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Local Coordination versus Overall Topology in Crystal Structures: Deriving Knowledge from Crystallographic Databases Alexander P. Shevchenko,† Igor A. Blatov,‡ Elena V. Kitaeva,† and Vladislav A. Blatov*,†,§ †

Samara Center for Theoretical Materials Science (SCTMS), Samara University, Ac. Pavlov Street 1, Samara 443011, Russia Volga State University of Telecommunications and Informatics, Moskovskoe sh. 77, Samara 443010, Russia § School of Materials Science and Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, People’s Republic of China ‡

S Supporting Information *

ABSTRACT: We propose three independent methods for determining geometrical shape and topological type of local atomic environment in crystals that is represented as a coordination figure. The methods are based on comparison of the edge graphs of the simplified Voronoi polyhedra, maximizing volume of intersection of polyhedra, and comparison of angular fingerprints of polyhedra. We implemented the methods to the program package ToposPro and tested them on 9642 crystal structures of five-coordinated copper and zinc complexes. The methods showed good selectivity and coincidence. We discuss correlations of the coordination figure type with other structure descriptors and perspectives to use the obtained results in knowledge databases.

1. INTRODUCTION The crystallographic information stored in the worldwide databases1−3 has now exceeded one million records, which invokes at least two interrelated challenges. First, such a huge amount of data certainly contains a lot of regularities and even quite general laws to be discovered, but how to find them? Second, the information should be arranged in many different ways to let the scientist easily screen it for the crystal properties. While the first challenge is quite general, the second one, having its own significance, outlines the route to extract regularities. Both problems are parts of a more general task of deriving knowledge from the initial experimental data and creating knowledge databases from current information storages.4 In crystal chemistry, the solution of this task is already in progress, and the first knowledge databases are being created.5−7 However, they are still far from completion because the second challenge mentioned above has not been fully handled. A crucial point in arranging information is the problem of descriptors: the more robust descriptors we invent, the more ways to derive correlations we have. The descriptors should be proper, independent, and strictly defined to be implemented into database management systems for automated processing of the information. In crystal chemistry, the structural descriptors can be separated into three groups: (i) geometrical descriptors, which describe the spatial distribution of atoms and other structural units; (ii) topological descriptors, which deal with connectivity of the structure; and (iii) physical descriptors, which include the information on physical properties of the crystal and its constituents. The set of crystallochemical © 2016 American Chemical Society

descriptors has been developing since the experimental methods of determining atomic structure appeared.8 Traditionally, the geometrical descriptors (interatomic distances, angles, packings, point and space symmetries, etc.) composed the main part of this set because the mathematical base (Euclidean geometry and space group theory) was developed primarily for them.6,9,10 When experimental technique became rather precise, physical descriptors such as electron density distribution and other derived parameters were being explored more often.11−13 A significant progress in DFT methods attracted even stronger attention to this group of descriptors.14−16 At the same time, topological descriptors such as atomic net topology, tiling parameters, and entanglement class remained less developed or even unknown until the end of the 1990s. However, in the past 15−20 years, we saw a great progress in the so-called topological crystal chemistry or reticular chemistry.17−20 The number of topological descriptors was essentially increased,21−23 and special software18,19 and databases24−26 have been developed, which has already resulted in the appearance of the first topological knowledge storages.27 This study is devoted to detailed exploration of just one of the topological descriptors, the so-called coordination figure. According to ref 28, a coordination figure is an object that is formed around a given point by all adjacent points. Thus, it can be referred to both topological and geometrical descriptors as it characterizes both local topology of the point environment and Received: November 9, 2016 Revised: December 15, 2016 Published: December 16, 2016 774

DOI: 10.1021/acs.cgd.6b01630 Cryst. Growth Des. 2017, 17, 774−785

Crystal Growth & Design

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its spatial embedding. It is known17,24,29 that the coordination figure significantly affects the topology of the crystal as a whole. However, to the best of our knowledge, there is no strict definition of this descriptor that could be used in computer algorithms to process the crystallographic information; in the quoted papers, the authors determined coordination figures visually. We examine possible definitions and test them with a large sample of crystal structures. This sample is composed by coordination polymers with five-coordinated copper or zinc atoms. Our choice was determined by at least two reasons: (i) coordination figures with five neighboring atoms are easy to draw but often difficult for classification, and (ii) this group of compounds is numerous and well-investigated due to their unusual optical,30 magnetic,31,32 adsorption,33 and catalytic properties.34,35

Table 1. Distributions of the Crystal Structures Containing Five-Coordinated Atoms of Cu or Zn Surrounded by N and/ or O Atoms Depending on the Type and Composition of the Coordination Figure, Dimensionality of Coordination Groups, and Overall Topology of the Coordination Network square pyramid value 0D 1D 2D 3D O5 NO4 N2O3 N3O2 N4O N5

2. EXPERIMENTAL SECTION 2.1. Objects of the Analysis. Crystallographic information on the copper and zinc coordination polymers was taken from the Cambridge Structure Database (CSD, release 5.36) and Inorganic Crystal Structures Database (ICSD, release 2015/2). We considered the structures that contained only one independent metal center (Cu or Zn) bonded to five oxygen and/or nitrogen atoms. The resulting sample contained 7241 and 2401 crystal structures of copper and zinc complexes, respectively. Coordination of the atoms was determined by the Domains method.36 In addition to the descriptors that characterize coordination figure of the metal atom, we determined other local (Tables 1, S1, and S2) and overall (Tables 1, S1, and S3) topological parameters: coordination formula, which characterizes the local coordination topology,37 dimensionality of the polymeric group, and topological type of the underlying nets (nets of structural units) in the standard and cluster representations.18 The types of local and overall topologies are designated in accordance with the corresponding nomenclatures.18,20,37 2.2. Methods and Descriptors for Recognizing the Shape of Coordination Figures. The concept of coordination figure is closely related to the concept of the coordination polyhedron, which is widely used in coordination and crystal chemistry.38 The convex coordination figure predetermines the coordination polyhedron, which can be treated as the space confined by the convex hull wrapped over the coordination figure (Figure 1). Unlike the coordination polyhedron, which is supposed to be convex, the coordination figure can be nonconvex (Figure 2); such cases are quite unusual in coordination chemistry, however. The crucial problem for the computer data analysis is the classification of the shape of the coordination polyhedron, i.e., assigning it to some reference solids (types). In particular, five-vertex coordination figures, or the corresponding coordination polyhedra, can be assigned to the triangular-bipyramid or square-pyramid types (Figure 1). There are several methods for determining the shape of coordination polyhedra. Most often, they use an iterative algorithm for superposing the solid under consideration with a reference polyhedron.40 The maximal matching is determined by the minimum value of the sum of the distances between the corresponding pairs of vertices of the two polyhedra. The topology of the edge net of the coordination polyhedron41 or its point symmetry40 can also be taken into account. The angular distribution relative to the central atom can be characterized by invariants of spherical harmonics.42 However, all of these methods are applicable only for polyhedra with the same number of vertices, and the computer time sharply increases as this number increases. To overcome these problems, we developed and tested other approaches that are described below. All of them have been implemented into the ToposPro program package.18 2.2.1. Topology of Voronoi Polyhedra. This approach uses the edge graph of the simplified Voronoi polyhedron of the central atom of the coordination figure for its classification. For example, geometrically regular trigonal bipyramid and square pyramid have

sql hcb xah 4,4L1 fes dia srs 3,5L2 4,5L51 3,4L13 4,8T24 4,5T4 4,4L47 4,6T4 3,4L83 kdd 5,5T7 ins fet utk others

N

w, %

trigonal bipyramid N

w, %

others N

dimensionality of coordination groups 4770 67.5 797 72.7 983 1237 17.5 152 13.9 261 532 7.5 77 7.0 129 527 7.5 71 6.5 106 composition of coordination figures 795 11.3 52 4.7 79 1224 17.3 70 6.4 136 2111 29.9 263 24.0 429 1056 14.9 215 19.6 303 1092 15.5 292 26.6 275 788 11.2 205 18.7 257 overall topology of coordination networks 132 12.46 18 12.16 48 113 10.67 16 10.81 32 106 10.01 0 1 104 9.82 0 0 40 3.78 10 6.76 15 16 1.51 5 3.38 14 28 2.64 4 2.70 1 10 0.94 15 10.14 6 29 2.74 0 1 19 1.79 5 3.38 4 28 2.64 0 0 26 2.46 0 0 24 2.27 0 0 22 2.08 0 0 8 0.76 3 2.03 10 12 1.13 0 0 12 1.13 0 0 11 1.04 1 0.68 0 3 0.28 7 4.73 1 0 0 11 316 29.84 64 43.24 91

w, % 66.5 17.6 8.7 7.2 5.3 9.2 29.0 20.5 18.6 17.4 20.43 13.62 0.43 6.38 5.96 0.43 2.55 0.43 1.70

4.26

0.43 4.68 38.72

Figure 1. Five-vertex coordination figures: trigonal bipyramid (left) and square pyramid (right). clearly different forms of the corresponding Voronoi polyhedra: trigonal prism and tetragonal pyramid, respectively (Figure 3). The advantage of using Voronoi polyhedra instead of coordination polyhedra for classification of coordination figures is that geometrical distortions of the coordination figure result in changes of the edge graph of the Voronoi polyhedron and can be easily estimated (Figure 775



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Figure 2. An example of nonconvex coordination figure in the cluster representation of the structure [Zn2(OH)(btc)]2(4,4′-bipy) (btc = 1,3,5benzenetricarboxylate; 4,4′-bipy = 4,4′-bipyridine) (CSD reference code: ESEVIH)39 (left) and its convex hull (right). Some atoms of the 4,4′bipyridine molecules are disordered. The vertices of the coordination figure are highlighted in red or green if they lie on or inside its convex hull, respectively. the edge is estimated by the angle γ, at which the edge is seen from the central atom (Figure 4, top). Topologically, the triangular prism differs from square pyramid by an additional edge (Figure 4). Analysis of the distribution of the edge angles of the Cu or Zn Voronoi polyhedra shows that we can contract the edge if γ < 30°, which corresponds to the first minimum of the distribution (Figure 5). This algorithm is very fast to compare coordination figures with both the same and different (even high) numbers of vertices (Figure 6).

Figure 3. Coordination figures and Voronoi polyhedra of the trigonal bipyramid (left, topological type {5/9/6-1}) and square pyramid (right, topological type {5/8/5-1}).

4, bottom). Only valence bonded atoms should be taken into account in the construction of Voronoi polyhedra. To compare such Voronoi polyhedron with the reference topological types, we simplify its topology by subsequent contracting of the shortest edges. The size of Figure 5. Distribution of angles at the metal atom, which are based on the edges of the Voronoi polyhedra. 2.2.2. Maximizing the Volume of the Intersection of Polyhedra. This approach is based on an iterative superposing of the coordination

Figure 6. Voronoi polyhedron built with valence bonded neighbors (left) and its simplified version obtained by contraction of the polyhedron edges with γ < 30° (right) for the zinc atom in the structure Zn(ATIBDC)(bpy)·3H 2 O (bpy = 4,4′-bipyridine; H2ATIBDC = 5-amino-2,4,6-triiodoisophthalic acid; CSD reference code: VOMSEW01).44

Figure 4. Voronoi polyhedron (trigonal prism) of the Cu atom built with only valence bonded neighbors (top) and its simplified form (tetragonal pyramid) obtained by contraction of the polyhedron edge with γ = 8.07° in the structure of [Cu{C2(COO)2}(H2O)3]·H2O; CSD reference code: MACGAZ).43 776



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(ii). Normalizing the polyhedron volume V per unit by dividing the coordinates of the polyhedron vertices by V1/3. (iii). Superposing the centroids of the polyhedra and aligning axes of their inertia tensors. All six possible orientations of the axes are considered. As a result, we obtain a good initial approximation to find the required minimum. (iv). Moving the polyhedron under consideration relative to the reference one by changing one of the six arguments of f. The optimal value of each argument is determined by the golden section method,47 and the function value is calculated by constructing the intersection of the two polyhedra with the subsequent determination of the intersection volume. The golden section method of one-dimensional minimization is very economical, and its application reduced the computation time from tens of minutes to a few seconds compared to εenumerating algorithms.45,46 This step repeats until the objective function f differs by less than a given accuracy after successive minimization of all six arguments. We tested the algorithm for polyhedra with the number of vertices up to 20 and revealed its high efficiency compared to that of the enumeration algorithms.45,46 2.2.3. Comparison of Angular Fingerprints. The last approach that we used is based on the well-known algorithm for comparing angular fingerprints of polyhedra.48,49 The fingerprints were calculated according to the following algorithm: (i). Angles at the central atom were calculated for all pairs of the neighboring atoms of the coordination figure. (ii). Angle values were arranged over the intervals of width δ in the range 0−180°. (iii). Occurrences for each interval were smoothed by a Gaussian function with a standard deviation σ and an average value μ, which is equal to the midinterval.

and reference polyhedra to be compared by translation, rotation, or scaling of the coordination polyhedron. Unlike the superposing methods mentioned above,40 this approach uses the part of maximal intersection of the two polyhedra (ω) as the similarity criterion (ω = 0−100%; the larger its value, the closer the two polyhedra are to each other, Figure 7). The best known algorithms of this kind45,46 are based

Figure 7. Superimposing of trigonal bipyramid and square pyramid, which results in the maximum volume of their intersection ωtbsp = 75.4%.

f (x) =

2 2 1 e−(x − μ) /2σ σ 2π

(1)

(iv). The total contribution Fi was calculated as a centuplicate sum of function values in each interval, which is normalized to the sum of the values of these functions at all intervals.

on enumeration of n points of the O(ε) net in the set of parameters defining the isometric three-dimensional space, and they have the computational complexity of at least O(ε−3nlog3.5 n), where ε is the accuracy of the volume calculation. As a result, these algorithms are improper for processing large (>1000 polyhedra) samples. In this paper, we propose a new algorithm which is based on the coordinate descent in the space of isometries of the three-dimensional space, and the search for the minimum for each coordinate with the golden section method. We look for a minimum of the function f(φ, ψ, θ, x, y, z), whose value is equal to −ω; φ and ψ angles define the axis of rotation of the polyhedron; θ is the angle of rotation around the axis, and x, y, z are the coordinates of the centroid of the moving polyhedron relative to the centroid of the stationary (reference) polyhedron. The algorithm includes the following steps: (i). Equalizing the distances from the vertices of the coordination figure to its central atom by multiplying the vectors connecting the vertices with the centroid by the corresponding factors. The resulting polyhedron is always convex, i.e., this procedure allows one to classify even nonconvex coordination figures.

180/ δ

Fi = 100

∑k = 1 fk (μi ) 180/ δ

180/ δ

∑m = 1 ∑k = 1 fk (μm )

(2)

As a result, each coordination figure corresponds to a fingerprint vector F, whose dimension is determined by δ and equal to the number of intervals; F is normalized to 100. The distance r between F and F′ fingerprints of two coordination figures is used to compare them: 180/ δ

r=

∑ i=1

⎛ Fi − F ′i ⎞2 ⎟ ⎜ ⎝ 2 ⎠

(3)

The coordinating figure is related to the closest type to which it has the minimal distance. By definition, this fingerprint is sensitive only to the angular distribution of atoms of the coordination figure. One can also consider the radial distribution of atoms using the radial fingerprints,49 but in our opinion (see the Results below), the angular

Table 2. Angular Fingerprints for Four Regular Coordination Figures with δ = 18° and σ = 9° angle range (deg)a 72 ± 9 tetrahedron trigonal bipyramid square pyramid base-centered square pyramid body-centered a

0 6.5 8.7 7.0

90 ± 9 7.6 48.1 64.3 45.5

108 ± 9 77.8 16.4 8.7 26.8

126 ± 9 14.6 19.2 0 0.8

144 ± 9 0.1 0.7 0 5.3

162 ± 9 0 1.1 2.2 14.0

180 ± 9 0 8.0 16.1 0.7

The fingerprint value is equal to zero for all other intervals. 777



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component is sufficient to determine the shape of the coordination figure. Because the method is sensible to the angles at the central atom, we considered two kinds of square pyramid: base-centered and bodycentered with the atom in the center of the basal face or in the centroid of the pyramid, respectively. These two kinds correspond to different hybridization of the central atom. To determine the shape of the coordination figures, we calculated the fingerprints with δ = 18° and σ = 9° (Table 2) and then computed the distances r between the figures (Table 3). The maxima of the fingerprint distributions indicate most typical bond angles for the coordination figures (Figure 8).

previous section, which we call Voronoi, Intersection, and Fingerprint for short. Voronoi Approach. The topological types {5/8/5-1} and {5/9/6-1} of Voronoi polyhedra, which correspond to squarepyramidal and triangular-bipyramidal coordination figures, were found in 7707 (79.9%) and 1927 (20.0%) crystal structures, respectively. In the remaining eight (0.1%) cases, the simplified Voronoi polyhedra correspond to tetrahedral coordination figures with four vertices as shown in Figure 6. Intersection Approach. The results of classification with this method are presented in Table 4. The digit in the reliability rating symbol characterizes robustness of the method in distinguishing the polyhedron under consideration between the given pair of regular polyhedra α = |ωtb − ωsp|, where ωtb and ωsp are the volumes of intersection of the polyhedron under consideration with regular triangular bipyramid and square pyramid, respectively: 1: α ≤ 10%; 2: 10 < α ≤ 20%; 3: 20 < α ≤ 30%; 4: 30 < α ≤ 40%. The closer is α to the volume of noncoinciding parts of the regular polyhedra αbest = 1 − ωtbsp = 24.6% (Figure 7), the more reliable are the results of determination of the coordination figure. In this case, the best robustness is associated with rating 3. The letter in the reliability rating symbol shows closeness of the polyhedron to one of the regular polyhedra β= ωtb or ωsp; the closer is β to 100%, the closer the polyhedron is to the reference one. We use the following ranking: A: 90 < β ≤ 100%; B: 80 < β ≤ 90%; C: 70 < β ≤ 80%; D: 60 < β ≤ 70% (Table 4, Figure 9). Thus, the best recognition corresponds to the best robustness (rank 3) and high closeness (ranks A or B) to the green zones on Figure 9. The yellow and orange zones in Figure 9 correspond to the coordination figures that cannot be assigned to any of the reference polyhedra. We emphasize that the high β value itself does not mean good recognition; if α = 0 (red line on Figure 9), the polyhedron is close to both reference polyhedra, and the method does not distinguish them. Table 4 shows that in most cases the coordination figures are distorted; the best rating 3A is realized in no more than 2% of structures. At the same time, in most cases the coordination figures are distinguishable: only 1.2−1.3% of them have rank C or D. The distortions are caused by either a special configuration of the ligand (Figure 10, left) or additional specific interactions of the metal atom (Figure 10, right). Fingerprint Approach. The method gives the following distribution of the main geometrical types of the Cu and Zn five-coordinated figures: body-centered square pyramid (SPY5b, 6430, 67%), base-centered square pyramid (SPY-5a, 1854, 19%), and trigonal bipyramid (TBPY-5, 1198, 12%). In addition, the method has distinguished a type that can be called capped tetrahedron (TF-5, 160, 2%). For a more detailed classification, we use two ranks as in the Intersection approach. The digital ranking evaluates the difference (α′) between the two shorter distances r. We considered four α′ levels: 1: α′ ≤ 5; 2: 5 < α′ ≤ 10; 3: 10 < α′ ≤ 15; 4: 15 < α′ ≤ 20. The larger α′ is, the more reliable the classification of the coordination figure. When distinguishing square pyramid and trigonal bipyramid, the best value α′best = 13.8 (Table 3) corresponds to rank 3. For example, the distances are equal to 12.96 (SPY-5a), 10.28 (SPY-5b), 3.28 (TBPY-5), and 26.31 (TF-5) for the coordination figure of the copper atom in the structure of Cu(bdmmp)2(H2O) (THF), where bdmmpH = 2,6-bis[(dimethylamino)methyl)-4-methylphenol (CSD reference code: YIKYAS),52 which unambigu-

Table 3. Distances r between Pairs of Regular Coordination Figures coordination figurea tetrahedron (T-4) trigonal bipyramid (TBPY-5) square pyramid base-centered (SPY-5a) square pyramid body-centered (SPY-5b) a

T-4

TBPY-5

SPY-5a

SPY-5b

0 37.2 46.2

37.2 0 13.8

46.2 13.8 0

33.5 13.2 16.5

33.5

13.2

16.5

0

We use the notation from ref 50.

Figure 8. Fingerprints corresponding to four regular coordination figures. The total area under each curve is equal to 100.

3. RESULTS AND DISCUSSION 3.1. Classification of Coordination Figures. Below we present the classification of coordination figures of metal atoms obtained within the three approaches considered in the Table 4. Occurrences of the Coordination Figures of Cu or Zn According to the Intersection Approach. The coordination figures are assumed unrecognized for ratings 1C, 1D, 2C and 2D square pyramid

trigonal bipyramid

rating

N

w, %

N

w, %

reliability

1A 1B 1C 1D 2A 2B 2C 2D 3A total

938 1966 90 1 3772 493 1 1 95 7357

12.7 26.7 1.2 0.0 51.3 6.7 0.0 0.0 1.3

230 1140 30 0 738 102 0 0 45 2285

10.1 49.9 1.3

+ + − − + + − − +

32.3 4.5

2.0

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Figure 9. Rating scheme (left) and scatter plot (right) of parts of the intersection volume of the coordination figure and regular triangular bipyramid (orange dots) or regular square pyramid (blue dots). The red bisecting line divides all the coordination figures to two kinds of shapes. Green, blue, and light blue areas on the left plot correspond to the best, good, and satisfactory agreement with the reference polyhedron, respectively, while yellow and orange areas correspond to a bad recognition.

Figure 10. Strongly distorted coordination figures of copper and zinc atoms characterized by rating 1C in (left) [(tBuN4)CuI(MeCN)]·OTf, where tBuN4 = N,N′-di-tert-butyl-2,11-diaza[3.3](2,6)pyridinophane and OTf = trifluoromethanesulfonate (CSD reference code: HICQUH)51 and (right) [Zn2(L)]·4H2O, where L = tetraanion of 4,4′-bipyridine-2,6,2′,6′-tetracarboxylic acid (H4L) (CSD reference code: HENYUV).52

ously indicates the type of trigonal bipyramid; α′ = 10.28−3.28 = 7. The letter of the ranking characterizes the closeness of the coordination figure to a particular reference one. It is defined by the minimal distance to the reference figures (β′): A: β′ ≤ 5; B: 5< β′ ≤ 10; C: 10 < β′ ≤ 15; D: 15 < β′ ≤ 20. For example, for YIKYAS53 β′ = rTBPY‑5 = 3.28, and the rank is 2A. The ranks A, B, and C correspond to reliable recognition of the coordination figure, while D and E do not allow classification of the figure type. Most of the coordination figures in the structures under consideration are well-recognized by this method (Table 5). Figure 11 shows a two-dimensional projection of the fourdimensional distribution of distances r from the Zn or Cu coordination figures to the reference polyhedra from Table 3. As expected, the observed αbest values, which correspond to the points where the distribution crosses the coordinate axes, are equal to 13.8, which is the distance between the reference polyhedra TBPY-5 and SPY-5a (Table 3). 3.2. Comparison of the Three Methods. All three methods have independent sets of descriptors as well as their own advantages and disadvantages. Thus, the Voronoi

approach is the fastest but does not discriminate degree of the figure distortion. The Intersection approach is the most time-consuming but allows one to compare any coordination figures irrespective of their shape and number of vertices. The Fingerprint approach is as fast as the Voronoi approach but also takes into account the position of the central atom, although being applicable only to the figures with an equal number of vertices. However, in general, the methods give similar results (Table 6) that let us use them in combination to distinguish coordination figures more reliably. To adjust other methods to the Fingerprint approach, we grouped the base- and bodycentered square pyramid types as well as the capped tetrahedron and triangular bipyramid types. In most cases, the conclusions of all three methods coincide (Table 6, Figure 12, left) but the part of contradictive conclusion is also significant and reaches 23% (Table 6, Figure 12, right). This reflects the fact that the two types of coordination figures are geometrically very close to each other: even small deformations can result in an unambiguity of their classification. In our 779



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Table 5. Occurrences of Coordination Figures of Cu or Zn According to the Fingerprint Approacha SPY-5b

a

SPY-5a

TBPY-5

TF-5

rating

N

w, %

N

w, %

N

w, %

1A 1B 1C 1D 1E 2A 2B 2C 2E 3A 3B 3C 4A 4B total

1 2962 428 11

0.0 46.1 6.7 0.2

1260 4

68.0 0.2

19 656 230 22

1.6 54.8 19.2 1.8

1079 1389 24 1 532 3

16.8 21.6 0.4 0.0 8.3 0.0

296 223

16.0 12.0

151 91 2

12.6 7.6 0.2

71

3.8

27

2.3

6430

1854

1198

N

w, %

71 34 1

44.4 21.3 0.6

6 19

3.8 11.9

14 1 8 6 160

8.8 0.6 5.0 3.8

reliability + + + − − + + + − + + + + +

The coordination figures were assumed as unrecognized for ratings 1D, 1E, and 2E.

Figure 11. Distributions (scatter plots) of distances between the angular fingerprints of the coordination figure under consideration and (left) regular triangular bipyramid or base-centered square pyramid or (right) trigonal bipyramid, body-centered square pyramid, base-centered square pyramid, or capped tetrahedron (orange, gray, blue, or yellow regions, respectively).

subsequent analysis, we assume the coordinating figure was unambiguously determined only if the results of all methods are identical; otherwise, the type is considered not assigned. Because in the most reliable conclusion all three methods should give the same results, we recommend to start with the fastest methods, Voronoi and Fingerprint, in any order, and if their conclusions coincide, then finish with the most timeconsuming Intersection method. If the conclusions of the Voronoi and Fingerprint methods are different, the shape of the coordination figure should already be considered uncertain, and there is no need to apply the Intersection method. 3.3. Coordination Figure as a Descriptor for Knowledge Databases. Having the coordination figures determined for most of the compounds from our sample, we can now consider how this new descriptor correlates with other structural descriptors (Tables 1, S4−S8) and how it can be used to derive new knowledge for future expert systems. Note that the correlations discussed below are conditional because all of them take into account the nature of the central atom (Zn or Cu) of the coordination figures as well as the nature of the environment of the atoms (O or N).

Table 6. Results of Recognition of the Shape of Cu and Zn Coordination Figures by Different Methods N coordination figure

Cu

square pyramid trigonal bipyramid others

6340 900 1

square pyramid trigonal bipyramid others square pyramid trigonal bipyramid others square pyramid trigonal bipyramid others

Zn

w, % all

Cu

Voronoi 1367 7707 87.6 1027 1927 12.4 7 8 0.0 Intersection 6022 1242 7264 83.2 1144 1111 2255 15.8 75 48 123 1.0 Fingerprint (δ = 18; σ = 9) 6687 1585 8272 92.3 545 756 1301 7.5 9 60 69 0.1 total result 5906 1160 7066 81.6 430 667 1097 5.9 905 574 1479 12.5

Zn

all

56.9 42.8 0.3

79.9 20.0 0.1

51.7 46.3 2.0

75.3 23.4 1.3

66.0 31.5 2.5

85.8 13.5 0.7

48.3 27.8 23.9

73.3 11.4 15.3

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Figure 12. (Left) A fragment of the 2D structure of [CuII(C6H6O4)H2O], where C6H6O4 = dianion of 1,1-cyclobutanedicarboxylic acid (CSD reference code: LEHVAW),54 with the coordination formula AK21 M1 and square-pyramidal coordination figure of the Cu atom with the ratings 2A and 3A for the Intersection and Fingerprint methods, respectively. (Right) The molecular structure of [CuII(C25H38N6O5)]·H2O, where C25H38N6O5 = 4-nitroso-pyridine-capped dioxocyclam (CSD reference code: IMOJID),55 with the coordination formula AP00001 and an ambiguous type of the coordination figure.

Figure 13. Scatter plots wSPY‑5 − wTBPY‑5 for occurrence of coordination figures on the dimensionality of complex groups (left), chemical composition of coordination figure (middle), or overall network topology (right).

Chemical Environment of Metal Atoms versus Coordination Figure. The first coordination sphere of the Cu and Zn atoms can contain nitrogen and oxygen in any ratio (Table 1). The most preferable composition of the NnOm coordination figure is N2O3 for both metals. The following rule holds (Table 1, Figure 13, middle): if the nitrogen atoms dominate in the coordination polyhedron (n > m), the trigonal bipyramidal coordination is more probable; otherwise, square-pyramidal coordination prevails. Local Coordination Topology versus Coordination Figure. The distribution of local coordination topologies, which are described by coordination formulas, is quite different for trigonal bipyramid and square pyramid. In total, there are 227 different local coordination topologies, 59 of which were realized only one time (Table S1). The square-pyramidal coordination figure can combine with all typical local topologies, while the triangular-bipyramidal one fits only some of them. In particular, trigonal bipyramid is preferable for the AB012M1, AK0001 M1, AT001B01, and AP00001 local topologies and is very rare for AB22M1, A2K42B2, AT001 M2, and A2B25 (Tables S4, S5, and S7). Overall Network Topology versus Coordination Figure. Expectedly, the shape of the coordination figure influences the overall network topology, but not in all cases (Tables 1, S4, S6, and S8). Such overall network topologies as 3,5L2, fet, fes are preferred for the trigonal-bipyramidal type, while the topological types xah and 4,4L1 are allowable only for the square-pyramidal type. The topological types located close to the bisecting line of the scatter plot (sql, hcb, srs, and utk; Figure 13, right) are insensitive to the coordination figure type. Multiple Correlations. The coordination figure can play a crucial role in complex relations between several descriptors.

Figure 14. Typical 2D underlying nets in the copper coordination polymers in the standard representation (metal atoms and ligands are the nodes of the underlying net). Most of the structures (94%) have the sql overall network topology (shown in the center) in the cluster representation; the nets on the left contain paddle-wheel cluster groups. The brown and gray balls designate copper atoms and centers of mass of ligands, respectively.

Dimensionality of Crystal Structure versus Coordination Figure. In both Cu and Zn compounds, the coordination groups are mostly isolated; the second place is occupied by one-periodic coordination polymers, while two- and threeperiodic coordination groups are almost equally frequent (Table 1). This statement can be written in the form of a rule as the occurrence of coordination groups decreases as their dimensionality increases. The type of coordination figure does not influence the dimensionality of the complex group: all points of the scatter plot lie close to the bisecting line (Figure 13, left). 781



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Figure 15. Pairs of typical 3D underlying nets in the copper coordination polymers in the standard (left) and cluster (right) representations. In all cases, the cluster group is of the paddle-wheel type. The brown and gray balls designate copper atoms and centers of mass of ligands, respectively.

14 and 15), which are leaders in the corresponding distribution (Table S9). This establishes an additional correlation between the local connectivity of coordination groups and the overall network topology, i.e., the multiple correlation “coordination figure−secondary building unit−local connectivity−overall topology” holds. Using the decision tree (Scheme 1), we can compute probabilities of various multiple correlations. In particular, we see from the path “Coordination figure−trigonal bipyramid− without paddle wheel” that trigonal bipyramid cannot be combined with the paddle-wheel structural fragment. The most complicated correlations hold for the topological type of the

Obviously, the more descriptors are involved into such multiple correlations, the more detailed the prediction is. Below, we consider just one example of multiple correlation; all such correlations can be derived from the table of descriptor values in an automated way. Many underlying nets mentioned above (Figures 14 and 15) contain a “paddle wheel” fragment56−58 (Figure 16). This fragment occurs in both copper and zinc complexes with the same probability of ca. 13%. In all cases, it is formed by two metal centers in a tetragonal-pyramidal coordination. If such a group is considered as a whole, i.e., the cluster representation is applied, and then these nets transform to simpler ones (Figures 782



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(iii). Coordination figure → trigonal bipyramid → without paddle wheel → AT3B2 → 3,5L2 (sql) (P = 0.10·1.00· 0.21·0.48 = 0.010). The probabilities P of each combination are found as products of the probabilities of the corresponding pair correlations in the decision tree. Such decision trees, although without the coordination figure branch, were successfully used to predict the local and overall connectivity of coordination polymers60,61 and molecular crystals.13,26

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CONCLUSION Inventing new robust descriptors is a crucial problem of materials design: each new descriptor allows us to find a lot of new correlations with other parameters of the substance and to extend essentially the corresponding knowledge database. Here, we have shown how complicated and unclear such development could be even for a seemingly simple geometrical descriptor, the coordination figure. To make this descriptor more robust, we implemented three independent methods of its assessment (Voronoi, Intersection, and Fingerprint) and checked them on a large sample of crystal structures of coordination compounds. We discussed the advantages and disadvantages of the methods and proposed a scheme of their combined use: applied together, they make the most reliable conclusion on the coordination figure type. With them, we found many reliable correlations between coordination figure shape, chemical composition of coordination shell, local connectivity of complex groups, and overall topological architecture of the whole coordination polymer. We will include these correlations into the knowledge database on coordination compounds 60 as a part of its chemical composition−structure prediction scheme.

Figure 16. A fragment of the crystal structure of [CuII(C8F4O4)(MeOH)]·MeOH, where C8F4O4 = 2,4,5,6-tetrafluorobenzene-1,3dicarboxylato (CSD reference code: PAYLOS),59 with the 4,4L1 underlying net. The copper dimers form paddle-wheel groups.

underlying net. For example, the following three combinations are the most probable: (i). Coordination figure → square pyramid → with paddle wheel → A2K42B2 → xah (pcu) (P = 0.73·0.48·0.35·0.59 = 0.072). (ii). Coordination figure → square pyramid → without paddle wheel → AB22M1 → sql (sql) (P = 0.73·0.52· 0.17·0.83 = 0.053).

Scheme 1. Multiple Correlations “Coordination Figure (Orange)−Secondary Building Unit (Yellow)−Local Connectivity (Green)−Overall Topology (Cyan)” are Shown for the 2D and 3D Examined Crystal Structuresa

Correlations are presented as a decision tree. The root node is highlighted in red. The cyan level “overall topology” shows the overall network topologies in the standard and cluster (in parentheses) representations. a

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.cgd.6b01630. Table S1, distributions of local and overall topologies; Tables S2 and S3, distributions of local coordination and overall network topologies, respectively; Table S4, distributions of local and overall topologies depending on the type of coordination figure; Tables S5 and S6, distributions of local coordination or overall network topologies, respectively; Table S7, the strongest quaternary correlations between the dimensionality of coordination groups, metal atom type, local coordination topology, and shape of coordination figure; Table S8, the strongest quaternary correlations between the dimensionality of coordination groups, metal atom type, overall network topology and shape of coordination figure; Table S9, correlations between shape of coordination figure and overall network topologies in the cluster representation (PDF)

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AUTHOR INFORMATION

Corresponding Author

*Phone: +7-8463356798; Fax: +7-8463345417; E-mail: [email protected]. ORCID

Vladislav A. Blatov: 0000-0002-4048-7218 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors are grateful to the Russian government (Grant 14.B25.31.0005) for support. REFERENCES

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