# Deconstructing the Crystal Structures of MetalOrganic Frameworks

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Deconstructing the Crystal Structures of MetalÀOrganic Frameworks and Related Materials into Their Underlying Nets

Michael O’Keeﬀe*,†,‡ and Omar M. Yaghi*,‡,§

†Department of Chemistry and Biochemistry, Arizona State University, Tempe, Arizona 85287, United States ‡Center for Reticular Chemistry, Center for Global Mentoring, Department of Chemistry and Biochemistry, University of CaliforniaÀLos Angeles, 607 Charles E. Young Dr. East, Los Angeles, California 90095, United States §Graduate School of EEWS, Korea Advanced Institute of Science and Technology, Daejeon, Korea

CONTENTS

1. Introduction

675

2. Identiﬁcation, Description, and Characterization of

Nets

676

3. Edge Nets, Augmented Nets, and the Underlying

Topology

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4. The Deconstruction of Crystal Structures

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4.1. Crystals with Corundum Net (cor)

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4.2. Some Symmetrical Metal-Containing SBUs

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4.3. Some Simple Organic SBUs

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4.4. Some Structures with the pts Topology

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4.5. Two MOFs Whose Preferred Description Is Not

the pts Topology

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4.6. MOFs with Multiple Links between SBUs

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4.7. Examples of Lower-Symmetry Metal-

Containing SBUs

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5. Some Case Studies

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5.1. A MOF with ubt Topology

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5.2. MOFs with Hexatopic Carboxylate Linkers

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5.3. MOFs with Octatopic Linkers

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5.4. More on Metal Cluster SBUs

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5.5. More Structures with Linked MOPs

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5.6. A Cyclodextrin MOF

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5.7. The Hierarchical Underlying Nets of MIL-101

and MIL-100

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6. MOFs with Rod SBUs

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6.1. SBUs as Zigzag Ladders

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6.2. A MOF with a Twisted Ladder Rod SBU

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6.3. A MOF with Rod SBUs of Linked Tetrahedra 697

6.4. Two-Way Rod SBUs of Linked Tetrahedra

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6.5. MOFs with Rod SBUs of Linked Octahedra

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6.6. Rod SBUs That Resist Simpliﬁcation

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7. MOFs with Ring SBUs

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7.1. Coda

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8. Concluding Remarks

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Author Information

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Biographies

699

Acknowledgment

700

References

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1. INTRODUCTION

The synthesis and characterization of metalÀorganic frameworks (MOFs) is one of the most rapidly developing areas of chemical science. These materials have unquestionably enormous potential for many practical applications, as detailed elsewhere in this issue, but they also often have exceptionally beautiful structures. It is the identiﬁcation and description of the nets that describe the underlying topology of these structures that is the main topic of this review. In particular we emphasize that this is not a review of MOF structures per se.

Why should we care about nets and related structural aspects of crystals? First and foremost, as chemists we recognize that the very core of our science lies in describing, and perhaps understanding, how atoms organize themselves, sometimes with our help, in chemical compounds. Such knowledge is also essential to designed (“rational”) synthesis of MOFs and related materials from component parts, as has been stressed recently.1 For this, of course, one needs to know the principal possibilities, which, as discussed below, have been established systematically only in the past few years. By deconstruction, we mean simply the reverse of the thought process that goes into designed synthesis, that is, breaking down a complex structure into its fundamental units without losing their chemical signiﬁcance. One can think of it as reverse engineering.

Another reason for knowing about nets and their occurrences is a result of the dramatic advances in methods of computer simulation of MOFÀadsorbate interactions, especially calculated adsorption isotherms, which makes the computer prescreening of potential materials an attractive procedure.2 Of course, to do this usefully it must be performed for materials for which there is a reasonable prospect of actual synthesis, which in turn will be done by design.

From the very earliest days of crystallography, simple inorganic structures were shown as “ball-and-stick” models in which the balls were the atoms and the sticks corresponded to bonds presumed to exist between nearest-neighbor atoms.3 It was early realized, particularly by Wells,4 that such models could be considered as representations or embeddings of special kinds of abstract graphs called nets (deﬁned below) with the vertices of the graph corresponding to the atoms and the edges (links) of the

Special Issue: 2012 Metal-Organic Frameworks

Received: June 6, 2011 Published: September 15, 2011

r 2011 American Chemical Society

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graph corresponding to the bonds. Wells devoted much eﬀort to enumerating nets, but he focused almost entirely on structures with three- and/or four-coordinated vertices and placed special emphasis on structures with shortest cycles (closed paths around the net) of all the same size; he called these structures uniform nets. Although he correctly recognized the importance of structures with symmetry-related vertices and edges, in fact, he found only very few of those now known.

It also became apparent that the same topology (net) was found in many diﬀerent chemical contexts. It was also realized that the edges and vertices of the net could be respectively polyatomic linkers and clusters. The work of the Iwamoto group on cyanides is notable in this respect. Complex cyanides with nets of forms of silica (cristobalite, tridymite and keatite) of other binary compounds such as rutile (TiO2), pyrite (FeS2), and cooperite (PtS) were prepared and their nets identiﬁed.5 The term “mineralomimetic” was coined to describe this kind of chemistry.

An important next step was the realization that, in fact, certain topologies could be targeted, especially for cyanides, by assembling appropriately shaped components.6,7 The wide variety of chemical compounds amenable to this approach was subsequently emphasized in several reviews.7,8 It should be emphasized, however, that it was in general rare for an underlying net to be identiﬁed in the older literature, and furthermore, when a net was identiﬁed, it was often done incorrectly. This last criticism applies far too often also to recent work. Indeed, the diligent reader will ﬁnd that some of the examples adduced in this review were originally assigned either to no topology or to an incorrect one. However, it is less the purpose here to correct errors than to point the way to better analyses in the future.

The discovery of MOFs, a term used here particularly to describe robust and highly porous metalÀorganic frameworks, led to the recognition that, in order to truly obtain structures by design, one had ﬁrst to identify the principal topological possibilities for nets. These, which were termed default structures,9,10 were identiﬁed as those with high point symmetry at the vertices and with a small number of diﬀerent kinds of vertex and edge— two conditions that are, of course, highly correlated. Subsequent analysis of published structures conﬁrmed the predominance of these default topologies.11

The discipline of preparing materials of targeted geometry by design is termed reticular chemistry10 and a series of compounds with the same underlying topology (net) is called an isoreticular series.12 As already mentioned, for successful reticular chemistry one needs to know the principal topologies, and a concerted eﬀort was made to enumerate them.13 The most important of the these are nets with one kind of edge (edge transitive) most of which were unknown prior to this work but are now realized to be of special importance. A review under the rubric “Taxonomy of Nets and the Design of Materials” has been published.14

Data for many of the nets most important for reticular chemistry are collected in a searchable database known as the Reticular Chemistry Structure Resource (RCSR).15 There nets are assigned three-letter symbols such as abc, or symbols with extensions as in abc-d (see below). This database is rather small (about 2000 entries). A much larger database is being developed in the EPINET project, which currently contains about 15 000 three-periodic nets.16 The computer program TOPOS recognizes even more, about 70 000.17 In this connection, mention should also be made to the extensive enumerations of sphere packings by Fischer and associates.18 The nets of these structures

have just one kind of vertex and have an embedding in which all the shortest (and equal) intervertex distances correspond to edges of the net. Most of these are incorporated in the RCSR.

2. IDENTIFICATION, DESCRIPTION, AND CHARACTERIZATION OF NETS

A net is just a special sort of graph. It is simple, meaning that there is at most one undirected edge that links any pair of vertices, and there are no loops (edges linking a vertex to itself). A net is also connected, meaning that every vertex is linked to every other by a continuous path of edges. The net of a polyhedron is ﬁnite. In crystals we will have inﬁnite nets that are one-, two-, or threeperiodic (“dimensional”). The emphasis here will be on threeperiodic nets. Graph-theoretical aspects, in particular terminology and deﬁnitions, have been given elsewhere.19

By “underlying topology” we mean the innate structure of the net associated with the crystal structure. Topology is really a branch of mathematics (or several branches as some would have it). However, here we use the term, following common usage, to refer to the combinatorial structure of a graph that is invariant in diﬀerent embeddings.20

In this review we show how a net is extracted from a crystal structure. The ﬁrst question is, what is the identity of the net? This can be answered in a meaningful way only by saying that it is identical to a previously known net that has an identiﬁer (such as a RCSR symbol). Otherwise, the net is new. The only algorithm devised to do this in a mathematically rigorous way is realized in Olaf DelgadoÀFriedichs’ program Systre.21 It should be mentioned, though, that in practice the program TOPOS17 also solves this problem with a high degree of certainty.

The second question to ask of the net is, what is the symmetry? By this we mean the combinatorial symmetry which, for the nets discussed here, is isomorphic with a space group and is the maximum possible symmetry of an embedding. As far as we know only Systre21 answers this question. As a bonus, Systre computes an embedding (a realization with space group, unit cell parameters, vertex coordinates, and edge speciﬁcation) in that symmetry. Occasionally, in practice very rarely, one encounters nets that have nonrigid body symmetries; we give an example below (section 5.2). Such nets are not currently considered by Systre.

The local topology of a vertex in a net is sometimes characterized by a point symbol or a vertex symbol. The point symbol, introduced by Wells,4 gives information about the shortest cycles at each angle of a vertex. The point symbol is of the form Aa.Bb... and signiﬁes that there are a angles at which the shortest cycle is an A-cycle, b angles at which the shortest cycle is a B-cycle, etc. By convention A < B < ... and a + b + ... = z(z À 1)/2, where z is the coordination number of the vertex and z(z À 1)/2 is the number of angles at that vertex. Point symbols for nets are conveniently obtained from TOPOS.

Our preference is for vertex symbols (also given by TOPOS), which give information about the number of rings (cycles that are not the sum of two shorter cycles) at each angle in a vertex.22 However, these become cumbersome for vertices of higher than six-coordination. These symbols are used to characterize the nets in the RCSR and in the Atlas of Zeolite Framework Types.23

Nets with one kind of vertex (i.e., those for which all the vertices are related by symmetry operations in their most symmetrical embeddings) are often called uninodal, those with two kinds of vertex binodal, etc.

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There are two common practices that we would like to discourage. The ﬁrst is that of calling a point symbol a “Schl€aﬂi symbol”. In universally accepted mathematical usage the latter is

a symbol for a regular tiling. In three-dimensional Euclidean space, there is only one such tiling—a face-to-face tiling by cubes for which the Schl€aﬂi symbol is {4,3,4} (the point symbol for the net is 412.63). The distinction between various symbols for vertex conﬁgurations and tilings has been discussed fully recently.24 In older papers, vertex symbols were also incorrectly called Schl€aﬂi symbols.22

The second usage we would like to discourage is that of referring to a point symbol or a vertex symbol as a “topology”. It is not; many nets with diﬀerent topologies have vertices with the

same point symbol. For example, the RCSR contains 14 distinct uninodal nets with point symbol 66, and if one includes polytypes of the diamond/lonsdaleite type, there is an inﬁnite number of nets in which all vertices have point symbol 66. By far the best way

of specifying a net, particularly a new, or previously undocumented, one is by a Systre-readable ﬁle. TOPOS can export Systre-readable ﬁles. For nets in the RCSR database, the RCSR symbol should be an adequate identiﬁer.

Recent advances in our knowledge of three-periodic nets have

come from tiling theory. In a tiling, space is divided into generalized polyhedra (cages) sharing faces (a “face-to-face” tiling). In a natural

tiling, the tiling has the same symmetry as the intrinsic symmetry of

the net, and no one face of a tile is bigger (has more edges) than the

rest. Subject to these constraints, the natural tiling consists of the smallest possible tiles.13a For some low symmetry nets, additional rules may be needed to obtain a unique tiling for a net.25 A simple

polyhedron is one in which exactly three edges meet at each vertex.

A simple tiling is a tiling by simple polyhedra in which exactly four

tiles meet at a vertex and exactly three meet at an edge. Foams and

cellular materials are simple tilings. A convenient measure of “regularity” of a net is the transitivity,

a set of four integers pqrs that states that a tiling has p kinds of vertex, q kinds of edge, r kinds of face, and s kinds of tile. The ﬁve regular nets13a are the only ones with a natural tiling with

transitivity 1111. Edge-transitive nets have transitivity p1rs with

p = 1 or 2.

It is common in chemistry to refer to nets in which k edges

meet at every vertex as k-connected. However, in graph theory k-connectivity has a quite diﬀerent meaning,19 so we prefer to use

k-coordinated or k-c for short. Nets with vertices with two or more diﬀerent coordinations are written as (k1,k2,...)-c.

For linkers with respectively two, three, four, ﬁve, six, ...

coordinating groups, we use the accepted terminology of ditopic,

tritopic, tetratopic, pentatopic, hexatopic, ....

This review discusses how one goes about abstracting the

underlying topology from an experimental crystal structure of

materials like MOFs. It is hoped that this will complement a

recent article in which an analysis of the structures of the more

than 6000 such materials in the Cambridge Structural Database was undertaken in an automated manner using TOPOS.26

Clearly only a few illustrative examples of those 6000 structures

can be presented here. These have been chosen to illustrate

general principals, to point up occasions where there may be no

clear choice of a unique underlying topology, and also to illustrate some minor diﬀerences of opinion on how best to deconstruct

crystal structures.

Readers interested in the variety of known three-periodic nets are referred to a recent comprehensive review.27 For the deconstruction of zero-periodic metalÀorganic polyhedra and descriptions

Figure 1. Nets derived from the net of the primitive cubic lattice (pcu). of their underlying topology, reference is also made to a recent review.28

3. EDGE NETS, AUGMENTED NETS, AND THE UNDERLYING TOPOLOGY

In Figure 1 the net (RCSR symbol pcu) of the primitive cubic lattice is shown. Also shown is the edge net, in which new vertices are placed in the middle of each original edge and vertices in edges with a common original vertex are joined together to form an octahedron around the original vertex. One can also think of the new net as an expansion of the original net.9 In any event, the new net is symbolized pcu-e. In this case the net is simple and “important” enough to merit its own RCSR symbol, which is reo (the O net in ReO3 has this structure). Note that the new net is not an edge graph in the mathematical sense, as in that case the new vertices would form a complete graph around the original vertex (thus including in this case edges linking opposite corners of the octahedron).29

Zeolite frameworks have stoichiometry TX2. Here T is a tetrahedrally coordinated atom and the X atoms form a net of cornersharing tetrahedra. In characterizing zeolite topology, the net is considered as four-coordinated with T atoms at the vertices and ÀXÀ as the edges. Thus, for faujasite with zeolite framework type23 FAU (RCSR symbol fau), the framework is said to have the 4-c fau topology. If we want to explicitly describe the net of X atoms we use the 6-c net fau-e. The important point is that we consider the expanded and unexpanded structures to have the same underlying net (“underlying topology”).

Figure 1 also shows an augmented net derived from pcu. Now the vertices of the original net are replaced by their vertex ﬁgures—polyhedron or polygon—in this instance an octahedron. The symbol for the new net is pcu-a. Again in this case there is an alternative symbol—cab—reﬂecting the fact that the net is the net of the B atoms in CaB6. For ﬁnite polyhedra the process of augmentation has long been termed truncation, but clearly the process of truncation (amputation of part of a polyhedron containing that vertex) cannot be carried out for two- or three-periodic nets.

The process of augmentation, like that of forming edge nets, is not based on graph theoretical foundation as again, in general, the new sets of vertices and edges do not form complete graphs. In fact, the new net derives essentially from a symmetric embedding of the original net. This is illustrated in Figure 2 for

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augmented net has vertices replaced by “vertex ﬁgures” with additional or missing edges. These are considered to have the same underlying topology.

Notice that we generally avoid multiple generations of edgeor augmented-nets. Thus, the zeolite framework lta may be seen (Figure 3) to be reo-a = pcu-e-a. Reducing lta down to an underlying topology of pcu would eliminate vital information about the cages (tiles) of this structure. Likewise, reducing pts to cds-e (see Figure 2), although it leads to interesting insights, would obscure the basic nature of materials based on the pts-a structure. The drawings in Figure 3 also illustrate a tiling. In this case it is a simple tiling.

Figure 2. Nets related to the PtS (cooperite) net (pts).

Figure 3. The net (lta) of the zeolite with framework type LTA. It should be clear from Figure 1 that this net could be assigned RCSR symbol reo-e. On the right the tiles of the structure are slightly shrunken to make the tiling clearer.

the pair pts, pts-a in a maximum-symmetry embedding. Notice that the two four-coordinated vertices are treated diﬀerently; in particular, the one with four coplanar edges lying in a mirror plane is replaced by a rectangle (square) in the augmented net.

In a net with more than one kind of vertex, there is the possibility that not all vertices are augmented. Thus, for pts one could leave the tetrahedral vertex but replace the square coordinated vertex by a square of vertices producing the structure identiﬁed as pts-f in Figure 2. Alternatively, leaving the square vertex alone and augmenting the tetrahedral vertex produces the pattern identiﬁed as pts-g in Figure 2. Both of these “halfaugmented” nets, as well as the edge net and the augmented net, are all considered to have the same underlying pts topology.

We will see later (section 4.7) that we may have “augmented” nets with lower intrinsic symmetry than the parent net if the

4. THE DECONSTRUCTION OF CRYSTAL STRUCTURES

MOFs, by deﬁnition, are made up of two kinds of secondary building unit (SBU). One kind is organic linkers that, as shown below, may be ditopic or polytopic. The second kind of SBU may a metal atom or (most commonly) a ﬁnite polyatomic cluster containing two or more metal atoms or an inﬁnite unit such as a one-periodic rod of atoms. The two types of SBU are treated slightly diﬀerently in a way that reﬂects their diﬀerent roles in the design and synthesis process.

Metal-containing SBUs are formed at the time of synthesis using conditions (e.g., temperature, pH) designed to produce just that SBU. Their shape is deﬁned by points of extension10a where they connect to organic linking components. This shape is generally a polygon, polyhedron, or a rod that often does not reveal the full internal structure of the SBU (we give examples below).

On the other hand, organic SBUs are preformed to a custom shape. The essence of systematic MOF chemistry (reticular chemistry) is the combination of a given metal-containing SBU with a variety of organic SBUs. In particular, the latter may have the same topology but a diﬀerent metric, producing, one anticipates, an isoreticular series of structures with the same underlying net. Because of the ﬂexibility of design of these organic components, it is important to identify all the branching points (vertices) and individual links (edges) rather than just identifying the envelope (points of extension).

The deconstructive procedure we follow when confronted with a new MOF structure is as follows. First, the diﬀerent vertices of the net are identiﬁed, as shown in the examples below. Next, the coordinates of one of each crystallographic type of vertex and those of its neighbors are presented to Systre together with the crystal symmetry. Systre will then identify the true combinatorial symmetry of the net (except for the rare cases of noncrystallographic symmetry as discussed in § 5.2). If the net is in the RCSR database, Systre will identify it. For nets new to Systre, TOPOS can be consulted for point and vertex symbols and for tiling data. Several detailed examples are given in the Supporting Information.

The crystal structure drawings in this paper are designed to illustrate the deconstruction process rather than to illustrate structures themselves. In particular, atoms irrelevant to topology, such as H, methyl groups, and monotopic coordinating groups, are usually omitted. We use the color codes C, black; O, red; N, green; and metal, blue.

4.1. Crystals with Corundum Net (cor)

Corundum, Al2O3, has a simple crystal structure with one kind of octahedral Al and one kind of tetrahedral O; i.e., it has a binodal (4,6)-c net, symbol cor, and is commonly found as the

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Figure 4. Examples of materials with the underlying net cor (corundum). See also Figure 5.

Figure 6. SBUs with two, three, or four square planar or square pyramidal units.

Figure 5. (a) The Re6Se8(CN)6 unit. (b) The same abstracted as an octahedral SBU.

structure of sesquioxides, sesquisulﬁdes, etc. The same topology is found in compounds like Fe2(SO4)3, now with ÀOÀ links as edges. The O atom net is now cor-e. In compounds like K2Zn3[Fe(CN)6]2 = K2Fe2[Zn(NC)4]3, there is again the same underlying topology.30 Now Fe and Zn are joined by ÀCÀNÀ links to form the cor net, or alternatively, ZnN4 tetrahedra and FeC6 octahedra are joined in the cor-a net as shown in Figure 4. K ions are in cavities of the structure.

Interesting cyanide compounds based on the cor net were prepared more recently.31 In these, the octahedral cation is replaced by an SBU with composition Re6Se8 with CN groups attached to the Re so that the octahedral group is Se8Re6C6, as shown in Figure 5. The C atoms of the SBU are the points of extension and deﬁne the (octahedral) shape of the SBU. This SBU is linked to ZnN4 tetrahedra by CÀN bonds. Na ions and water molecules are in the large cavities. The increase in unit cell volume over that of the original Al2O3 is a factor of 49. It is this dramatic change in scale that makes MOF structures intrinsically open and provides the basis for their applications in gas storage, among others.

4.2. Some Symmetrical Metal-Containing SBUs

In Figure 5 it was shown how the Se8Re6C6 unit could be abstracted as an octahedron and ultimately as an octahedrally coordinated vertex of the cor net. Here some other examples of symmetrical metal-cluster SBUs are shown and similarly abstracted as regular geometrical shapes formed by their points of extension. In MOFs they are linked by organic linkers such as carboxylates. Some have been known for a long time in molecules such as copper acetate, basic zinc acetate, and basic chromium acetate; a review describing 131 of these molecular clusters has been given recently and this can be consulted for references.32

Figure 7. (a) The basic zinc acetate SBU OZn4(CO2)6. (b) The basic chromium acetate SBU OCr3(CO2)X3.

Figure 6 shows SBUs consisting of two, three, or four square planar or square pyramidal units. The two-unit SBU is the “paddle wheel” motif associated especially with compounds like copper acetate. The zinc analog is a structural component of one of the earliest porous MOFs.33 The four C atoms are the points of extension and are at the vertices of a square. Usually, as prepared, the Cu atoms have an additional ligand such as a water molecule forming square pyramidal coordination. This extra ligand can subsequently be removed, leaving “open metal sites”.34

The three-unit SBU is found in UMCM-150.35 Here the points of extension form a trigonal pyramid.

An example with four metal-containing units is Cd4(CO2)8X4 (here X is the O atom of DMF coordinated in the pyramid apical position).36 Now the points of extension form a cube (perhaps better a tetragonal prism).

Figure 7 shows two SBUs long known as acetates, basic zinc acetate and basic chromium acetate.32 They are now ubiquitous in MOF chemistry; early examples are in MOF-537 and in MIL-88, respectively.38 For carboxylates, the composition is OZn4(CO2)6 and OCr3(CO2)6X3 (X = OH/H2O). The points of extension form respectively an octahedron and a trigonal prism.

Figure 8 shows some SBUs with higher-coordination regular shapes. They are OCo4(CO2)8;39 note the rather unusual squareplanar coordination for the central O atom, a related SBU

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Figure 8. SBUs with (from top) eight, eight, and 12 points of extension.

Figure 10. Examples of tetratopic linkers: (a and b) tetrahedral and (c) square.

Figure 9. (a) A ditopic linker and (b and c) two tritopic linkers.

ClMn4(tta)8 (here tta is a ﬁve-membered CN4 tetrazole ring part of a linker)40 and O4(OH)4Zr8(CO2)12,41 in which the 12 C atoms that deﬁne the points of extension are at the vertices of a cuboctahedron.

4.3. Some Simple Organic SBUs Figures 9 and 10 show some commonly used carboxylate

linkers. Notice that the two tritopic linkers in Figure 9 are considered to have the same “underlying” topology as do the two tetrahedral linkers in Figure 10 (recall the discussion in section 3). It is noted that linking the octahedral Zn4 unit of Figure 7 with the tritopic linker N(C6H4CO2)3 (Figure 9b) yields MOF-150 with the expected edge-transitive (3,6)-c net

Figure 11. Two examples of octahedra or octahedral SBUs linked by tetratopic linkers in structures discussed in the text. Large spheres indicate 3-c branching points.

pyr.42 However, as a relatively rare exception to the predictability of linking simple shapes that is the basis for reticular chemistry, linking with the other tritopic linker in Figure 9c and with their longer derivatives produces an isoreticular series of MOFs (MOF-177, MOF-180, MOF-200) having the edge 5-transitive (3,6)-c topology qom. These are particularly attractive candidates for practical gas-storage applications.43

A tetratopic organic unit joined to octahedrally coordinated metal atoms is illustrated in Figure 11. The authors44 recognized that the linker could be considered either as a single tetrahedral unit or as two triangular nodes; again the latter is preferred, as shown in the ﬁgure. It is noted in passing that the (4,6)-c net is iac net rather than what was described as (4,6)-c net “corundum” cor. The corresponding (3,6)-c net has the RCSR symbol act.

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Figure 12. (4,6)-c (iac) and (3,6)-c (act) nets discussed in the text shown in their augmented forms.

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Figure 14. (a) MOF-505. (b) The underlying (3,6)-c net fof. (c) Showing how merging the 3-c vertices of fof produces the nbo topology.

Figure 13. (a) a MOF (named “In MOF”) with a metal SBU with six points of extension linked by a tetratopic linker. (b) The preferred (3.6)-c net edq describing the underlying topology. (c) A (4,6)-c net (soc) alternatively used to describe the topology.

Curiously, the same misassignment (to cor rather than iac) was made in another example with a tetratopic organic linker (also shown in Figure 11) linked now to the Zn4 octahedral cluster of Figure 7.45 Here again we prefer to consider the underlying

topology as the (3,6)-c net act. The iac and act nets are compared in augmented form (iac-a and act-a) in Figure 12.

Another example of a MOF with the same kind of tetratopic linker is the “In MOF” shown in Figure 13.46 The authors described the topology as that of the (4,6)-c net soc, but the description preferred here is of the (3,6)-c net edq (see Figure 13). The metal cluster SBU, which is composed of three InO6 octahedra, is discussed further below.

The same tetratopic linker was used to generate the structure of MOF-505 by linking Cu2 paddle wheels (Figure 14).47 If the linker were to be considered as derived from one 4-c branching point (vertex) linked to the 4-c paddle wheel vertex, the underlying net would be the cubic 4-c net nbo, as shown in the ﬁgure. But again we prefer to consider the organic linker to have two 3-c vertices and the structure is then based on the (3,4)-c net fof. The

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Figure 17. (a) The (3,6)-c net wbl. (b) The (3.5)-c net lqm.

Figure 15. (a) Linker used in MOF-505 (Figure 14). (b) Linker in an isoreticular MOF. On the right is shown a primitive cell that would be cubic if the net were nbo.

Figure 16. An example of a tetratopic linker. Large green balls indicate the location of 3-c branching points (vertices of the underlying net). The magenta balls are 4-c vertices of that net.

validity of the latter description becomes apparent when examining an isoreticular series derived from linkers with greater spacing between the 3-c vertices, as shown in Figure 15.48 In these last examples, the structure deviates markedly from cubic metrics; the structures are actually rhombohedral and the axial ratios c/a are respectively for cubic nbo, MOF-505, and the two isoreticular analogs 1.22, 1.34, 2.07, and 2.84. Clearly, the last in particular is very far from cubic.

Our last example is a tetracarboxlate linker, which again we prefer to consider as having two 3-c branching point comes from an isoreticular series of chiral materials.49 The linker is shown in

Figure 18. Some structures discussed in the text with the pts underlying net.

Figure 16. The authors preferred to consider the linker as a single 4-c vertex, and together with the 4-c paddle wheel vertex, the underlying net was described as a 4-c binodal net to which the RCSR symbol wbl has been assigned. However, in our

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Figure 20. (a) A fragment of the structure of zinc dimethyl benzene dicarboxylate. (b) The (3,4)-c net that would result if the Zn atoms were taken as tetrahedral vertices and the tetratopic linker taken as having two 3-c vertices (see the text).

Figure 19. Fragments of the structure of hydrated zinc acetylene dicarboxylate. (a) Zn coordination to carboxylate O (coordinating water molecules not shown). (b) The acetylene dicarboxylate linker. (c) The resulting (3,4)-c net.

interpretation the underlying net is (3,4)-c (symbol lqm). Interestingly, both these nets are intrinsically chiral (symmetry I4122); they are compared in Figure 17. Note that the use of enantiopure ligands, as in this study, necessitates formation of enantiopure chiral crystals, regardless of the ideal symmetry of the underlying net. See section 5.6 for an example of chiral crystals produced with chiral ligands but with an achiral underlying net.

4.4. Some Structures with the pts Topology Figure 18 illustrates some structures with the 4-c net pts as

underlying topology. CuO (tenorite) has a structure that is a monoclinic distortion of the tetragonal PtS (cooperite) structure. This allows the Cu atoms to have two more-distant O neighbors in addition to the four nearest neighbors completing an distorted octahedron. PdSO4 also has a monoclinic structure, but the PdS arrangement, with ÀOÀ links, is topologically again pts.50 Cyanides AB(CN)4 with neutral or charged frameworks with the same (augmented) net have long been known and repeatedly rediscovered.51 A atoms are tetrahedral cations such as Cu(I), Zn, Cd, and B atoms such as Cu(II), Ni(II), Pd(II), and Pt(II) with square-planar coordination.

An example of a compound (shown in Figure 18), again monoclinic, with tetrahedral metal and planar tetratopic linker is Na2Zn(pm) 3 xH2O, where pm = pyromellitate = benzene-1,2,4,5tetracarboxylate (linker shown in Figure 10).52 The Na ions are in the interstices of the charged Zn(pm) framework. Notice that only one of each two carboxylate linkers is bonded to Zn. Compounds of this sort (reported in 1882) with charged frameworks and counterions in the channels may be considered as forerunners to the later MOFs with neutral frameworks and permanent porosity. The author did note that the structure was “zeolite-like”.

Also shown in Figure 18 are early examples, MOF-1134a and MOF-36,34b of neutral-framework MOFs with the pts topology. These combine the paddle wheel metal-containing SBU in Figure 6 with one or other of the tetrahedral linkers of Figure 10. The volume increase from PdO to MOF-36, which have the same symmetry, is a factor of 66.

4.5. Two MOFs Whose Preferred Description Is Not the pts Topology

Here we discuss two MOFs to which Alexandrov et al.26 assign the pts topology but for which we prefer an alternative description. The ﬁrst is a hydrated zinc acetylene dicarboxyllate.53 The zinc ions are six-coordinated, to two water molecules and to four carboxylate O atoms from diﬀerent carboxylate groups, so the latter form four points of extension at the vertices of a square (Figure 19). The organic linker has a tetrahedral envelope, so Alexandrov et al.26 identify the underlying net as pts. But as described in section 4.3, we prefer to consider such a linker as having two 3-c vertices and the net to be the (3,4)-c net sur, as shown in the ﬁgure.

The second example is a zinc salt of dimethyl benzenedicarboxylic acid54 shown in Figure 20. Now the Zn is tetrahedrally four-coordinated, and the tetratopic linker has a planar envelope, so taking Zn atoms and the center of the linker as vertices, Alexandrov et al.26 again get the 4-c pts net. But, as in the previous case, we would prefer to consider the linker as having two 3-c vertices. If the Zn atoms are retained as 4-c nodes, we would again get a (3,4)-c net, now dmd (Figure 20). But in this case we note that the Zn atoms share points of extension so in fact we really prefer a third description in which the metal cluster SBU is an inﬁnite rod of Zn atoms linked by carboxylates. The linker from this point of view is ditopic, more consistent with the earlier discussion. We give this description in section 6.3

4.6. MOFs with Multiple Links between SBUs Occasionally one ﬁnds structures in which two or more linkers

run parallel and join the same pair of vertices. Three examples are illustrated here. It is generally agreed that crystal nets do not admit multiple edges, so it is probably best in most cases to consider these as just one edge, although in some instances a pair of edges joining two SBUs can be abstracted as part of a rectangle, as shown here.

The ﬁrst involves an elegant SBU with composition SiO4Zn8(CO2)12 with the 12 C atoms acting as points of extension.55 In one example of this SBU the linkers are bdc = terephthalate = benzene-1,4-dicarboxylate, so the framework has composition SiO4Zn8(bdc)6.55 As shown in Figure 21, pairs of parallel linkers join pairs of SBUs, so each SBU is linked to six others as in a primitive cubic lattice (pcu net), and this is one way of describing the topology. However, if we consider the 12 points of extension separately, it may be seen in the ﬁgure that the topology is that of icosahedra linked to six others by rectangles (squares in the illustration). This second description corresponds to the uninodal 6-c net snf.

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Figure 21. MOFs with multiple links between pairs of SBUs. (a) An SBU linked to six others by pairs of links. (b) The net obtained if pairs of linkers are considered sides of quadrangles. (c) An SBU linked to four others by pairs of links. (d) The net obtained if pairs of linkers are considered sides of quadrangles.

Figure 24. A MOF with trigonal-prism SBUs linked with primitive cubic (pcu) topology: (a) a unit cell of the structure and (b) the trigonal prisms linked in pcu topology. The net of the vertices (red) is the uninodal 4-c unp.

extension, again carboxylate C atoms, now of biphenyl dicarboxylate (bpdc).57 The Cd cluster, formed from 11 CdO6 octahedra, also contains formate (HCO2) groups that serve to hold the cluster together, and the framework is formulated as Cd11(HCO2)6(bpdc)9. In this structure, each cluster is linked to eight neighboring clusters—to each of two by three bpdc linkers and to the other six by pairs of linkers (Figure 22). Considering the multiple links between a given pair of SBUs as one edge and the clusters as 8-c vertices, the net is body-centered cubic (bcu). In this case, there is no obvious alternative description. Notice that the cluster is chiral and has ideal symmetry 32 (D3); the crystal symmetry is the achiral R3c, which means that clusters of both hands occur equally.

Figure 22. A MOF with an SBU with 18 points of extension and multiple links between pairs of SBUs. Each SBU is linked to eight others.

The second example has a related structure. Now the metal cluster SBU with stoichiometry Zn7O4(CO2)10 has 10 points of extension and is linked to six other SBUs—to four by double links and to two by single links.56 Again the double links may be abstracted as rectangles and the net described as the 5-c net fqr, as shown in Figure 21.

A third example is a structure in which the metal-containing SBU consists of 11 Cd atoms and which has 18 points of

4.7. Examples of Lower-Symmetry Metal-Containing SBUs

It was remarked earlier that the augmented net derived from an underlying net was not strictly deﬁned in a mathematical sense. For example, if one takes any polyhedron with eight vertices and links it to eight neighbors in the same way topologically as in bcu (the net of the body-centered cubic lattice), the derived net will have bcu as its underlying net. This is illustrated in Figure 23 using the net bcu-i, in which square antiprisms are linked in such a way. Clearly, the net is diﬀerent (5-c, symmetry I422) from bcu-a (4-c, symmetry Im3m), but the underlying net is the same.

Such examples are quite common in crystals. Figure 24 shows an example of an SBU that has points of extension at the vertices of a trigonal prism linked with the pcu topology by bent ditopic linkers. The SBU is in fact the same as that shown in Figure 13 and the authors correctly identiﬁed the underlying topology as pcu.46 The “augmented” 4-c net shown in Figure 24 has the RCSR symbol unp.

An example with a square antiprism replacing a cube is found in the structure of a cyanide with framework composition Fe2(H2O)4Mo(CN)8, with Fe bonded in a planar fashion to four

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Deconstructing the Crystal Structures of MetalÀOrganic Frameworks and Related Materials into Their Underlying Nets

Michael O’Keeﬀe*,†,‡ and Omar M. Yaghi*,‡,§

†Department of Chemistry and Biochemistry, Arizona State University, Tempe, Arizona 85287, United States ‡Center for Reticular Chemistry, Center for Global Mentoring, Department of Chemistry and Biochemistry, University of CaliforniaÀLos Angeles, 607 Charles E. Young Dr. East, Los Angeles, California 90095, United States §Graduate School of EEWS, Korea Advanced Institute of Science and Technology, Daejeon, Korea

CONTENTS

1. Introduction

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2. Identiﬁcation, Description, and Characterization of

Nets

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3. Edge Nets, Augmented Nets, and the Underlying

Topology

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4. The Deconstruction of Crystal Structures

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4.1. Crystals with Corundum Net (cor)

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4.2. Some Symmetrical Metal-Containing SBUs

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4.3. Some Simple Organic SBUs

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4.4. Some Structures with the pts Topology

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4.5. Two MOFs Whose Preferred Description Is Not

the pts Topology

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4.6. MOFs with Multiple Links between SBUs

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4.7. Examples of Lower-Symmetry Metal-

Containing SBUs

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5. Some Case Studies

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5.1. A MOF with ubt Topology

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5.2. MOFs with Hexatopic Carboxylate Linkers

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5.3. MOFs with Octatopic Linkers

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5.4. More on Metal Cluster SBUs

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5.5. More Structures with Linked MOPs

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5.6. A Cyclodextrin MOF

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5.7. The Hierarchical Underlying Nets of MIL-101

and MIL-100

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6. MOFs with Rod SBUs

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6.1. SBUs as Zigzag Ladders

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6.2. A MOF with a Twisted Ladder Rod SBU

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6.3. A MOF with Rod SBUs of Linked Tetrahedra 697

6.4. Two-Way Rod SBUs of Linked Tetrahedra

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6.5. MOFs with Rod SBUs of Linked Octahedra

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6.6. Rod SBUs That Resist Simpliﬁcation

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7. MOFs with Ring SBUs

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7.1. Coda

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8. Concluding Remarks

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Author Information

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Biographies

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Acknowledgment

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References

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1. INTRODUCTION

The synthesis and characterization of metalÀorganic frameworks (MOFs) is one of the most rapidly developing areas of chemical science. These materials have unquestionably enormous potential for many practical applications, as detailed elsewhere in this issue, but they also often have exceptionally beautiful structures. It is the identiﬁcation and description of the nets that describe the underlying topology of these structures that is the main topic of this review. In particular we emphasize that this is not a review of MOF structures per se.

Why should we care about nets and related structural aspects of crystals? First and foremost, as chemists we recognize that the very core of our science lies in describing, and perhaps understanding, how atoms organize themselves, sometimes with our help, in chemical compounds. Such knowledge is also essential to designed (“rational”) synthesis of MOFs and related materials from component parts, as has been stressed recently.1 For this, of course, one needs to know the principal possibilities, which, as discussed below, have been established systematically only in the past few years. By deconstruction, we mean simply the reverse of the thought process that goes into designed synthesis, that is, breaking down a complex structure into its fundamental units without losing their chemical signiﬁcance. One can think of it as reverse engineering.

Another reason for knowing about nets and their occurrences is a result of the dramatic advances in methods of computer simulation of MOFÀadsorbate interactions, especially calculated adsorption isotherms, which makes the computer prescreening of potential materials an attractive procedure.2 Of course, to do this usefully it must be performed for materials for which there is a reasonable prospect of actual synthesis, which in turn will be done by design.

From the very earliest days of crystallography, simple inorganic structures were shown as “ball-and-stick” models in which the balls were the atoms and the sticks corresponded to bonds presumed to exist between nearest-neighbor atoms.3 It was early realized, particularly by Wells,4 that such models could be considered as representations or embeddings of special kinds of abstract graphs called nets (deﬁned below) with the vertices of the graph corresponding to the atoms and the edges (links) of the

Special Issue: 2012 Metal-Organic Frameworks

Received: June 6, 2011 Published: September 15, 2011

r 2011 American Chemical Society

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graph corresponding to the bonds. Wells devoted much eﬀort to enumerating nets, but he focused almost entirely on structures with three- and/or four-coordinated vertices and placed special emphasis on structures with shortest cycles (closed paths around the net) of all the same size; he called these structures uniform nets. Although he correctly recognized the importance of structures with symmetry-related vertices and edges, in fact, he found only very few of those now known.

It also became apparent that the same topology (net) was found in many diﬀerent chemical contexts. It was also realized that the edges and vertices of the net could be respectively polyatomic linkers and clusters. The work of the Iwamoto group on cyanides is notable in this respect. Complex cyanides with nets of forms of silica (cristobalite, tridymite and keatite) of other binary compounds such as rutile (TiO2), pyrite (FeS2), and cooperite (PtS) were prepared and their nets identiﬁed.5 The term “mineralomimetic” was coined to describe this kind of chemistry.

An important next step was the realization that, in fact, certain topologies could be targeted, especially for cyanides, by assembling appropriately shaped components.6,7 The wide variety of chemical compounds amenable to this approach was subsequently emphasized in several reviews.7,8 It should be emphasized, however, that it was in general rare for an underlying net to be identiﬁed in the older literature, and furthermore, when a net was identiﬁed, it was often done incorrectly. This last criticism applies far too often also to recent work. Indeed, the diligent reader will ﬁnd that some of the examples adduced in this review were originally assigned either to no topology or to an incorrect one. However, it is less the purpose here to correct errors than to point the way to better analyses in the future.

The discovery of MOFs, a term used here particularly to describe robust and highly porous metalÀorganic frameworks, led to the recognition that, in order to truly obtain structures by design, one had ﬁrst to identify the principal topological possibilities for nets. These, which were termed default structures,9,10 were identiﬁed as those with high point symmetry at the vertices and with a small number of diﬀerent kinds of vertex and edge— two conditions that are, of course, highly correlated. Subsequent analysis of published structures conﬁrmed the predominance of these default topologies.11

The discipline of preparing materials of targeted geometry by design is termed reticular chemistry10 and a series of compounds with the same underlying topology (net) is called an isoreticular series.12 As already mentioned, for successful reticular chemistry one needs to know the principal topologies, and a concerted eﬀort was made to enumerate them.13 The most important of the these are nets with one kind of edge (edge transitive) most of which were unknown prior to this work but are now realized to be of special importance. A review under the rubric “Taxonomy of Nets and the Design of Materials” has been published.14

Data for many of the nets most important for reticular chemistry are collected in a searchable database known as the Reticular Chemistry Structure Resource (RCSR).15 There nets are assigned three-letter symbols such as abc, or symbols with extensions as in abc-d (see below). This database is rather small (about 2000 entries). A much larger database is being developed in the EPINET project, which currently contains about 15 000 three-periodic nets.16 The computer program TOPOS recognizes even more, about 70 000.17 In this connection, mention should also be made to the extensive enumerations of sphere packings by Fischer and associates.18 The nets of these structures

have just one kind of vertex and have an embedding in which all the shortest (and equal) intervertex distances correspond to edges of the net. Most of these are incorporated in the RCSR.

2. IDENTIFICATION, DESCRIPTION, AND CHARACTERIZATION OF NETS

A net is just a special sort of graph. It is simple, meaning that there is at most one undirected edge that links any pair of vertices, and there are no loops (edges linking a vertex to itself). A net is also connected, meaning that every vertex is linked to every other by a continuous path of edges. The net of a polyhedron is ﬁnite. In crystals we will have inﬁnite nets that are one-, two-, or threeperiodic (“dimensional”). The emphasis here will be on threeperiodic nets. Graph-theoretical aspects, in particular terminology and deﬁnitions, have been given elsewhere.19

By “underlying topology” we mean the innate structure of the net associated with the crystal structure. Topology is really a branch of mathematics (or several branches as some would have it). However, here we use the term, following common usage, to refer to the combinatorial structure of a graph that is invariant in diﬀerent embeddings.20

In this review we show how a net is extracted from a crystal structure. The ﬁrst question is, what is the identity of the net? This can be answered in a meaningful way only by saying that it is identical to a previously known net that has an identiﬁer (such as a RCSR symbol). Otherwise, the net is new. The only algorithm devised to do this in a mathematically rigorous way is realized in Olaf DelgadoÀFriedichs’ program Systre.21 It should be mentioned, though, that in practice the program TOPOS17 also solves this problem with a high degree of certainty.

The second question to ask of the net is, what is the symmetry? By this we mean the combinatorial symmetry which, for the nets discussed here, is isomorphic with a space group and is the maximum possible symmetry of an embedding. As far as we know only Systre21 answers this question. As a bonus, Systre computes an embedding (a realization with space group, unit cell parameters, vertex coordinates, and edge speciﬁcation) in that symmetry. Occasionally, in practice very rarely, one encounters nets that have nonrigid body symmetries; we give an example below (section 5.2). Such nets are not currently considered by Systre.

The local topology of a vertex in a net is sometimes characterized by a point symbol or a vertex symbol. The point symbol, introduced by Wells,4 gives information about the shortest cycles at each angle of a vertex. The point symbol is of the form Aa.Bb... and signiﬁes that there are a angles at which the shortest cycle is an A-cycle, b angles at which the shortest cycle is a B-cycle, etc. By convention A < B < ... and a + b + ... = z(z À 1)/2, where z is the coordination number of the vertex and z(z À 1)/2 is the number of angles at that vertex. Point symbols for nets are conveniently obtained from TOPOS.

Our preference is for vertex symbols (also given by TOPOS), which give information about the number of rings (cycles that are not the sum of two shorter cycles) at each angle in a vertex.22 However, these become cumbersome for vertices of higher than six-coordination. These symbols are used to characterize the nets in the RCSR and in the Atlas of Zeolite Framework Types.23

Nets with one kind of vertex (i.e., those for which all the vertices are related by symmetry operations in their most symmetrical embeddings) are often called uninodal, those with two kinds of vertex binodal, etc.

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There are two common practices that we would like to discourage. The ﬁrst is that of calling a point symbol a “Schl€aﬂi symbol”. In universally accepted mathematical usage the latter is

a symbol for a regular tiling. In three-dimensional Euclidean space, there is only one such tiling—a face-to-face tiling by cubes for which the Schl€aﬂi symbol is {4,3,4} (the point symbol for the net is 412.63). The distinction between various symbols for vertex conﬁgurations and tilings has been discussed fully recently.24 In older papers, vertex symbols were also incorrectly called Schl€aﬂi symbols.22

The second usage we would like to discourage is that of referring to a point symbol or a vertex symbol as a “topology”. It is not; many nets with diﬀerent topologies have vertices with the

same point symbol. For example, the RCSR contains 14 distinct uninodal nets with point symbol 66, and if one includes polytypes of the diamond/lonsdaleite type, there is an inﬁnite number of nets in which all vertices have point symbol 66. By far the best way

of specifying a net, particularly a new, or previously undocumented, one is by a Systre-readable ﬁle. TOPOS can export Systre-readable ﬁles. For nets in the RCSR database, the RCSR symbol should be an adequate identiﬁer.

Recent advances in our knowledge of three-periodic nets have

come from tiling theory. In a tiling, space is divided into generalized polyhedra (cages) sharing faces (a “face-to-face” tiling). In a natural

tiling, the tiling has the same symmetry as the intrinsic symmetry of

the net, and no one face of a tile is bigger (has more edges) than the

rest. Subject to these constraints, the natural tiling consists of the smallest possible tiles.13a For some low symmetry nets, additional rules may be needed to obtain a unique tiling for a net.25 A simple

polyhedron is one in which exactly three edges meet at each vertex.

A simple tiling is a tiling by simple polyhedra in which exactly four

tiles meet at a vertex and exactly three meet at an edge. Foams and

cellular materials are simple tilings. A convenient measure of “regularity” of a net is the transitivity,

a set of four integers pqrs that states that a tiling has p kinds of vertex, q kinds of edge, r kinds of face, and s kinds of tile. The ﬁve regular nets13a are the only ones with a natural tiling with

transitivity 1111. Edge-transitive nets have transitivity p1rs with

p = 1 or 2.

It is common in chemistry to refer to nets in which k edges

meet at every vertex as k-connected. However, in graph theory k-connectivity has a quite diﬀerent meaning,19 so we prefer to use

k-coordinated or k-c for short. Nets with vertices with two or more diﬀerent coordinations are written as (k1,k2,...)-c.

For linkers with respectively two, three, four, ﬁve, six, ...

coordinating groups, we use the accepted terminology of ditopic,

tritopic, tetratopic, pentatopic, hexatopic, ....

This review discusses how one goes about abstracting the

underlying topology from an experimental crystal structure of

materials like MOFs. It is hoped that this will complement a

recent article in which an analysis of the structures of the more

than 6000 such materials in the Cambridge Structural Database was undertaken in an automated manner using TOPOS.26

Clearly only a few illustrative examples of those 6000 structures

can be presented here. These have been chosen to illustrate

general principals, to point up occasions where there may be no

clear choice of a unique underlying topology, and also to illustrate some minor diﬀerences of opinion on how best to deconstruct

crystal structures.

Readers interested in the variety of known three-periodic nets are referred to a recent comprehensive review.27 For the deconstruction of zero-periodic metalÀorganic polyhedra and descriptions

Figure 1. Nets derived from the net of the primitive cubic lattice (pcu). of their underlying topology, reference is also made to a recent review.28

3. EDGE NETS, AUGMENTED NETS, AND THE UNDERLYING TOPOLOGY

In Figure 1 the net (RCSR symbol pcu) of the primitive cubic lattice is shown. Also shown is the edge net, in which new vertices are placed in the middle of each original edge and vertices in edges with a common original vertex are joined together to form an octahedron around the original vertex. One can also think of the new net as an expansion of the original net.9 In any event, the new net is symbolized pcu-e. In this case the net is simple and “important” enough to merit its own RCSR symbol, which is reo (the O net in ReO3 has this structure). Note that the new net is not an edge graph in the mathematical sense, as in that case the new vertices would form a complete graph around the original vertex (thus including in this case edges linking opposite corners of the octahedron).29

Zeolite frameworks have stoichiometry TX2. Here T is a tetrahedrally coordinated atom and the X atoms form a net of cornersharing tetrahedra. In characterizing zeolite topology, the net is considered as four-coordinated with T atoms at the vertices and ÀXÀ as the edges. Thus, for faujasite with zeolite framework type23 FAU (RCSR symbol fau), the framework is said to have the 4-c fau topology. If we want to explicitly describe the net of X atoms we use the 6-c net fau-e. The important point is that we consider the expanded and unexpanded structures to have the same underlying net (“underlying topology”).

Figure 1 also shows an augmented net derived from pcu. Now the vertices of the original net are replaced by their vertex ﬁgures—polyhedron or polygon—in this instance an octahedron. The symbol for the new net is pcu-a. Again in this case there is an alternative symbol—cab—reﬂecting the fact that the net is the net of the B atoms in CaB6. For ﬁnite polyhedra the process of augmentation has long been termed truncation, but clearly the process of truncation (amputation of part of a polyhedron containing that vertex) cannot be carried out for two- or three-periodic nets.

The process of augmentation, like that of forming edge nets, is not based on graph theoretical foundation as again, in general, the new sets of vertices and edges do not form complete graphs. In fact, the new net derives essentially from a symmetric embedding of the original net. This is illustrated in Figure 2 for

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augmented net has vertices replaced by “vertex ﬁgures” with additional or missing edges. These are considered to have the same underlying topology.

Notice that we generally avoid multiple generations of edgeor augmented-nets. Thus, the zeolite framework lta may be seen (Figure 3) to be reo-a = pcu-e-a. Reducing lta down to an underlying topology of pcu would eliminate vital information about the cages (tiles) of this structure. Likewise, reducing pts to cds-e (see Figure 2), although it leads to interesting insights, would obscure the basic nature of materials based on the pts-a structure. The drawings in Figure 3 also illustrate a tiling. In this case it is a simple tiling.

Figure 2. Nets related to the PtS (cooperite) net (pts).

Figure 3. The net (lta) of the zeolite with framework type LTA. It should be clear from Figure 1 that this net could be assigned RCSR symbol reo-e. On the right the tiles of the structure are slightly shrunken to make the tiling clearer.

the pair pts, pts-a in a maximum-symmetry embedding. Notice that the two four-coordinated vertices are treated diﬀerently; in particular, the one with four coplanar edges lying in a mirror plane is replaced by a rectangle (square) in the augmented net.

In a net with more than one kind of vertex, there is the possibility that not all vertices are augmented. Thus, for pts one could leave the tetrahedral vertex but replace the square coordinated vertex by a square of vertices producing the structure identiﬁed as pts-f in Figure 2. Alternatively, leaving the square vertex alone and augmenting the tetrahedral vertex produces the pattern identiﬁed as pts-g in Figure 2. Both of these “halfaugmented” nets, as well as the edge net and the augmented net, are all considered to have the same underlying pts topology.

We will see later (section 4.7) that we may have “augmented” nets with lower intrinsic symmetry than the parent net if the

4. THE DECONSTRUCTION OF CRYSTAL STRUCTURES

MOFs, by deﬁnition, are made up of two kinds of secondary building unit (SBU). One kind is organic linkers that, as shown below, may be ditopic or polytopic. The second kind of SBU may a metal atom or (most commonly) a ﬁnite polyatomic cluster containing two or more metal atoms or an inﬁnite unit such as a one-periodic rod of atoms. The two types of SBU are treated slightly diﬀerently in a way that reﬂects their diﬀerent roles in the design and synthesis process.

Metal-containing SBUs are formed at the time of synthesis using conditions (e.g., temperature, pH) designed to produce just that SBU. Their shape is deﬁned by points of extension10a where they connect to organic linking components. This shape is generally a polygon, polyhedron, or a rod that often does not reveal the full internal structure of the SBU (we give examples below).

On the other hand, organic SBUs are preformed to a custom shape. The essence of systematic MOF chemistry (reticular chemistry) is the combination of a given metal-containing SBU with a variety of organic SBUs. In particular, the latter may have the same topology but a diﬀerent metric, producing, one anticipates, an isoreticular series of structures with the same underlying net. Because of the ﬂexibility of design of these organic components, it is important to identify all the branching points (vertices) and individual links (edges) rather than just identifying the envelope (points of extension).

The deconstructive procedure we follow when confronted with a new MOF structure is as follows. First, the diﬀerent vertices of the net are identiﬁed, as shown in the examples below. Next, the coordinates of one of each crystallographic type of vertex and those of its neighbors are presented to Systre together with the crystal symmetry. Systre will then identify the true combinatorial symmetry of the net (except for the rare cases of noncrystallographic symmetry as discussed in § 5.2). If the net is in the RCSR database, Systre will identify it. For nets new to Systre, TOPOS can be consulted for point and vertex symbols and for tiling data. Several detailed examples are given in the Supporting Information.

The crystal structure drawings in this paper are designed to illustrate the deconstruction process rather than to illustrate structures themselves. In particular, atoms irrelevant to topology, such as H, methyl groups, and monotopic coordinating groups, are usually omitted. We use the color codes C, black; O, red; N, green; and metal, blue.

4.1. Crystals with Corundum Net (cor)

Corundum, Al2O3, has a simple crystal structure with one kind of octahedral Al and one kind of tetrahedral O; i.e., it has a binodal (4,6)-c net, symbol cor, and is commonly found as the

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Figure 4. Examples of materials with the underlying net cor (corundum). See also Figure 5.

Figure 6. SBUs with two, three, or four square planar or square pyramidal units.

Figure 5. (a) The Re6Se8(CN)6 unit. (b) The same abstracted as an octahedral SBU.

structure of sesquioxides, sesquisulﬁdes, etc. The same topology is found in compounds like Fe2(SO4)3, now with ÀOÀ links as edges. The O atom net is now cor-e. In compounds like K2Zn3[Fe(CN)6]2 = K2Fe2[Zn(NC)4]3, there is again the same underlying topology.30 Now Fe and Zn are joined by ÀCÀNÀ links to form the cor net, or alternatively, ZnN4 tetrahedra and FeC6 octahedra are joined in the cor-a net as shown in Figure 4. K ions are in cavities of the structure.

Interesting cyanide compounds based on the cor net were prepared more recently.31 In these, the octahedral cation is replaced by an SBU with composition Re6Se8 with CN groups attached to the Re so that the octahedral group is Se8Re6C6, as shown in Figure 5. The C atoms of the SBU are the points of extension and deﬁne the (octahedral) shape of the SBU. This SBU is linked to ZnN4 tetrahedra by CÀN bonds. Na ions and water molecules are in the large cavities. The increase in unit cell volume over that of the original Al2O3 is a factor of 49. It is this dramatic change in scale that makes MOF structures intrinsically open and provides the basis for their applications in gas storage, among others.

4.2. Some Symmetrical Metal-Containing SBUs

In Figure 5 it was shown how the Se8Re6C6 unit could be abstracted as an octahedron and ultimately as an octahedrally coordinated vertex of the cor net. Here some other examples of symmetrical metal-cluster SBUs are shown and similarly abstracted as regular geometrical shapes formed by their points of extension. In MOFs they are linked by organic linkers such as carboxylates. Some have been known for a long time in molecules such as copper acetate, basic zinc acetate, and basic chromium acetate; a review describing 131 of these molecular clusters has been given recently and this can be consulted for references.32

Figure 7. (a) The basic zinc acetate SBU OZn4(CO2)6. (b) The basic chromium acetate SBU OCr3(CO2)X3.

Figure 6 shows SBUs consisting of two, three, or four square planar or square pyramidal units. The two-unit SBU is the “paddle wheel” motif associated especially with compounds like copper acetate. The zinc analog is a structural component of one of the earliest porous MOFs.33 The four C atoms are the points of extension and are at the vertices of a square. Usually, as prepared, the Cu atoms have an additional ligand such as a water molecule forming square pyramidal coordination. This extra ligand can subsequently be removed, leaving “open metal sites”.34

The three-unit SBU is found in UMCM-150.35 Here the points of extension form a trigonal pyramid.

An example with four metal-containing units is Cd4(CO2)8X4 (here X is the O atom of DMF coordinated in the pyramid apical position).36 Now the points of extension form a cube (perhaps better a tetragonal prism).

Figure 7 shows two SBUs long known as acetates, basic zinc acetate and basic chromium acetate.32 They are now ubiquitous in MOF chemistry; early examples are in MOF-537 and in MIL-88, respectively.38 For carboxylates, the composition is OZn4(CO2)6 and OCr3(CO2)6X3 (X = OH/H2O). The points of extension form respectively an octahedron and a trigonal prism.

Figure 8 shows some SBUs with higher-coordination regular shapes. They are OCo4(CO2)8;39 note the rather unusual squareplanar coordination for the central O atom, a related SBU

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Figure 8. SBUs with (from top) eight, eight, and 12 points of extension.

Figure 10. Examples of tetratopic linkers: (a and b) tetrahedral and (c) square.

Figure 9. (a) A ditopic linker and (b and c) two tritopic linkers.

ClMn4(tta)8 (here tta is a ﬁve-membered CN4 tetrazole ring part of a linker)40 and O4(OH)4Zr8(CO2)12,41 in which the 12 C atoms that deﬁne the points of extension are at the vertices of a cuboctahedron.

4.3. Some Simple Organic SBUs Figures 9 and 10 show some commonly used carboxylate

linkers. Notice that the two tritopic linkers in Figure 9 are considered to have the same “underlying” topology as do the two tetrahedral linkers in Figure 10 (recall the discussion in section 3). It is noted that linking the octahedral Zn4 unit of Figure 7 with the tritopic linker N(C6H4CO2)3 (Figure 9b) yields MOF-150 with the expected edge-transitive (3,6)-c net

Figure 11. Two examples of octahedra or octahedral SBUs linked by tetratopic linkers in structures discussed in the text. Large spheres indicate 3-c branching points.

pyr.42 However, as a relatively rare exception to the predictability of linking simple shapes that is the basis for reticular chemistry, linking with the other tritopic linker in Figure 9c and with their longer derivatives produces an isoreticular series of MOFs (MOF-177, MOF-180, MOF-200) having the edge 5-transitive (3,6)-c topology qom. These are particularly attractive candidates for practical gas-storage applications.43

A tetratopic organic unit joined to octahedrally coordinated metal atoms is illustrated in Figure 11. The authors44 recognized that the linker could be considered either as a single tetrahedral unit or as two triangular nodes; again the latter is preferred, as shown in the ﬁgure. It is noted in passing that the (4,6)-c net is iac net rather than what was described as (4,6)-c net “corundum” cor. The corresponding (3,6)-c net has the RCSR symbol act.

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Figure 12. (4,6)-c (iac) and (3,6)-c (act) nets discussed in the text shown in their augmented forms.

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Figure 14. (a) MOF-505. (b) The underlying (3,6)-c net fof. (c) Showing how merging the 3-c vertices of fof produces the nbo topology.

Figure 13. (a) a MOF (named “In MOF”) with a metal SBU with six points of extension linked by a tetratopic linker. (b) The preferred (3.6)-c net edq describing the underlying topology. (c) A (4,6)-c net (soc) alternatively used to describe the topology.

Curiously, the same misassignment (to cor rather than iac) was made in another example with a tetratopic organic linker (also shown in Figure 11) linked now to the Zn4 octahedral cluster of Figure 7.45 Here again we prefer to consider the underlying

topology as the (3,6)-c net act. The iac and act nets are compared in augmented form (iac-a and act-a) in Figure 12.

Another example of a MOF with the same kind of tetratopic linker is the “In MOF” shown in Figure 13.46 The authors described the topology as that of the (4,6)-c net soc, but the description preferred here is of the (3,6)-c net edq (see Figure 13). The metal cluster SBU, which is composed of three InO6 octahedra, is discussed further below.

The same tetratopic linker was used to generate the structure of MOF-505 by linking Cu2 paddle wheels (Figure 14).47 If the linker were to be considered as derived from one 4-c branching point (vertex) linked to the 4-c paddle wheel vertex, the underlying net would be the cubic 4-c net nbo, as shown in the ﬁgure. But again we prefer to consider the organic linker to have two 3-c vertices and the structure is then based on the (3,4)-c net fof. The

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Figure 17. (a) The (3,6)-c net wbl. (b) The (3.5)-c net lqm.

Figure 15. (a) Linker used in MOF-505 (Figure 14). (b) Linker in an isoreticular MOF. On the right is shown a primitive cell that would be cubic if the net were nbo.

Figure 16. An example of a tetratopic linker. Large green balls indicate the location of 3-c branching points (vertices of the underlying net). The magenta balls are 4-c vertices of that net.

validity of the latter description becomes apparent when examining an isoreticular series derived from linkers with greater spacing between the 3-c vertices, as shown in Figure 15.48 In these last examples, the structure deviates markedly from cubic metrics; the structures are actually rhombohedral and the axial ratios c/a are respectively for cubic nbo, MOF-505, and the two isoreticular analogs 1.22, 1.34, 2.07, and 2.84. Clearly, the last in particular is very far from cubic.

Our last example is a tetracarboxlate linker, which again we prefer to consider as having two 3-c branching point comes from an isoreticular series of chiral materials.49 The linker is shown in

Figure 18. Some structures discussed in the text with the pts underlying net.

Figure 16. The authors preferred to consider the linker as a single 4-c vertex, and together with the 4-c paddle wheel vertex, the underlying net was described as a 4-c binodal net to which the RCSR symbol wbl has been assigned. However, in our

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Figure 20. (a) A fragment of the structure of zinc dimethyl benzene dicarboxylate. (b) The (3,4)-c net that would result if the Zn atoms were taken as tetrahedral vertices and the tetratopic linker taken as having two 3-c vertices (see the text).

Figure 19. Fragments of the structure of hydrated zinc acetylene dicarboxylate. (a) Zn coordination to carboxylate O (coordinating water molecules not shown). (b) The acetylene dicarboxylate linker. (c) The resulting (3,4)-c net.

interpretation the underlying net is (3,4)-c (symbol lqm). Interestingly, both these nets are intrinsically chiral (symmetry I4122); they are compared in Figure 17. Note that the use of enantiopure ligands, as in this study, necessitates formation of enantiopure chiral crystals, regardless of the ideal symmetry of the underlying net. See section 5.6 for an example of chiral crystals produced with chiral ligands but with an achiral underlying net.

4.4. Some Structures with the pts Topology Figure 18 illustrates some structures with the 4-c net pts as

underlying topology. CuO (tenorite) has a structure that is a monoclinic distortion of the tetragonal PtS (cooperite) structure. This allows the Cu atoms to have two more-distant O neighbors in addition to the four nearest neighbors completing an distorted octahedron. PdSO4 also has a monoclinic structure, but the PdS arrangement, with ÀOÀ links, is topologically again pts.50 Cyanides AB(CN)4 with neutral or charged frameworks with the same (augmented) net have long been known and repeatedly rediscovered.51 A atoms are tetrahedral cations such as Cu(I), Zn, Cd, and B atoms such as Cu(II), Ni(II), Pd(II), and Pt(II) with square-planar coordination.

An example of a compound (shown in Figure 18), again monoclinic, with tetrahedral metal and planar tetratopic linker is Na2Zn(pm) 3 xH2O, where pm = pyromellitate = benzene-1,2,4,5tetracarboxylate (linker shown in Figure 10).52 The Na ions are in the interstices of the charged Zn(pm) framework. Notice that only one of each two carboxylate linkers is bonded to Zn. Compounds of this sort (reported in 1882) with charged frameworks and counterions in the channels may be considered as forerunners to the later MOFs with neutral frameworks and permanent porosity. The author did note that the structure was “zeolite-like”.

Also shown in Figure 18 are early examples, MOF-1134a and MOF-36,34b of neutral-framework MOFs with the pts topology. These combine the paddle wheel metal-containing SBU in Figure 6 with one or other of the tetrahedral linkers of Figure 10. The volume increase from PdO to MOF-36, which have the same symmetry, is a factor of 66.

4.5. Two MOFs Whose Preferred Description Is Not the pts Topology

Here we discuss two MOFs to which Alexandrov et al.26 assign the pts topology but for which we prefer an alternative description. The ﬁrst is a hydrated zinc acetylene dicarboxyllate.53 The zinc ions are six-coordinated, to two water molecules and to four carboxylate O atoms from diﬀerent carboxylate groups, so the latter form four points of extension at the vertices of a square (Figure 19). The organic linker has a tetrahedral envelope, so Alexandrov et al.26 identify the underlying net as pts. But as described in section 4.3, we prefer to consider such a linker as having two 3-c vertices and the net to be the (3,4)-c net sur, as shown in the ﬁgure.

The second example is a zinc salt of dimethyl benzenedicarboxylic acid54 shown in Figure 20. Now the Zn is tetrahedrally four-coordinated, and the tetratopic linker has a planar envelope, so taking Zn atoms and the center of the linker as vertices, Alexandrov et al.26 again get the 4-c pts net. But, as in the previous case, we would prefer to consider the linker as having two 3-c vertices. If the Zn atoms are retained as 4-c nodes, we would again get a (3,4)-c net, now dmd (Figure 20). But in this case we note that the Zn atoms share points of extension so in fact we really prefer a third description in which the metal cluster SBU is an inﬁnite rod of Zn atoms linked by carboxylates. The linker from this point of view is ditopic, more consistent with the earlier discussion. We give this description in section 6.3

4.6. MOFs with Multiple Links between SBUs Occasionally one ﬁnds structures in which two or more linkers

run parallel and join the same pair of vertices. Three examples are illustrated here. It is generally agreed that crystal nets do not admit multiple edges, so it is probably best in most cases to consider these as just one edge, although in some instances a pair of edges joining two SBUs can be abstracted as part of a rectangle, as shown here.

The ﬁrst involves an elegant SBU with composition SiO4Zn8(CO2)12 with the 12 C atoms acting as points of extension.55 In one example of this SBU the linkers are bdc = terephthalate = benzene-1,4-dicarboxylate, so the framework has composition SiO4Zn8(bdc)6.55 As shown in Figure 21, pairs of parallel linkers join pairs of SBUs, so each SBU is linked to six others as in a primitive cubic lattice (pcu net), and this is one way of describing the topology. However, if we consider the 12 points of extension separately, it may be seen in the ﬁgure that the topology is that of icosahedra linked to six others by rectangles (squares in the illustration). This second description corresponds to the uninodal 6-c net snf.

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REVIEW Figure 23. Two nets with the same underlying topology (bcu).

Figure 21. MOFs with multiple links between pairs of SBUs. (a) An SBU linked to six others by pairs of links. (b) The net obtained if pairs of linkers are considered sides of quadrangles. (c) An SBU linked to four others by pairs of links. (d) The net obtained if pairs of linkers are considered sides of quadrangles.

Figure 24. A MOF with trigonal-prism SBUs linked with primitive cubic (pcu) topology: (a) a unit cell of the structure and (b) the trigonal prisms linked in pcu topology. The net of the vertices (red) is the uninodal 4-c unp.

extension, again carboxylate C atoms, now of biphenyl dicarboxylate (bpdc).57 The Cd cluster, formed from 11 CdO6 octahedra, also contains formate (HCO2) groups that serve to hold the cluster together, and the framework is formulated as Cd11(HCO2)6(bpdc)9. In this structure, each cluster is linked to eight neighboring clusters—to each of two by three bpdc linkers and to the other six by pairs of linkers (Figure 22). Considering the multiple links between a given pair of SBUs as one edge and the clusters as 8-c vertices, the net is body-centered cubic (bcu). In this case, there is no obvious alternative description. Notice that the cluster is chiral and has ideal symmetry 32 (D3); the crystal symmetry is the achiral R3c, which means that clusters of both hands occur equally.

Figure 22. A MOF with an SBU with 18 points of extension and multiple links between pairs of SBUs. Each SBU is linked to eight others.

The second example has a related structure. Now the metal cluster SBU with stoichiometry Zn7O4(CO2)10 has 10 points of extension and is linked to six other SBUs—to four by double links and to two by single links.56 Again the double links may be abstracted as rectangles and the net described as the 5-c net fqr, as shown in Figure 21.

A third example is a structure in which the metal-containing SBU consists of 11 Cd atoms and which has 18 points of

4.7. Examples of Lower-Symmetry Metal-Containing SBUs

It was remarked earlier that the augmented net derived from an underlying net was not strictly deﬁned in a mathematical sense. For example, if one takes any polyhedron with eight vertices and links it to eight neighbors in the same way topologically as in bcu (the net of the body-centered cubic lattice), the derived net will have bcu as its underlying net. This is illustrated in Figure 23 using the net bcu-i, in which square antiprisms are linked in such a way. Clearly, the net is diﬀerent (5-c, symmetry I422) from bcu-a (4-c, symmetry Im3m), but the underlying net is the same.

Such examples are quite common in crystals. Figure 24 shows an example of an SBU that has points of extension at the vertices of a trigonal prism linked with the pcu topology by bent ditopic linkers. The SBU is in fact the same as that shown in Figure 13 and the authors correctly identiﬁed the underlying topology as pcu.46 The “augmented” 4-c net shown in Figure 24 has the RCSR symbol unp.

An example with a square antiprism replacing a cube is found in the structure of a cyanide with framework composition Fe2(H2O)4Mo(CN)8, with Fe bonded in a planar fashion to four

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