CRYSTAL GROWTH & DESIGN 2003 VOL. 3, NO. 4 453-465
Articles Parametric Approach To Predicting Two-Dimensional Crystal Structures Scott W. Gordon-Wylie*,† and George R. Clark‡ Department of Chemistry, Cook Physical Sciences Building, The University of Vermont, Burlington, Vermont 05405, and Department of Chemistry, University of Auckland, Private Bag 92019, Auckland, New Zealand Received February 27, 2003
W This paper contains enhanced objects available on the Internet at http://pubs.acs.org/crystal. ABSTRACT: The prediction of 3-D solid-state structures using only a knowledge of the molecular geometry of the component molecules poses a difficult challenge. Here, we present details of a symmetry-based parametric approach to solving the 1- and 2-D versions of the 3-D structure prediction problem. The method developed here generates, from molecular building blocks, a small number of possible structures for which the supramolecular bonding interactions obey very specific symmetry-based rules. We have successfully applied the method to predicting the possible supramolecular structures for the high- and low-temperature polymorphs of acetylene and a metal oxalate layer structure ([M(ox)3/2]m-)n. The procedure provides an important first step toward solving or at least substantially simplifying the full 3-D supramolecular structure prediction problems encountered in protein folding, rational drug design, solid-state materials synthesis, and crystal engineering. Introduction It is extremely difficult to predict the 3-D structures of large biomolecules or crystalline solids commencing only with knowledge of the properties of individual low molecular weight components. At the present time, there is tremendous interest in this problem because of potential applications in fields as diverse as materials design, biology, and medicine. For example, in structural biology, predicting the 3-D folded structure of a given protein starting only from the primary nucleic acid sequence has assumed particular relevance now that the human genome has been unraveled.1 The rest of this paper focuses specifically on the solidstate structure prediction problem. Attempts at predicting crystal structures are traditionally carried out by a computational search for the global minimum-energy conformation. This approach usually founders because of an inadequate understanding of the following: the relative effects of the various intermolecular atom-atom forces; the subtle consequences of kinetic versus thermodynamic factors; the way in which nucleation influences the formation of different polymorphs; attachment * Corresponding author. Fax: (802) 656-8705. E-mail: swgordon@ zoo.uvm.edu. † The University of Vermont. ‡ University of Auckland.
energies; and the intricacies of weakly bonded cooperative structures.2-4 There are frequently only very small differences in energies between possible solid-state polymorphs, and conformational space may present many local minima that are nearly as low in energy as the global best solution. Alternative predictions can be obtained depending upon the relative weighting of the energy terms used in the calculations and which programs are employed. A recent survey showed that many theoretically derived structures resided in computed local minima rather than the global minimum energy configuration(s)5 found crystallographically. Most investigations to date have been performed on molecules that possess known types of intermolecular interactions and/or predictable hydrogen bonding patterns. Even so, it remains a significant challenge to identify and quantify which types of inter- or intramolecular interactions play a dominant role in directing structure formation. Several recent blind tests of crystal structure prediction have confirmed that slow progress is being made but also show that there is still a long way to go before structures can be predicted with any certainty.6,7 It would be of great benefit to have a generally applicable tool capable of relating a particular molecular building block to one of a small number of possible
10.1021/cg030010m CCC: $25.00 © 2003 American Chemical Society Published on Web 06/03/2003
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Figure 1. Generalized 3-D crystal structure prediction problem proceeding from a pyramidal building block and employing supergroup symmetry relationships.
supramolecular structures. As the 32 crystallographic point groups engender 230 space groups, knowledge of the point group of a molecule alone is insufficient to determine the space group of the resulting solid-state material.8 We can only assign some number of possible supergroups (i.e., space groups) to any given point group. Crystallographers and synthetic materials designers would also like to be able to predict the precise arrangement of molecules within a crystal unit cell, but this is difficult to determine a priori even when the specific point group/supergroup relationship is known. There is growing interest in developing an alternative approachsthe use of topological principlessto guide the design of new solid-state materials.9-12 A flow diagram for predicting 3-D solid-state space groups using a purely symmetry-based (as opposed to an energy-based) approach is shown in Figure 1. We have addressed the problem in a new way by using parametric means of generating possible supramolecular structures in combination with topological/ symmetry based constraints. The work is an offshoot of a previous investigation in which specific bonding constraints were employed to allow the parametric generation of all possible 3-D structures for a particular molecular building block.13 Unfortunately, the critical question of which of the multitude of possible 3-D structures were allowed and furthermore which were preferred was not able to be answered in that study. It was necessary to resort to experimental X-ray evidence to confirm the existence or lack thereof of specific predicted structures.14,15 We have since re-examined the basic problem of determining which of the multitude of possible structures able to be generated from a single building block are allowed. Here, we are pleased to report definitive answers as to which specific 1- and 2-D parametric structures are allowed crystallographically. Not too surprisingly, allowed solutions directly correspond to particular space group descriptions. Notably, the parametric approach yields the precise supramolecular arrangement within the unit cell, something that cannot be determined by point group/supergroup relationships alone. Determining which of the possible allowed structures is energetically preferred remains an elusive problem. Extending the symmetry-based parametric approach to 3-D crystal structure prediction remains an ongoing area of investigation. Parametric Approach. We postulated that for the 1- and 2-D cases it should be possible to employ parametric methods in conjunction with symmetry considerations to determine all allowable structures that can be derived from a specific molecular building block. The key feature of our approach is the use of symmetry, rather than energy, as the ultimate arbiter of both molecular and supramolecular connectivity. This approach is shown schematically in Figure 2.
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Figure 2. Overview of the parametric approach for crystal structure generation/prediction.
Figure 3. Two possible 1-D crystal structures able to be constructed from the 1-D building block, an arrow. Structure a is head-to-tail, and b is head-to-head/tail-to-tail.
Figure 4. Parametric generation/prediction of allowed 2-D crystal structures proceeding from a triangular building block.
In the 1-D case, the building block is defined as an arrow, and only two distinct supramolecular structures are possible.16 The arrows assemble into either a headto-tail structure or a head-to-head/tail-to-tail structure as shown in Figure 3. From a symmetry perspective, the connectivity between adjacent building blocks (arrows) can be via a mirror reflection or a 2-fold rotation to yield the headto-head/tail-to-tail structure, or there can be no symmetry (other than translation connecting the building blocks) to yield the head-to-tail structure. The significance of Figure 3 for the molecular designer is that molecules that prefer to interact head-to-tail will yield structure a, while molecules that prefer to interact headto-head and tail-to-tail will yield structure b. This demonstrates a satisfying 1:1 correspondence between the geometric configuration/supramolecular bonding properties of the building block employed and the specific crystal structure that results. The question we then asked ourselves was “Does a similar 1:1 correspondence between building block geometric configuration/supramolecular connectivity and resulting crystal structure hold true in more than one dimension”? In the remainder of this paper, we describe our attempts to apply a parametric approach to generating/ predicting allowed solid-state structures in the 2-D case (see Figure 4) and determining the precise correspondences that must exist between building block geometry, parametrically generated supramolecular structures, and the resulting allowed space group descriptions. To the best of our knowledge, this approach has not previously been formulated or successfully applied. However, the inspiration for this approach is clearly traceable to known topological concepts.9-12,14,15,17 To simplify the calculations, we considered only situations where building blocks are connected via mirror or 2-fold rotations (as opposed to pure translations). The basic triangular building block used for the 2-D case is defined as shown in Figure 5. Triangles can only interact at complementary edges (i.e., a with a, b with b, and c with c). Permitted
Parametric Approach to Predicting 2-D Crystal Structures
Figure 5. Definition of the 2-D building block employed, a triangle.
Figure 6. Symmetry operators: 2 is a 2-fold rotation about the midpoint of a given edge, m is a mirror reflection through a given edge, g1/2 is a mirror reflection through a given edge followed by translation along that edge by half of the edge distance, and g is a mirror reflection through a given edge followed by translation along that edge by all of the edge distance.
interactions are defined according to the symmetry operations shown in Figure 6. These are defined as follows: 2, a 2-fold rotation perpendicular to the plane defined by the triangle and bisecting the appropriate edge; m, a reflection perpendicular to the plane of the triangle and lying along the appropriate edge; and a glide operation of reflection followed by translation along the same edge utilized for the mirror operation. The basic glide operations can correspond either to reflection followed by translation along half the edge distance, g1/2, or the entire edge distance, g. Glides with translations along greater than one edge distance are also theoretically possible but are not considered further as they would result in unconnected solids. The m, 2, g1/2, and g operators act as fundamental generators of supramolecular structures. Calculations Calculations were carried out using the high level programming language Mathematica.18 In each run, we commenced with a single triangular building block, defined according to its R and θ values. Each of the 64 possible combinations of fundamental m, 2, g1/2, and g operators were applied to the a, b, and c edges of the triangular building block in an ordered manner. Calculations were performed over five generations of the building process to construct a planar array consisting of many triangles. A large number of runs were computed, systematically incrementing the values of R and θ for each run. No attempt was made during these calculations to restrict the overlapping of adjacent building blocks, and as a result, many of the arrays consisted of an impossible jumble of overlapping and misaligned triangles. Some such structures are evident in Figure 7. The full data set of patterns was then subdivided into two groups according to whether the number of unique vertexes was less than or greater than
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a specific cutoff number. Those below the cutoff were provisionally allowed, those above were labeled as forbidden. This screening of supramolecular arrangements employed a very simple vertex count algorithm. The vast majority of forbidden structures showed substantially higher numbers of unique vertexes than the allowed structures. This is best explained conceptually by considering a jumble of randomly scattered tiles on a floor as compared to a precisely laid tiled floor. The carefully tiled floor has far fewer vertexes than the jumbled tiled floor because in the ordered system multiple vertexes repetitively coincide to form single points. The same types of vertex/symmetry concepts are the basis for the beautifully elegant topological descriptions of crystal structures in terms of connected nets, first put forward by Wells.9,10 The simple vertex count method employed here provided an easily coded starting point for the general parametric approach but is not proposed as an ideal screening algorithm, and improved algorithms are currently under development. Many structures that should have been forbidden leaked through our crude vertex count screening algorithm, and these had to be visually identified as false positives and manually rejected. False positives arise from two main sourcessfrom structures that are nearly correct (quasicrystalline) and from intrinsically 3-D solutions being projected into the 2-D plane. Although the false positives are of significant theoretical interest, they remain beyond the scope of this paper and of our current methods of investigation. Clearly, there are interrelationships between our method and (a) other topological approaches aimed at determining possible tilings of the plane (penrose tiling, graph theory,17 basic topology) and (b) computational methods aimed at determining the low(est) energy conformations of molecules residing within a particular space group.7 The original source codes in Mathematica for our symmetrybased parametric construction and screening of allowed 2-D structures are freely available upon request, provided they are not used as part of any commercial product without express written consent of the authors. Results and Discussion All of the information necessary to generate the allowed structures in the 2-D case is contained in the geometric parameters of the triangular building block (i.e., R and θ) with connectivity information between building blocks being provided exclusively by the m, 2, g1/2, and g operators acting on the appropriate edges listed in order as {a,b,c}. Since there are four symmetry operators able to act at each edge, and three edges, there are precisely 43 ) 64 distinct possible connectivity generators that can act on a single building block: {{2,2,2}, {2,2,m}, {2,2,g1/2}, {2,2,g}, {2,m,2}, {2,m,m}, {2,m,g1/2}, {2,m,g}, {2,g1/2,2}, {2,g1/2,m}, {2,g1/2,g1/2}, {2,g1/2,g}, {2,g,2}, {2,g,m}, {2,g,g1/2}, {2,g,g}, {m,2,2}, {m,2,m}, {m,2,g1/2}, {m,2,g}, {m,m,2}, {m,m,m}, {m,m,g1/2}, {m,m,g}, {m,g1/2,2}, {m,g1/2,m}, {m,g1/2,g1/2}, {m,g1/2,g}, {m,g,2}, {m,g,m}, {m,g,g1/2}, {m,g,g}, {g1/2,2,2}, {g1/2,2,m}, {g1/2,2,g1/2}, {g1/2,2,g}, {g1/2,m,2}, {g1/2,m,m}, {g1/2,m,g1/2}, {g1/2,m,g}, {g1/2,g1/2,2}, {g1/2,g1/2,m}, {g1/2,g1/2,g1/2}, {g1/2,g1/2,g}, {g1/2,g,2}, {g1/2,g,m}, {g1/2,g,g1/2}, {g1/2,g,g}, {g,2,2}, {g,2,m}, {g,2,g1/2}, {g,2,g}, {g,m,2}, {g,m,m}, {g,m,g1/2}, {g,m,g}, {g,g1/2,2}, {g,g1/2,m}, {g,g1/2,g1/2}, {g,g1/2,g}, {g,g,2}, {g,g,m}, {g,g,g1/2}, and {g,g,g}}.
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Figure 7. Examples of fourth generation plots in two dimensions for geometric parameters of R ) 2, θ ) π/4, and the specific connectivity generators shown. Note that only four connectivity generators lead to allowed structures.
All possible solid-state structures can be derived from the above connectivity generators and the two geometric parameters, R and θ. Figure 7 displays the eight patterns of triangles resulting from constructing four generations of the simplified cases where no glide operators are present and R ) 2, θ ) π/4. Note that four of the connectivity generators lead to forbidden supramolecular arrangements for the particular geometric parameters of R ) 2 and θ ) π/4. Forbidden patterns exhibit overlapping or misaligned triangles that result in many collisions between building blocks, a lack of periodicity, and a high number of distinct vertexes. The remaining four patterns correspond to allowed solutions or hits. Sorting Figure 7 manually into allowed and forbidden structures is easily accomplished. Still, as the number of possible geometric parameters and connectivity generators increases, the hand sorting method rapidly becomes unwieldy. To automate the process, we wrote a program that samples a range of shapes and connectivities in an incremental and systematic manner (see Calculations for details), from which we could compile representative crystal phase diagrams that map the occurrences of possible hits. Figure 8A shows the one crystal phase diagram that results for the connectivity patterns {2,2,2}, {2,2,g1/2}, {2,2,g}, {2,2,m}, {2,g1/2,2}, {2,g,2}, {2,m,2}, {g1/2,2,2}, {g,2,2}, {m,2,2}. These connectivity operators yield hits for all values of R, θ, where a hit is defined for a fifth generation plot as a structure possessing 90 or fewer unique vertexes. Figure 8B shows the much smaller number of hits obtained for the connectivity pattern {2,g,m} as R and θ are incremented over a sampling of all possible values, and the resulting structures are sorted only according to the vertex count cutoff value. Several movies showing how the vertex count correlates with possible parametric structures for the connectivity pattern {2,m,m} are available on the In-
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Figure 8. (A) Crystal phase diagram showing hits obtained for the connectivity generators {2,2,2}, {2,2,g1/2}, {2,2,g}, {2,2,m}, {2,g1/2,2}, {2,g,2}, {2,m,2}, {g1/2,2,2}, {g,2,2}, and {m,2,2}. These operators yield hits for all values of R, θ with a specific cutoff value of 90 unique vertexes per structure. (B) Crystal phase diagram showing hits obtained for the connectivity generator {2,g,m} using the same search values as in panel A. The vertex count algorithm can yield false positives, so the real hits remaining after a more stringent second screening process are shown as filled points. The pattern of some of the filled points suggests a parametric representation, R ) sec θ or θ ) arccos(1/R) depending on boundary values, that allows other missing solutions to be predicted (curved line).
Chart 1a
a Different trajectories across parametric space for {2,m,m} connectivity presented as movies on the web. Black filled circles correspond to real 2-D structures, gray filled circles are quasi 2-D structures, and unfilled circles are false positives for the vertex count algorithm with an arbitrary cutoff value of 90 unique vertices per structure after five generations of parametric growth. Most of the false positives can be removed by setting the vertex cutoff slightly lower.
W Three movies (W radius.mov, W angrad.mov, and W angle.mov) are available in .mov format and can be viewed using QuickTime.
ternet at http://pubs.acs.org/crystal. The movies trace particular trajectories across the crystal phase diagram as R and θ are varied and show how the resulting parametric structures vary between crystalline, noncrystalline, and quasi-crystalline outcomes (see Chart
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Table 1. Specific Relationships between Geometric Parameters, Connectivity Generators, and Allowed 2-D Space Groupsa parameters
non-chiral
chiral
parameters
non-chiral
chiral
{2,2,2}{π/3, 1} {2,2,2}{θ, R} {2,2,g1/2}{θ, R} {2,2,g}{θ, R} {2,2,m}{θ, R} {2,g1/2,2}{θ, R} {2,g,2}{θ, R} {2,m,2}{θ, R} {g1/2,2,2}{θ, R} {g,2,2}{θ, R} {m,2,2}{θ, R} {2,m,m}{π/6, x3} {2,m,m}{π/6, 2/x3} {2,m,m}{π/4, x2} {2,m,m}{π/3, 1} {2,m,m}{π/3, 2} {2,m,m}{θ < π/2, >R ) sec[θ]} {2,m,m}{θ ) arccos[1/R], R > 1} {m,2,m}{π/3, 1} {m,2,m}{2π/3, 1} {m,2,m}{π/2, R} {m,m,2}{π/6, 1/x3} {m,m,2}{π/4, 1/x2} {m,m,2}{π/3, 1/2} {m,m,2}{π/3, 1} {m,m,2}{θ < π/2, R ) cos[θ]} {m,m,2}{θ ) arccos[R], R < 1} {2,m,g}{π/6, 2/x3} {2,m,g}{π/4, x2} {2,m,g}{π/3, 2} {2,m,g}{θ < π/2, R ) sec[θ]} {2,m,g}{θ ) arccos[1/R], R > 1} {2,g,m}{π/6, 2/x3} {2,g,m}{π/4, x2} {2,g,m}{π/3, 2} {2,g,m}{θ < π/2, R ) sec[θ]} {2,g,m}{θ ) arccos[1/R], R > 1} {m,2,g}{π/2, R} {g,2,m}{π/2, R}
p6 p2, p6 p2, pg pmg pmg p2, pg pmg pmg p2, pg pmg pmg p6 cmm p4 p6 cmm cmm, p4 cmm, p4 p6 p6 cmm, p4 p6 p4 cmm p6 cmm, p4 cmm, p4 pmg pmg pmg pmg pmg pmg pmg pmg pmg pmg pmg pmg
p6 p2, p6 p2, pg pmg pmg p2, pg pmg pmg p2, pg pmg pmg forbidden cmm cmm forbidden cmm cmm cmm forbidden forbidden cmm forbidden cmm cmm forbidden cmm cmm pmg pmg pmg pmg pmg pmg pmg pmg pmg pmg pmg pmg
{g,m,2}{π/4, 1/x2} {g,m,2}{π/3, 1/2} {g,m,2}{θ < π/2, R ) cos[q]} {g,m,2}{θ ) arccos[R], R < 1} {2,g,g}{π/6, 2/x3} {2,g,g}{π/4, x2} {2,g,g}{π/3, 2} {2,g,g}{θ < π/2, R ) Sec[θ]} {2,g,g}{θ ) arccos[1/R], R > 1} {g,2,g}{π/2, R} {g,g,2}{π/4, 1/x2} {g,g,2}{π/3, 1/2} {g,g,2}{θ < π/2, R ) cos[θ]} {g,g,2}{θ ) arccos[R], R < 1} {m,m,m}{π/6, 1/x3} {m,m,m}{π/6, 2/x3} {m,m,m}{π/6, x3} {m,m,m}{π/4, 1/x2} {m,m,m}{π/4, x2} {m,m,m}{π/3, 1/2} {m,m,m}{π/3, 1} {m,m,m}{π/3, 2} {m,m,m}{π/2, 1/x3} {m,m,m}{π/2, 1} {m,m,m}{π/2, x3} {m,m,m}{2π /3, 1} {g,m,m}{π/6, x3} {g,m,m}{π/4, x2} {g,m,m}{π/3, 1} {m,g,m}{π/3, 1} {m,g,m}{π/2, 1} {m,g,m}{2π/3, 1} {m,m,g}{π/6, 1/x3} {m,m,g}{π/4, 1/x2} {m,m,g}{π/3, 1} {m,g,g}{π/4, x2} {g,m,g}{π/2, 1} {g,g,m}{π/4, 1/x2} {g,g,g}{π/3, 1}
pmg pmg pmg pmg pgg p4 pgg pgg, p4 pgg, p4 pgg, p4 p4 pgg pgg, p4 pgg, p4 p6m p6m p6m p4m p4m p6m p6m p6m p6m p4m p6m p6m p6 p4 p6 p6 p4 p6 p6 p4 p6 p4 p4 p4 p6
pmg pmg pmg pmg pgg pgg pgg pgg pgg pgg pgg pgg pgg pgg forbidden p6m forbidden p4 p4 p6m p31m p6m p6m p4 p6m forbidden forbidden p4g p31m p31m p4g forbidden forbidden p4g p31m forbidden forbidden forbidden p31m
a
That is, assignment of which parametric structures are allowed in two dimensions.
1). The movies use color to highlight how the parametric crystal is grown in a treelike or generational manner starting from a single central building block according to the connectivity and geometric parameters indicated. The same general approach was applied to all 64 connectivity generators, using patterns of triangles computed out to five generations. As we were using incremental sampling, and to broaden the coverage of parametric space, two separate runs were performed using different starting points and increments. In the first pass, sample points were chosen so as to evenly screen conformational space. In the second pass, sample points were chosen to more rigorously sample those specific angles that correspond to 3-, 4-, and 6-fold rotational axes (see Figure 8A). The first pass spanned R ) 1/4 f 4 in steps of 1/4 and θ ) π/36 f 35π/36 in steps of π/36 for a total of 64 × 16 × 35 ) 35 840 parametric structures. The second pass spanned R2 ) 1/6 f 4 in steps of 1/6 and θ ) π/12 f 11π/12 in steps of π/24 for a total of 64 × 24 × 21 ) 32 256 parametric structures. A small number of points were common to the two passes, and the total number of unique parametric supramolecular structures generated was 64 × 1042 ) 66 688. Each unique structure was then subjected to a vertex count with a best guess cutoff limit of 90 unique vertexes per structure. Any structure displaying 90 or fewer vertexes
was classed as a possible hit. The highest observed vertex count was 672, and the lowest was 26. Two failings of the approach are that the two passes of structure generation employed only finite increments in R and θ (therefore, the coverage of sample space is by definition incomplete) and that the vertex counting algorithm, although highly efficient and fast, is patently imperfect. Nevertheless, the approach shown schematically in Figure 8A,B provides a powerful first attempt at determining the range of allowed structures. As can be seen from Figure 8A, certain connectivity generators yield hits regardless of the geometric parameters employed. It is assumed that testing of additional points would continue to generate hits for all values of R and θ investigated for these specific connectivity generators. A more stringent second screening by hand in a manner similar to that shown in Figure 7 confirms that these connectivity generators always yield allowed structures. The 10 heavy hitter connectivity generators all contain two or more 2 operators (i.e., {2,2,2}, {2,2,g1/2}, {2,2,g}, {2,2,m}, {2,g1/2,2}, {2,g,2}, {2,m,2}, {g1/2,2,2}, {g,2,2}, and {m,2,2}). Twenty-three of the 64 connectivity generators yielded no hits for any value of R and θ tested. These no hitter cases would appear as a blank slate if crystal phase diagrams were mapped as in Figure 8A,B. They are {2,g1/2,g1/2}, {2,g1/2,g}, {2,g,g1/2}, {g1/2,2,g1/2}, {g1/2,2,g},
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{g1/2,g1/2,2}, {g1/2,g1/2,g1/2}, {g1/2,g1/2,g}, {g1/2,g1/2,m}, {g1/2,g,2}, {g1/2,g,g1/2}, {g1/2,g,g}, {g1/2,g,m}, {g1/2,m,g1/2}, {g,2,g1/2}, {g,g1/2,2}, {g,g1/2,g1/2}, {g,g1/2,g}, {g,g,g1/2}, {g,m,g1/2}, {m,g1/2,g1/2}, {m,g1/2,g}, and {m,g,g1/2}. It is assumed that testing of additional points for these connectivity generators would produce no new hits. All of the no hitters contain at least one g1/2 operator in combination with at least one other g or g1/2 operator. For the remaining 31 connectivity generators, each of the computer-allowed hits with a vertex count 1. The acetylene is aligned at an arbitrary angle φ relative to the c face.
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an intriguing result. If it holds true for further examples, it seems to suggest that despite significant space group differences, polymorphs are structurally related in a fundamental way via symmetry-based supramolecular connectivity relationships. Note that not all of the 17 crystallographic plane groups are recovered in Table 1. Presumably, only some of the 17 possible plane groups are symmetry connected in the manner described here for the assembly of a single unique building block according to the particular geometric and symmetry derived constraints of 2, m, g, and g1/2 operators acting in concert with strict selfcomplementarity of building blocks (i.e., only head-tohead/tail-to-tail connectivity has been considered along each of the three unique triangular edges so the resulting connectivity is always a with a, b with b, and c with c). The preference of acetylene for a specific connectivity pattern and choice of angles from among the possibilities shown in Figures 15-36 strongly suggests that acetylene in the solid state is correctly viewed as a symmetryconnected solid (i.e., that it is supramolecularly bonded together to form a crystal according to rules that can be described explicitly using the symmetry-based machinery developed here). Conclusion
Figure 37. Matching both polymorphs of the experimentally observed 2-D unit cell(s) of acetylene (hashed lines) to the parametrically derived {g1/2,2,2} structure with θ ) π/2.
The slab structure of the higher temperature form of acetylene can be projected onto two dimensions in one of two ways. Either each slab can be treated separately, or the entire unit cell can be flattened. Since a variety of structures with overlapping building blocks were rejected as false positives in the construction of Table 1, it seems appropriate to consider only the slab-like structures at this point. Expansion of Table 1 to allow for 2-D projections of 3-D unit cells provides a logical extension of the work presented here. The 2-D projection of a single slab of the higher temperature form also corresponds to θ ) π/2 and a connectivity pattern of {2,2,g1/2} (){g1/2,2,2}). For acetylene, the different crystal polymorphs are found to share the same supramolecular connectivity, at least in two dimensions. This is
A parametric approach for generating 2-D crystal structures has been developed and in combination with a primitive pattern recognition algorithm, has allowed the definitive enumeration of essentially all possible specifically connected 2-D structures. The forward mapping from building block to resulting crystal structure is shown to be 1:1. Examples are presented to confirm the validity of this approach for predicting certain types of 2-D layered crystal structures. The concept of a supramolecular bond as a symmetry-derived entity is put forward. The approach explicitly solves the problem of determining which structures are allowed, but since it is based purely on symmetry considerations, it cannot address the question of the relative energetic orderings of the possible allowed structures. Therefore, it cannot determine the global ground state of the system. Nevertheless, the parametric approach provides a powerful new tool for reducing a multitude of possible supramolecular arrangements down to just a handful of allowed structures. Our purpose in developing the model was to provide a practical approach to assessing how changes in building block geometry and modes of supramolecular connectivity affect the resulting solid-state structures. Future efforts will be aimed at extending the model to 3-D structures. Acknowledgment. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for support of this research. Supporting Information Available: Structures found in Table 1 that require special values of R, θ. This material is available free of charge via the Internet at http://pubs.acs.org.
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