such elongated components. (4). Collision Detection (Optional). Our algorithm also provides for an optional collision-detection routine, which rejects framework geometries resulting in atoms positioned too close to one another; this feature is optional since some degree of atomic collision may be resolvable with subsequent application of framework relaxation techniques. The above algorithm has been implemented in Zeo++, an opensource, high-throughput porous structure assembly, characterization, and comparison tool. The Results section provides some specific examples of this computational framework in application to various types of crystalline porous polymer assembly and analysis tasks. Characterization. The porosity of structure models assembled by the above approach can be characterized by a suite of algorithms recently developed by our group and also implemented within Zeo++. 2433

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Briefly, Zeo++ uses a computational geometry technique called the Voronoi decomposition to translate structural information (e.g., position of atoms and size and shape of the periodic unit cell) into a periodic graph representation of the spaces between atoms and their connectivity (essentially, a map of the material’s void space). This graph representation (Voronoi network) provides an elegant abstraction of void space, which can be efficiently analyzed to identify probe-accessible regions of void space (with respect to particular probe sizes) and their related properties. For example, recently developed algorithms allow rapid calculation of restricting pore diameters,35 pore size distributions,39 and accessible pore volume and surface area with respect to particular guests.35 For cases where high-precision calculation of geometric properties such as restricting pore diameters is required (e.g., screening materials for use as shape-selective membranes), a controllable accuracy mode is also provided which can reduce geometric errors in pore size calculations to below 0.02 Å without sacrificing the high-throughput character of the code.40 Rapid computation of these descriptors enables the efficient comparison and classification of materials, and Zeo++ also provides novel descriptors specifically designed for the comparison of materials’ pore networks and structures, enabling sophisticated functionalities for navigating large sets of materials such as similarity searching and diversity selection,41 including cavity connectivity-describing Voronoi holograms29 and stochastic ray tracing histograms,42 and provides functionality for the rapid identification of guest-accessible binding sites.31

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be positioned along the edges to bridge these vertices. It is important to note that the rotation of 2-c linkers about the edge is ill-defined (in general, three positions are required to unambiguously define an orientation in 3D space). To position the linkers, a direction must be chosen; the algorithm aligns the linker such that its greatest extent or “width” is directed away from adjacent linkers (see Figure 1, right). We note that it is easy to accommodate specific alternative linker orientations. Having aligned the building blocks appropriately, the stored symmetry operations are performed, resulting in appropriately oriented building blocks for each vertex of the net. This set of aligned, but not translated building blocks, is the input to the next stage. (3). Assembling the Framework. Both methods described above for positioning the building units relative to one another are explored for each material. In this specific case, since the copper paddlewheel exhibits a perfect square coordination while the net vertex is not perfectly square, the net-based assembly method results in a structure where coordinating sites are slightly offset from one another (the worst case detected being a 0.427 Å distance). By contrast, the connection-based assembly method is robust under these specific circumstances and results in a perfect worst case site distance of zero. The structure produced by the connection-based method is propagated to the next step. (4). Collision Detection (Optional). In this case, we perform the optional collision detection step; no collision is detected since the 2-c linker is not sufficiently wide for any two instances of it to overlap. The structure from the previous step is output as a successful model. On a standard workstation, less than half a second has elapsed since the program was initiated; we note, however, that at the time of writing our code has not been optimized, and substantial gains in performance are envisioned in subsequent releases. The ability to build models exhibiting a prespecified net allows for easy enumeration of large sets of isoreticular materials based on a selection of appropriate linkers. This approach to combinatorial library design is demonstrated in our recent work44 in which commercially available dicarboxylic acid molecules were utilized for the design of pcu-net MOFs, isoreticular with the well-known MOF-5 system,43 and commercially available organic dibromides and related compounds were used to enumerate a large set of hypothetical porous polymer network (PPN) structures.45 Furthermore, we have exploited the capability of this tool to easily construct models with various nets to explore the achievable surface area across many topological classes of MOF.32−34 Quality of Assembled Models with Respect to Experimental Structures. An important step in the validation of our structure assembly algorithm is comparison of the resulting materials with known experimental structures. We provide an example comparing the crystallographic structure of a MOF (HKUST-146) with its counterpart model built using our method. We compare the built and experimental frameworks based on various structural descriptors calculated using Zeo++. For the purposes of comparison, we emphasize that the structures built using our method were achieved only with the aid of standard chemical drawing software for generating linker molecule geometry, and neither the linkers nor resulting frameworks were relaxed. Relaxing the building blocks and/or resulting structures can be expected to improve the match to experiment. Finally, we note that the connectionbased method was chosen by our algorithm to produce the

RESULTS

Material Assembly Given a Net. Our structure-building algorithm is demonstrated using the assembly of a metal− organic framework (MOF) structure model which exhibits the rhr net. The below steps, introduced in the Assembly of Structure Models section, illustrate and further discuss the process of assembling a model exhibiting a particular net: (1). Interpreting the Net. The rhr net can be seen to be comprised of one kind of vertex, repeated by symmetry operations (Figure 1, left). This vertex has four edges (i.e., it is a 4-c vertex) with a tetragonal arrangement. In modularly constructed advanced porous materials such as MOFs, one can also consider the constituent metal and organic building units to have a particular coordination. For instance, a copper paddlewheel is a metal building unit which can coordinate in a 4-c, tetragonal arrangement to four organic linkers. Accordingly, the rhr net is a possible topology for MOFs based on copper paddlewheels, bridged by linear ditopic (2-c) organic molecules, in this example benzene 1,4-dicarboxylic acid or BDC (Figure 2); we note that this choice of net is just one of several possibilities43 chosen for illustration purposes. Our algorithm stores the vertices and edges read from the input topology file and performs the symmetry operations corresponding to the space group read from the same file. The result is a periodic graph (net) comprising vertices and edges between them; within this graph, information on the symmetry operation and initial position which resulted in each vertex is preserved. This graph is the input to the next stage. (2). Aligning the Building Blocks. To build an rhr-net structure model using copper paddlewheels and the selected linear ditopic linker, we must also determine the position in space and orientation of these building blocks. Each building block is rotated to be aligned to the appropriate initial (presymmetry operation) vertex; it is not yet translated to overlay the given vertex. Only 4-c vertices are present in the rhr net (2-c vertices are typically not provided explicitly in the topology definition); the alignment of 4-c copper paddlewheels to these vertices is trivial, while the 2-c linkers must somehow 2434

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guaranteed to be equivalent topologically to the experimental structures which exhibit the same net and therefore, as previously stated, a further refinement of the structure (e.g., through force-field-based relaxation) can be expected to lead to models with an improved match to experimental structures. Assembly of Distinct Topologies with the Same Building Blocks. When combining building blocks of particular shape, there are a small number of high-symmetry crystal topologies that may arise. The highest symmetry possibility for the combination of trigonal and octahedral building blocks is the pyr net, as observed in its interpenetrated form (pyr-c) in MOF-150.47 However, the same combination, albeit with a different trigonal linker, has also been seen in MOF-17748 to give rise to the related but lower symmetry qom net (note that qom is not self-dual, and so is mathematically precluded from self-interpenetration). Here we explore how our structure building algorithm can help investigate scenarios such as these where multiple topologies may arise from the assembly of equivalent SBU shapes. The octahedral unit utilized in both structures is the Zn4O SBU, shown in Figure 4. The tricarboxylic acid linkers for

highest quality model (as in the rhr example), requiring a run time of 0.3 s. HKUST-1 is a metal−organic framework exhibiting the tbo net (comprising 4-c and 3-c vertices). It is comprised of copper paddlewheels (4-c square, Figure 2, above) and benzene 1,3,5tricarboxylic acid or BTC linkers. The resulting framework is illustrated in Figure 3; the experimental structure (red) and structure built with our algorithm (blue) are shown in Figure 3 for comparison.

Figure 3. tbo-net model constructed using the BTC linker and a copper paddlewheel (Figure 2, above) (center). Experimental structure of HKUST-1 (red) and built structure (blue) are shown at the same level of zoom for ease of comparison.

Figure 4. The octahedral Zn4O building unit, shown with markers indicating connection sites (black) superimposed with the carbon atoms of the coordinating carboxylic groups. Purple represents Zn atoms and red O atoms.

Figure 3 allows us to illustrate the differences between assembled and experimental structures. A very good qualitative match to the experimental structure of HKUST-1 is observed, and a quantitative comparison of the structures with respect to various descriptors is provided in Table 1. The properties of the model are shown to be within 10% of those of the experimental structure. Finally, we emphasize that despite the various deviations in geometry, assembled structure models are

MOF-150 and MOF-177, with carboxylic acid groups represented as connection site markers for clarity, are shown in Figure 5 (left). In the experiment, the pyr-c net occurs for MOF-150 because of the greater ease of achieving the required torsion angle of 45 degrees between the carboxylic groups and the plane, facilitated by the nitrogen atom at the center of the linker. By contrast, the planar linker for MOF-177 more easily forms the qom net.48 Our algorithm allows for structure modeling based on any valid net, and hence, both the pyr-c and qom nets can be modeled with each linker; we note that the abstraction of connection sites such as carboxylic acid groups into simple position markers enables the easy assembly of nets which require distinct torsion angles. The corresponding structures are illustrated in Figure 5 (center for pyr-c and right for qom). Our algorithm selected the net-based method for assembly of these structures; in this case, the irregularity of the 6-c vertices causes some deviation in the position of connecting sites, while the connection-based method results in building block collisions. The models require between approximately 0.15 (pyr) and 1.0 (qom) s to generate. With the capability provided by our algorithm, multiple candidate nets for a particular set of building blocks can readily be compared. Given these possible structure models for each

Table 1. Structural Descriptors for Experimental and Built Frameworks (to Two Decimal Places).a HKUST-1

a = b = c parameter (Å) density (g/cm3) surface area (m2/g)b,c void fractionb,d largest included sphere (Å) largest free sphere (Å)

experimental

built

overestimation (%)

26.34 0.88 2656.24 0.36 13.20 6.75

27.27 0.79 2916.91 0.38 13.82 6.94

3.50 −9.78 9.81 7.28 4.74 2.88

a

All descriptors were calculated using the Zeo++ software package. Calculated with respect to a probe of radius 1.4 Å. cCalculated using ten thousand random samples per framework atom. dCalculated using one hundred thousand random samples over the unit cell. b

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Figure 5. Left: building blocks, shown with connection sites as black markers, for MOF-150 (above) and MOF-177 (below). Center: pyr-c net structures built using each linker and the Zn4O metal building block. Right: qom net structures built similarly. The correct structure for the MOF150 linker is the pyr-c net (i.e., above center), while for the MOF-177 linker, the qom net is correct (below, right). Purple represents Zn atoms, gray C atoms, red O atoms, blue N atoms, and white H atoms.

Figure 6. Predicted powder X-ray diffraction patterns for each of the four structures in Figure 5, along with corresponding patterns obtained for experimental structures.

linker, one can proceed to determine the likelihood of either occurring, through evaluating the properties of each structure (for example, relative energetics). As an illustration, we have predicted the XRD patterns of each of these four structure models using Mercury from the Cambridge Crystallographic Data Centre,49 and these are illustrated in Figure 6. Comparison to patterns obtained through Mercury for experimental structures reveals a good match for MOF-177 exhibiting the qom net rather than the pyr-c net as is found to be the case experimentally. Similarly, the predicted XRD pattern for MOF-150 matches the XRD pattern obtained for the pyr-c model. We again note that the models have not been relaxed in any way and that subsequent relaxation may improve these comparisons. One can follow this same approach to identify phases of newly synthesized materials. For example, knowing the constituent building blocks, one can build structure models

exhibiting the various permissible topologies arising from the building blocks’ connectivity, predict their XRD patterns, and identify the one which best matches the experimental XRD pattern. Examples of More Complex Nets and Special Cases. Finally, we would like to present some examples which illustrate the capabilities of our algorithm to assemble more challenging material models, namely, materials based on nonsymmetric building blocks, nets with more than one vertex of the same connectivity degree, bipartite and 2D nets (i.e., layers), and the ntt net (wherein the linker is described by multiple coplanar vertices). The examples presented so far involved molecular building blocks with rotational symmetry; however, it is also possible to assemble frameworks from nonsymmetric building blocks (even C1 symmetry). For instance, the nonsymmetric organic building block pictured in Figure 7 (left) assembled into the rhr 2436

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Our material assembly tool can also construct models of materials that exhibit two-dimensional layered structures. For example, Figure 9 illustrates the building blocks of a

Figure 7. A linear (2-c) dicarboxylic acid linker which is nonsymmetric due to the presence of a functional group (left). The rhr framework constructed from this linker (shown without perspective, for clarity) and a copper paddlewheel; notice that the positioning of the functional groups follows the underlying symmetry of the rhr net (right). Brown represents Cu atoms, gray C atoms, red O atoms, blue N atoms, and white H atoms.

Figure 9. Molecule C (above left), also shown with connection site markers superimposed upon the N atoms (above right). Molecule 3 (below left), also shown with markers replacing O atoms (below right). Gray represents C atoms, red O atoms, blue N atoms, and white H atoms.

net with the addition of copper paddlewheels, will lead to the model shown in Figure 7 (right). In practice, materials built from nonsymmetric building blocks may exhibit inherent disorder.50 Our algorithm typically imposes the underlying symmetry operations of the selected net, and so generates only a single framework, corresponding to the highest symmetry arrangement of building blocks. However, by disabling symmetry, our algorithm can produce all possible orientations of linkers within single unit cells, allowing techniques such as diversity selection29 to be employed to identify sufficiently distinct molecular arrangements. The run time and construction method for this structure are the same as for the initial rhr net example. Another special case involves nets exhibiting more than one vertex of the same connectivity degree, such as the pts net (see Figure 8, left); in this case, the net exhibits a 4-c tetragonal as well as a 4-c tetrahedral vertex. Accordingly, two distinct molecules are required to build a framework with such a net, e.g., a 4-c tetrahedral linker (see Figure 8, center) along with a 4-c tetragonal copper paddlewheel (Figure 2, above). Our algorithm identifies all potential matches between molecules and vertices and investigates each combination of assignments. Since the incorrect assignment (tetrahedral linker to tetragonal vertex and tetragonal paddlewheel to tetrahedral vertex) can produce no valid framework, only one model is achieved, which exhibits the correct net and building block arrangement (see Figure 8, right). In this case, the connection-based method achieves the best quality model, with run time of less than one second.

hypothetical covalent organic framework (COF) known as “C3” in the work of Trewin and Cooper;28 the constituent building blocks are two distinct trigonal molecules denoted as “C” and “3” (Figure 9, above and below, respectively). Graph representations of two-dimensional topologies are referred to as layers rather than nets; the underlying layer of C3 is comprised of trigonal components arranged in the only permissible 2D manner (a hexagonal arrangement) and is known as hcb, illustrated in Figure 10 (above left). This layer exhibits only one type of vertex; however, the structure requires two distinct molecules. In situations where more than one molecule is required to fit one vertex, our algorithm partitions the net/layer into a bipartite arrangement (i.e., one where there are two kinds of vertex A and B) and the only edges permitted are A−B edges (i.e., no A−A or B−B edges). Of course, not all nets and layers permit such a partition. Finally, when assembling twodimensional, layered structures, the separation between layers is not predefined; our algorithm permits a desired separation distance to be provided as an additional argument. The resulting hcb structure for molecules C and 3, with a 5 Å layer separation, is illustrated in Figure 10. We note that the layers in this example are assumed to be stacked (vertically above one another, i.e. alpha and beta are 90 degrees) in accordance with

Figure 8. The pts net, which exhibits both 4-c square and 4-c tetrahedral vertices; they are colored similarly (left). A simple tetrahedral organic linker (center). A framework built from the illustrated linker (occupying the tetrahedral vertex position) and a copper paddlewheel (square) exhibiting the pts net; a supercell, without perspective, is shown for clarity (right). Brown represents Cu atoms, gray C atoms, red O atoms and white H atoms. 2437

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Figure 10. The two-dimensional hcb layer (left). The corresponding two-dimensional structure of C3, built from the two molecules in Figure 9 (center). An orthogonal view to illustrate the layers with separation of 5 Å (right). Gray represents C atoms, red O atoms, blue N atoms, and white H atoms.

Figure 11. The (3,3,4)-coordinated ntt net (above left) and the simplified (4,6)-coordinated version (above right). An example hexacarboxylic acid, which exhibits the irregular hexagonal arrangement of connection sites common to linkers in MOFs of this topology (below left). The framework model arising from assembly of the linker with a copper paddlewheel according to the simplified ntt net (below right). Brown represents Cu atoms, gray C atoms, red O atoms, blue N atoms, and white H atoms.

making the assignment of the constituent 4- and 6-c SBUs to net vertices challenging. For this net, the challenge can be overcome simply by collapsing adjacent and coplanar 3-c vertices into a single 6-c vertex (Figure 11, above right). The new 6-c vertex is at the crystallographic position previously occupied by the centermost 3-c vertex (hence representing the linker centroid) and coordinates to six 4-c vertices. We note that subsuming vertices in this manner does involve the loss of some information; however, this net definition permits the (irregular hexagonal) hexacarboxylic acid linker type common to these MOFs (Figure 11, below left) to be aligned to a single vertex, permitting easy framework construction (Figure 11, below right). This final example utilizes the connection-based assembly method and requires approximately 1.5 s on a standard workstation to generate the model; the large number of atoms in this exercise compared to previous examples accounts for the longer run

the hcb layer definition provided by the RCSR; models where the distinct layers are sheared relative to one another (as in the work of Trewin and Cooper28) can easily be achieved by providing layer topology information with non-90 degree alpha and/or beta angles. Finally, we note that in this example the connection-based method is selected, and the run time is less than one thirtieth of a second. The final example involves construction of a framework model based on the net ntt (Figure 11, above left). Notable MOFs exhibiting the ntt net have recently been created,37 including the current highest Brunauer−Emmett−Teller surface area porous material.38 The topology of these materials is sometimes described by the (3,24)-c rht net; however, the (3,3,4)-c ntt net is preferred,15 as it more accurately reflects the connectivity of the individual constituent SBUs. However, we note that neither description explicitly includes a 6-c vertex, 2438

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time. We again note that our tool has at present not been optimized for performance, and we foresee significant improvements in the algorithmic implementation in the future.

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DISCUSSION Our topology-specific structure assembly algorithm is a powerful tool for rapidly constructing structure models. Our approach differs significantly from recursion-based assembly methods previously introduced in the literature.26 In recursive algorithms, each step comprises adding a new building block to a vacant coordination position in the structure model, in one of the various ways in which it can connect. Such an approach has advantages and disadvantages: information on a specific topology is not required beforehand in a recursive strategy; however, a list of the acceptable methods by which building blocks may coordinate to one another (in terms of directions and torsion angles, etc.26) must be prespecified. Furthermore, recursive algorithms can be inherently expensive, provide little control over the resulting model’s topology and symmetry, and the building block coordination parameters must be precisely calculated in the case of more intricate topologies. Our approach therefore overcomes these disadvantages, at the expense of requiring topology information; however, with the recent and ongoing developments in cataloguing as well as mathematically predicting crystal topologies (such as the RCSR), it is now easy to identify permissible topologies and enumerate the corresponding models.

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CONCLUSIONS The reticular chemistry underlying advanced crystalline materials such as metal−organic frameworks and covalent organic frameworks enables the design of material structures from a set of chemical building blocks and a particular topological blueprint. We have developed algorithms that allow easy and rapid construction of crystal structure models of desired topologies, and described their implementation. We have demonstrated various applications of this new tool in a number of crystal design and discovery tasks, and provide this tool to the materials research community as part of our open source Zeo++ software suite.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Fax: (+1) 510 486 5812. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Center for Applied Mathematics for Energy Research Applications (CAMERA), which is a partnership between Basic Energy Sciences (BES) and Advanced Scientific Computing Research (ASCR) at the U.S Department of Energy. Lawrence Berkeley National Laboratory is supported by the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC0205CH11231.

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