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RCSR

** **

Suggestions, corrections, additions to **mokeeffe@asu.edu**

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You
are looking at a **R**eticular **C**hemistry **S**tructure **R**esource
constructed especially to be useful in the design of new structures of
crystalline materials and in the analysis of old ones (cf. The discussion of
reticular chemistry by Yaghi, O. M. et al. Nature **2003**, 423, 705-714). It is worth quickly reading
through this page once, as in the spirit of modern software design we
leave the user to discover how it really works.

To reference the database please cite:

O'Keeffe, M.; Peskov, M. A.; Ramsden, S. J.; Yaghi, O. M. *Accts.
Chem. Res. ***2008**, *41*, 1782-1789.

The
most important part is a searchable database of 3-periodic nets. These are
presented for the most part in maximum symmetry embeddings in which edge
lengths are constrained to be equal to unity and the volume is maximized
subject to that constraint. For these the "embed type" is 1. Some
nets do not allow such embeddings and other embeddability types are given
below.

Clicking
the "nets" link leads to a search page where searches can be done by *symbol* (every structure in this database has a unique
symbol), by *name* (some
structures have names) by *keyword*
and by *properties*. The terms
are all explained in popups accessed by clicking the appropriate link on the
search page. As well as crystallographic data, vertex symbols and coordination
sequences are given. For an explanation of these see reference [1] below.
Tilings are generally given as natural tilings [2] with face symbols [*N ^{n}*.

Other
databases are of 2-periodic structures ("layers") and of
"polyhedra". At present we have a fewer entries for these. Users
familiar with the "nets" search should find these self-explanatory,
but read on after "embeddings" on how to use the coordinate data for
polyhedra and some other notes.

The
symbol is either a three-letter symbol **pqr** or a three letter symbol with an extension as in **pqr-a**. It is not necessary to know the meanings of the
extensions but it can be useful and clicking on the "symbol" on the
"nets" search page will provide an explanation of the extensions.

Much
of the data (structures with embed type not 1) have been entered by hand, so
there doubtless errors, please let us know if you identify any.

The
structures are nets (special kinds of graphs) and to use the data one needs to
know the edges ("bonds"). For "embed type" 1 (most of the
data) the shortest distances are the edges. For other structures it is
generally sufficient to put a vertex of valence 2 at the midpoints of the edges
(these data are given). Structures where this won't work (one has to explicitly
use the symmetry) include **elv**, **mab**, **tcb**,
**tcc**, **ten**. If in doubt use the Systre file export at the
bottom of the page of the net.

At
the end of this page are references to some of our recent work relevant to the
database.

**Credits**

The
origin of this project was a National Science Foundation small grant for
exploratory research (SGER) DMR-0243082 to **Omar
Yaghi**. and **Michael O'Keeffe**
who collected the data. **Maxim Peskov**
prepared the tiling data, the Systre files and the ist of occurrences. The
current realization is implemented by **Stuart
Ramsden**. Software and other support is being generously provided by **Vladislav Blatov**, **Olaf
Delgado-Friedrichs, Stephen Hyde** and **Davide M. Proserpio** who have corrected
numerous errors and contributed in many other ways.

Users
familiar with the *Atlas of Zeolite Structure Types* (see "links") will recognize our debt to
that valuable work.

See
the references at the end for the **Fischer-Koch**
sphere packings.

**Embeddings of nets** [10]

Usually
in describing a net, parameters are given appropriate to a maximum-symmetry
embedding. Ideally also the shortest distances are all equal (for convenience
unity in the units of the lattice parameters) and correspond to the edges of
the net. This ideal situation is not always possible however, as can seen by
considering a generic infinite net without symmetry. Let the average
coordination number be *Z*, then
there are *Z*/2 edges per vertex.
Each vertex has three coordinates (*x*, *y*, *z*), so if *Z* > 6 there are more constraints (edge lengths) than degrees of
freedom (coordinates) and there is no solution for equal edges possible. In
fact, as the equations for edge lengths are non-linear in coordinates, there is
no guarantee that there is a solution even for *Z* <= 6. Of course we are not dealing with generic
structures, but rather with translational and, usually, other symmetries, so
the problem is quite complicated. There are several cases to consider:

**1**. For the simple nets most common in crystal
chemistry the maximum symmetry embedding is a sphere packing. There are three
cases:

**a**. The coordinates and unit cell shape are fixed by
symmetry in the maximum-symmetry embedding.

**b**. In other cases there are still degrees of freedom
remaining. It is common in this case to report a structure that is a minimum of
density (maximum cell volume) subject to the constraint of equal edge lengths
which can be taken as unity.

**c**. Again there are free parameters, but the minimum
density configuration corresponds to a structure with more contacts. This is
almost the same as 2a (not exactly as minimum density may not be maximum
symmetry)

**2**. The maximum symmetry embedding corresponds to a
sphere packing with more contacts than the coordination number, *n*, but *n*
of those contacts correspond to edges; there are two cases:

**a**.
There is an embedding as an *n*-coordinated sphere packing of lower symmetry in which the *n* edges correspond to edges.

**b** There is not an embedding as above (strictly we
should say that such an embedding cannot be found, *proving* one does not exist might be difficult). This case
is more common than might be supposed.

**3**. The maximum symmetry embedding is a sphere
packing with fewer contacts than the coordination number, *n*; but the *n* shortest distances correspond to the edges of the *n*-coordinated net; there are three cases:

**a** There is a lower-symmetry embedding with equal
edge lengths and edges still corresponding to shortest distances between
vertices.

**b**. There is no embedding with equal edge lengths.

**c.** There is an embedding with shortest distances
equal to edges, but they cannot all be made equal and remain shortest
distances.

**4**. There may be no embedding with shortest distances
corresponding to edges. This is in fact the case for the vast majority of nets.
Until recently they were not recognized in crystal chemistry, as normally bonds
(corresponding to edges of nets) are formed to nearest neighbors, but, with
flexible linkers instead of bonds as the edges of nets, they are beginning to
appear (see e.g. **tcb**). Most
catenated nets (see e.g. **pcu-c**)
are like this.

**5**. There may be no faithful embedding in the maximum
symmetry conformation because the symmetry will require edges to intersect.
There is always a lower-symmetry embedding – all graphs have a faithful
embedding in 3-dimensional Euclidean space. These lower symmetry embeddings may
be any one of the types described above in 1-4, (but of course not of maximum
symmetry) and there may be several equally "good". These structures
can be recognized by the fact that: (a) They have symbol **abc-z** and there is no entry **abc**. (b) They have embed type 5. The maximum symmetry
for these nets is given in the same box as the embed type.

**Collisions**

In
computing properties of nets, for example in the program **Systre**, it is convenient to use center-of-mass
(barycentric) coordinates (all vertices assigned the same mass - see reference
[4]). For some exceptional structures one may find that two vertices then have
identical coordinates. Such structures are identified by an asterisk in embed
type (e.g. *1b). **Systre** won't
recognize these structures. **Systre**
can be obtained from gavrog.org.

**Density**

The
density is the number of vertices per unit volume for the given embedding. For
structures of embed, type 1 or 2 (all edges of equal length and no shorter
intervertex distance shorter) on can calculate the space-filling fraction of
equal spheres in contact for that structure by multiplying the density by pi/6.

**Genus**

A
net can be considered a periodic surface of genus *g* if one imagines the edges inflated to have finite
width. If there are *e* edges and
*v* vertices in the primitive
cell then *g* = 1 + *e* -*v*.
See the pop-up for 'density' on the search page.

**Tiling
data**

Data
for tilings are for the so-called "natural" tilings (reference [2]
and [13]). The D-symbols are what were originally called "Delaney"
symbols by Dress and "Delaney-Dress" symbols by subsequent workers.
The economical one-line encoding is due to Delgado-Friedrichs so
"D-symbol" seems appropriate. The size of the D-symbol is what is
known as "flag transitivity" to mathematicians. One may use the
search to find the only regular tiling (D-symbol size = 1) and all natural
tilings with size up to 4. Regular *nets* are defined independently of tilings but are the only structures with
natural tilings that have one kind of vertex, edge, face and tile (transitivity
1111 - see reference [2]).

Most
of the tiling data come from TOPOS (see "links") and were prepared by
Maxim Peskov.

**Layers
and polyhedra**

*Layers* are three dimensional 2-periodic structures and
their symmetries are layer groups. Strictly 2-dimensional 2-periodic structures
have symmetries that are plane groups. Structures that are not 2-dimensional
have the keyword "layer" We give also a 3-dimensional space group
which can be used for entering the structure in a crystal-drawing program such
as CrystalMaker. Where possible, coordinates are given for unit edge length.

Notice
that for 2-dimensional structures, vertex symbols denote the polygons meeting
at that vertex in cyclic order (compare 3^3.4^2 and 3^2.4.3.4); for layers
vertex symbols are the same as for nets.

See
reference [12] below for an account of plane nets in crystal chemistry. For
nets named OKHnn, nn is the net number of that work.

*Polyhedra* are 0-periodic, but of course 3-dimensional. We
use the term to include generalized polyhedra ("cages") such as the
adamantane cage (see **ada**). The
coordinates of the vertices are cartesian coordinates but can be used with the
given space group to again draw as if it were part of a crystal structure. To
do this set the unit cell of the given space group to *d* *d d*
90.0 90.0 90.0 with *d *some
convenient number (say 10.0) and divide the given coordinates by *d* and use them as crystal coordinates. One will then
get the polyhedron centered on 0, 0, 0.

Notice
that for polyhedra with non-crystallographic symmetries more than one set of
coordinates may be given for a single vertex type - see e.g. **dod** with icosahedral symmetry but described with *m*-3 symmetry - the single vertex type splits into
V1a and V1b. Hexagonal polyhedra are described using an orthorhombic space
group – see e.g. **hxb**. A
square antiprism is another example of a polyhedron with non-crystallographic
symmetry - see **sap**.

For
convenience the coordinates of the centers of faces are also given.

In
the strict definition adopted by many mathematicians *polyhedra* have a planar 3-connected graph and can be
realized as a strictly convex solid. We refer to structures that do not have
this property as *cages* for the
purpose of the database (generally elsewhere the term "cage" includes
polyhedra as in "sodalite cage" or "faujasite cage", but
these structures are polyhedra *sensu stricto*).

Notice
that generally we give vertices for equal edge; some polyhedra do not then have
planar faces (see e.g. the faujasite cage **fac** which is given as it appears in the crystal
structure). For *Archimedean*
polyhedra (one kind of vertex) and the full symmetry the requirement of equal
edges uniquely defines the embedding; for their duals (*Catalan* polyhedra with one kind of face) there are
generally more degrees of freedom and there appears to be no definitive
embedding agreed upon - the most pleasing have unequal edges. [Exceptions are **rdo** and **trc** which are the duals of the two quasiregular (one kind of edge)
Archimedean polyhedra **cbo** and **ido**]

*Simple
*polyhedra have exactly three faces
meeting at a vertex; *simplicial*
polyhedra have all triangular faces. The dual of a simple polyhedron is
simplicial and *vice versa*.

**References**

References
give in the database are to descriptions of nets, and not, except incidentally,
to occurrences in crystals. [references to occurrences will be added later]

Some
of our work relevant to 3-periodic nets and tilings:

1. *Crystal Structures I: Patterns and Symmetry*. M. O'Keeffe, B. G. Hyde. Min Soc. Amer.,
Washington D.C. (available at http://www.public.asu.edu/~rosebudx/okeeffe.htm)

2.
*Three-periodic nets and tilings: regular nets*. O. Delgado-Friedrichs, M. O'Keeffe, O. M. Yaghi,
Acta Crystallogr. A 59, 22-27 (2003).

3. *Reticular synthesis and the design of new materials.*
O. M. Yaghi, M. O'Keeffe, N. W. Ockwig, H. K. Chae, M. Eddadoudi, J. Kim.
Nature **423**, 705-714 (2003).

4. *Identification and symmetry computation for crystal nets*. O. Delgado-Friedrichs, M. O'Keeffe, Acta
Crystallogr. A 59 351-360 (2003). (This gives an informal account of how **Systre** works)

5. *Three-periodic nets and tilings: semiregular nets*. O. Delgado-Friedrichs, M. O'Keeffe, O. M. Yaghi,
Acta Crystallogr. **A 59**, 515-525
(2003).

6. *Three-periodic nets and tilings: minimal nets*. C. Bonneau, O. Delgado-Friedrichs, M. O'Keeffe,
O. M. Yaghi, Acta Crystallogr. **A 60**,
517-520 (2004).

7. *Crystal nets as graphs: terminology and definitions. *O. Delgado-Friedrichs, M. O'Keeffe, J. Solid State
Chem. **178**, 2480-2485 (2005).

8.
*Reticular chemistry: occurrence and taxonomy of nets and grammar for the
design of frameworks.* N. W.
Ockwig, O. Delgado-Friedrichs, M. O'Keeffe & O. M. Yaghi. Accts. Chem. Res.
**38**, 176-182 (2005).

9.*Three-periodic nets and tilings: face transitive tilings and edge
transitive nets*. O.
Delgado-Friedrichs & M. O'Keeffe. Acta Crystallogr. **A63**, 344-357 (2007).

10. *What do we know about 3-periodic nets? *O. Delgado-Friedrichs, M. D. Foster, M. O'Keeffe,
D. M. Proserpio, M. M. J. Treacy
& O. M. Yaghi, J. Solid State Chem. **178**,
2533-2554 (2005).

11. *Taxonomy of periodic nets and the design of materials.* O. Delgado-Friedrichs,* *M. O'Keeffe & O. M. Yaghi, Phys. Chem. Chem.
Phys. **9**, 1035-1943 (2007).

12. *Plane nets in crystal chemistry*. M. O'Keeffe & B. G. Hyde. Phil. Trans. Roy.
Soc. **A295**, 553-623 (1980).

13. *Three-periodic nets and tilings: natural tilings for nets*. V. A. Blatov, O. Delgado-Friedrichs, M. O'Keeffe
& D. M. Proserpio, Acta Crystallogr. **A 63**, 418-425 (2007).

A recent book that uses the RCSR system:

14. *Molecule-Based materials. The Structural Network Approach*. L. Öhrstrom & K. Larsson. Elsevier (2005).

The classics (note that "three-dimensional" here means
"three-periodic"):

15. *Three-dimensional nets and polyhedra*. A. F. Wells. Wiley (1977).

16. *Further studies of three-dimensional nets.* A. F. Wells. ACA Monograph **8**. American Crystallographic Association (1979).

17. An enormously valuable set of data on homogenous sphere packings
come from Werner Fischer and Elke Koch and collaborators (3-coordinated: E.
Koch and W. Fischer, *Z. Kristallogr. ***210**, 407-414 (1995).
Cubic: W. Fischer, Acta Cryst. A**60**, 246-249 (2004). Tetragonal: W. Fischer, Acta Cryst. A**61**,
435-441 (2005). Hexagonal: H. Sowa and E. Koch, Acta Cryst. A**62**, 379-399 (2006). Orthorhombic: H. Sowa, E. Koch
& W, Fischer, Acta Cryst. A**63**,
354-364 (2007). Triclinic: W. Fischer & E. Koch, Acta Cryst. A**58**, 509-513 (2003). These papers give references to
earlier work.

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