The Chemistry of Metal-Organic Frameworks: Synthesis, Characterization, and Applications

By Stefan Kaskel

Providing vital knowledge on the design and synthesis of specific metal-organic framework (MOF) classes as well as their properties, this ready reference summarizes the state of the art in chemistry.
Divided into four parts, the first begins with a basic introduction to typical cluster units or coordination geometries and provides examples of recent and advanced MOF structures and applications typical for the respective class. Part II covers recent progress in linker chemistries, while special MOF classes and morphology design are described in Part III. The fourth part deals with advanced characterization techniques, such as NMR, in situ studies, and modelling. A final unique feature is the inclusion of data sheets of commercially available MOFs in the appendix, enabling experts and newcomers to the field to select the appropriate MOF for a desired application.
A must-have reference for chemists, materials scientists, and engineers in academia and industry working in the field of catalysis, gas and water purification, energy storage, separation, and sensors.

• Language: English
• Publisher: Wiley
• Release date: Jun 14, 2016
• ISBN: 9783527693085

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List of Contributors

Mark D. Allendorf

Sandia National Laboratories

7011 East Avenue

Livermore

CA 94551-0969

USA

Debasis Banerjee

Rutgers University

Department of Chemistry and Chemical Biology

610 Taylor Road

Piscataway

NJ 08854

USA

Elisa Barea

Universidad de Granada

Departamento de Química Inorgánica

Av. Fuentenueva S/N

18071 Granada

Spain

Michael Beetz

University of Munich (LMU)

Department of Chemistry and Center for NanoScience (CeNS)

Butenandtstraße 11 (E)

81377 München

Germany

Volodymyr Bon

Technische Universität Dresden

Department of Inorganic Chemistry

Bergstraße 66

01062 Dresden

Germany

Francesca Bonino

University of Torino

Department of Chemistry

NIS and INSTM Reference Centers

Via Quarello 15

I10135 Torino

Italy

Silvia Bordiga

University of Torino

Department of Chemistry

NIS and INSTM Reference Centers

Via Quarello 15

I10135 Torino

Italy

Mathieu Bosch

Texas A&M University

Department of Chemistry

3255 TAMU

College Station

TX 77843

USA

Stephan I. Brückner

Technische Universität Dresden

Department of Chemistry

Institute of Inorganic Chemistry

Chair of Inorganic Chemistry I

Bergstr. 66

01062 Dresden

Germany

Eike Brunner

Technische Universität Dresden

Fachrichtung Chemie und Lebensmittelchemie

Bergstrasse 66

01062 Dresden

Germany

Jérôme Canivet

University Lyon 1

IRCELYON-CNRS

2 Avenue Albert Einstein

69626 Villeurbanne

France

Jong-San Chang

Research Center for Nanocatalysts

Korea Research Institute of Chemical Technology (KRICT)

141 Gajeong-ro

305-600 Yuseong-gu

Daejeon

Korea

and

Sungkyunkwan University

Department of Chemistry

2066 Seobu-ro

Gyenggido

440-476 Jangan-gu

Suwon-si

Korea

Xiao-Ming Chen

Sun Yat-Sen University

MOE Key Laboratory of Bioinorganic and Synthetic Chemistry

School of Chemistry and Chemical Engineering

135 XinGang Rd. W.

510275 Guangzhou

China

Katie A. Cychosz

Quantachrome Instruments

1900 Corporate Drive

Boynton Beach

FL 33426

USA

Benjamin J. Deibert

Rutgers University

Department of Chemistry and Chemical Biology

610 Taylor Road

Piscataway

NJ 08854

USA

Thomas Devic

Université de Versailles

St-Quentin en Yvelines

UMR CNRS 8180

Institut Lavoisier

45 Avenue des Etats-Unis

78035 Versailles Cedex

France

David Farrusseng

University Lyon 1

IRCELYON-CNRS

2 Avenue Albert Einstein

69626 Villeurbanne

France

Roland A. Fischer

Technical University Munich

Chair of Inorganic and Metal-Organic Chemistry

Lichtenbergstraße 4

85748 Garching

Germany

Michael Fröba

University of Hamburg

Department of Chemistry

Institute of Inorganic and Applied Chemistry

Martin-Luther-King-Platz 6

20146 Hamburg

Germany

Zhi Hua Fu

Fujian Institute of Research on the Structure of Matter

Chinese Academy of Sciences

State Key Laboratory of Structural Chemistry

155 Yangqiao Road West

350002 Fuzhou

PR China

Hiroyasu Furukawa

University of California–Berkeley

Department of Chemistry

Lawrence Berkeley National Laboratory

Materials Sciences Division

Berkeley

CA 94720

USA

and

Center of Research Excellence in Nanotechnology (CENT)

King Fahd University of Petroleum and Minerals

34464 Dhahran

Saudi Arabia

Hartmut Gliemann

Karlsruhe Institute of Technology (KIT)

Institute of Functional Interfaces (IFG)

Hermann-von-Helmholtz-Platz 1

76344 Eggenstein-Leopoldshafen

Germany

Guo Cong Guo

Fujian Institute of Research on the Structure of Matter

Chinese Academy of Sciences

State Key Laboratory of Structural Chemistry

155 Yangqiao Road West

350002 Fuzhou

PR China

Olesia Halbherr

Ruhr-University Bochum

Inorganic Chemistry II–Organometallics and Material Chemistry

Universitätsstrasse 150

44801 Bochum

Germany

Chun-Ting He

Sun Yat-Sen University

MOE Key Laboratory of Bioinorganic and Synthetic Chemistry

School of Chemistry and Chemical Engineering

135 XinGang Rd. W.

510275 Guangzhou

China

Lars Heinke

Karlsruhe Institute of Technology (KIT)

Institute of Functional Interfaces (IFG)

Hermann-von-Helmholtz-Platz 1

76344 Eggenstein-Leopoldshafen

Germany

Frank Hoffmann

University of Hamburg

Department of Chemistry

Institute of Inorganic and Applied Chemistry

Martin-Luther-King-Platz 6

20146 Hamburg

Germany

Young Kyu Hwang

Research Center for Nanocatalysts

Korea Research Institute of Chemical Technology (KRICT)

141 Gajeong-ro

305-600 Yuseong-gu

Daejeon

Korea

and

University of Science and Technology (UST)

Department of Green Chemistry

217 Gajeong-ro

305-350 Yuseong-gu

Daejeon

Korea

Stefan Kaskel

Technische Universität Dresden

Department of Chemistry

Institute of Inorganic Chemistry

Bergstr. 66

01062 Dresden

Germany

Christel Kutzscher

Technische Universität Dresden

Faculty of Science

Department of Chemistry and Food Chemistry

Inorganic Chemistry I

Bergstraße 66

01069 Dresden

Germany

Carlo Lamberti

University of Torino

Department of Chemistry

NIS and INSTM Reference Centers

Via Quarello 15

I10135 Torino

Italy

and

Southern Federal University

IRC ``Smart Materials''

Zorge Street 5

344090 Rostov-on-Don

Russia

U-Hwang Lee

Research Center for Nanocatalysts

Korea Research Institute of Chemical Technology (KRICT)

141 Gajeong-ro

305-600 Yuseong-gu

Daejeon

Korea

and

University of Science and Technology (UST)

Department of Green Chemistry

217 Gajeong-ro

305-350 Yuseong-gu

Daejeon

Korea

Alexandre Legrand

University Lyon 1

IRCELYON-CNRS

2 Avenue Albert Einstein

69626 Villeurbanne

France

Kirsty Leong

Sandia National Laboratories

7011 East Avenue, MS 9161

Livermore CA 94550

USA

Jing Li

Rutgers University

Department of Chemistry and Chemical Biology

610 Taylor Road

Piscataway

NJ 08854

USA

Pei-Qin Liao

Sun Yat-Sen University

MOE Key Laboratory of Bioinorganic and Synthetic Chemistry

School of Chemistry and Chemical Engineering

135 XinGang Rd. W.

510275 Guangzhou

China

Philip Llewellyn

Aix-Marseille University

MADIREL (UMR CNRS 7246)

Gas Storage and Separations Group

Centre de St. Jérôme

13397 Marseille Cedex 20

France

Guillaume Maurin

CNRS UM ENSCM Université Montpellier

ENSCM

Institut Charles Gerhardt Montpellier

Institut Universitaire de France

UMR 5253

Place E. Bataillon

34095 Montpellier

Cedex 05

France

Matthias Mendt

Leipzig University

Faculty of Physics and Earth Sciences

Institute of Experimental Physics II

Linnéstr. 5

04013 Leipzig

Germany

Franck Millange

Université de Versailles-St-Quentin-en-Yvelines

Département de Chimie

45 Avenue des états-Unis

78035 Versailles Cedex

France

Philipp Müller

Technische Universität Dresden

Faculty of Science

Department of Chemistry

Institute of Inorganic Chemistry I

Bergstraße 66

01062 Dresden

Germany

Klaus Müller-Buschbaum

Universität Würzburg

Institut für Anorganische Chemie

Am Hubland

97074 Würzburg

Germany

Jorge A. R. Navarro

Universidad de Granada

Departamento de Química Inorgánica

Av. Fuentenueva S/N

18071 Granada

Spain

Julia Pallmann

Technische Universität Dresden

Fachrichtung Chemie und Lebensmittelchemie

Bergstrasse 66

01062 Dresden

Germany

Andreas Pöppl

Leipzig University

Faculty of Physics and Earth Sciences

Institute of Experimental Physics II

Linnéstr. 5

04013 Leipzig

Germany

Silvia Raschke

Technische Universität Dresden

Faculty of Science

Department of Chemistry

Institute of Inorganic Chemistry I

Bergstraße 66

01062 Dresden

Germany

Helge Reinsch

Christian-Albrechts-Universität

Institut für Anorganische Chemie

Max-Eyth-Straße 2

24118 Kiel

Germany

L. Marleny Rodríguez-Albelo

Universidad de Granada

Departamento de Química Inorgánica

Av. Fuentenueva S/N

18071 Granada

Spain

Lars-Hendrik Schilling

Christian-Albrechts-Universität

Institut für Anorganische Chemie

Max-Eyth-Straße 2

24118 Kiel

Germany

Alexander Schoedel

University of California-Berkeley

Department of Chemistry

Materials Sciences Division

Lawrence Berkeley National Laboratory

and Kavli Energy NanoSciences Institute

602 Latimer Hall

Berkeley

CA 94720

USA

Irena Senkovska

Technische Universität Dresden

Department of Chemistry

Institute of Inorganic Chemistry I

Bergstraße 66

01062 Dresden

Germany

Christian Serre

Université de Versailles

St-Quentin en Yvelines

UMR CNRS 8180

Institut Lavoisier

45 Avenue des Etats-Unis

78035 Versailles Cedex

France

Mantas Šimėnas

Vilnius University

Faculty of Physics

Radiophysics department

Sauletekio 9

LT10222 Vilnius

Lithuania

Norbert Stock

Christian-Albrechts-Universität

Institut für Anorganische Chemie

Max-Eyth-Straße 2

24118 Kiel

Germany

Xixi Sun

University of California–Berkeley

Department of Chemistry

Lawrence Berkeley National Laboratory

Materials Sciences Division

Berkeley

CA 94720

USA

Matthias Thommes

Quantachrome Instruments

1900 Corporate Drive

Boynton Beach

FL 33426

USA

Pierre Tremouilhac

Karlsruhe Institute of Technology (KIT)

Institute of Functional Interfaces (IFG)

Hermann-von-Helmholtz-Platz 1

76344 Eggenstein-Leopoldshafen

Germany

Anil H. Valekar

Research Center for Nanocatalysts

Korea Research Institute of Chemical Technology (KRICT)

141 Gajeong-ro

305-600 Yuseong-gu

Daejeon

Korea

and

University of Science and Technology (UST)

Department of Green Chemistry

217 Gajeong-ro

305-350 Yuseong-gu

Daejeon

Korea

Richard I. Walton

University of Warwick

Department of Chemistry

Coventry CV4 7AL

UK

Guan E. Wang

Fujian Institute of Research on the Structure of Matter

Chinese Academy of Sciences

State Key Laboratory of Structural Chemistry

155 Yangqiao Road West

350002 Fuzhou

PR China

Hao Wang

Rutgers University

Department of Chemistry and Chemical Biology

610 Taylor Road

Piscataway

NJ 08854

USA

Christof Wöll

Karlsruhe Institute of Technology (KIT)

Institute of Functional Interfaces (IFG)

Hermann-von-Helmholtz-Platz 1

76344 Eggenstein-Leopoldshafen

Germany

Stefan Wuttke

University of Munich (LMU)

Department of Chemistry and Center for NanoScience (CeNS)

Butenandtstraße 11 (E)

81377 München

Germany

Gang Xu

Fujian Institute of Research on the Structure of Matter

Chinese Academy of Sciences

State Key Laboratory of Structural Chemistry

155 Yangqiao Road West

350002 Fuzhou

PR China

Omar M. Yaghi

University of California-Berkeley

Department of Chemistry

Materials Sciences Division

Lawrence Berkeley National Laboratory

and Kavli Energy NanoSciences Institute

602 Latimer Hall

Berkeley

CA 94720

USA

Ming Shui Yao

Fujian Institute of Research on the Structure of Matter

Chinese Academy of Sciences

State Key Laboratory of Structural Chemistry

155 Yangqiao Road West

350002 Fuzhou

PR China

Shuai Yuan

Texas A&M University

Department of Chemistry

3255 TAMU

College Station

TX 77843

USA

Ryan A. Zarkesh

Sandia National Laboratories

7011 East Avenue, MS 9161

Livermore, CA 94550

USA

Jie-Peng Zhang

Sun Yat-Sen University

MOE Key Laboratory of Bioinorganic and Synthetic Chemistry

School of Chemistry and Chemical Engineering

135 XinGang Rd. W.

510275 Guangzhou

China

Dong-Dong Zhou

Sun Yat-Sen University

MOE Key Laboratory of Bioinorganic and Synthetic Chemistry

School of Chemistry and Chemical Engineering

135 XinGang Rd. W.

510275 Guangzhou

China

Hong-Cai Zhou

Texas A& M University

Department of Chemistry

3255 TAMU

College Station

TX 77843

USA

Andreas Zimpel

University of Munich (LMU)

Department of Chemistry and Center for NanoScience (CeNS)

Butenandtstraße 11 (E)

81377 München

Germany

Chapter 1

Introduction

Stefan Kaskel

In 2002, Ferdi Schüth asked me to write a review chapter on metal–organic frameworks (MOFs) for the handbook of porous solids. Most of the MOFs known at that time would easily fit in one chapter of a book, and even some materials with relatively low porosity could be discussed. What a tremendous progress after a little more than one decade! By 2016 one could fill an encyclopedia with MOF structures alone. Significant progress has not only led to the discovery of materials with record surface areas exceeding that of traditional adsorbents. Major hurdles such as poor hydrothermal stability of the early MOFs have been overcome due to the development of highly stable aluminum and zirconium MOFs. Nowadays more than 10 000 MOFs are known, and MOF databases of hypothetical and real MOFs are being developed for managing the enormous amounts of data. They also give access to search more easily for desired properties, and MOFs with their tailorable pore size and functionality seem to be ideal for the computational design of novel functional materials. In that sense, MOFs are certainly a prototypical materials genome test case, as relation between structure and function is readily computed.

Due to the enormous number of MOF structures published today, often the question is asked: Do we need new MOFs? The answer is a question of imagination and creativity! The most exciting research nowadays addresses challenges imagining properties not foreseen 10 years ago. A good example is electronics. For device integration, a key target is the development of electron-conducting MOFs with specific response combining porosity and semiconducting behavior. The latter will enable sensor integration for various control operations. But also ion conductivity is important for fuel cell membranes and other physical properties become relevant, thus fostering the search for ever new materials with enhanced or customized performance.

MOFs have also contributed significantly to the fundamental understanding in the field of porous materials, adsorption phenomena, separation technology, and catalysis. Especially flexibility (switchability) of some MOF materials is a unique feature not observed in other porous solids.

Innovation cycles in materials science may take 20 years or even longer. In the case of polyacrylonitrile (PAN)-based carbon fibers, after their discovery in the early 1960s in Japan, it took several innovation cycles, and only today's huge demand in energy savings resulting from the use of lightweight composites changed the scene to a situation where carbon fibre (CF) composites are ubiquitous in airplanes and enter even the car manufacturing sector.

As for any new material, prices are high at an early discovery stage due to the economy of scale. However, such high prices inhibit innovation severely since industry will not consider their use as long as they are not cost competitive. In general the limited availability of even kilogram MOF amounts is an innovation blocker because prototype demonstration requires often a few kilogram material. In this context, start-ups and research institutes offering MOFs for demonstration purposes play a key role. Currently there are already a few sources available offering kilogram MOF samples such as MOF Technologies, metal-organic-frameworks.eu, and ImmondoTech, while some smaller companies offer development services (MOFapps, NuMat, framergy). The biggest company involved in MOF commercialization from early on is BASF. Several larger companies in the United States and Japan have considered MOF commercialization but seem to be waiting how the market develops and for the early patents to run out. For larger-scale production regulatory issues such as REACH, EPA-regulations, toxic substances control act (TSCA), ASEAN regulations, and so on, need to be addressed, an issue that seems impossible to be handled by small and medium sized enterprises (SMEs).

Roadmaps can help to identify promising development targets, and expert workshops are a useful instrument to match technology push with market needs by bringing experts from research institutes and industry together. The outcome of such an expert workshop held late 2013 was published early 2015 as a Roadmap by DECHEMA (Roadmap MOF). However, industrial application targets are changing from year to year.

The chemistry of MOFs as the underlying technological basis has now reached a mature standard, and the book may serve as a basic reference describing the most important compositions, technologies, analysis techniques, and basic properties.

The aim of this book was to organize MOF compounds according to their main components: (i) clusters (or better multinuclear complexes) and (ii) linker systems. Following a more general structural view of networks and their topologies, PART I largely follows the organization of metals in the periodic table, without aiming at a classification since chemical similarity within the periodic table generates fluent passage between certain groups nor aiming for an exhaustive description of all known structures or metals used.

Essential functions in MOFs are generated nowadays by specific functional groups in the linkers, as outlined in PART II. This is not only relevant to achieve the highest degree of porosity by rigid and extended linker design but becomes especially important when it comes to optical or electronic functionality such as fluorescence or charge transport. But also catalysis and separation profit from the modular integration of enantioselective or other functional groups introducing a high degree of selectivity.

Since shaping of MOFs into granules, monoliths, and nanostructures is a prerequisite for any application, PART III addresses those topics specifically. While the chapter on nanoparticles is mostly addressing modern medicinal targets, the chapters on shaping on a macroscale and thin films mostly address technical applications such as heat storage and electronic integration.

The progress in specific characterization methods for MOFs outlined in PART IV is impressive, illuminating long-range structural features, local structural features, reactivity of open sites, and the importance of defect chemistry. Especially the development of in situ monitoring techniques in combination with spectroscopies and diffraction gives deep insight into flexibility phenomena and crystallization mechanisms, emphasizing that crystallization is a much more complex phenomenon, and more needs to be understood to achieve a truly rational design of MOFs and other networks. The importance of theoretical modeling and prediction is also pointed out in a chapter by Maurin included in this section.

The appendix contains brief descriptions of a few MOFs that are commercially available, since they are either relatively easy to upscale and/or highly stable chemically. The information summarized shall be a basis for industrial users who are interested to test MOFs in applications as a first choice testing set. This somewhat arbitrary selection is based on the editor experience. Further information is available on request.

We hope the book will be a good starting point for readers as an introduction into MOF chemistry and for experts as a standard reference book.

Chapter 2

Network Topology

Frank Hoffmann and Michael Fröba

2.1 Introduction

More and more frequently one can find publications in the area of metal–organic frameworks (MOFs)/coordination polymers (CPs) that are enriched with topological considerations, that is, descriptions of the structure not only in classical chemical terms (atoms connected by bonds, crystal structure) but also in terms of (periodic) networks. This follows the concept that on a slightly more abstract level, the chemical species involved do build – or can at least be regarded as – a special kind of graph, that is, nodes/vertices that are connected by lines/edges. However, also the reverse is true: Based on this concept more and more papers emphasize the (crystal) engineering aspect, claiming that a particular framework structure was intellectually anticipated and accordingly designed by assembling appropriate (supra)molecular building blocks with a certain connectivity that give a specific net. This approach is known as reticular¹ chemistry and was introduced by Yaghi and coworkers for MOFs already 12 years ago [1]. However, the concept of the description and systematization of classical crystal structures as nets is, of course, known for a much longer time, and its foundations had been led by the impressive work of A.F. Wells, beginning with a two-part paper in Acta Crystallographica from 1954 [2]. His ongoing studies on this subject were summarized in his book Three-dimensional Nets and Polyhedra published in 1977 [3], and a short version has also been included as Chapter 3, Polyhedra and nets, in his classical textbook Structural Inorganic Chemistry [4]. A related monograph from 1996 titled Crystal Structures: I. Patterns and Symmetry by O'Keeffe and Hyde should also be mentioned in this context [5]. A book that accounted for the more recent developments in the area of molecular networks has been published by Öhrström and Larsson [6].

With the boost in the area of periodic scaffold- or framework-like materials (MOFs, COFs, ZIFs, supramolecular H-bonded networks, etc.) over the last 20 years, additional effort has been invested into developing a respective solid theoretical foundation of the description of crystals as nets, allowing at the same time a more systematic enumeration of nets. This theoretical base belongs to the truly very complicated mathematical disciplines algebraic topology and graph theory. It is neither intended nor possible to give an appropriate introduction into these fields in the context of this chapter. Actually, if a topologist was asked to describe the mathematical nature of crystal structures, his/her answer could be that topologically viewed crystals are infinite-fold abelian covering graphs over finite graphs and that crystal nets are their periodic realizations, to cite from the introduction chapter of Sunada's monograph Topological Crystallography [7] – and probably hardly any chemist would understand that. Therefore, the aim of the present chapter is rather to get the reader acquainted with all relevant aspects and terms that are present in the respective literature, in which MOFs and related compounds are treated as nets. It should serve as a non-mathematical entry point for everyone who would like to be able to understand the very basic concepts behind this network-like view on this fascinating class of materials. Introductions into network topology with a very similar scope can also be found in several chapters of books dealing with crystal design [8, 9], CPs [10], and MOFs [11].

2.2 Crystal Structures and MOFs Regarded as Nets

A first question coming to a chemist's mind, who is not a specialist in the field of structural chemistry of the solid state when confronted with the description of crystal structures and MOFs as nets, is: Why should we do this? Isn't it sufficient to specify the crystal structure, that is, the space group, the unit cell dimensions, and the fractional coordinates and type of all atoms? Well, on the one hand, the answer is yes. But this is only one of several ways of describing a structure, and there are plenty of other possibilities to do this, all of which have certain advantages and disadvantages. The specific representation of a structure is dependent on the concrete aspect we want to emphasize, and this, in turn, is driven mostly by a certain motivation, which has always been present in structural chemistry: the establishment of structural categories, the identification of prototypical structures, and the classification of structures.

The structure of classical, relatively simple and compact inorganic and mostly ionic solid-state compounds like rock salt, fluorite, and so on is usually described in terms of two competing but otherwise complementing concepts. One is based on (densest) sphere or rod packings, the other on (mostly regular or only slightly distorted) corner-, edge-, or face-connected coordination polyhedra. However, it should be clear that with an increasing (i) level of irregularity, (ii) amount of covalent or H-bonded species that are involved in the compound, and (iii) degree of porosity, these concepts are becoming less applicable. The reader could think of zeolites: The basic building blocks are Al/SiO4 tetrahedra, and all tetrahedra are exclusively corner-connected to each other. Clearly, this description is far away from being sufficient to describe all the different zeolite types, which are hitherto known or which should be possible in theory (see the Hypothetical Zeolite Database [12]). Zeolites are, of course, characterized by their channels and cages, which are built by rings of different sizes. With the description of the size of the rings and the manner in which the different types of rings are spatially connected, one is already deeply immersed in the domain of network topology, even if one is actually unaware of this circumstance. In this sense, network topology is virtually a quasi-natural approach of describing and systematizing certain classes of chemical compounds. This is, in particular, comprehensible for MOFs, being composed of two different building blocks, the inorganic metal or metal oxide clusters (the secondary building units, SBUs), and the organic linkers, which are covalently bonded to each other, showing degrees of free volume, which never have been achieved before. The deconstruction of the myriad of MOFs to an array of connected points and subsequently into fundamental types of different nets is very similar to the reduction of the abundance of inorganic compounds onto certain structure types. This chapter aims at explaining how nets can be described and what language network topologists use for this purpose.

2.3 Some Introductory Remarks about Graphs, Topology, and Symmetry

Networks, or short: nets, are not only frequently used by chemists but are at the same time omnipresent in everyday life: railway networks, electronic circuits, wireless local area networks, neural networks, social networks, and so on. The common feature of all these nets is that there are some objects that can be identified as nodes and that these nodes have connections to other nodes. These kinds of interconnected objects can be represented as a special kind of graph, which in turn is a mathematical object, which consists of abstractions called vertices connected by edges.² Generally one discriminates between various classes of graphs which are characterized by certain properties, for instance, a graph can have loops (vertices that are connected to themselves) or a subset of vertices of a graph can have multiple connections to other vertices and it may have "loose ends" or – like in a flow diagram – it may have a certain direction, and there are finite and infinite graphs (see Figure 2.1).

c02fgz001

Figure 2.1 Some examples of different kinds of graphs. (a) A finite, non-simple graph with directed edges, a loose end, and a loop; (b) a finite, non-simple graph with multiple connections between some vertices, and (c) an infinite, 2-periodic, simple graph; the black line indicates the repeating unit cell.

If we consider crystalline MOFs as nets, we are exclusively faced with the so-called simple, infinite, and 2- or 3-periodic graphs, and this means that the abstraction process leads to graphs, which contain no loops and no loose ends, are non-directional, have no multiple connections between vertices, and have translational symmetry in two or three spatial directions. Note the distinction between dimensional and periodic: A spatially extended object like a cube in real space is per se a 3D object, but it is not a periodic object.

The very first step of considering an MOF as a net consists of the identification of its vertices, that is, breaking down the MOF into its fundamental building blocks, which then are regarded as vertices with a certain connectivity. For this process O'Keeffe and Yaghi coined the term deconstruction of MOFs [13, 14]. While this simplification process is not in every case unique, they gave reasonable recommendations how to proceed in cases of uncertainty (see also Section 2.7). The outcome of this process are the SBUs or tertiary building units (TBUs), which often can be described as polygons or polyhedra, with their corners being connection points to other entities (so-called points of extension),³ and the organic linkers, likewise with a number of certain branching points of a certain connectivity. The connectivity of the linker is referred to as the topicity. The linker can have two, three, four, and more connections to other units and is therefore classified as being a ditopic linker, tritopic linker, tetratopic linker, and so on. Note that a ditopic, non-branched linker is topologically viewed not as a set of two vertices but only as an edge. Here, we already linguistically touch the area of topology for the first time.

The next step consists of the description of the net, that is, describing in which way the different kinds of vertices are connected to each other: How many different kinds of vertices/edges are there, what do we actually mean by different kind? What is the coordination number (CN) of the vertices, which types of vertices are bonded to which other types? For convenience, throughout this chapter we will always use the net identifiers of the Reticular Chemistry Structure Resource (RCSR) [15], that is, the three-letter codes, typeset in bold like dia, which allow to unambiguously refer to a unique net (for full details, see below). This is also in accordance with the recently published IUPAC recommendations concerning the terminology of MOFs and CPs [16].

In order to characterize a net, it is not sufficient to specify the coordination number. This is illustrated in Figure 2.2a, where three geometrically different but topologically identical 2-periodic nets (hcb) with three-coordinated⁴ (3-c) vertices are shown. It is likewise not sufficient to specify which vertices are connected to which other vertices: Two graphs can be identical (isomorphic) while being topologically different as shown in Figure 2.2b.

c02fgz002

Figure 2.2 (a) Three representations of a 3-c net that are geometrically different but topologically identical, because they can be transformed into each other without breaking (and reconnecting) edges. (b) Three identical graphs in which the vertices have identical connections to other vertices (1 is always connected to 2 and 12 and so on) but that have different topological realizations.

Topology (from Greek τóπoς, place, and λóγoς, study) is a mathematical discipline concerned with the properties of space or objects in space that are preserved under continuous deformations including stretching and bending but not tearing or gluing. Topologically it does not matter if two connected vertices have distances to each other of 1 Å or 1000 miles; it does not matter either if the neighboring vertices are symmetrically arranged or if they form a completely distorted coordination figure. As long as two concrete arrangements of vertices can be continuously deformed into another, without the need of breaking and reconnecting bonds, they are topologically identical – as in the example of Figure 2.2a.

This leads to two further questions which are strongly related to each other: (i) How should we represent a certain net, and (ii) on which basis can we answer the question if two nets are identical, that is, isoreticular? Answering the latter question is the crucial step, before a proper classification of nets (sorting the nets according to their different types) can be carried out. Topologically, the vertices do not have a concrete place in space at all; they are abstract, infinitely small objects, which can be placed at an arbitrary spot in Euclidean space, as long as the connection scheme between the other vertices remains the same. The answer is that we should consider all nets in their most symmetrical form, that is, if we assign the vertices concrete coordinates in Euclidean space, we should do this in a way that the smallest number of symmetrically inequivalent vertices and also the smallest number of symmetrically inequivalent edges are used to realize a map of that net, on the constraint that the connection scheme remains intact. Crystallographically speaking, the vertices and midpoint of the edges should be placed at coordinates with a maximum site symmetry. And different kinds of vertices/edges mean that they are not symmetry-related to each other. The procedure to assign coordinates to the vertices (and hence edges) of an abstract graph is referred to as an embedding or also a realization – vertices and edges (without physical properties) become nodes and links (with physical properties). If we now refer to Figure 2.2a again, we see that all vertices in each representation are 3-c, and we see also that the ring sizes (a proper definition of the term ring will follow) in all three cases are identical (six-membered rings). However, if we consider the symmetry, we immediately see that the left-hand representation of the net is the realization with the highest symmetry (compare, for instance, the number of vertices in the unit cell). The representations in the middle and on the right are embeddings of lower symmetry.

The process of maximizing the symmetry of a spatial arrangement of interconnected vertices as well as the characterization of the vertices and edges – which will be the topic of the next section – and in turn the classification of the net is not possible without using modern software packages. Two of the most widely used and equipped with the richest capabilities should be mentioned already here: Systre, which was developed by Olaf Delgado-Friedrichs and that is part of the wider GAVROG package [17], and ToposPro (formerly TOPOS 4.0) by Blatov, Shevchenko, and Proserpio [18].

2.3.1 Interplay of Local Geometry and Topology

In the first instance vertices of a certain coordination number are topologically identical whatsoever their geometrical coordination geometry to their neighboring vertices might look like. For instance, both a tetrahedrally and a square-planar coordinated vertex are topologically viewed simply 4-c vertices. But how is it then possible that two arrangements of vertices (nets) as a whole can have different topologies even if they were built from topologically identical vertices? In the RCSR database alone, more than 180(!) different nets are listed which are built by only 4-c vertices. Although the local geometry of the single vertices involved in the net is not a topological feature per se, it is exactly this geometry which gives rise to the different possibilities to arrange a set of interconnected vertices in space, which are topologically distinguishable (cannot be transformed into each other without breaking bonds). It is the interplay of the local geometry of the topological neighborhood of the vertices together with their relative orientation to each other that leads to globally different topological arrangements of the vertices (see Figure 2.2b). It is obvious that different nets result if we take two vertices with different geometrical environments but identical coordination numbers, for instance, in the one case only tetrahedrally coordinated and in the other case only square-planar coordinated vertices (see Figure 2.3a,b).

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Figure 2.3 Three different ways of building 3-periodic nets with uninodal 4-c vertices only: (a) the dia net, with tetrahedrally coordinated vertices, (b) the nbo net, and (c) the lvt net, the latter two with square-planar coordinated vertices; nbo and lvt are represented in their augmented versions (-a).

But even if we consider only one distinct type of coordination surroundings, it is still possible to construct different nets: Take the two nets of Figure 2.3b,c, in which all 4-c vertices are square-planar coordinated – respectively, represented as filled solid squares⁵; they both are 4-c uninodal nets, meaning that there is only one kind of vertices present, which are all symmetry-related to each other. In which way they are topologically different – that will be the topic of the following sections. Clearly and consequently, we need additional descriptors that characterize the vertices of a net and the net itself (and its topology) in an unambiguous way. Network topology in its notion of analyzing nets can be regarded as a way to infer how different sorts of coordination polyhedra are connected to each other in a given net. Its reverse engineering aspect also strongly relies on the knowledge of how nets of a certain topology can be constructed by putting together the respective appropriate building units and hence is important for future targeted design and synthesis of new MOFs.

2.4 Nomenclature of and Symbols for Nets – or What Does 4.4.4.4.4.4.4.4.4.4.4.4.*.*.* Mean?

Unfortunately, in the present literature of chemistry-related topological descriptions, a great variety of different terms, names, and symbols are used for the characterization of vertices/faces in nets, tiles, and polyhedra. These include Schläfli symbols, extended or long Schläfli symbols, circuit symbols, point symbols (PS), vertex symbols (VS), vertex figures, face symbols, the Delaney (or also Delaney–Dress) symbols, and the already mentioned three-letter codes of the RCSR database. What makes it confusing is that some of these symbols refer to the same thing but have different underlying concepts or recipes for deriving the respective symbol, while in other instances the different symbols refer to different things! Both these circumstances contribute to the present confusion in this area. Therefore it seems to be helpful to shortly recapitulate how the most important descriptors are derived and to which objects they are appropriately applied; thereby, we will follow the recommendations given by Blatov et al. [19]. At the end of this section, a table with an overview of the different symbols is given.

2.4.1 Schläfli Symbols

Often the terms Schläfli symbol and PS (see in the following) are used synonymously, but they do not refer to the same thing. The Schläfli symbol is a symbol of the form {p,q,r,…} that is used to describe regular polygons, polyhedra (including starlike bodies), and their higher-dimensional counterparts (which we will not cover here). In strict mathematical terms a Schläfli symbol with n entries refers to tilings or tessellations of an n-dimensional space. This sounds rather abstract; however with the following examples, their meaning should become clear:

A one-dimensional regular tile is nothing more than a regular p-sided polygon (regular means that all edges have the same length and are symmetrically placed about a common center); this means that {3}, {4}, {5}, {6}, … describe an equilateral triangle, a square, a regular pentagon, a regular hexagon, and so on.

Schläfli symbols with two entries {p,q} are used to describe either regular polyhedra (made of regular polygons) or regular 2D tilings, that is, tilings of the plane. While p stands again for a regular polygon with p edges, q expresses how many polygons meet at each corner/vertex. There are only five regular polyhedra, which are also called Platonic solids, and the names and Schläfli symbols are as follows: tetrahedron {3,3}, octahedron {3,4}, cube {4,3}, icosahedron {3,5}, and finally, the dodecahedron {5,3}. To give also an example in the form of its translation to everyday language: A cube is a polyhedron composed of squares, where three squares meet at each corner. Now it should be easy to transfer this concept to tilings of the plane. There are only three regular tilings: A floor of a bathroom covered with square tiles would be described as {4,4} (sql = square lattice), a pavement in a pedestrian area covered with six-sided stones can be characterized with the Schläfli symbol {6,3} (hcb = honeycomb), and finally, a wallpaper made of regular triangles has the Schläfli symbol {3,6} (hxl = hexagonal) (see Figure 2.4).

Schläfli symbols with three entries {p,q,r} refer to tilings of the 3D space, and it gives the answer to the question how many regular polyhedra of type {p,q} meet at each edge. The most descriptive example is probably the tiling of an ensemble of joined cubes {4,3}, where four of such cubes meet at every edge, that is, {4,3,4}, the tiling of the net pcu (see again Figure 2.4).

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Figure 2.4 The five Platonic solids (a), the three 2-periodic nets sql, hcb, and hxl as well as the tiling of the pcu net (b), together with their respective Schläfli symbols.

Note that the main underlying conceptual object of Schläfli symbols is in all cases faces, be it a polygon, be it a face of a polyhedron, or be it a tile of a 2- or 3-periodic tiling. Note further that Schläfli symbols can (and should) only be used for regular objects.

2.4.2 Vertex Symbols

In contrast to Schläfli symbols, VS designate topological descriptions of vertices. They can be applied to the vertices of polyhedra, and in fact not only regular ones, and to those of 2- and 3-periodic nets of arbitrary complexity. Unfortunately, the notation system for polyhedra and 2-periodic nets, on the one hand, and 3-periodic nets, on the other, are slightly different, and for 3-periodic nets it is additionally dependent on the coordination number of the vertices. These circumstances probably account partially for the confusion in the usage and reception of net notations.

The general recipe in order to achieve the symbol for a given vertex is to look at all possible angles – this means pairs of two edges that meet at a corner (for an n-coordinated node there are n angles for polyhedra and 2-periodic nets and n(n − 1)/2 such angles for 3-periodic nets) – and inspect in which shortest rings (for the difference between a ring and a cycle, see below) the vertex is involved and specify the ring size, that is, the number of vertices in the ring. Do this for all angles, and separate each ring-size specification with a point . So, the general VS has the form a.b.c.d…, and if rings of the same size appear directly consecutively, they can be grouped with the help of a superscript index, giving an.bm.co.dp. This procedure has to be applied for each type of vertex present.

One question remains: In which order do we inspect the angles? For polyhedra and 2-periodic nets (which, obviously, both have only one ring per angle), we do this in cyclic order, meaning that we begin with one angle, which is involved in the smallest ring, and then we follow a steady direction (clock- or counterclockwise) and determine the ring sizes of the other angles. The direction has to be chosen in such a way that a sequence of smallest possible ring sizes results (see also Figure 2.5).

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Figure 2.5 The vertex symbols of (a) a cuboctahedron, (b) a square pyramid, and (c) the 2-periodic tetranodal (3,3,3,3)-c net hnc. Note that the VS 5.5.7 (not 5.7.5) is the result of the cyclic order, which ensures that the direction with a sequence of smallest possible numbers is chosen.

Again, some examples should be given for the illustration of this concept. All vertices of a cube have the VS 4.4.4 = 4³ and all vertices of the octahedron 3.3.3.3 = 3⁴, and for a cuboctahedron (see Figure 2.5a) the descriptor is 3.4.3.4 (not 4.3.4.3!). This concept can, of course, also be applied to objects with more than one kind of vertex, for instance, the apical vertex at the tip of a square pyramid has the symbol 3.3.3.3 = 3⁴, while the four vertices and the ground plane have VS 3.3.4 = 3².4, which in total gives in a stoichiometric-like notation (3⁴)(3².4)4 (Figure 2.5b). The beautiful 2-periodic tiling hnc with 5- and 7-rings has four different vertices, three of them with the VS 5.7.7 and one with 5.5.7 (Figure 2.5c).

If we now switch to 3-periodic nets, we have to be aware of the following three facts:

First, an angle can be involved in more than one ring. For example, the vertices of the dia net are involved in 6-rings, and for each angle there are two such 6-rings (Figure 2.6a). Rings of the same size at the same angle are indicated with a subscript index, that is, for the striated red-gray angle in Figure 2.6a, it gives 62. However, in contrast to VS of polyhedra and 2-periodic tilings, angles are never grouped with an additional superscript index, and all angles are separated with a point ..

Second, for vertices with a connectivity higher than three, a cyclic order is no longer possible, and the question arises, in which order the angles should be inspected. For four-coordinated vertices O'Keeffe [20] proposed to group the six angles in three pairs of opposite angles, that is, those without a common edge (see Figure 2.6b). The three pairs of two angles and the rings in which they are involved in the dia net are highlighted in Figure 2.7; the VS is 62.62.62.62.62.62. If an angle is part of rings of different sizes, then for each pair of angles, the smaller ring size is specified first. For instance, the four-coordinated vertices of the zeolite net lta have the VS 4.6.4.6.4.8 (see Figure 2.10).

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Figure 2.6 In 3-periodic nets an angle can be involved in more than one ring. (a) The red–gray striated angle of the depicted section of the dia net is involved in two six-membered rings simultaneously, highlighted in red. (b) The six angles of 4-c vertices are grouped in three pairs of opposite angles (those that do not share an edge).

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Figure 2.7 The three pairs of opposite angles together with their rings in which they are involved in the dia net. For each angle there are two rings that meet at one vertex.

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Figure 2.10 The tiling of Zeolite A (lta), which is composed of double 4-rings (green; face symbol [4⁶]), truncated octahedra (orange; face symbol [4⁶.6⁸]), and truncated cuboctahedra (blue; face symbol [4¹².6⁸.8⁶]).

The angles of five- and higher-coordinated nodes cannot be reasonably grouped anymore, so the VS is written according to increasing ring sizes and increasing subscripts. For instance, the VS of the six-coordinated vertices (which means that there are already 15 angles) of the uninodal net hxg is 4.4.4.4.4.4.64.64.64.66.66.66.66.66.66.

Third, sometimes for certain angles of a given vertex, no rings can be found. Take a look at the six-coordinated vertices of the pcu net: There are twelve 4-rings (four for each of the three perpendicular planes; see Figure 2.8a–c); however, the three pairs of edges with angles of 180° are not involved in rings. Of course, it is possible to find a cycle (a closed or cyclic path of arbitrary length) for these angles, but these are not rings, because there is a shortcut (of the length of exactly one edge) to the home vertex present. This is exemplified for one angle (edge a,b) in Figure 2.8d. Rings – also called fundamental circuits – are closed paths without such a shortcut. For a more formal description of the difference between cycles, rings, and the so-called strong rings, see [21]. The presence of an angle, which is not involved in a ring is denoted with an asterisk *, so the full VS of the vertices of the pcu net is 4.4.4.4.4.4.4.4.4.4.4.4.*.*.*.

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Figure 2.8 Part of the pcu net. (a–c) The 12 four-membered rings that meet at the orange vertex. For the three 180° angles of that vertex, there are no shortest rings, as exemplified for one 180° angle (edge a,b) in (d) because the purple 6-cycle contain a potential shortcut to the home vertex.

In nets with six-coordinated nodes, we have to specify already 15 angles, and it is easy to see that VS can become quickly cumbersome for nodes with even higher connectivity. Therefore, it was recommended to use VS not for vertices with coordination number of 7 and higher [19].

Are VS unique? Unfortunately, this is not the case. For instance, the vertices of the 4-c vertices of both the dia and lon nets have the VS 62.62.62.62.62.62. This means that the specification of the VS of a net is not sufficient in order to specify a certain topology, and we need additional descriptors, such as the coordination sequence (see below).

2.4.3 Point Symbols

Point symbols were introduced by Wells for non-uniform 3-periodic nets, that is, those nets in which not only rings of the same size meet at all vertices. The basic principle of deriving PS is very similar to that of VS, namely, that we again inspect all angles of a given vertex. However, the difference between VS and PS is that in VS (shortest) rings are specified for the angles of a vertex, but PS specifies shortest cycles instead. This means, we look for each pair of edges around a vertex for the shortest possible cyclic paths, even if there are shortcuts to the home vertex present. If we take up the example of the pcu net of Figure 2.8d, then the purple six-membered cycle is indeed the shortest cycle for the angle of edge pair (a,b). The 12 four-membered rings in Figure 2.8a–c are also shortest cycles, and as there are three 180° angles, the PS is therefore 4¹².6³. The PS is sorted according to increasing cycle sizes and indicate with a superscript index how many times a cycle with a certain size is present: an.bm.co…, with a ≤ b ≤ c; again, different cycle sizes are separated with a point. In comparison to VS, PS are much more compact for vertices with high coordination numbers, due to this grouping of cycles with the same size, and this is also why some publications report only PS.

A good example to illustrate the difference between vertex and PSs is the 2-periodic net fxt, which has the VS 4.6.12 but the PS 4.6.8 (see Figure 2.9). The angle (b,c) is involved in a 12-membered ring, but if we are free to choose a closed path of shortest length to get back to the home vertex, then we see that there is the shorter eight-membered cycle, shown in green.

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Figure 2.9 Part of the uninodal 2-periodic net fxt. The vertex symbol is based on shortest rings (a), which gives 4.6.12, while the point symbol is derived from shortest cycles; for the angle comprising edge b,c, the shortest possible closed cyclic path is an eight-membered cycle – shown in green; therefore the point symbol is 4.6.8 (b).

2.4.4 Face Symbols

For polyhedra and cages covered by tiles, the face symbol describes the number of a certain polygon type that is present and specifies of how many vertices these polygons are made up. Polygons/tiles which have the same number of vertices can be again grouped by using a superscript index. In order to prevent danger of confusion with PS or VS, it is recommended to place the face symbol in square brackets. So a face symbol in the form [an.bm.co…] means that the object is made up of n pieces of faces/tiles that are a-sided faces, m faces that are b-sided faces, and o faces that are c-sided faces, with a < b < c… An octahedron has the face symbol [3⁸], a cube and also the 3-periodic tiling of the net pcu has the face symbol [4⁶], and a cuboctahedron has the face symbol [3⁸.4⁶]. For objects that are made of two or more types of cages, the ratio of their occurrence can be included in a stoichiometric-like notation: Zeolite A (Linde Type A, lta) is composed of double 4-rings (cubes), β- or sodalite cages (truncated octahedra), and α- or supercages (truncated cuboctahedra) in the ratio 3:1:1 giving the face symbol [4⁶]3[4⁶.6⁸][4¹².6⁸.8⁶] (see Figure 2.10). In the software package ToposPro, this symbol would be written as 3[4^6] + [4^6.6^8] + [4^12.6^8.8^6].

2.4.5 RCSR Symbols

The RCSR [15] is a database, which is the result of a highly admirable venture⁶ to collect and enumerate nets of MOFs (and other much longer known classical crystalline phases). At the date of writing this chapter (February 2015), it contained an incredible number of 2205 different 3-periodic nets, 33 different layers (2-periodic nets), and 47 polyhedra. For each net a single and unique three-letter code is assigned, for which the respective practice of the database of zeolite structures (like SOD for sodalite, FAU for faujasite, MOR for mordenite) based on the Atlas of Zeolite Framework Types [22] of the International Zeolite Association (IZA) was the antetype. The only formal difference is that the framework types of the zeolites are typeset in upper case, while the codes of the RCSR are always in lower case and typeset in bold. Some of the three-letter codes of the nets and layers of the RCSR database are abbreviations either of a structure type or mineral name (like dia = diamond, pcu = primitive cubic, sql = square lattice, qtz = quartz, ant = anatase, tbo = twisted boracite), some are a short or contracted form of the formula of chemical compounds in which one or more atom sorts build this specific net (like cds = the net of the cadmium and sulfur atoms in CdSO4, crb = the net of boron atoms in CrB4, srs = the net of the silicon atoms in SrSi2, ubt = UB-twelve = UB12, pts = the net of the platinum and sulfur atoms in PtS), but most of the codes are randomly assigned and do not refer to any particular word or structure type.

As these three-letter codes are unique, they are sufficient to describe a certain net (and its topology). These RCSR symbols work as identifiers for nets, and their underlying structural and topological attributes are listed for each entry in the RCSR database, for which of the most important ones we will give a short explanation in the next section. This database has a user-friendly web-based graphical user interface (GUI), and since its last update it is also incredibly fast.

We want to close this section with a table-like overview of the different symbols we discussed so far (Table 2.1).

Table 2.1 Overview of the most important and common symbols for the characterization of polyhedra and 2- and 3-periodic nets/tilings

Adapted and modified from [19].

2.5 Characterization of Nets in the Spirit of the RCSR

In this section we want to explain the most important specifications, which are listed under a certain net in the RCSR database. For most of them short explanations can also be found on the About page of the RCSR [15b]. The following screenshot shows a part of the entries for the net dia (Figure 2.11), and the reader is encouraged to access the web page when reading the following explanations.

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Figure 2.11 Screenshot of the dia entry of the RCSR.

2.5.1 Pictures, Names, Keywords, and References

For many nets right under the symbol, a pictorial representation in various styles (for instance, as ball and stick or tiling) is depicted, followed by a direct access link to the respective web page. If the three-letter code is derived from a name, it is given (here diamond), and sometimes other names that refer to the same net are listed (here D for the so-called D minimal surface and the respective Fischer symbol, which we will not explain here). A line with keywords and a literature reference may follow.

2.5.2 Embedding and Realization: How to Assign Coordinates to Vertices?

The next three tables specify properties, which are related to two things: (i) the difference between an abstract, mathematical graph, and a real object with a certain extension made up of building units with certain distances to each other and so on and (ii) which guidelines should be followed, if we are forced to assign real-space coordinates to the vertices (which represent either a linker or an SBU/SBB) that are specifically connected by edges. The latter procedure is referred to as an embedding (see also Section 2.1). Remember that the length of the edges between the vertices plays no role in determining whether two graphs are topologically identical (strictly speaking, an edge does not have a property like length). Furthermore, a graph also has a priori no symmetry (at least not in the classical crystallographic sense). In simple terms, the assignment of vertices to real (fractional) coordinates is carried out according to the following four basic principles: (i) all edges should be equal in length and have the value of 1 (in units of the lattice parameters), (ii) the shortest intervertex distances should correspond to edges (bonds!), (iii) the realization should result in a structure with maximum symmetry, and (iv) the volume of the unit cell should be maximized, subject to the constraint of equal edge lengths, which means – vice versa – that the density (vertices per volume unit) should be minimized. There are many cases for which the fulfillment of all criteria simultaneously is not possible, and certain compromises have to be made. This is expressed as the embed type, and a further explanation of these types can be found at the About page of the RCSR and in [23]. According to these criteria the crystallographic specifications are defined: the space group (always origin 2 according to the International Tables for Crystallography is chosen [24]), the metric, and the volume of the unit cell (a, b, c, α, β, γ, V). The density is derived by dividing the number of vertices per cell (see also below) by the volume; in our dia example it is 8/12.3168 = 0.6495.

2.5.3 Genus

In a strict mathematical sense, the genus is quite a complex property of a surface of a topological object, which has a connection to the so-called Euler characteristics χ of a surface. However, in simpler, non-mathematical terms it can be perceived as the number of holes in or handles of an object. The genus is another interesting property on the basis of which a further classification of nets is possible, in particular if we are dealing with porous MOFs. To give a few examples, a football and a sphere but also a wine glass have the genus 0; a finger ring, a coffee cup, or a torus the genus 1; a double torus the genus 2; a pretzel the genus 3; and so on. The genus can also be defined as the maximal number of continuous cuts that are possible on the condition that afterward the surface is still composed of one continuous piece. Cutting the sphere would result in two pieces – ergo the genus is zero. A complete cut along the radial direction of a bicycle tube would not break it into two pieces as the surface is still composed of one continuous piece, while a further slice, regardless of its direction, would divide the tube into two pieces, which means the genus is 1.

But as we are concerned with nets, how can we apply the concept of the genus to mathematical objects which are composed only of infinite small vertices and edges having no surface at all? First, we have to inflate the edges and vertices to finite width, at least virtually. Second, as we are faced with infinite 2- or 3-periodic nets, they necessarily do also have an infinite number of holes; therefore, the genus of a net refers to the smallest possible repeating unit of the net, that is, the primitive unit cell (not the face- or body-centered ones). The procedure to obtain the genus consists of the following steps: Take the vertices of the repeating unit together with their first topological neighbors. Then connect each topological neighbor vertex that emerged from the vertex of the repeating unit in the crystallographic +[uvw] direction with its vertex counterpart of the −[uvw] direction with a handle (see Figure 2.12). The number of newly created handles is then equivalent to the genus. Keeping in mind that the premise was to assign a finite width to the edges and vertices then we see indeed that this genus is equivalent to the number of holes in a continuous surface.

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Figure 2.12 The repeating unit (blue vertices) and their pairs of topological neighbors in the +[uvw] and −[uvw] directions of the net pcu (a), dia (b), and pts (c). If the topological partners are joined by an arc, then an n-handled body is formed. n is the genus of that net.

There is also a related procedure to obtain an equivalent 2D representation of the genus: Here, the repeating unit is drawn as a 2D graph together with the edges that emerge from the vertices of this graph, that is, as a graph with loose ends. Now, we connect the vertices with the vertices of the same type, but not to the next neighboring repeating unit, instead within this same repeating unit, within this 2D graph. We do this until no loose ends remain. The number of new arcs/edges is the genus (Figure 2.13). In graph theory, the repeating unit together with the loose ends is called the spanning tree, the closed graph after joining the vertices with arcs is termed the quotient graph, and the number of new arcs is referred to as the cyclomatic number of the quotient graph. Bonneau et al. worked out an elegant respective mathematical formulism for that issue, and it turns out that the genus can be simply calculated by the formula g = 1 + e − v, where e is equivalent to the number of edges and v to the number of vertices of the primitive unit cell [25]. For 3-periodic nets the minimal number of g is three, and there are exactly eight nets, which have a genus of three, the so-called minimal nets (pcu, dia, srs, ths, cds, tfa, hms, tfc) [25].

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Figure 2.13 Spanning trees (vertices plus blue edges plus blue loose ends) that correspond to the repeating unit of a net, the red handles, and the resulting quotient graphs of pcu (a), dia (b), and pts (c). The number of holes (closed areas bordered by lines/arcs) is the genus of the net.

2.5.4 Topological Density

The td10 value represents the topological density as the cumulative sum of the first 10 shells of topological neighbors (i.e., the first 10 terms of the coordination sequence; see below) of a vertex, plus the one reference vertex from which we started (coordination shell zero). For structures with more than one kind of vertex, the value given is a weighted average over the vertices. It is obvious that this value is usually high for vertices