Flexibility of ideal zeolite frameworks

V. Kapko, C. Dawson, M. M. J. Treacy* and M. F. Thorpe

Received 9th March 2010, Accepted 19th June 2010

First published as an Advance Article on the web 29th June 2010

DOI: 10.1039/c003977b

We explore the flexibility windows of the 194 presently-known zeolite frameworks. The flexibility

window represents a range of densities within which an ideal zeolite framework is stress-free. Here,

we consider the ideal zeolite to be an assembly of rigid corner-sharing perfect tetrahedra. The

corner linkages between tetrahedra are hard-sphere oxygen atoms, which are presumed to act as

freely-rotating, force-free, spherical joints. All other inter-tetrahedral forces, such as coulomb

interactions, are ignored. Thus, the flexibility window represents the null-space of the kinematic

matrix that governs the allowable internal motions of the ideal zeolite framework. We show that

almost all of the known aluminosilicate or aluminophosphate zeolites exhibit a flexibility window.

Consequently, the presence of flexibility in a hypothetical framework topology promises to be a

valuable indicator of synthetic feasibility. We describe computational methods for exploring the

flexibility window, and discuss some of the exceptions to this flexibility rule.

I. Introduction

Zeolites are remarkable oxide framework nanomaterials that

have important applications in the petrochemical and fine

chemical industries. Zeolites are structurally related to quartz

in the sense that they are periodic frameworks comprising

corner-sharing SiO4 tetrahedra. Each tetrahedron is connected

to four others via the four oxygen atoms that are located at the

tetrahedral apices. Zeolite frameworks are distinguished from

quartz, and other dense silicates, by their microporosity. Their

frameworks contain periodic arrays of channels and pores that

allow small molecules, such as water and light hydrocarbons,

to diffuse through them. Consequently, zeolites have large

internal surface areas accessible by such molecules.

Zeolites are not constrained to a pure SiO2 composition.

Typically, they are aluminosilicates, with trivalent aluminium

substituting isomorphously for a tetravalent silicon, with the

charge being balanced by extra-framework cations, such as

Na+ and H+, which are not physically part of the framework.

Other elements such as beryllium, boron, iron, phosphorous,

germanium, cobalt and zinc can also be substituted into the

framework tetrahedral position. These cations influence the

structure and chemistry of the framework. For example, for a

T–O distance d (where T represents the tetrahedral atom, Si,

Al etc.), the tetrahedron edge length L is the nearest-neighbour

O–O distance given by L ¼ffiffiffiffiffiffiffiffi8=3

pd. Thus, for a charge-neutral

SiO4 tetrahedron, we have d E 0.161 nm and L E 0.263 nm,

whereas for an anionic AlO�4 tetrahedron we have dE 0.173 nm

which corresponds to a larger tetrahedron with edge length

L E 0.282 nm. This mixture of tetrahedron sizes, and their

distribution, modifies the precise details of the framework

structure. Additionally, the extra-framework cations will affect

the chemistry, and the associated coulomb forces will, to a

small extent, also influence the structure. In particular, the

presence of protons (H+) renders a zeolite framework into a

solid Brønsted acid, a most valuable property for the zeolite-

based catalytic cracking of hydrocarbons.

Because of their usefulness, there are considerable ongoing

efforts to make new zeolites, and to better understand their

properties. Furthermore, the recent development of large

databases containing millions of hypothetical zeolite frameworks

demands improved computational tools that will rapidly and

reliably sift through these vast databases to identify the

potentially useful frameworks.1–3 Perhaps of even more

importance, but seemingly more difficult to achieve, is the

development of computational tools that generate recipes for

the selective synthesis of targeted hypothetical frameworks.

The ZEBEDDE program has been an important development

in this area.4

In this paper we examine a remarkable mechanical property

of ideal zeolite frameworks—their flexibility. We define an

ideal zeolite framework as one that is built using rigid, perfect,

corner-connected tetrahedra with freely rotating spherical

joints at the vertices. Here we show that almost all of the

known zeolite frameworks (those that have been synthesized

or occur as minerals) lack mechanical rigidity as ideal zeolite

frameworks over a range of densities. Within this range of

densities, which is referred to as the flexibility window,5 there is

no stress within the idealized framework. However, many

hypothetical zeolite frameworks lack this flexibility, at least

when constructed with only one type of tetrahedron, such as in

a pure SiO2 composition. Thus, many hypothetical frameworks

contain stressed tetrahedra, even when the nominal energy of

formation, as calculated by GULP6 for example, is low. This

raises the interesting possibility that the existence of a flexibility

window in an idealized hypothetical zeolite framework is a

strong indicator that the framework is realisable. This has

important implications in ascertaining which new zeolites are

most likely to be synthesized in the future.

It has been long known that some idealized zeolite frame-

works lack rigidity. Pauling, in his 1930 paper on the structureArizona State University, Department of Physics, P.O. Box 871504,Tempe, AZ 85287-1504, USA. E-mail: treacy@asu.edu

This journal is �c the Owner Societies 2010 Phys. Chem. Chem. Phys., 2010, 12, 8531–8541 | 8531

PAPER www.rsc.org/pccp | Physical Chemistry Chemical Physics

of sodalite, remarked that the SOD framework lacked rigidity

over a wide range of densities.7 In fact, the SOD framework

has one of the largest flexibility windows obtained for an ideal

zeolite framework. The experimental flexibility of zeolite rho

(RHO framework) is also well known,8 and its range is

comparable to that for SOD. However, many zeolite frame-

works are found to be rigid when constrained to their nominal

space group symmetry. For example, in the cubic space group

Pm�3m, the flexibility window of the SOD framework has zero

width, with a unique solution for the atom coordinates and cell

dimension. More recently, Sartbaeva et al.5 showed that when

the symmetry is reduced to P1, and as pure SiO2 composition,

many of the known zeolite frameworks exhibit such a flexibility

window. In addition, they noted that real zeolite materials

tend to adopt a structure that is close to the low density end of

the window. This preference for low density configurations

was ascribed to coulomb repulsions, mainly between frame-

work oxygen atoms.9 Simple models that provide insight and

unifying principles are particularly important when trying to

make sense of large amounts of data; in this case structural

data associated with real and hypothetical zeolites. It is in this

spirit that this paper is written.

The flexibility of zeolites arises from the fact that their

frameworks are locally isostatic; that is, the six degrees of

freedom per tetrahedron (three translational, three rotational)

are exactly matched by the six constraints imposed by the

sharing of the oxygen atoms (three constraints for each of four

oxygen atoms, divided by two since every oxygen is shared

between two tetrahedra). It would seem at first that the

equality of degrees of freedom and constraints would mean

that zeolite frameworks are fully determined geometrically.

However, the periodicity introduces six additional degrees of

freedom globally to the framework.10 Three of these can be

discarded as they correspond to simple translations of the

structure along the three orthogonal axes of the underlying

lattice. The remaining three must correspond to internal

mechanisms of the periodic structure. (They cannot be

three trivial rotations, since such motions are generally

incommensurate with the underlying periodic lattice.) The

presence of additional crystallographic symmetries (mirror

and glide planes, rotation and screw axes) also adjusts the

numbers of degrees of freedom as well as the constraints,

potentially increasing the surfeit of degrees of freedom over

constraints. (We are always guaranteed at least three more

degrees of freedom than constraints in three dimensions.10)

This excess of degrees of freedom over constraints is then

responsible for the generally under-determined nature of

zeolite structures, creating a null-space of solutions to the

kinematic matrix that are stress-free.

The computational algorithms needed to identify and

systematically explore the flexibility window cannot use the

ubiquitous gradient methods that minimize energy or cost

functions. Since we ignore coulomb terms in our ideal zeolite

framework, the potential energy is zero and flat within the

window. Within the flexibility window the framework is

stress-free and there are no restoring forces to guide any

search of this space. Rigid unit modes (RUM model) have

been studied extensively in tetrahedral frameworks by Dove

and co-workers.11,12 The rigid-unit modes are closely related

to the flexible folding modes studied here. Generally speaking,

rigid-unit modes exist as states at the surface of the flexibility

window where the spring forces between tetrahedra vanish. In

principle, such modes can be used to explore the denser states

of the flexibility window. The GASP program by Wells13 uses

geometric algebra to identify stress-free frameworks, and

explores the flexibility window by incrementally modifying

the structure while constraining those configurations to lie

within the window. Depth-first-search methods using GASP

within the flexibility window allow some systematic exploration.

However, neighbouring points in the flexibility window, based

on those structures having closely similar densities, do not

guarantee that those structures are topological neighbours. In

addition, multiple distinct structures can correspond to the

same density, yet necessitating large cooperative motions in

order to transform one structure into the other. Essentially,

most neighbours can be reached only by ‘unfolding’ the

framework, and then initiating a new, topologically distinct,

folding mechanism. Consequently, such searches of the

flexibility window tend to be hit-or-miss and incomplete.

A promising approach to the study of framework flexibility

has been developed for mechanical engineering applications by

Pellegrino14 and Guest and Hutchinson10 to study rigidity in

deployable structures and periodic trusses. The method uses

singular value decomposition (SVD) to systematically explore

the null-space of flexible systems. This method is readily

adapted to the study of ideal zeolite frameworks.

In this paper, we examine the flexibility characteristics of all

of the 194 zeolite frameworks that have been approved to date

by the Structure Commission of the International Zeolite

Association. We present results using both the GASP program

of Wells13 and a newly-implemented program ZeNuSpEx

(Zeolite Null Space Explorer) that is based on the SVD

methods. We discuss the flexibility characteristics of the

SOD framework, as well as examples of frameworks that

appear to be intractably rigid. This work establishes the

validity of this approach and sets the stage for future work

which will apply the principles learnt from real zeolites as a

sieve to sort through hypothetical zeolites for promising new

zeolite structures.

II. Computational methods

Zeolite frameworks were initially modeled using the GASP

computational tool (Geometric Analysis of Structural

Polyhedra), which was developed by Wells et al.13 This tool

had been used successfully in a previous study of zeolite

flexibility.5 Frameworks are treated as periodic networks of

rigid, corner-connected tetrahedral units that are formed by a

T-atom (such as silicon, aluminium, etc.) that has four oxygen

atoms at the regular tetrahedral vertices to form a TO4 unit.

To find relaxed atom configurations satisfying constraints that

are consistent with rigid tetrahedra, GASP attempts to match

oxygen atom positions to a template (represented by the

appropriate number of regular tetrahedra with an implied

T-atom at the center) within the unit cell. Periodic boundary

conditions are applied, but no further symmetry constraints

are included in order to allow maximal exploration of the

configurational space. Each oxygen atom is tethered to two

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corresponding template tetrahedra by a fictitious harmonic

spring with a natural length of zero. Hard-sphere repulsion

between oxygen atoms that are on different tetrahedra is

imposed to prevent steric clashes. We used an oxygen radius

of 0.135 nm, which is the value used by the Atlas of Zeolite

Framework Types.15 A cost function is then generated that

penalizes both the bending and stretching of the springs, as

well as any intertetrahedral oxygen atom overlaps. GASP uses

the limited memory Broyden–Fletcher–Goldfarb–Shanno

(BFGS) algorithm, which is a quasi-Newton method,16 to

minimize this function. A configuration is considered relaxed

when all T–O bond lengths have reached 0.161 � 0.0001 nm (a

typical silicon-oxygen bond distance in zeolites) and all

O–T–O bond angles have reached 109.471 � 0.0011 with no

oxygen overlaps.

GASP attempts to explore the flexibility window by

incrementally varying unit cell parameters, thereby adjusting

cell volume and density, and using the fractional coordinates

of the previously-obtained relaxed configuration as an initial

guess. The primary input is usually the cell dimensions and

coordinates obtained from the International Zeolite Association

database.17 The oxygen atoms are then moved a small distance

(typically, 0.01 nm or less) from their initial positions and each

tetrahedral template is placed at the geometric center of the

four corresponding oxygen atoms. The positions of the atoms

and the template are then refined until conditions for relaxation

are met or the minimization algorithm fails. This process is

repeated numerous times at each given density to determine

whether relaxed configurations can be found for a specific set

of unit cell parameters.

An alternative approach to exploring the flexibility window

using GASP is based on a breadth-first-search algorithm. This

algorithm explores the flexibility window by exploring a

finely-spaced grid of unit cell parameters, starting from a

known relaxed configuration and checking whether or not its

neighbouring configurations can also be relaxed. Once all of

the neighbours of the starting configuration have been

checked, then those neighbours that remain flexible are also

checked for their flexible neighbours. The algorithm stops

when it has exhausted all relaxed configurations accessible

from the starting point. This method has been used to build

two-dimensional projections of the flexibility window for

the frameworks MTN and SOD, which are presented and

discussed in the following sections.

A third method of probing the flexibility window involves

modification of the GASP approach. In this approach,

neighbouring oxygen atoms and T-atoms are attached to

one another by springs of the correct natural length for

forming unstressed perfect tetrahedral units. For each

tetrahedron, there a four springs associated with the T–O

bond distances, and six springs associated with the O–O edge

lengths. No angular constraints are necessary, since we are

allowing the tetrahedra to rotate about the oxygen force-free

‘spherical joints’. As before, periodic boundary conditions are

applied without any additional internal symmetry constraints.

A cost function is produced that includes each spring term as

well as hard-sphere repulsion terms between oxygen atoms,

and minimization of this function is undertaken by refining

input coordinates, using the same quasi-Newton method used

by GASP, until tetrahedral distance constraints are satisfied to

within 0.0001 nm. This stringent tolerance is needed in order

to find the near-perfect geometry. The results of these simulations

were used to check the GASP simulations, and also to generate

new relaxed configurations from which to restart the GASP

program.

While these methods have proven effective in finding

numerous relaxed conformations for almost all known zeolite

structures, the configuration space to be explored within the

flexibility window is enormous and it is difficult to ascertain

the extent to which our simulations efficiently explore this

space. Almost all structures exhibit multiple folding paths

from a low-density, maximum symmetry configuration. As a

result, we notice variations at the high-density end of the

window depending on the folding path followed. It is clear

that multiple relaxed configurations with significantly different

folded topologies can exist at any given density and even for a

given set of unit cell parameters. There is also the possibility

that ranges of relaxed configurations are topologically isolated

from one another when constraints are placed on the unit cell

parameters. Despite the power of the above tools, it is

apparent that a more systematic exploration of the flexibility

window is necessary to make final, definitive statements in this

regard.

A promising approach, which has been developed in the

field of mechanical engineering for the analysis of deployable

structures, such as spacecraft antennae, involves analysis

of the compatibility (or kinematic) matrix C.14 This is a

mathematical construct that relates the vector d, formed from

the generalized atomic displacements (displacements of the

‘joints’), to the vector e, which is the vector formed from the

extensions of the bonds (‘bars’),18

Cd = e. (1)

The compatibility matrix is found from the set of constraint

conditions (rj � ri)2 = L2

ij, where Lij is the bond length. It

has b rows, one for each bond in the unit cell (allowing for

periodicity), and, for zeolites, b+ 6 columns. Although zeolite

frameworks are locally isostatic, that is, b = 3N for N oxygen

vertices, there are six additional degrees of freedom associated

with the unit cell dimensions. It is important to include the

effects of small changes in the unit cell parameters in the

displacements (see ref. 10 for details). Within the flexibility

window, the extension vector is zero, e = 0. Displacements

that lie within this strain-free flexible regime are the null

eigenvectors of the eigenvalue equation Cd = ld. The null

eigenvectors are those solutions with eigenvalue l = 0, and

correspond to the strain-free folding mechanisms of the

framework. The kinematic matrix C for periodic locally

isostatic frameworks, such as zeolites, has more degrees of

freedom than constraints. This is because the generalized

displacement vector not only represents the displacements of

all the atomic coordinates, but also includes the six components

corresponding to changes in the unit cell parameters.

Consequently, C is not a square matrix, having six more

columns than it has rows. Frequently, topological symmetry

within the framework renders some of the constraints

degenerate, and consequently there can be more than six null

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eigenvalues. This tends to happen at the maximum symmetry

condition of the unit cell, which almost always corresponds to

the maximum unit cell volume. Three of the null eigenvectors

correspond to trivial translations along the lattice axes. Thus,

there remain at least three null eigenvectors for every zeolite

framework that correspond to strain-free displacements.10 Of

interest to us are the finite-amplitude null eigenvectors that

define the flexibility window.

Our newly-developed program ZeNuSpEx (Zeolite Null

Space Explorer) uses singular value decomposition (SVD)

methods to find non-trivial null vectors of the kinematic

matrix.14 These null vectors can be added, with small amplitude,

to the original coordinate vector space to produce a new,

nearly-relaxed configuration. This configuration is then

relaxed again by energy minimization (to eliminate second

and third order deformations), resulting in a unique set of

relaxed coordinates that is a topological neighbour of the

original set. By following the null eigenvectors in this way,

we can, at least in principle, explore systematically the

flexibility window, eliminating the need for random moves.

In ZeNuSpEx, we have adopted the form of the kinematic

matrix introduced by Guest and Hutchinson10 for studies

of periodic trusses in mechanical engineering applications.

Using a hybrid of energy minimization and SVD techniques,

ZeNuSpEx allows systematic exploration of the flexibility

window. The issues still confronting our implementation of

this new approach will be discussed later.

III. Results and discussion

A Two-dimensional example; the kagome lattice

As a two-dimensional example of framework flexibility, we

first examine the kagome lattice (Fig. 1). This lattice has been

studied in more detail elsewhere,10,19,20 and comprises a plane

tiling of corner-connected equilateral triangles. The lattice is

equivalent to that found in a graphite sheet, where each

carbon atom has been replaced by an equilateral triangle. In

Fig. 1a, the primitive unit cell contains two triangles, and the

unit cell volume is the maximum allowed that is consistent

with perfect, unstrained, equilateral triangles. If we assume

that the triangle corners are interconnected by force-free pin

joints, then the triangles can rotate cooperatively to produce

higher-density configurations, such as those shown in Fig. 1b.

These configurations are still stress-free, and represent states

that lie within the flexibility window. In this particular folding

mechanism, the symmetry constraints dictate that alternate

triangles rotate in opposite directions. The internal degrees of

freedom of the structure can be augmented by increasing the

size of the unit cell so that there are more triangles per unit

cell. In Fig. 1c, the cell is doubled in area by transforming the

cell in Fig. 1a to a non-primitive rectangular unit cell to

enclose four triangles per cell. A second alternative folding

mode emerges as a result of this cell-doubling. The earlier

mode in Fig. 1b is still available, but can only be accessed by

expanding back to the maximum volume (Fig. 1a) and then

initiating the alternative folding mechanism corresponding to

Fig. 1b. Fig. 1d shows the new mode that occurs when the cell

is quadrupled to enclose 8 triangles per unit cell. This increase

in modes occurs because the number of symmetry constraints

per triangle is being reduced as the unit cell area is increased.

Zeolite frameworks are three-dimensional versions of such

four-valent frameworks. The four corners of each tetrahedron

are interconnected by force-free ‘spherical joints’, enabling

a more complex class of folding mechanisms to evolve.

Alternatively, zeolites can be viewed as six-valent networks

of oxygen atoms, where each oxygen atom is connected to six

others in an approximate octahedral arrangement. This latter

approach, while unconventional, is useful for exploring the

flexibility of these frameworks. We examine several zeolite

examples below.

B Examination of the known zeolites

Table 1 presents the lowest and the highest relative framework

densities, rmin and rmax (in units of T-atoms per nm3),

bounding the flexibility window for all of the 194 known-to-

date zeolite frameworks. The results are for frameworks

represented as pure SiO2, and we do not include the results

when different T-atoms (such as boron, aluminium or germanium)

are substituted at various sites. Different T-atoms have different

T–O and O–O tetrahedral distances, and this is found to alter

the flexibility window, as we shall discuss later. The flexibility

parameter,19 FO = Vmax /Vmin R rmax / rmin, is a simple

measure of the relative width of the flexibility window. For

the results presented in this table, at each density the unit cell

volume is held constant while the six unit cell parameters (cell

edge lengths a, b and c, and cell interaxial angles a, b and g) areallowed to vary. This introduces five additional degrees of

freedom which were included as adjustable parameters

along with the list of coordinate variables in the kinematic

matrix. This Table represents results from both GASP and

Fig. 1 Four configurations of the two-dimensional kagome lattice.

(a) Minimum density state (i.e.maximum unit cell volume), with p6mm

plane group symmetry. (b) A snapshot of the collapse mode when

there are two triangles per unit cell (p31m plane group symmetry). (c)

Snapshot of an alternative collapse mode when there are four triangles

per rectangular unit cell (p2gg plane group symmetry). (d) Snapshot of

an additional collapse mode when there are eight topologically distinct

hexagons per oblique unit cell (p2 plane group symmetry). If we allow

overlap of the equilateral triangles, the density can become infinite.

Adopted from ref. 19.

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ZeNuSpEx, and we report the extremes of density found

by either program. (As mentioned earlier, with improved

eigenvalue-following procedures, the ZeNuSpEx program will

be the more reliable tool.) The Table reveals that 182 of the

194 known zeolite frameworks are flexible when formed with a

pure SiO2 composition. The 12 inflexible frameworks tend to

have features such as double 4-rings, and more than one 5-ring

meeting at a vertex. The residual deformations are always

small, B0.01 nm or less.

C Shapes of the flexibility windows

The flexibility window exists in a six-dimensional space that is

defined by the six unit cell parameters, and is hard to represent

graphically. Nevertheless, it is instructive to examine the

boundaries of two-dimensional projections of the flexibility

windows (Fig. 2). The results for three well known cubic

zeolites, FAU, SOD and MTN are presented in Fig. 2a–d.

The yellow areas represent the flexibility windows. Fig. 2a–b

show the variation in relative density r/r0 of the FAU and

SOD frameworks as a function of the axial ratio c/a. r0 is thenominal density of the pure silica material as determined by

distance-least-squares fitting by the DLS76 program.21 The c/a

ratio represents a tetragonal distortion of the topologically

cubic cells. We have held b= a and a= b= g= 901 for these

2D plots. The numbers decorating the boundaries of the

windows are the number of extra-tetrahedral (non-codimeric)

oxygen–oxygen contacts that define the boundary. Such

contacts almost always delimit the high-density boundaries

Table 1 Minimum (rmin) and maximum (rmax) framework densities (T-atoms per nm3) for idealized zeolites with SiO2 composition. The flexibilityparameter FO is the ratio rmax/rmin. Entries with a dash ‘‘—’’ indicate that there is no flexibility window for this framework as a pure SiO2

composition. In many cases, flexibility is restored with a mixed composition, e.g. as an alumino-silicate or germano-silicate etc

CODE rmin rmax FO CODE rmin rmax FO CODE rmin rmax FO CODE rmin rmax FO

ABW 17.49 30.06 1.72 DDR 17.54 18.91 1.08 LTN 16.49 19.79 1.19 SAT 16.33 19.14 1.17ACO 16.40 25.89 1.58 DFO 14.78 16.53 1.12 MAR 16.70 27.51 1.65 SAV 15.20 17.24 1.13AEI 14.97 18.41 1.23 DFT 17.57 22.71 1.29 MAZ 16.63 18.65 1.12 SBE 13.61 16.05 1.18AEL 18.03 24.45 1.36 DOH 17.80 20.76 1.17 MEI 14.56 15.42 1.06 SBN 16.60 19.02 1.15AEN 20.88 22.36 1.07 DON 16.18 23.18 1.43 MEL 16.94 20.66 1.22 SBS 13.66 15.64 1.14AET 17.49 22.79 1.30 EAB 15.92 18.82 1.18 MEP 18.41 21.09 1.15 SBT 13.67 15.63 1.14AFG 16.73 26.17 1.56 EDI 16.26 19.24 1.18 MER 16.08 25.29 1.57 SFE 16.96 19.72 1.16AFI 16.67 21.49 1.29 EMT 13.13 16.83 1.28 MFI 16.94 21.21 1.25 SFF 18.03 19.99 1.11AFN 17.80 20.00 1.12 EON 16.78 18.74 1.12 MFS 17.66 18.78 1.06 SFG 17.44 19.38 1.11AFO 18.19 24.44 1.34 EPI 18.21 21.31 1.17 MON 17.60 23.33 1.33 SFH 16.33 18.14 1.11AFR 14.99 18.31 1.22 ERI 15.94 18.80 1.18 MOR 16.94 19.58 1.16 SFN 16.41 17.87 1.09AFS 14.49 16.99 1.17 ESV 16.37 21.99 1.34 MOZ 17.00 18.66 1.10 SFO 15.15 17.25 1.14AFT 14.76 17.58 1.19 ETR 15.91 18.20 1.14 *MRE 18.27 26.41 1.45 SFS 16.36 18.62 1.14AFX 15.03 17.46 1.16 EUO — — — MSE 16.17 17.91 1.11 SGT 17.58 19.38 1.10AFY 14.45 16.06 1.11 EZT 16.90 20.28 1.20 MSO 17.74 22.64 1.28 SIV 16.09 24.49 1.52AHT 19.38 28.26 1.46 FAR 16.82 26.12 1.55 MTF 20.95 22.61 1.08 SOD 16.58 27.98 1.69ANA 19.20 27.01 1.41 FAU 13.13 16.85 1.28 MTN 18.18 20.87 1.15 SOF 16.99 17.41 1.02APC 17.49 22.47 1.29 FER 17.51 22.64 1.29 MTT 18.17 20.94 1.15 SOS 16.58 21.39 1.29APD 17.93 27.69 1.54 FRA 16.63 25.66 1.54 MTW 18.29 20.24 1.10 SSF 16.36 21.23 1.30AST 15.78 27.38 1.74 GIS 16.08 26.94 1.67 MVY — — — SSY 16.85 19.91 1.18ASV 19.80 21.58 1.09 GIU 16.84 27.79 1.65 MWW 16.30 17.33 1.06 STF 17.00 19.88 1.17ATN 17.38 29.94 1.72 GME 15.02 17.46 1.16 NAB 16.25 25.15 1.55 STI 16.63 20.54 1.24ATO 18.46 22.16 1.20 GON 17.86 23.55 1.32 NAT 16.89 19.00 1.12 *STO 17.95 21.86 1.22ATS 15.91 26.85 1.69 GOO — — — NES 18.08 18.08 1.00 STT 17.89 17.89 1.00ATT 16.86 25.43 1.51 HEU 17.18 20.67 1.20 NON 18.87 19.93 1.06 STW — — —ATV 18.81 25.06 1.33 IFR 16.84 19.43 1.15 NPO 16.02 19.00 1.19 –SVR 16.48 18.43 1.12AWO 18.15 22.25 1.23 IHW 18.59 20.31 1.09 NSI 18.98 22.18 1.17 SZR 17.45 24.08 1.38AWW 16.52 25.60 1.55 IMF 17.42 19.76 1.13 OBW 12.97 17.19 1.32 TER 17.02 20.77 1.22BCT 18.89 36.82 1.95 ISV — — — OFF 15.96 18.35 1.15 THO 16.37 18.10 1.11*BEA 15.27 16.36 1.07 ITE 15.78 21.08 1.34 OSI 17.86 25.44 1.42 TOL 16.73 24.36 1.46BEC 16.03 17.13 1.07 ITH 17.78 19.18 1.08 OSO — — — TON 18.08 24.29 1.34BIK 18.61 34.97 1.88 ITR — — — OWE 17.03 19.43 1.14 TSC 13.15 16.17 1.23BOF 17.69 26.06 1.47 ITW 17.49 19.59 1.12 �PAR — — — TUN 17.38 19.69 1.13BOG 15.46 18.24 1.18 IWR 15.52 18.35 1.18 PAU 15.81 17.64 1.12 UEI 18.18 21.13 1.16BPH 14.49 16.82 1.16 IWS 10.06 11.27 1.12 PHI 16.09 24.47 1.52 UFI 15.03 18.00 1.20BRE 17.97 20.65 1.15 IWV — — — PON 17.95 20.91 1.16 UOS 17.23 19.59 1.14BSV 18.68 19.86 1.06 IWW 16.49 18.39 1.11 PUN — — — UOZ 19.74 21.44 1.09CAN 16.72 24.67 1.47 JBW 18.52 25.66 1.39 RHO 14.30 24.26 1.70 USI 16.05 17.98 1.12CAS 18.70 26.42 1.41 JRY 18.70 22.94 1.23 �RON 19.22 20.56 1.07 UTL 16.54 16.69 1.01CDO 20.40 20.40 1.00 KFI 14.93 17.99 1.20 RRO — — — VET 22.00 22.00 1.00CFI 17.44 19.10 1.10 LAU 17.75 20.95 1.18 RSN 16.87 20.40 1.21 VFI 14.80 17.83 1.20CGF 18.81 20.85 1.11 LEV 15.90 18.32 1.15 RTE 17.05 19.56 1.15 VNI — — —CGS 16.49 20.31 1.23 LIO 16.75 22.63 1.35 RTH 16.05 18.73 1.17 VSV 16.96 20.90 1.23CHA 15.01 17.19 1.15 �LIT 16.71 26.20 1.56 RUT 17.70 21.94 1.24 WEI 16.07 21.73 1.35�CHI 19.90 24.42 1.23 LOS 16.84 27.15 1.61 RWR 19.34 23.57 1.22 �WEN 16.84 20.52 1.22�CLO 10.89 13.69 1.26 LOV 16.84 20.60 1.22 RWY 8.31 14.03 1.69 YUG 18.13 20.82 1.15CON 15.85 17.90 1.13 LTA 13.95 17.33 1.24 SAF 17.96 18.75 1.04 ZON 17.53 21.84 1.25CZP 21.95 26.42 1.20 LTF 16.86 18.24 1.08 SAO 14.20 16.18 1.14DAC 17.43 22.69 1.30 LTL 16.77 19.23 1.15 SAS 15.23 18.69 1.23

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of the window. The low-density boundaries tend to be

defined by the onset of T–O stretching. As noted earlier

by Sartbaeva et al.,5 real materials adopt densities close

to the minimum allowed by the window. This inflation was

explained in terms of coulomb repulsions between non-

codimeric oxygen atoms, that is, those oxygens beyond the

six immediate oxygen neighbours. This inflationary mechanism

had been postulated earlier by O’Keeffe for all oxide frame-

work materials.9

From these two-dimensional plots, the flexibility parameters

for both FAU and SOD are greatest when the unit cell is cubic.

The low density cubic structure for SOD is portrayed in

Fig. 3a. For SOD a second high density state can occur when

the cell has a tetragonal shape with c/a= 1.27 (Fig. 3b). When

all six unit cell dimensions are allowed to vary, higher density

states are accessible. A remarkable example is shown in

Fig. 3c, which depicts a cell in lowest symmetry, P1, with a

density of 29.67 T-atoms per nm3, a value that exceeds that of

quartz (27.71 T-atoms per nm3).

An alternative two-dimensional representation of the

flexibility of SOD is to examine the flexibility as a function

of one of the cell angles (Fig. 2c). As the relative density

increases towards 1.27, the cell angle is increasingly

constrained to lie close to 901. For r/r0 = 1, the range of cell

angles is widest, with a = 90 � 8.51.

Fig. 2 Two dimensional representation of flexibility windows for three topologically cubic zeolite framework types. The flexibility regions are

depicted by the yellow areas. The numbers along the boundary are the number of non-codimeric oxygen contacts that define the boundary. (a) r/r0versus c/a for FAU. (b) r/r0 versus c/a for SOD. (c) r/r0 versus cell angle a for SOD. Windows for different fixed c/a ratios are shown. Note the

discontinuities within the flexibility window for c/a = 1.05. The window region is continuous with narrow channels connecting the three main

areas. (d) c versus a for MTN. Constant density contours are shown. The real material lies at the intersection of the lines c = a and r = r0(black dot).

Fig. 3 Representations of the SOD framework as a pure SiO2

composition at different densities. (a) Minimum density in cubic space

group Pm�3m, 16.61 T-atoms per nm3. (b) Maximum density in cubic

space group I�43m (a subgroup of Pm�3m), 29.67 T-atoms per nm3. (c)

Maximum density in triclinic space group P1, 30.39 T-atoms per nm3.

A hard-sphere radius of 0.135 nm for oxygen has been assumed.

8536 | Phys. Chem. Chem. Phys., 2010, 12, 8531–8541 This journal is �c the Owner Societies 2010

Data for the MTN framework is presented in a slightly

different format (Fig. 2d). Here, we plot the c-axis length

against the a-axis length. Contours of constant density are

overlaid. In addition, a diagonal line corresponding to a= c is

indicated. This line represents those structures adopting a

cubic cell shape, but does not guarantee that the underlying

space group symmetry of the cell contents are cubic. As a rule,

the structural symmetry in this plot is everywhere triclinic, P1,

even though the topological symmetry is cubic Fd�3m. The

MTN framework is classified as a clathrate, comprising a high

density of 5-rings, with a small fraction of 6-rings (see Fig. 4).

The structure of ZSM-39, which is a pure silica material with

the MTN framework, is reported to have the full Fd�3m space

group symmetry. That structure is indicated by the dark spot

near the center of the plot.MTN is unusual among the zeolites

studied in that it adopts a structure that lies outside the

flexibility window. One possibility is that the pure-silica

ZSM-39 material is actually a mosaic of cubic cells, whose

internal arrangements are not strictly periodic. The topology

remains periodic of course, it is just the precise atom locations

that do not repeat periodically. In this model, a grain of

ZSM-39 is not strictly crystalline, but is instead one large

multiple unit cell. The loss of strict periodicity, reduces the

constraints on the system, potentially extending the flexibility

window. The lack of strict periodicity would be expected to

show as an increase in diffuse scattering and a lengthening of

the atomic displacement factors, but no such effects have been

reported.

Even in two-dimensional plots, the flexibility windows

reveal complex shapes. The boundary for MTN (Fig. 2d)

exhibits pronounced involuted forbidden regions. Not evident

in such plots is the fact that within the flexibility window,

neighbouring points may be topologically remote from each

other. To transform between neighbouring points, while

remaining within the flexibility window at all times, the frame-

work needs to unfold to a high symmetry state (almost always

the maximum cell volume) and then to initiate a new folding

mechanism back to the desired point.

Some zeolite networks are extraordinarily flexible.

Frameworks ABW and AST (see the Table) can be collapsed

to 60% of their maximum volume before non-codimeric

oxygen atoms start to overlap. The most flexible zeolites

we have found, BCT and BIK, have flexibility parameters

close to 2, which almost reaches the theoretical upper limit

for face-centered cubic packing for oxygen atoms, 35.93

T-atoms per nm3. In real materials, interatomic forces will

prevent such high densities being reached at atmospheric

pressure.

There is a weak correlation between flexibility parameter

and homogeneity of ring size. A ring is a closed path through

connected tetrahedra within the framework, and has the

property that there is no shorter path between any two

tetrahedra on the ring than the path on the ring itself. The

ring size is the number of tetrahedral atoms visited by the ring.

The average ring size in tetrahedral zeolitic frameworks is

B5.5.22 The most flexible frameworks tend to have rings that

are all near this value (5, 6 and 8; 7-rings are rare in zeolites).

This observation is perhaps counterintuitive, since isolated

large rings are generally more flexible than small rings. For

example, a 3-ring is isostatically rigid, whereas an 18-ring

‘necklace’ is flexible in isolation. It appears that if a zeolite

contains large size rings (such as 10, 12 and larger) then it also

tends to include rings of small sizes (3, 4 and 5 rings).23 As a

rule, it appears that the combination of large and small rings

renders frameworks less flexible. One notable exception to this

generalization is the RWY framework. It has 3-, 8- and

12-rings, yet also has a remarkably large flexibility parameter

of 1.69, equal to that for SOD. This equality may be no

coincidence. RWY can be generated from the SOD framework

by the isomorphic substitution of each T-atom site by a

(locally rigid) ‘super-tetrahedron’ built from a tetrahedral

arrangement of four T-atoms. Such a substitution expands

some of the ring sizes, while introducing 3-rings that are

entirely within the super-tetrahedra, yet maintains the under-

lying framework flexibility.

Twelve of the 194 zeolites failed to relax as pure silica

structures. Those frameworks are indicated in the Table by

the dashed entries, since the window boundaries are undefined.

Most of these frameworks become flexible when some of the Si

T-atoms are substituted by other elements, such as boron

(dTO E 0.148 nm), aluminium (dTO E 0.173 nm), germanium

(dTO E 0.176 nm) or phosphorous (dTO E 0.154 nm). We

briefly discuss examples of this below. Substituting tetrahedra

of different size into the framework introduces additional

degrees of freedom, whilst the number of length constraints

remains unchanged, allowing those frameworks a chance to

Fig. 4 Three variations of the dodecahedral cage that occurs in the MTN framework for a cubic shaped cell, c = a. The unit cell is outlined.

(a) Structure in standard cubic space group Fd�3m. (b) Minimum density flexible structure, with c= a. (c) Maximum density flexible structure, with c= a.

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become flexible. However, some structures, such as goosecreekite,

GOO, resolutely remain inflexible (see discussion below).

It is worth remarking at this juncture that in an earlier

study,2 during the course of optimising the structures of

hypothetical frameworks, it was found that a simple harmonic

potential acting on T–O distances, O–T–O and T–O–T angles

alone, would allow non-codimeric oxygen atoms to overlap.

This tended to happen in the more complex topologies, those

with more than three unique T-atoms per fundamental region.

It was necessary to introduce repulsion terms between the

non-codimeric oxygen atoms in order to remove the flexibility,

and to inflate the unit cells towards their maximum volume.2

A similar coulomb repulsion force was also invoked in a study

on zeolite flexibility to explain the preference for maximum

unit cell volume.5

In Fig. 5 we present maps of the T–O–T angle and its mean

square root deviation from the preferred value (here, taken to

be 1451) for the FAU and DFT frameworks. Contours of

constant angle are shown. In the case of FAU, the configuration

with the maximum volume reproduces the experimental

preferred angles. However, it is a different state of affairs

for the DFT framework. It adopts a state close to the

maximum volume, despite the fact that a denser state with

mean Si–O–Si angle of 1451 is available within the flexibility

window.24 As a rule, pure silica zeolite frameworks tend

to adopt a structure that maximally inflates the unit cell,

whilst retaining perfect tetrahedra. This is consistent with

the argument that non-codimeric oxygen atoms in these

frameworks tend to repel each other, thereby expanding the

framework.

D Mixed tetrahedral types

Most zeolite materials have compositions with more than one

type of tetrahedral atom. For example, natural zeolites tend to

be aluminosilicates. A significant family of aluminophosphates

have been synthesized, and some unusual frameworks

have been synthesized with cobalt, gallium and arsenic atoms

in the tetrahedral positions. In addition, we have found

that some frameworks will only exhibit flexibility when tetra-

hedra of different sizes are present at specific sites in the

framework.

The recently-reported IWS framework has seven crystallo-

graphically distinct T-atoms.25 Experimental data reports that

sites 2, 4 and 6 are occupied by Ge with probability 32%

(Fig. 6). (Here, we are following the site-ordering reported in

ref. 25.) The Ge–O bond is longer (B0.176 nm) than the Si–O

bond (B0.161 nm). We find that IWS cannot be relaxed as a

pure silica (or germania) composition. By exploring all

27 = 128 possible combinations of Ge and Si tetrahedra

on the seven sites we find that the framework can become

flexible only if a Ge atom resides on site 4. The flexibility

window is largest if there is Ge on both sites 2 and 4. This

is consistent with the experimental evidence, and implies that

the composition of the IWS type-material, ITQ-26, is optimized

for flexibility.

Goosecreekite presents an interesting exception to the

flexibility rule. It has an aluminosilicate composition, and is

found as a mineral.26 The framework, GOO, can not be

relaxed as a pure silicate, or as an aluminosilicate. Examination

of the reported structure26 reveals that all of the SiO4 and

AlO�4 tetrahedra are slightly distorted, with the Si3 site being

distorted the most. Si3 is in close proximity to a divalent Ca2+

cation, which balances the framework charges associated with

the neighbouring Al1 and Al2 framework atoms (Fig. 7). Most

aluminosilicate zeolites are charge balanced by monovalent

cations such as Na+ and K+. The GOO example indicates

that strong cationic charges may stabilize distortions in the

framework tetrahedra. Thus, divalent cations may be helpful

for stabilizing frameworks that are not too far from being

flexible.

The recently-approved framework MVY offers a similar

example. The charge-balancing K+ extra-framework cations

in the type material, ZSM-70, cannot be ion-exchanged.27

They adhere tenaciously to the framework. Possibly, this

strong adherence is a result of their important role in stabilizing

the framework.

Fig. 5 Maps of the mean T–O–T angle and its mean square root

deviation d from the preferred value, which here is assumed to be

1451). (a) FAU framework. The outer perimeter is the boundary of the

flexibility window when c/a is the only unit cell degree of freedom. The

contours within the window are the lines of constant mean T–O–T

angle. (b) Contours of constant standard deviation in T–O–T angle

relative to 1451. (c) DFT framework. Both the mean and standard

deviation contours are shown on the same plot. The experimental

structure, DAF-2, is indicated by the black dot.

8538 | Phys. Chem. Chem. Phys., 2010, 12, 8531–8541 This journal is �c the Owner Societies 2010

Zeolite beta has long been a zeolitic material of great

interest because of its three-dimensional intersecting 12-ring

channels. Further, one of the polymorphs, *BEA is chiral, with

left- and right-handed helical 12-ring channels.28–30 Despite

concerted efforts, neither of the end-member polymorphs has

yet been synthesized in topologically pure form. Zeolite beta

can be thought of as an intergrowth of two end-member

framework types, polymorph A (*BEA) and polymorph B.

Polymorph B does not yet have an official framework code

assigned, and so here we will refer to it by an unofficial

lowercase code, beb. Crystallographic studies of zeolite beta

materials, which examine the stacking sequences carefully,

suggest that there is a slight preference for the beb

polymorph.29 Our own studies of flexibility find that the

flexibility parameter for pure silica form of beb is FO = 1.13,

which is bigger than that for *BEA, FO = 1.07. This flexibility

only emerges when we lower the space group symmetry from

(chiral) tetragonal P4122, to triclinic P1 (the chiral topology,

however, is still present in the lower-symmetry representation).

There are nine crystallographically unique T-sites in each

polymorph. We find that adding a single aluminium atom to

the *BEA framework makes the flexibility window disappear,

whereas for beb, five of the nine sites allow, or extend, the

flexibility. When substituting two aluminium atoms, relaxed

configurations were found in 2 out of 32 cases for *BEA and

12 out of 32 cases for beb. It appears that the beb framework

type is slightly more flexible than the *BEA framework type.

The detailed role of Al T-atoms on the flexibility of the zeolite

beta end-member frameworks is still not fully surveyed. It is

clear that for materials that are about 10% Al composition,

the flexibility argument suggests that the beb framework is the

preferred configuration, whereas *BEA will be more flexible as

a highly siliceous material. One question yet to be resolved

concerns the flexibility of the disordered intergrowth structure.

So far, we have examined only the pure end-member

polymorphs. This will entail examination of large structures

with extended cells along the c-axis to accommodate random

intergrowths.

Interrupted frameworks. The interrupted frameworks, those

with a ‘‘–’’ dash in front of the three-letter framework type

code, are frameworks containing tetrahedra with at least one

vertex that is terminated by a hydrogen atom and not

connected to another tetrahedron. The oxygen atom at the

terminated vertex is unconstrained by any neighbouring

tetrahedron. Fully-tetrahedral (tectosilicate) zeolites are

everywhere locally isostatic, that is the number of coordinate

degrees of freedom is equal to the number of constraints.

Interrupted frameworks (inosilicates) are locally hypostatic at

the interrupted tetrahedra, that is the number of coordinate

degrees of freedom is greater than the number of constraints.

Intuitively, one would expect interrupted frameworks to

exhibit increased flexibility. Surprisingly, the Table shows that

the seven interrupted frameworks have a spread of flexibility

parameters similar to those for the regular frameworks. At the

high end is -LIT with a flexibility parameter of 1.56, whereas

-PAR does not show flexibility in the pure silica form.

Thermodynamics of flexible frameworks. Since the presence

of a flexibility window indicates that the framework can be

formed by perfect regular tetrahedra, then the framework

energy of formation will also be low. This follows because

most zeolite structures are known to comprise near-perfect

tetrahedra. Tetrahedral distortions will cost energy. We would

expect a siliceous framework that possesses a flexibility

window to also have a low framework energy. As we have

found, a low framework energy does not necessarily guarantee

the presence of a flexibility window. Goosecreekite (GOO) is

such an example.

The large databases of hypothetical zeolites that are now

emerging tend to be sorted according to framework energy.1–3

However, a low framework energy does not guarantee that the

framework is synthetically viable. The presence of a flexibility

Fig. 6 Unit cell of the germanosilicate framework of IWS. The pure

SiO2 composition cannot be formed without straining T sites T2 and

T4. The material ITQ-26 has germanium atoms at these sites. The

larger GeO4 tetrahedra at these particular sites restore flexibility to the

IWS framework.

Fig. 7 Fragment of the Goosecreekite structure. This aluminosilicate

zeolite framework is unusual in that it exhibits no flexibility window.

The framework can not be formed from perfect silicon and aluminium

tetrahedra, without forcing the oxygen atoms to overlap (the oxygen

atoms have a nominal radius of 0.135 nm, and are not shown to scale

here). It is likely that the divalent calcium cations and trivalent Al

cations in the vicinity of each Si3 tetrahedron serve to stabilize the

tetrahedral distortions inevitably present in the GOO framework.

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window for a given zeolite topology indicates that there is a

potentially large configuration space that corresponds to

the same zeolite topology. It is known that vibrational

entropy, which explores local configurations, is an important

consideration in zeolite formation.31

However, in our model of an ideal zeolite framework that is

in a configuration inside its flexibility window, the framework

energy is zero. Apart from the bond length constraints, there

are no forces acting on an ideal zeolite framework that is in a

flexible state. There are no forces guiding it towards any

preferred configuration. The energy landscape within the

window is a flat null space. The absence of harmonic forces

means that if the framework is given kinetic energy, the null

eigenvectors do not oscillate about a mean position, but can

grow in amplitude as the motion progresses. Vibrational

modes will still be associated with the stiff springs within the

tetrahedra, but we are assuming that the springs are too rigid

to activate such modes. The growing modes are related to the

flexible folding modes. At a non-zero temperature, there will

be a kinetic energy per unit cell associated with the framework.

The resulting motion allows the system to explore the available

states within the window. This is analogous to an ideal

gas, except that a zeolite framework has significantly more

constraints on the atomic motions.

At maximum volume, the unit cell is almost always in its

maximum symmetry state. There is only one known exception

to this rule in crystallography,32 and no exceptions have

yet been found for zeolite frameworks. At this maximum

symmetry state, many of the distance constraints become

degenerate and there is an increase in the number of null

eigenvectors in this maximum symmetry state. Once one of the

null eigenvectors is excited at large amplitude, symmetry is

lost, and most of the null states vanish. For example, the FAU

framework has 39 null eigenvectors at maximum volume

where it has Fd�3m symmetry. The number of null modes

drops significantly once any one of these states is excited with

a large amplitude.

IV. Conclusions

We have examined the flexibility of ideal zeolite frameworks,

by treating them as idealized assemblies of perfect, rigid,

tetrahedra that are interconnected at their corners via

force-free spherical joints. The flexibility of 194 known zeolite

frameworks has been analyzed, and we find that 182 of them

are flexible when modeled as a pure SiO2 composition. Most of

the 12 exceptions become flexible once a mixed-tetrahedron

composition is allowed. This establishes an important new

characteristic of zeolite frameworks that further unifies zeolites

as an important and unusual group of similar framework

structures.

Despite the fact that the most of the configurations inside

the flexibility window are not accessible in real zeolites due to

their high energy, zeolites with larger flexibility windows have

a much higher possibility of appearing during synthesis, since

most currently known zeolites (SOD, FAU etc) do indeed have

large flexibility windows.

Singular value decomposition of the kinematic matrices is

emerging as a powerful tool for exploring the flexibility

window. Reliable eigenvalue-following algorithms are needed

in order to navigate efficiently through the bifurcation points

encountered while following folding mechanisms within the

flexibility window.

We emphasize that the ideal zeolite frameworks studied here

are idealized geometric representations of the real structures.

Even in quartz, the SiO4 tetrahedra deviate slightly from

perfection. The quartz framework is isohedral, possessing

one topologically unique intertetrahedral angle (T–O–T) that

adopts an angle around 1441 at room temperature. The

maximum volume state for quartz would adopt an angle of

154.61, which elevates the framework energy. Most zeolite

frameworks are not isohedral, possessing several topologically

nonequivalent T–O–T angles, not all of which can settle at the

preferred angle of B1441. Such geometrically conflicted

structures can express a wide range of T–O–T angles, along

with small distortions of the tetrahedra. These deviations are

caused primarily by electronic interactions. In addition, alumino-

silicate compositions, in addition to introducing tetrahedra of

different sizes, also introduce framework charge that is

balanced by extra-framework cations, such as H+, Na+ and

K+. These additional electrostatic terms introduce opportunities

for additional framework distortions. Nevertheless, the structural

differences between ideal zeolite frameworks and the real

zeolite materials are remarkably small.

The near-universal flexibility of ideal zeolite frameworks is

emerging as an important topological property of their

structures. Flexibility provides us with a useful selection

criterion for evaluating whether or not a hypothetical zeolite

framework is realisable. A desirable extension of this work is

the development of a method to infer flexibility from the graph

of the framework.

This is a relatively unexplored area that is highly relevant to

all oxide framework materials, and promises to be an exciting

area of research.

Acknowledgements

The authors are grateful to Asel Sartbaeva (University of

Oxford), Stephen Wells (University of Warwick) and Simon

Guest (University of Cambridge) for stimulating discussions.

We also acknowledge the financial support of the National

Science Foundation grants Nos. DMR-0703973 and DMS-

0714953, and the Donors of the American Chemical Society

Petroleum Research Fund for partial support of this research.

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