ISSN 163-7745, rystallography Reports, 29, Vol. 54, No. 5, pp. 727 733. Pleiades Publishing, Inc., 29. Original Russian Text N.V. Somov, E.V. huprunov, 29, published in Kristallografiya, 29, Vol. 54, No. 5, pp. 773 779. RYSTLLOGRPHI SYMMETRY edicated to the 1th birthday of G.. okiі The Translational and Inversion Pseudosymmetry of the tomic rystal Structures of Organic and Organometallic ompounds N. V. Somov and E. V. huprunov Nizhni Novgorod State University, pr. Gagarina 23, Nizhni Novgorod, 6395 Russia e-mail: somov@phys.unn.ru Received ecember 18, 28 bstract The translational and inversion pseudosymmetry of 211162 atomic crystal structures from the ambridge Structural atabase have been investigated. PS numbers: 61.5.h OI: 1.1134/S16377459522 The main geometric feature characteristic of all crystals is symmetry, i.e., the invariance of atomic structure with respect to the transformations of one of the 23 symmetry space groups. However, some crystals are also characterized by the approximate symmetry of their atomic structure; in this case, a sufficiently large atomic fragment is invariant with respect to a certain supergroup of the symmetry space group of the crystal. Such crystals are generally referred to as pseudosymmetric. Pseudosymmetric features of atomic crystal structures are responsible for the special types of diffraction patterns and features of pyroelectric (and other properties) of crystals [1 3]. It is well known that some atomic structures of low-symmetry crystalline phases that are characterized by secondorder phase transitions have pseudosymmetry properties [4, 5]. The additional symmetry of such an atomic fragment can be noncrystallographic; in this case, supersymmetry or hypersymmetry are generally mentioned. ifferent types of hypersymmetry of molecular crystals were investigated in detail by Zorkiі and colleagues [6, 7]. They described hypersymmetry groups (non- Fedorov space groups), which, along with conventional symmetric operations, include hypersymmetric ones. Let us consider the case where a large part of a crystal structure is invariant with respect to the latticecompatible symmetry operations. Such a pseudosymmetry will be referred to as Fedorov to indicate the specifically crystallographic (Fedorov) character of the increase in the symmetry of the crystal structure fragments [1]. It is of interest to perform a systematic quantitative analysis of the pseudosymmetric features of atomic crystal structures with Fedorov pseudosymmetry for which reliable structural information exists. In this paper we report the results of an analysis of the atomic structures included in one of the versions of ambridge Structural atabase [8]. The atomic structure of a pseudosymmetric crystal can be described by the symmetry space group G (with respect to which the entire atomic structure is invariant (the total function of electron density and nucleus of atoms)) and its supergroup T (to which the structural fragment is invariant). In the case of Fedorov pseudosymmetry, the group T is one of 23 space groups. The group T can be represented as a combination of adjacent classes with respect to its subgroup G. Let us expand group T in left-handed adjacent classes relative to G: T = G t 1 G t 2 G. (1) Here, the elements t 1, t 2, belong to the group T and do not belong to the group G; they are operations with respect to which the high-symmetry fragment of atomic crystal structure is invariant. epending on the type of the operations t 1, t 2, which help to form adjacent classes (1), translational, inversion, rotational, and other types of pseudosymmetry are distinguished. QUNTITTIVE ESTIMTION OF THE EGREE OF STRUTURL INVRINE To analyze the pseudosymmetric features of atomic crystal structures, it is necessary to introduce the quantitative characteristics of pseudosymmetry. Many physical properties are determined by the electrondensity distribution in a specific crystal; therefore, to quantitatively estimate the degree of pseudosymmetry of atomic crystal structures, it is necessary to use quantities fit for describing the pseudosymmetry of functions. To quantitatively estimate the degree of struc- 727
728 SOMOV, HUPRUNOV tural invariance with respect to an operation t, it is convenient to use the functional [9] η t ] ρ( x)ρ( tx) dv = ----------------------------. V ρ 2 ( x) dv V (2) Here, ρ(x) is the electron density function of a crystal and integration is over the unit-cell volume V. For nonnegative functions (such as the electron density function), the dimensionless quantity η t [ρ(x)] takes values from to 1. If the electron density function of a crystal is completely invariant with respect to operation t, i.e., if operation t belongs to the symmetry space group of the crystal structure, η t [ρ(x)] = 1. If η t [ρ(x)] =, one can consider the crystal structure completely noninvariant (asymmetric) with respect to this operation. For each specific structure, η t [ρ(x)] depends not only on the type of operation t but also on the position of the symmetry element corresponding to this operation in the unit cell. s a characteristic of atomic structure pseudosymmetry with respect to t, we will take the maximum degree of invariance for all possible positions of the corresponding symmetry element in the unit cell: η t ] max = max( η ti ]). Pseudosymmetric features can noticeably affect the physical properties of crystals at a sufficiently high degree of invariance. Strictly speaking, each individual physical property is characterized by its own minimum degree of invariance, beginning with the degree with which the crystal properties become significantly affected. previous analysis showed that, for a number of properties, the minimum degree of invariance (at which the structure can be considered pseudosymmetric) is in the range from.4 to.6 [1]. Therefore, we will arbitrarily assume a structure to be pseudosymmetric if η t ] max >.5. (3) For each crystal structure from the ambridge Structural atabase, we calculated the degree of invariance (2) for translation and inversion operations. The translation operations under study were described in a general form by the parallel-translation vector t = a b c --i, -j, -k, where m, n, p, i, j, and k are m n p integers. For each symmetry group, we determined an eigenset of numbers m, n, p, i, j, and k, which makes it possible to exclude calculating the degree of invariance with respect to the translations that are either lattice translations or obviously cannot lead to a Fedorov increase in symmetry [1]. When calculating the inversion pseudosymmetry for each crystal, we previously determined the possible points with rational coordinates at which inversion centers can be located (for example, particular positions and points of extraordinary regular systems [1]). The nearest vicinities of these points were also analyzed. The maximum calculated η t [ρ(x)] values with respect to translations and inversions were put into correspondence to each crystal. Functional (2) was calculated as follows. The crystal electron density function can be expanded in a Fourier series whose coefficients are structural amplitudes F(hkl). When substituting the Fourier expansions into the expression for functional (2), one can derive a formula for calculating η t [ρ(x)] by summing the series in reciprocal space. The formulas for calculating the degree of invariance of the electron density function with respect to translation a and inversion at a point with the radius vector τ have the form ] = η a F( hkl) 2 cos( 2π( ha x + ka y + la z )) --------------------------------------------------------------------------------------, h k l h k η 1 l ] F( hkl) 2 F( hkl) 2 exp( 4πi( hτ x + kτ y + lτ z )) = -------------------------------------------------------------------------------------. h k l h k l F( hkl) 2 (4) (5) The finiteness of the Fourier series in (4, 5), along with the error in representing the atomic factor when calculating structural amplitudes, determines the main error in calculating η t [ρ(x)], which does not exceed ±.2 in our calculations for most atomic structures. RESULTS OF LULTION OF STRUTURES WITH TRNSLTIONL N INVERSION PSEUOSYMMETRY The symmetry of the atomic fragments of crystal structures can be increased more so than the symmetry group of the crystal as a whole due to the location of one or several atoms over special regular systems of points, which are characterized by a higher symmetry [1, 1]. In addition, the symmetry of atomic fragments can also be increased due to the rise in the symmetry of a set (superposition) of regular systems. The values of unit-cell parameters, which are special for a given system, can also lead to pseudosymmetry. If pseudosymmetry arises due to an increase in the symmetry of one special regular system of points, the
THE TRNSLTIONL N INVERSION PSEUOSYMMETRY 729 atoms with the largest atomic number (i.e., the heaviest ones) in pseudosymmetric structures should be located over this regular system. We will characterize such crystal structures by the quantity r, defined as described by the space groups (a) P1 and (b) P 2 1 ---. The b degree of invariance of crystals with respect to translations (η t [ρ(x)] max ) is plotted over the vertical axis, and the r values calculated from formula (6) are on the horizontal axis. For the convenience of analysis, each diagram is divided into four parts. Part (.5 η t [ρ(x)] max 1; r.5) corresponds to atomic structures with high pseudosymmetry and small r. This part contains crysr = Z T N 1-2 2 Z j j = 1 ----------------, (6) where Z T is the atomic number of the heaviest atom in the structure and N is the number of atoms per unit cell. The quantity r characterizes the contribution that the heaviest atoms make to the electron-density distribution in a given atomic structure. For different crystal structures, r may vary from to 1. epending on the number of heavy atoms in the crystal space and their atomic numbers, r may tend to different limit values. The parameter r tends to zero for structures with a large (infinite) number of atoms with approximately identical atomic numbers per unit cell and is unity for crystals of chemical elements with one atom per unit cell. For other numbers n of heavy atoms per unit cell, r max can be written as r 1 max = -----. (7) n For example, for structures with two heavy atoms per unit cell, r max.71; with four heavy atoms, r max =.5; etc. It is convenient to calculate the pseudosymmetry and perform a comparative analysis of the results for many crystal structures using special diagrams. Generally, one can consider multidimensional diagrams with r values plotted over one axis and the degrees of invariance of atomic structures with respect to isometric operations plotted over other axes. Each crystal is presented by a point in the diagram. 2 cross sections of such diagrams can also be considered. In total, we found 211162 crystals with a complete set of characteristics necessary for calculations within the analyzed version of ambridge Structural atabase. The degree of invariance with respect to translations was calculated for all structures, and the degree of invariance with respect to inversion was calculated for 6 78 crystals (the symmetry of which is described by noncentrosymmetric space groups). Figure 1 shows the η r diagrams for the crystals η α [ρ] max (a) 1. 1.8.6.4 1..8.6.4 (b) 2 1 r Fig. 1. η r diagram of translational pseudosymmetry of crystals of organic and organometallic compounds described by the symmetry space groups (a) P1 and (b) P2 1 /b. tals formed by atoms with similar atomic numbers. In such structures, translational pseudosymmetry is due to the mutual ordering of different regular systems of the points over which atoms are located. Part (.5 η t [ρ(x)] max 1;.5 r 1) contains pseudosymmetric atomic structures formed by heavy atoms. These atoms make the largest contribution to the electron density function of the crystal and most often determine the pseudosymmetric properties of given crystals. Part ( η t [ρ(x)] max.5; r.5) contains atomic structures composed of atoms with approximately equal serial numbers and low pseudosymmetry. Part ( η t [ρ(x)] max.5;.5 r 1) contains nonpseudosymmetric crystals that have one or several heavy atoms per unit cell. On the whole, Fig. 1 shows 6226 and 13647 structures with the symmetries P1 and P2 1 /b, respectively.
73 SOMOV, HUPRUNOV η 1 [ρ] max (a) 1. 1.8.6.4 1..8.6.4 (b) 1 r Fig. 2. η r diagram of inversion pseudosymmetry of crystals of organic and organometallic compounds described by the symmetry space groups (a) P2 1 2 1 2 1 and (b) P2 1. For the group P1, the numbers of pseudosymmetric structures in parts and are, respectively, 673 and 198, i.e., about 1.4% of the total number of structures with this symmetry group. For the group P2 1 /b, the numbers of pseudosymmetric structures in parts and are, respectively, 3955 and 931, i.e., about 4.7% on the total number of structures with such a symmetry. The distribution of pseudosymmetric structures over parts and of the diagram indicates that the translational pseudosymmetry in these space groups for the structures from the ambridge Structural atabase is most often due to the increase in symmetry of the superposition of different regular systems of points over which atoms are located in the unit cell. The diagrams exhibit characteristic sequences, which include several points. These sequences correspond to the atomic crystal structures that are characterized by different numbers of regular systems of points occupied by heavy atoms. Let us consider sequence 1 in Fig. 1a. It can be seen from the diagram that the crystals forming this 2 2 sequence have the maximum r value r max.71 (shown by a dotted line) and η t [ρ(x)] max tends to unity. It follows from formula (7) that this sequence is formed by the atomic crystal structures with two heavy atoms per unit cell. This conclusion was confirmed by a direct analysis of a large number of structures forming this sequence. There are two versions of the location of two heavy atoms in structures with the symmetry P1 ; they are characterized by strong translational pseudosymmetry (part in Fig. 1a). In the first version, two symmetrically independent heavy atoms with approximately equal atomic numbers occupy particular positions of the group P1 ; i.e., they are located at inversion centers. The second version is implemented when two heavy atoms occupy a special regular system of points (for example, (1/4), (3/4)) or are located near such points. ll points in the diagram with r >.71 are related to crystals with one heavy atom per unit cell. This can only occur when a heavy atom occupies a particular position at the inversion center. Such a regular system of points does not have an additional symmetry; therefore, the pseudosymmetry of such structures is due to the ordering of positions of lighter atoms. similar η r diagram for the symmetry group P2 1 /b (Fig. 1b) also exhibits pronounced sequences. Sequences 1 and 2 correspond to atomic structures that have a pronounced translational pseudosymmetry and two and four heavy atoms per unit cell, respectively. This is evidenced by the fact that the limit r value for sequence 1 is 1 r max.71, while, for sequence 2 r max =.5. Sequence 1 is formed by crystals with heavy atoms occupying double particular regular systems of points, which are characterized by a reduction of translation (for example, 2a (,,), (,1/2,1/2)). Sequence 2 includes crystals with heavy atoms which occupy quartuple general regular systems of points that have coordinates characterized by reduced translation. One example of such a regular system of points is (x, y, 1/4), ( x, y, 3/4), ( x, 1/2 y, 3/4), (x, 1/2 + y, 1/4) [1]. The space group P2 1 /b does not have a regular system of points with a multiplicity of 1. The reason is that crystals with r >.71 are absent in the diagram. The inversion pseudosymmetry of crystals was analyzed similarly. For example, Fig. 2 shows the corresponding η r diagram for the noncentrosymmetric space groups (a) P2 1 2 1 2 1 and (b) P2 1. Figure 2a presents 23 347 atomic crystal structures described by the space group P2 1 2 1 2 1. This space group is characterized by one general regular system of points with a multiplicity of 4 and the coordinates {(x, y, z); (1/2 x, y, 1/2 + z); (1/2 + x, 1/2 y, z ); ( x, 1/2 + y, 1/2 z)} [11]. Specific regular systems of points are absent. Hence, the number of atoms of any type per
THE TRNSLTIONL N INVERSION PSEUOSYMMETRY 731 Table 1. Results of a calculation of the translational pseudosymmetry of crystals of organic and organometallic compounds in the most widespread symmetry groups Symmetry group according to [1] Symbol of symmetry group Number of pseudosymmetric structures (% of the total number of structures in parentheses) Numbers of structures in the parts of η r diagrams 1 P1 2347 (1.2%) 23 5 2 P1 6226 (1.4%) 673 198 4 P2 1 1523 (.5%) 61 18 5 2 1951 (7.9%) 147 7 7 Pb 1137 (5.7%) 52 13 9 b 341 (7.6%) 231 11 P2 1 /m 1647 (3.3%) 49 6 12 2/m 1359 (21.1%) 28 7 13 P2/b 168 (7.6%) 12 21 14 P2 1 /b 13647 (4.7%) 3955 931 15 2/b 21916 (9.8%) 2154 18 P2 1 2 1 2 119 (1.9%) 18 3 19 P2 1 2 1 2 1 2153 (.4%) 75 2 222 1 259 (2.8%) 54 29 Pca2 1 262 (3.6%) 74 33 Pna2 1 4252 (3.3%) 142 36 mc2 1 475 (9.7%) 46 41 ba2 275 (22.2%) 61 43 Fdd2 393 (9.9%) 39 unit cell (including heavy atoms) cannot be smaller than 4, and r.5. The atomic structures of this symmetry group contain only 6.2% of pseudosymmetric crystals. Part of the diagram contains 1441 atomic structures, some of which fall in sequence 1. The increase in the symmetry of structures forming sequence 1 is due to the location of heavy atoms over the regular systems of points, which are characterized by an additional inversion center. For example, the regular system of points with the coordinates {(,,,); (1/2,, 1/2); (1/2, 1/2, ); (, 1/2, 1/2)} is invariant with respect to inversion at the point (,,). Sequence 2, with a limit invariance of.5, is also formed by crystal structures with four heavy atoms per unit cell. In contrast to the structures of sequence 1, sequence 2 contains crystals where only two of the four heavy atoms are invariant with respect to the center of symmetry. This property is typical of any regular system of points of the group P2 1 2 1 2 1, with one of the coordinates equal to 1/4 (for example, the regular system of points {(x, y, 1/4); (1/2 x, y, 3/4); (1/2 + x, 1/2 y, 3/4); ( x, 1/2 + y, 1/4)}. Indeed, the first two and last two points are pairwise invariant with respect to inversion at the point (1/4,,1/2). However, this regular system is, on the whole, noncentrosymmetric. The space group P2 1 is presented in the diagram by 14122 atomic structures, about 22.8% of which are pseudosymmetric. The limit r value for the atomic structures of this symmetry group is.71. The reason is that the group P2 1 contains no particular regular systems of points, and the multiplicity of the general regular system of points is 2. Part of the diagram contains 173 atomic structures; in most of them, symmetry increases due to the ordering of a certain part of the regular system of points occupied by atoms. Part contains 1485 atomic structures, many of which belong to sequence 1. The inversion pseudosymmetry of the crystal structures described by the space group P2 1 is mainly determined by the following. Note [12] that any regular system of points of this group is exactly invariant with respect to its local inversion center. In this case, the entire structure obviously remains noncentrosymmetric. If there is a regular system in the structure occupied by a heavy atom, a large part of electron density is invariant with respect to the local inversion center corresponding to this regular system. In other words, all structures described by the space group P2 1 and containing heavy atoms are pseudosymmetric to some extent. mong the 23 symmetry space groups, there are 48 groups for which any regular system of points is
732 SOMOV, HUPRUNOV Table 2. Results of a calculation of the inversion pseudosymmetry of crystals of organic and organometallic compounds in the most widespread symmetry groups Symmetry group according to [1] Symbol of symmetry group Number of pseudosymmetric structures (% of the total number of structures in parentheses) Numbers of structures in the parts of η r diagrams 1 P1 185 (32%) 55 13 4 P2 1 3215 (23%) 173 1485 5 2 719 (32%) 641 78 7 Pb 285 (29%) 163 122 8 m 8 (59%) 44 36 9 b 187 (39%) 187 18 P2 1 2 12 231 (19%) 167 64 19 P2 1 2 1 2 1 1442 (6%) 1441 2 222 1 136 (28%) 136 29 Pca2 1 521 (27%) 521 31 Pmn2 1 129 (62%) 81 48 33 Pna2 1 1658 (4%) 1658 36 mc2 1 269 (55%) 269 Table 3. Numbers of pseudosymmetric structures of crystals of organic and organometallic compounds in different systems System Triclinic Monoclinic Orthorhombic Tetragonal Hexagonal ubic Total number of investigated noncentrosymmetric crystals Number of pseudosymmetric crystals (% of structures in parentheses) inversion pseudosymmetry 576 2362 33794 3426 253 496 185 (32.1%) 546 (26.5%) 519 (15.1%) 717 (2.9%) 51 (24.8%) 91 (18.3%) translational pseudosymmetry Total number of investigated structures 64371 151739 49935 4676 4522 34 Number of pseudosymmetric structures 899 (1.4%) 868 (5.3%) 322 (6.1%) 584 (12.5%) 33 (6.7%) 36 (1.6%) centrosymmetric (locally centrosymmetric groups). list of them can be found in [12]. Thus, sequences 1 and 2 in Fig. 2b determine crystals with, respectively, two and four heavy atoms per unit cell. The additional increase in the symmetry (points above these sequences) is due to the fact that several regular systems of points arranging atoms are simultaneously invariant with respect to the same local center. Tables 1 and 2 contain the results of a pseudosymmetry calculation for the most widespread symmetry groups of crystals of organic and organometallic compounds. Table 3 contains generalized data on the numbers of pseudosymmetric crystal structures in different systems. The data of the tables indicate that pseudosymmetry is a widespread phenomenon among crystals of organic and organometallic compounds. Note that inversion pseudosymmetry is much more widespread than translational. This is especially true for crystals whose symmetry is described by locally centrosymmetric groups (P1, P2 1, b, Pna2 1, etc.). KNOWLEGMENTS This study was supported in part by Grant NSh- 2192.28.5 for Support of Leading Scientific Schools. REFERENES 1. E. V. huprunov, Kristallografiya 52, 1 (27) [rystallogr. Rep. 52, 1 (27)]. 2. E. V. huprunov, T. N. Tarkhova, and I. P. Makarova, Kristallografiya 26, 1177 (1981) [Sov. Phys. rystallogr. 26, 667 (1981)].
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