# Template-Based Geometric Simulation of Flexible Frameworks

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Comparison to Conventional Simulation Methods

## 3. Templates for Geometric Analysis and Simulation

#### 3.1. Polyhedral Alignment and Residual Mismatch

#### 3.2. Force Models in Geometric Simulation

## 4. Applications of Geometric Analysis and Simulation

#### 4.1. Dynamic Disorder in Frameworks

#### 4.2. Framework Flexibility and Motion

#### 4.3. Compression Mechanisms in Zeolites

**EDI**) [29] and levyne (

**LEV**) [30] have revealed subtle connections between compression of the unit cell and the collective response of the framework. For example, in the levyne framework

**LEV**, a form of “internal auxetic” effect couples compression of the cell to variations in channel profile in a non-obvious way—a uniform contraction of a channel profile is produced by a non-uniform compression of the cell.

#### 4.4. Generation of Hypothetical Zeolites

#### 4.5. Application to Manganites: Modelling of Jahn–Teller Distortion

#### 4.6. The Flexibility Window of Zeolites

#### 4.7. Phase Transitions in Zeolites

**ANA**framework—leucite, pollucite and wairakite [42]—and in wairakite, Al/Si ordering in the framework is significant [43].

#### 4.8. Application to Proteins

## 5. Availability of Geometric Simulation Codes

## Acknowledgements

## References

- Wragg, D.S.; Morris, R.S.; Burton, A.W. Pure silica zeolite-type frameworks: A structural analysis. Chem. Mater.
**2008**, 20, 1561–1570. [Google Scholar] [CrossRef] - Baur, W.H. Straight Si–O–Si bridging bonds do exist in silicates and silicon dioxide polymorphs. Acta Crystallogr.
**1980**, B36, 2198–2202. [Google Scholar] [CrossRef] - Dove, M.T.; Heine, V.; Hammonds, K.D. Rigid unit modes in framework silicates. Mineral. Mag.
**1995**, 59, 629–639. [Google Scholar] [CrossRef] - Dove, M.T.; Pryde, A.K.A.; Heine, V.; Hammonds, K.D. Exotic distributions of rigid unit modes in the reciprocal spaces of framework aluminosilicates. J. Phys. Condens. Matter
**2007**, 19, 275209. [Google Scholar] [CrossRef] - Hammonds, K.D.; Dove, M.T.; Giddy, A.P.; Heine, V. CRUSH: A FORTRAN program for the analysis of the rigid unit mode spectrum of a framework structure. Am. Mineral.
**1994**, 79, 1207–1209. [Google Scholar] - Giddy, A.P.; Dove, M.T.; Pawley, G.S.; Heine, V. The determination of rigid unit modes as potential soft modes for displacive phase transitions in framework crystal structures. Acta Crystallogr.
**1993**, A49, 697–703. [Google Scholar] [CrossRef] - Wells, S.A.; Dove, M.T.; Tucker, M.G.; Trachenko, K. Real-space rigid-unit-mode analysis of dynamic disorder in quartz, cristobalite and amorphous silica. J. Phys. Condens. Matter
**2002**, 14, 4645. [Google Scholar] [CrossRef] - Gale, J.D. GULP: A computer program for the symmetry-adapted simulation of solids. J. Chem. Soc. Faraday Trans.
**1995**, 93, 629–637. [Google Scholar] [CrossRef] - Bozin, E.S.; Sartbaeva, A.; Zheng, H.; Wells, S.A.; Mitchell, J.F.; Proffen, T.; Thorpe, M.F.; Billinge, S.J.L. Structure of CaMnO
_{3}in the range 10K–550K from neutron time-of-flight total scattering. J. Phys. Chem. Solids**2008**, 69, 2146–2150. [Google Scholar] [CrossRef] - Wells, S.A.; Dove, M.T.; Tucker, M.G. Finding best-fit polyhedral rotations with geometric algebra. J. Phys. Condens. Matter
**2002**, 14, 4567–4584. [Google Scholar] [CrossRef] - Sartbaeva, A.; Wells, S.A.; Redfern, S.A.T. Li
^{+}ion motion in quartz and β-eucryptite studied by dielectric spectroscopy and atomistic simulations. J. Phys. Condens. Matter**2004**, 16, 8173–8189. [Google Scholar] [CrossRef] - Sartbaeva, A.; Wells, S.A.; Redfern, S.A.T.; Hinton, R.W.; Reed, S.J.B. Ionic diffusion in quartz studied by transport measurements, SIMS and atomistic simulations. J. Phys. Condens. Matter
**2005**, 17, 1099–1112. [Google Scholar] [CrossRef] - Bornhauser, J.B.; Bougeard, D. Intensities of the vibrational spectra of siliceous zeolites by molecular dynamics calculations II—Raman spectra. J. Raman Spectrosc.
**2001**, 32, 279–285. [Google Scholar] [CrossRef] - Baerlocher, C.; Hepp, A.; Meier, W.M. DLS-76, A FORTRAN Program for the Simulation of Crystal Structures by Geometric Refinement; Institut fur Kristallographie and Petrographie ETH: Zurich, Switzerland, 1978. [Google Scholar]
- Kapko, V.; Dawson, C.; Treacy, M.M.J.; Thorpe, M.F. Flexibility of ideal zeolite frameworks. Phys. Chem. Chem. Phys.
**2010**, 12, 8531–8541. [Google Scholar] [CrossRef] [PubMed] - Kapko, V.; Dawson, C.; Rivin, I.; Treacy, M.M.J. Density of mechanisms within the flexibility window of zeolites. Phys. Rev. Lett.
**2011**, 107, 164304. [Google Scholar] [CrossRef] [PubMed] - Dove, M.T.; Tucker, M.G.; Keen, D.A. Neutron total scattering method: Simultaneous determination of long-range and short-range order in disordered materials. Eur. J. Mineral.
**2002**, 14, 331–348. [Google Scholar] [CrossRef] - Wells, S.A.; Dove, M.T.; Tucker, M.G. Reverse Monte Carlo with geometric analysis—RMC + GA. J. Appl. Crystallogr.
**2004**, 37, 536–544. [Google Scholar] [CrossRef] - Keen, D.A.; Tucker, M.G.; Dove, M.T. Reverse Monte Carlo modelling of crystalline disorder. J. Phys. Condens. Matter
**2005**, 17, S15–S22. [Google Scholar] [CrossRef] - Kimizuka, H.; Kaburaki, H.; Kogue, Y. Molecular-dynamics study of the high-temperature elasticity of quartz above the alpha-beta phase transition. Phys. Rev. B
**2003**, 167, 024105. [Google Scholar] [CrossRef] - Goodwin, A.L.; Redfern, S.A.T.; Dove, M.T.; Keen, D.A.; Tucker, M.G. Ferroelectric nanoscale domains and the 905 K phase transition in SrSnO
_{3}: A neutron total-scattering study. Phys. Rev. B**2007**, 76, 174114. [Google Scholar] [CrossRef] - Hui, Q.; Tucker, M.G.; Dove, M.T.; Wells, S.A.; Keen, D.A. Total scattering and reverse Monte Carlo study of the 105 K displacive phase transition in strontium titanate. J. Phys. Condens. Matter
**2005**, 17, S111–S124. [Google Scholar] [CrossRef] - Tucker, M.G.; Goodwin, A.L.; Dove, M.T.; Keen, D.A.; Wells, S.A.; Evans, J.S.O. Negative thermal expansion in ZrW
_{2}O_{8}: Mechanisms, rigid unit modes, and neutron total scattering. Phys. Rev. Lett.**2005**, 95, 255501. [Google Scholar] [CrossRef] [PubMed] - Conterio, M.J.; Goodwin, A.L.; Tucker, M.G.; Keen, D.A.; Dove, M.T.; Peters, L.; Evans, S.O. Local structure in Ag
_{3}[Co(CN)_{6}]: Colossal thermal expansion, rigid unit modes and argentophilic interactions. J. Phys. Condens. Matter**2008**, 20, 255225. [Google Scholar] [CrossRef] - Goodwin, A.L. The crystallography of flexibility: Local structure and dynamics in framework materials. Z. Krist.
**2009**, 30, 1–11. [Google Scholar] [CrossRef] - Sartbaeva, A.; Redfern, S.A.T.; Lee, W.T. A neutron diffraction and Rietveld analysis of cooperative Li motion in beta-eucryptite. J. Phys. Condens. Matter
**2004**, 16, 5267–5278. [Google Scholar] [CrossRef] - Wragg, D.S.; Akporiaye, D.; Fjellvag, H. Direct observation of catalyst behaviour under real working conditions with X-ray diffraction: Comparing SAPO-18 and SAPO-34 methanol to olefin catalysts. J. Catal.
**2011**, 279, 397–402. [Google Scholar] [CrossRef] - Goodwin, A.L.; Wells, S.A.; Dove, M.T. Cation substitution and strain screening in framework structures: The role of rigid unit modes. Chem. Geol.
**2006**, 225, 213–221. [Google Scholar] [CrossRef] - Gatta, G.D.; Wells, S.A. Rigid unit modes at high pressure: An explorative study of a fibrous zeolite-like framework with
**EDI**topology. Phys. Chem. Miner.**2004**, 31, 1–10. [Google Scholar] [CrossRef] - Gatta, G.D.; Wells, S.A. Structural evolution of zeolite levyne under hydrostatic and non-hydrostatic pressure: Geometric modelling. Phys. Chem. Miner.
**2006**, 33, 1–10. [Google Scholar] [CrossRef] - Zwijnenburg, M.A.; Simperler, A.; Wells, S.A.; Bell, R.G. Tetrahedral distortion and energetic packing penalty in “zeolite” frameworks: Linked phenomena? J. Phys. Chem. B
**2005**, 109, 14783–14785. [Google Scholar] [CrossRef] [PubMed] - Treacy, M.M.J.; Randall, K.H.; Rao, S.; Perry, J.A.; Chadi, D.J. Enumeration of periodic tetrahedral frameworks. Z. Krist.
**1997**, 212, 768–791. [Google Scholar] [CrossRef] - Treacy, M.M.J.; Rivin, I.; Balkovsky, E.; Randall, K.H.; Foster, M.D. Enumeration of periodic tetrahedral frameworks. II. Polynodal graphs. Microporous Microporous Mater.
**2004**, 74, 121–132. [Google Scholar] [CrossRef] - Boisen, M.B.; Gibbs, G.V.; Bukowinski, M.S.T. Framework silica structures generated using simulated annealing with a potential energy function based on an H
_{6}Si_{2}O_{7}molecule. Phys. Chem. Miner.**1994**, 21, 269–284. [Google Scholar] [CrossRef] - Wells, S.A.; Foster, M.D.; Treacy, M.M.J. A simple geometric structure optimizer for accelerated detection of infeasible zeolite graphs. Microporous Microporous Mater.
**2007**, 93, 151–157. [Google Scholar] [CrossRef] - Sartbaeva, A.; Wells, S.A.; Thorpe, M.F.; Bozin, E.S.; Billinge, S.J.L. Quadrupolar ordering in LaMnO
_{3}revealed from scattering data and geometric modelling. Phys. Rev. Lett.**2007**, 99, 155503. [Google Scholar] [CrossRef] [PubMed] - Sartbaeva, A.; Wells, S.A.; Thorpe, M.F.; Bozin, E.S.; Billinge, S.J.L. Geometric simulation of perovskite frameworks with Jahn-Teller distortions: Applications to the cubic manganites. Phys. Rev. Lett.
**2006**, 97, 065501. [Google Scholar] [CrossRef] [PubMed] - Billinge, S.J.L. Nanoscale structural order from the atomic pair distribution function (PDF): There’s plenty of room in the middle. J. Solid State Chem.
**2008**, 181, 1695–1700. [Google Scholar] [CrossRef] - Sartbaeva, A.; Wells, S.A.; Treacy, M.M.J.; Thorpe, M.F. The flexibility window in zeolites. Nat. Mater.
**2006**, 5, 962–965. [Google Scholar] [CrossRef] [PubMed] - Rivin, I. Geometric simulations: A lesson from virtual zeolites. Nat. Mater.
**2006**, 5, 931–932. [Google Scholar] [CrossRef] [PubMed] - Sartbaeva, A.; Gatta, G.D.; Wells, S.A. Flexibility window controls pressure-induced phase transition in analcime. Europhys. Lett.
**2008**, 83, 26002. [Google Scholar] [CrossRef] - Gatta, G.D.; Sartbaeva, A.; Wells, S.A. Compression behaviour and flexibility window of the analcime-like feldspathoids: Experimental and theoretical findings. Eur. J. Mineral.
**2009**, 21, 571–580. [Google Scholar] [CrossRef] - Wells, S.A.; Sartbaeva, A.; Gatta, G.D. Flexibility windows and phase transitions of ordered and disordered
**ANA**framework zeolites. Europhys. Lett.**2011**, 94, 56001. [Google Scholar] [CrossRef] - Wells, S.A.; Menor, S.; Hespenheide, B.M.; Thorpe, M.F. Constrained geometric simulation of diffusive motion in proteins. Phys. Biol.
**2005**, 2, S127–S136. [Google Scholar] [CrossRef] [PubMed] - Jacobs, D.J.; Rader, A.J.; Kuhn, L.A.; Thorpe, M.F. Protein flexibility predictions using graph theory. Proteins Struct. Funct. Bioinfom.
**2001**, 44, 150–165. [Google Scholar] [CrossRef] [PubMed] - Farrell, D.W.; Speranskiy, K.; Thorpe, M.F. Generating stereochemically acceptable protein pathways. Proteins Struct. Funct. Bioinfom.
**2010**, 78, 2908–2921. [Google Scholar] [CrossRef] [PubMed] - Ahmed, A.; Rippmann, F.; Barnickel, G.; Gohlke, H. A normal mode-based geometric simulation approach for exploring biologically relevant conformational transitions in proteins. J. Chem. Inform. Model.
**2011**, 51, 1604–1622. [Google Scholar] [CrossRef] [PubMed] - Jimenez-Roldan, J.E.; Freedman, R.B.; Roemer, R.A.; Wells, S.A. Protein flexibility explored with normal modes and geometric simulation. Phys. Biol.
**2012**, 9, 016008. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**(

**a**) Part of a silicate mineral framework containing Si and O atoms, with the SiO${}_{4}$ units viewed as polyhedra; (

**b**) a rigid-unit model of the framework for calculation of RUMs. The polyhedra are considered as rigid units and their vertices are connected by harmonic constraints; (

**c**) templates and constraints for geometric simulation. A tetrahedral template is centred on each Si atom. Vertex O atoms are tethered by harmonic constraints to template vertices.

**Figure 2.**The mismatch between an atom and a template vertex is decomposed into components of bond-stretching, parallel to the bond vector in the template, and of bond-bending, perpendicular to the bond vector in the template.

**Figure 3.**Force model for geometric simulation. The oxygen atom at the centre of the diagram is in steric contact with a nearby atom (above) and is connected to two template vertices to right and left. The mismatch of the atom from a template position is resolved into bond-stretching and bond bending components. Harmonic penalties apply to the bond-bending distortions, bond-stretching distortions, and steric overlap of atomic spheres.

**Figure 4.**Structural models of domain walls in quartz, viewed down the crystallographic c axis (from [12]). The upper panel shows a geometrically relaxed configuration using $\alpha $-quartz cell parameters, in which the domain wall (indicated by line) shows an elliptical channel profile. The lower panel shows a geometrically relaxed configuration using cell parameters from the incommensurate phase near the $\alpha $–$\beta $ phase transition; the domain wall is less distinct.

**Figure 5.**Use of regular (dashed outline) and Jahn–Teller distorted (solid outline) octahedral templates to model a manganite perovskite framework. From [36].

**Figure 6.**Faujasite (

**FAU**) framework during an exploration by geometric simulation of the flexibility window, from the high-density limit at left to the low-density, expanded limit at right. The upper half of the framework is shown in a polyhedral view while the lower half is shown in a space-filling view to emphasise the significance of the oxygen atomic radii. From [39].

**Figure 7.**Use of overlapping templates (shown as red, green and yellow sticks) as constraints in the simulation of proteins. The tethering of atoms (grey) to these overlapping templates constrains interatomic bond lengths and angles but allows dihedral angles to vary.

© 2012 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/.)

## Share and Cite

**MDPI and ACS Style**

Wells, S.A.; Sartbaeva, A.
Template-Based Geometric Simulation of Flexible Frameworks. *Materials* **2012**, *5*, 415-431.
https://doi.org/10.3390/ma5030415

**AMA Style**

Wells SA, Sartbaeva A.
Template-Based Geometric Simulation of Flexible Frameworks. *Materials*. 2012; 5(3):415-431.
https://doi.org/10.3390/ma5030415

**Chicago/Turabian Style**

Wells, Stephen A., and Asel Sartbaeva.
2012. "Template-Based Geometric Simulation of Flexible Frameworks" *Materials* 5, no. 3: 415-431.
https://doi.org/10.3390/ma5030415