computer programs
Inorganic structure prediction with GRINSP
^{a}Université du Maine, Laboratoire des oxydes et Fluorures, CNRS UMR 6010, Avenue O. Messiaen, 72085 Le Mans Cedex 9, France
^{*}Correspondence email: alb@cristal.org
A new computer program is described, GRINSP (geometrically restrained inorganic structure prediction), which allows the exploration of the possibilities of occurrence of 3, 4, 5 and 6connected threedimensional networks. Hypothetical (as well as known structure) models for binary compounds are produced with exclusive connection of polyhedra by corners, such as [MX_{3}] triangles in M_{2}X_{3} formulation, [MX_{4}] tetrahedra in MX_{2} (zeolites or dense SiO_{2} polymorphs), [MX_{5}] polyhedra in M_{2}X_{5}, and finally [MX_{6}] octahedra in MX_{3} polymorphs. Moreover, hypothetical ternary compounds are built up by combinations of either two different polyhedra or two different radii for two different cations adopting the same coordination. The cost function is based on the agreement of the model interatomic distances with ideal distances provided by the user. The Monte Carlo algorithm first finds structure candidates selected after the verification of the expected geometry, and then optimizes the cell parameters and the atomic coordinates. A satellite software (GRINS) uses the predicted models and produces the characteristics of isostructural compounds which would be obtained by cationic substitutions. A huge list of files of hypothetical boron oxide polymorphs (including nanotubes), zeolites, aluminium and 3delement fluorides, fluoroaluminates, borosilicates, titanosilicates, gallophosphates etc., is freely available at the PCOD (Predicted Crystallography Open Database).
Keywords: structure prediction; inorganic compounds; Monte Carlo; crystallography database; nanotube; computer programs.
1. Introduction
The final aim of structure prediction should be to announce a Stuctural Aspects of Oxide and Oxysalt Crystals, Frank C. Hawthorne (1994) stated, ten years ago, that: `The goals of theoretical crystallography may be summarized as follow: (1) predict the stoichiometry of the stable compounds; (2) predict the bond topology (i.e. the approximate atomic arrangement) of the stable compounds; (3) given the bond topology, calculate accurate bond lengths and angles (i.e. accurate atomic coordinates and cell dimensions); (4) given accurate atomic coordinates, calculate accurate static and dynamic properties of a crystal. For oxides and oxysalts, we are now quite successful at (3) and (4), but fail miserably at (1) and (2)'. This seems in contradiction with a previous statement by Catlow & Price (1990), four years earlier, that `computational methods can now make detailed and accurate predictions of the structures of inorganic materials'. The fact is that predictions of inorganic compounds mentioned in a recent book about computer modelling in inorganic crystallography (Catlow, 1997) are very few, if one excludes hypothetical zeolites. Moreover, in the case of organic molecules, the predictions do not appear to be any more brilliant, based on the results of a recent blind test (Motherwell et al., 2002). If the state of the art had dramatically evolved in the past ten years, we should have a huge database of predicted compounds, and no new would surprise us since it would correspond to an entry in that database. Moreover, we would have obtained in advance the physical properties and we would have preferably synthesized those interesting compounds. Of course, this is absolutely not the case, unfortunately. However, two databases of hypothetical compounds were built in 2004. One is exclusively devoted to zeolites (Foster & Treacy, 2004); the other includes zeolites as well as other predicted oxides (borosilicates, titanosilicates, gallophosphates etc.) and fluorides (Le Bail, 2004). Such databases will play a role analogous to databases of actually existing structures: in principle they preclude the prediction of a structure that has already been predicted, or the redetermination/republishing of a known structure. Moreover, calculated powder patterns from these databases would be useful at the identification stage, provided that the accuracy level of prediction is high (observed and predicted cellparameter differences smaller than 2%).
before any confirmation by chemical synthesis or discovery in nature. In a lead article entitledLet us cite a few of the computer programs and methods producing predictions in the inorganic world. CASTEP uses the density functional theory (DFT) for ab initio modelling, applying a pseudopotential planewave code (Payne et al., 1992). The structures gathered in the database of hypothetical zeolites (Foster & Treacy, 2004) are produced from a 64processor computer cluster, grinding away nonstop, generating graphs and annealing them, the selected frameworks being then reoptimized using the General Utility Lattice Program, GULP (Gale, 1997), using atomic potentials. GULP itself is able to predict crystal structures (TiO_{2} polymorphs). Recently, a was implemented (Woodley, 2004) in GULP in order to generate crystal framework structures from the knowledge of only the unitcell dimensions and constituent atoms (however, according to the definitions above, this is not prediction); the structures of the better candidates produced are relaxed by minimizing the energy, which is based on the Born model of a solid. The concept of `energy landscape' of chemical systems is used by Schön & Jansen (2001a,b) for structure prediction with their computer program G42. Another package, SPuDS, is dedicated especially to the prediction of perovskites (Lufaso & Woodward, 2001). The AASBU method (automated assembly of secondary building units) is developed by MellotDraznieks et al. (2000, 2002), using Cerius2 (2000) and GULP in a sequence of simulatedannealing plus minimization steps for the aggregation of large structural motifs. This list of software is rather small considering the fact that structure and properties prediction is obviously an unavoidable part of our future in crystallography and chemistry.
Possibilities for structure prediction which would be easily available freely to academic users appear to be limited somewhat. Apart from the broadly explored zeolite subject, one cannot find many atomic coordinates of hypothetical compounds in databases. Moreover, it seems better to gather hypothetical compounds in a specific database, different from those of determined crystal structures, because predictions will be much more numerous than confirmations. This, combined with the fact that we ought no longer to `fail miserably' at predicting the stoichiometry and the approximate atomic arrangement of stable compounds (see above), prompted the development of new software, GRINSP (geometrically restrained inorganic structure prediction). This computer program is described below, enabling the exploration of hypothetical 3, 4, 5 and 6connected threedimensional networks, in binary and ternary inorganic compounds, using a Monte Carlo approach.
2. GRINSP algorithm
2.1. Monte Carlo generation of structure candidates
With GRINSP, the occurrence of M_{u}X_{w} or M_{v}X_{w} models depends on a drastic selection when trying to build the net of M/M′ atoms. First, a and the M/X or M/M′/X cornersharing system to be explored are chosen; then a single initial M or M′ atom (selected at random) is placed at random coordinates (at one itself selected at random) in a box, the dimensions of which are again selected at random. The next M or M′ atoms are placed randomly in delimited volumes close to the M or M′ atoms already positioned (these volumes are restricted by the range of provided interatomic distances). Generally, 300000 Monte Carlo tests for placing atoms are realised before a new series of tests is started with different cell parameters. At this stage, in order to be retained, an M/M′ model should exactly correspond to the geometrical specifications (with exact coordinations, though the distances can vary: for instance, if M is decided to be in sixfold coordination, one has to find six M or M′ atoms around it at the end of the process). The fact that distances are given a large tolerance range allows the capture of many solutions which may not correspond to regular polyhedra. In other words, the Monte Carlo random walker may stay far above the deep local minimum of interest. In this first step, atoms do not move: their possible positions are only tested and checked; then they are retained or discarded. If the process fails before the end of the allowed series of tests (the number of tests for positioning a new atom inside of the defined restricted volumes is limited by the use of `insistence factors'), a new initial M or M′ atom is placed without changing the cell etc. The cell is progressively filled up to respect the geometrical restraints completely, if possible. The number of M/M′ atoms placed is not predetermined. The process is thus different from the AASBU approach, or from the simulatedannealing approach used in pioneering studies on zeolites (Deem & Newsam, 1989, 1992; Newsam et al., 1992), since GRINSP explores a large range of cell parameters for a given instead of concentrating on known cell parameters with a given number of M atoms moving up to find some energy cost function minimum. It is, however, obvious that GRINSP can also be used as a structure solution tool for cornersharing systems of polyhedra, including zeolites, if the cell parameters are known (but such structure solution takes us beyond the realms of structure prediction).
2.2. Model optimization
In a second step, the X atoms are added between the (M/M′)–(M/M′) first neighbours, at the midpoints, and it is verified by distance and cell improvements (using a Monte Carlo approach as well) that regular [(M/M′)X_{n}] polyhedra can really be built, i.e. that there is a deep local minimum existing close to this previously selected rough arrangement of (M/M′) atoms. The cost function enabling the finding of a minimum R is based on the verification of ideal (M/M′)–(M/M′), (M/M′)–X and X–X firstneighbour distances, provided by the user. The total R factor is defined by
where R_{n} and R_{0n} for n = 1, 2, 3 are defined by the expressions
and
where the d_{0n} values for n = 1 to 3 are the ideal first interatomic distances (M/M′)–X (n = 1), X–X (n = 2) and (M/M′)–(M/M′) (n = 3), whereas the d_{n} values are the corresponding observed distances in the structure model for these atom pairs. The weights retained (w_{n}) are the same as those used in the DLS software (Baerlocher et al., 1978) for the calculation of idealized framework data (w_{1} = 2.0, w_{2} = 0.61 and w_{3} = 0.23). The ideal distances are to be provided by the user for pairs of atoms supposed to form polyhedra (for instance in the case of [SiO_{4}] tetrahedra, one expects to have d_{1} = 1.61 Å, d_{2} = 2.629 Å and d_{3} = 3.07 Å). The similarity of the cell parameters estimated by GRINSP for zeolites with the idealized cell constant listed at the official zeolite Web site (Database of Zeolite Structures, http://www.izastructure.org/databases/) is thus not fortuitous, since these idealized values are calculated by using the DLS software applying a similar cost function during the distance leastsquares refinements. Some differences may come from the spacegroup constraint (always P1 with GRINSP).
The strategy for the model optimization is first to allow 1/4 of the Monte Carlo events (NA × 10–20000 events, generally, where NA is the total number of atoms in the cell) only to move randomly the M/M′/X atoms; then another 1/4 are exclusively devoted to random cellparameter changes, and finally the remaining Monte Carlo events are used for both kind of changes, chosen randomly. A smooth quenching is imposed: the maximum amplitudes of the changes are progressively reduced during the optimization process.
For ternary compounds, the M–M′ ideal distances are calculated by GRINSP as being the average of the M–M and M′–M′ distances. During this second step of optimization, all the atoms can move, but no jump is allowed because a jump would break the coordinations established at the first step. The change in the cell parameters from the structure candidate to the final model may be quite considerable (up to 30%); this explains why some models may show parameters that are larger or smaller than the limits defined at the beginning of the runs, these limits being applied only to the results of the first step (when placing the M/M′ atoms). During the optimization, the original selected for placing the M atoms may not be conserved after having added the X atoms, so that the final structure is always proposed in the P1 The final cell characteristics and atomic coordinates are presented in a file. An ultimate check of the real symmetry has to be performed by using a program like PLATON (Spek, 2003).
The models produced by GRINSP may need further optimization by using bond valence rules, or energy calculations; however, in many cases the predicted cell parameters differ by less than 2% from the real ones, when the real compounds are built up from ideal polyhedra, which is the case with dense SiO_{2} polymorphs or zeolites (Table 1) and fluoroaluminate phases (Table 2). Choosing to use one precise ideal M–M firstneighbour distance, depending on the M–X–M angles (even if coming from an average value produced by data mining), will produce the smaller R values for particular models. In Table 1, the quartz structure is clearly favoured (R = 0.0006). In Table 2, the smaller R value corresponds to the HTB model, not to the perovskite one. Modifying the Al–Al distance in order to have an Al–F–Al angle of 180° would of course have favoured a small R value for the perovskite structure, but without obtaining cell parameters closer to the observed ones in the Rc (a cubic would have been obtained instead). This shows that no confidence can be given to a precise classification by R values in a range of, say 0 < R < 0.01. Moreover, values 0.01 < R < 0.02 may well correspond to existing compounds (R = 0.0159 for τAlF_{3} in Table 2, offering a large distribution of Al–F–Al angles).

2.3. The GRINS satellite program
Searching for the characteristics of isostructural hypothetical compounds obtained by cation substitution (FeF_{3} or GaF_{3} etc., instead of AlF_{3} for instance), it is not necessary to run again the structure prediction software GRINSP. A satellite program named GRINS was developed, including a modified version of the structure optimization part (Monte Carlo adjustment of the atomic coordinates and cell parameters). This software uses the desired starting M/M′ positions and cell parameters, and finds the minimum R factor corresponding to any new set of ideal interatomic distances for new cation/anion pairs selected by the user.
3. Results
3.1. Binary compounds
Formulations M_{2}X_{3}, MX_{2} and MX_{3} were partly examined (not yet M_{2}X_{5} which would occur for M cations in fivefold coordination, since [MX_{5}] polyhedra cannot be regular).
The complete exploration of the zeolites is still not finished. More than a thousand models are expected to be produced by GRINSP with R < 0.01 and cell parameters <16 Å. The PCOD database (Le Bail, 2004) already contains more than 300 models, mainly in cubic and hexagonal symmetry. Examples establishing the quality of the predictions are presented in Table 1, showing some of the already known zeotypes retrieved by the program. The files can be obtained by consulting the PCOD, giving the entry number provided with the figure captions, for instance PCOD1030081 (Fig. 1).
Not many crystalline varieties are known for the B_{2}O_{3} composition. Too many were proposed by GRINSP, even reducing the limit to R < 0.006 (see an example in Fig. 2).
Apart from the well known perovskite structure type, which can be retrieved in almost all space groups during the exploration of the 6connected threedimensional nets with GRINSP, all the known structure types with AlF_{3} formulation were retrieved (Table 2), including the most complex one recently discovered, τAlF_{3} (Le Bail et al., 1992). A series of `yet to be synthesized' AlF_{3} polymorphs were also proposed, one example being presented in Fig. 3. A detailed study of the hypothetical MF_{3} phases (M = Al, Cr, V, Fe, Mn, Ga) will be published elsewhere (Le Bail, 2005).
3.2. Ternary compounds
For ternary compounds, M and M′ cations are considered. They could have the same coordination but different ionic radii (enabling the exploration of ordered aluminosilicates or aluminophosphates etc.) or different coordination (exploring calcium–aluminium fluorides, titanosilicates, gallophosphates, borosilicates etc.), but the current limitation with GRINSP is that the connections by X atoms will only be by corner sharing: all X atoms should be connected to exclusively two M atoms or two M′ atoms or one M and one M′ atom. As a consequence, only some formulations can occur which fulfill these conditions. Moreover, if M or M′ are not able to form electrically neutral binary compounds with cornersharing only, then the built ternary compound will also not be electrically neutral. All the borosilicates formed with GRINSP are automatically electrically neutral. There is only one hit in the ICSD database for this kind of compound. A strange result is that GRINSP produces a huge quantity of hypothetical borosilicates, showing exclusively [BO_{3}] triangles and [SiO_{4}] tetrahedra linked by corners. Limiting R < 0.006, and working in cell symmetry higher or equal to monoclinic, but using the general of the P1 57 different models were found with SiB_{2}O_{5} formulation, 32 Si_{3}B_{4}O_{12} models, 28 Si_{2}B_{6}O_{13} and Si_{4}B_{2}O_{11} models, 24 Si_{2}B_{2}O_{7} models, 18 for SiB_{6}O_{11}, 17 for SiB_{4}O_{8}, 14 for Si_{3}B_{2}O_{9}, six for Si_{6}B_{2}O_{15}, and two Si_{3}B_{6}O_{15} models. Moreover, 369 different additional models were disclosed in triclinic symmetry! The number of these models would probably explode if a complete search was done in the 230 space groups, since the introduction of Wyckoff positions having more than one equivalent boosts the capacity of the GRINSP software when experiencing difficulties to find structures more complex than 10–20 independent M/M′ atoms in a triclinic cell. Those hypothetical borosilicates are not all yet included in the PCOD. One example is shown in Fig. 4.
Explorations in the titanosilicates domain (in fact a part of that domain where octahedra and tetrahedra are exclusively cornerlinked) are in progress. The models are not electrically neutral so that the frameworks would have to accept some additional cations or charged molecule to exist in reality. One example is shown in Fig. 5.
3.3. Byproducts of the search with GRINSP
Other sixfold polyhedra than octahedra can be obtained: trigonal prisms or pentagonalbased pyramids. Since they do not correspond to unique ideal X–X or M–X distances, they are ranked with high R values. Aluminium is not known in solid fluorides with coordination other than very regular octahedral, so that such predictions are very probably useless, at least for an AlF_{3} formulation. However, from the point of view of the structures, surprisingly some presented very small framework densities, showing large tunnels, and may be of interest. Two examples are shown in Figs. 6 and 7.
Moreover, many twodimensional compounds can be formed which correspond to all polyhedral connections being satisfied by corners. In such cases, GRINSP has no way of making any correct estimate of the intersheet distance at the optimization stage, and thus these models are not collected (they frequently correspond to extremely small FD values). Some onedimensional models have even been built (nanotubes with B_{2}O_{3} formulation for instance; Fig. 8), but again, the distances between the rods could not be estimated and the cell parameters are fanciful.
4. Prediction confirmation
More difficult even than structure prediction would be the prediction of the synthesis conditions for realising these hypothetical crystal structures. However, if the chemical composition is complex enough (at least ternary or quaternary), one may first try the battery of solidstate classical synthesis routes with the suggested compositions (this being of no help at all for binary compounds). For instance, the calcium and sodium fluoroaluminates were only partly explored by GRINSP up to now, combining octahedra with different sizes (AlF_{6} with CaF_{6} or NaF_{6}). Some known 6connected frameworks were retrieved, such as [Ca_{4}Al_{7}F_{33}]^{4−}, which actually exists as Na_{4}Ca_{4}Al_{7}F_{33} (Hemon & Courbion, 1990), or [NaAl_{6}F_{21}]^{2−}, known in Rb_{2}NaAl_{6}F_{21} (Le Bail et al., 1989). One of the latest discovered metastable τAlF_{3} variety (Le Bail et al., 1992) was obtained from the thermolysis of either an organometallic compound [(CH_{3})_{4}N]AlF_{4}.H_{2}O, or amorphous AlF_{3}.xH_{2}O (x < 0.5). Thus, if a GRINSP version had existed before 1990, it would possibly have helped to solve the τAlF_{3} structure, the solution of which was long delayed until a pure and sufficiently well crystallized powder could be obtained (no single crystal of suitable size available), or the synthesis of Na_{4}Ca_{4}Al_{7}F_{33} may have been suggested sooner. Another hypothetical framework suggested by GRINSP in this series, which could well be viable, is that of [Ca_{3}Al_{4}F_{21}]^{3−} (Fig. 9). Consequently, the idea was to try to synthesize compounds with formulations M_{3}Ca_{3}Al_{4}F_{21} (M = Li, Na, K, Rb, Cs). Unfortunately, attempts (using the solidstate route) have failed to produce the desired structure. Attempts at confirming the hypothetical titanosilicates predicted by GRINSP could be worth pursuing (for instance M_{2}Si_{3}TiO_{9} shown in Fig. 5).
We can already be sure that most predictions will be in vain, and never confirmed, because the synthesis route may depend on a precursor (organometallic, hydrate, amorphous compound) which itself is yet unknown, or because the prediction is simply false. For the confirmation of some of the predictions gathered in the PCOD database, we may have to wait for decades or centuries. Nevertheless, structure prediction is an unavoidable part of our future in crystallography and chemistry. A further prediction is that the accuracy of the structure prediction methods will considerably improve.
5. Further planned improvements
The introduction of more complexity in the predictions can be readily imagined, by authorizing the connection of polyhedra by corner, edge and facesharing, altogether, and by enabling the automatic reestablishment of electrical neutrality by the detection of holes and the filling of these holes by appropriate cations.
It is clear that the R factor considers only the X–X intrapolyhedra distances, neglecting any X–X interpolyhedra distances. This cost function, R, could possibly be better defined differently, for instance by using the bond valence sum rules, or energy calculations.
The way GRINSP recognizes an already existing or predicted structure is by comparison of the coordination sequence (Meier & Moeck, 1979) of any model with a list of previously established ones (as well as with the other coordination sequences already stored during the current run). This method is in fact insufficient because it may occur (scarcely) that the coordination sequences of two different models can be identical up to the tenth order. Therefore, other means are needed in order to differentiate structures (vertex symbol for instance).
A problem is the long calculation time. For instance, installed on a singleprocessor PC running at 2 GHz, the GRINSP software needs one day to examine one set of chemical elements in one (realising 20000 to 200000 runs of 300000 Monte Carlo tests in each run), for random search of composition and random cell parameters (<16 Å), so that the full exploration needs 230 days! Moreover, one given model can be retrieved in different space groups with slightly different R values. Exploring the 230 space groups is a tedious task. One can imagine using a parallel computer, or grid computing, with a GRINSP version which would allow also the random selection of a so that one run would provide the optimal model for each structure type, the best results being sorted out only at the end of such a global process.
6. Conclusions
Combining accurate structure and properties predictions would provide inorganic chemists with invaluable information enabling them to concentrate their synthesis efforts on compounds of interest. The GRINSP and GRINS computer programs are a small step in the direction of such an ambitious vision. They are potentially able to suggest thousands of hypothetical inorganic structures with complex formulations (ternary and quaternary compounds).
7. Program features
7.1. Hardware and software environment
The executable program was built by using the Compaq Visual Fortran compiler. It runs on a PC under Windows 9x/2000/Me/NT/XP. No DLL is necessary. There would be no serious problem in installing GRINSP on Unix platforms by using a different Fortran 77 compiler (though a few compilerdependent subroutines would have to be adapted, mainly those calculating the elapsed CPU time).
7.2. Program specifications
The maximum number of M/M′ atoms is 64. GRINSP explores the 3, 4, 5 or 6connected nets leading to cornersharing polyhedra, in binary (M_{2}X_{3}, MX_{2}, M_{2}X_{5}, MX_{3}) or ternary (M_{u}X_{w}) compounds. A file (Wyckoff.txt) contains the general and special positions of the 230 space groups. The user provides his or her own set of ideal interatomic distances inside of a text file (distgrinsp.txt). The coordination sequences avoiding the proposal of already predicted structures are gathered inside a text file (connectivity.txt). The parameters describing the conditions for a run need to be prepared in a short entry file (with .dat extension), containing a title, the the choice of M/M′/X atoms, the minimum and maximum cell parameters, the minimum and maximum framework density, the number of independent tests, the number of Monte Carlo events in each test, the maximum R value for retaining a model, the number of Monte Carlo events at the cell and atomic coordinate optimization stage, and the initial file name. Output files contain the atomic coordinates in the P1 in format as well as in a .dat file, the latter being directly readable by the structure drawing software STRUPLO/STRUVIR, producing VRML files which can be displayed in three dimensions by visualizer software (CosmoPlayer, VrWeb etc.). A series of test file examples are provided.
7.3. Program availability
GRINSP is available via http://www.cristal.org/grinsp/. The software is free of charge for nonprofit organizations, and is delivered with the Fortran source code under the GNU Public Licence. The installation instructions and the user manual are accessible via the Web in HTML format, as well as included in the package.
7.4. PCOD database
Most of the hypothetical structures predicted by GRINSP were included in the PCOD (Predicted Crystallography Open Database), freely available via http://www.crystallography.net/pcod/. The search by elements, formula or/and cell parameters is possible through an Apache/MySQL/PHP server, delivering directly the and VRML files. The search can also be performed by using the PCOD entry number, as given in the above figure captions. The database accepts the upload of any new hypothetical structure, organic as well as inorganic.
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