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research papers Completion of crystal structure by powder diffraction data: a new method for locating atoms with polyhedral coordination

Journal of

Applied Crystallography ISSN 0021-8898

Received 20 November 2001 Accepted 11 April 2002

Carmelo Giacovazzo,a,b* Angela Altomare,a Corrado Cuocci,b Anna Grazia Giuseppina Moliternia and Rosanna Rizzia a

IC c/o Dipartimento Geomineralogico, UniversitaÁ di Bari, Campus Universitario, Via Orabona 4, 70125 Bari, Italy, and bDipartimento Geomineralogico, UniversitaÁ di Bari, Campus Universitario, Via Orabona 4, 70125 Bari, Italy. Correspondence e-mail: [email protected]

# 2002 International Union of Crystallography Printed in Great Britain ± all rights reserved

Ab initio crystal structure solution via powder diffraction data is often incomplete: frequently, the heavy atoms are correctly located but the lightatom positions are usually unreliable. The recently developed procedure POLPO [Altomare et al. (2000). J. Appl. Cryst. 33, 1305±1310], implemented in the EXPO program [Altomare et al. (1999). J. Appl. Cryst. 32, 339±340], aims at completing a partial structure model provided by direct methods by exploiting the prior information on the polyhedral coordination of the located atoms (tetrahedral or octahedral) and their connectivity. The POLPO procedure requires that all the cations are correctly labelled and rightly located. This condition does not always occur, particularly when the data quality is poor. A new method is described which is able to locate missing cations and surrounding anions when the cation coordination is tetrahedral or octahedral. The procedure has been successfully checked on different test structures.

1. Introduction In spite of the continuous development of new procedures and strategies, ab initio crystal structure solution by powder diffraction data is still a challenge in many cases. The peak overlap, the background estimate and the preferred orientation are the main obstacles to the process of structure solution (see Giacovazzo, 1996, for a review). Different mathematical techniques have been developed to optimize the procedure of extraction of the integrated intensities from the pattern (Pawley, 1981; Baerlocher, 1982; Rodriguez-Carvajal, 1990; Jansen et al., 1992; Sivia & David, 1994; Altomare et al., 1995; Wessels et al., 1999), but they are not able to provide error-free intensities. In recent years, an alternative approach has been introduced: it avoids the step of the extraction of the integrated intensities so that data quality is not a strictly necessary condition for succeeding in the crystal structure solution. The new approach works in the direct space and exploits some additional information on the molecular geometry. Stereochemical parameters (bond distances, angles and torsion angles) are varied according to one of the following techniques: genetic algorithm (Harris et al., 1998), grid search (Chernyshev & Schenk, 1998), Monte Carlo (Harris et al., 1994), simulated annealing (Kirkpatrick et al., 1983). The selection of the most reliable model is carried out by comparing the observed and calculated pro®les. Of special interest for this paper is the POLPO procedure (Altomare et al., 2000), which combines direct-methods performance with the Monte Carlo technique. It may be usefully applied when

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Completion of crystal structure

the structure solution provided by direct methods is partial. This situation often occurs in the case of heavy-atom structures, where the heavy atoms are easily located but the light atoms are hardly recognized. The starting point for POLPO is the set of cations located via direct methods: the procedure tries to locate the missing anions by exploiting the available prior information about the polyhedral coordination of the cations and their connectivity. The selection of the most reliable model is carried out by comparing the observed and calculated pro®les. POLPO requires that all the cations are located. This request is not always met via direct methods. We describe here a method that is able to locate a cation and its bonded anions between two already positioned cations, provided that lowlevel crystallochemical information is known a priori. The method has been integrated into the POLPO procedure, so making possible the completion of a crystal structure starting from the knowledge of only some cation positions. The procedure has been tested using various experimental data.

2. The method 2.1. Overview

The new POLPO procedure requires the following information. (a) The positions of some cations in the asymmetric unit in addition to their correct labels. J. Appl. Cryst. (2002). 35, 422±429

research papers (b) The number of missing cations and, for each of them, the already located cations between which it must be positioned. (c) The polyhedral coordination information about each located and missed cation: the type of coordination (tetrahedral or octahedral) and the expected polyhedral distance (a tolerance on distances and angles must also be supplied; see x2.2). For each cation, the location of a suitable atomic chain containing both the already positioned and the missing atoms is automatically selected by the program and is described in terms of the internal coordinates: distances, angles and torsion angles. The stereochemical parameters, not restrained by the polyhedral coordination rules, are randomly varied and the trial providing the best agreement between the observed and the calculated pattern is chosen. The process is repeated for each chain necessary to construct the full structural model. Once all of the cations and the coordination anions have been located, the pro®le reliability parameter Rp corresponding to the complete model is calculated and the procedure starts again to ®nd alternative models, among which the ®ve with the lowest Rp values are selected and analysed.

2.2. The algorithm

A variety of cases can be studied according to the number of located cations, the number of cations to locate and the cation connectivity. The following notation will be used. Kpj, j = 1, . . . np, indicate the np already positioned cations. Kmi, i = 1, . . . nm, indicate the nm missed cations. We suppose that direct methods failed to provide their positions. Aij corresponds to the generic anion bonded both to Kmi and to Kpj. dKmi and Kmi refer to the Kmi cation. The ®rst is the expected cation±anion distance (it is supplied by the user). Kmi is the anion±cation±anion angle: it is equal to 109.47 in the case of a tetrahedron, and to 90.0 or 180.0 in the case of an octahedron. dKpj and Kpj are the corresponding quantities for the Kpj cation. Again the dKpj value is supplied by the user; Kpj is ®xed by the same geometrical criteria applied to de®ne Kmi . dpi, pj is the mutual distance between Kpi and Kpj. In this paper, we will describe three basic situations.

^

where 1 is the bond angle Km1A11Kp1; 1 is the torsion angle ^ de®ned by A12, Km1, A11 and Kp1; 2 is the angle A11Kp1Kp2; 2 is the torsion angle de®ned by Km1, A11, Kp1 and Kp2. (ii) Random values are assigned to the stereochemical parameters 1, 1 and 2. The parameter 2 is calculated from 1, 1 and 2 [see (iv) below]. Since the chain must bridge Kp1 and Kp2, not all the random choices for the stereochemical descriptors 1, 1 and 2 are permitted. The random choices that are not consistent with equation (2) (see Appendix A) are rejected. (iii) The coordinates of the atoms A12, Km1, A11, Kp1 are calculated in a local Cartesian frame (Arnott & Wonacott, 1966; Andreev et al., 1997) with the origin in the Kp1 position. The X axis of the frame lies along the bond between Kp1 and to the X axis and lies within A11, the Y axis is perpendicular ^ the obtuse angle A11K p1Kp2 and the Z axis completes the right-handed orthogonal set (see Fig. 1). (iv) The information about orientation and modulus of the interatomic vector A12±Kp2 is not included in the Z matrix but it will be taken into account later for determining the value of 2 (as obtained in Appendix A), from which the local coordinates of Kp2 will be deduced. (v) The local coordinates of all the atoms of the chain can be transformed into general Cartesian frame coordinates by using the Eulerian matrix. The general Cartesian frame has: the origin coincident with the origin of the crystallographic frame; its Y axis coincident with the Y axis of the crystallographic frame; its X axis perpendicular to the Y axis and lying in the plane formed by the two Y and Z crystallographic axes; its Z axis completing the right-handed orthogonal set. The coordinates of Kp2 in the general Cartesian frame are easily derived from the crystallographic coordinates (Kp2 being located). The Eulerian matrix is expected to transform the local coordinates of Kp2 into its known general coordinates. The angles of the Eulerian matrix satisfying such a condition cannot be unequivocally calculated (see Appendix

2.2.1. Case 1: np = 2, nm = 1, Kp1 and Kp2 have been located, Km1 has been missed together with the anions of its polyhedron. The following steps are executed:

(i) The atomic chain A12±Km1±A11±Kp1±Kp2 (see Fig. 1) is considered and the corresponding following Z-matrix description is adopted: A12 Km1 A11 Kp1 Kp2

0 dKm1 dKm1 dKp1 dp1;p2

J. Appl. Cryst. (2002). 35, 422±429

0 0

Km1 1 2

0 0 0 1 2

Figure 1

The atomic chain A12±Km1±A11±Kp1±Kp2 is considered. Carmelo Giacovazzo et al.


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research papers B). Therefore an empirical instead of an analytical approach is followed: random values are assigned to the three Eulerian angles. (vi) The crystallographic coordinates can be calculated from the general Cartesian coordinates by using the matrix whose elements are determined by the lattice constants (Giacovazzo et al., 1992). Such crystallographic coordinates satisfy the stereochemical requisites de®ned in the Z matrix but, because of the free Eulerian angle values, the position of the last atom in the chain is not constrained to coincide with the known 0 position. A position of Kp2 and it is actually located on a Kp2 rotation of the chain atoms around an axis passing through Kp1 0 and perpendicular to the plane formed by Kp2, Kp1 and Kp2 , by 0 an angle such that Kp2 is moved onto Kp2, is carried out. The crystallographic coordinates of the chain atoms bridging Kp1 and Kp2 are so obtained. (vii) The coordination polyhedron is built around Km1 by taking into account the A11 and A12 positions. (viii) The pro®le reliability parameter, P wijYo i ÿ Yc ij 100 Rp P Yo i ÿ Yb i is calculated, where Yo(i) and Yc(i) are the observed and the calculated counts, respectively, corresponding to the 2i angular position, Yb(i) is the background and w(i) is a suitable weight {w(i) = 1/[Yo(i)1/2]}. (ix) The process returns to step (ii) and is repeated about 5000 times. (x) The set of atomic positions corresponding to the lowest RP value is selected. An extension of case 1 occurs when (a) np > 2 (for example, a supplementary Kp3 cation is correctly located); (b) the connectivity analysis suggests that an anion is bonded both to one of two located cations of the chain (e.g. to Kp2) and to the third cation (e.g. to Kp3). Conditions (a) and (b) are checked automatically by the procedure. Besides the steps (i)±(vi) described for case 1, the polyhedron around Kp2 is built using the already de®ned A12 position. The chain atoms are accepted and the procedure passes to the successive steps only if one of the atoms of the polyhedron is bonded to Kp3 with a distance value inside a bond range determined by the expected bond distance and by the corresponding tolerance value provided by the user. Otherwise, the procedure starts again from step (ii).

A22 Km2 A21 Kp1 A11 Km1 A12 Kp2

0 dKm2 dKm2 dKp1 dKp1 dKm1 dKm1 dKp2

0 0

Km2 1 Kp1

Kp1 ;A11 ;Km1 A11 ;Km1 ;A12 Km1 ;A12 ;Kp2

0 0 0 1 2 3

Kp1 ;A11 ;Km1 ;A12 A11 ;Km1 ;A12 ;Kp2

^

1 is the bond angle Km2A21Kp1; 1 is the torsion angle de®ned by A22, Km2, A21, Kp1; 2 is the torsion angle de®ned by Km2, A21, Kp1, A11; 3 is the torsion angle de®ned by A21, Kp1, A11, Km1. (iii) Random values are assigned to 1, 1, 2 and 3. (iv) The coordinates of the atoms of the chain are calculated in a local Cartesian frame with the origin in the A12 position. The X axis lies along the bond between A12 and Km1; the Y axis ^is perpendicular to the X axis and lies within the obtuse Km1A12Kp2 angle; the Z axis completes a right-handed orthogonal set (see Fig. 2). (v) The Eulerian angle matrix which transforms the local coordinates into the general Cartesian coordinates can be unequivocally determined owing to the fact that the general coordinates of Kp1, A11, Km1 and Kp2 are known (being already located). (vi) The crystallographic coordinates are calculated from the general Cartesian coordinates. Since no restraint is set about the distance of A22±Kp2 and about the bonding angle ^ A22Kp2A12, the trial crystallographic position of A22 may be incompatible with the expected bond distance and angle of Kp2 (within a variability range, de®ned by the user). If so, the trial atomic positions of A22, Km2 and A21 are randomly rotated around the axis Kp1±A11 (such a rotation corresponds to modi®cation of the value of 3). The distance A22±Kp2 and ^ the angle A22K p2A12 are calculated for each rotation. If they are within the expected ranges, the set of coordinates of the

2.2.2. Case 2: np = 2, nm = 2, Kp1 and Kp2 have been located, Km1 and Km2 have been missed together with the anions of their polyhedra. This case corresponds to the case in

which the bonded atoms form an eight-sided ring (see Fig. 2). The following steps are carried out. (i) The ®rst chain of atoms A11±Km1±A12 and the corresponding polyhedron around Km1 are positioned via the steps of case 1 above. (ii) The chain A22±Km2±A21±Kp1±A11±Km1±A12±Kp2 is considered and the corresponding following Z-matrix description is adopted:

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Figure 2

The atomic chain A22±Km2±A21±Kp1±A11±Km1±A12±Kp2 is considered. J. Appl. Cryst. (2002). 35, 422±429

research papers Table 1

Code and crystallochemical information (space group, unit-cell content and non-hydrogen symmetry-independent atoms) for the test structures. X indicates local laboratory diffractometer data; S indicates synchrotron data. References: (1) Norby et al. (1991); (2) Baerlocher et al. (1994); (3) Meden et al. (1997); (4) Camblor et al. (1998); (5) Zah-Letho et al. (1992); (6) Jouanneaux, Verbaere, Piffard et al. (1991); (7) Estermann et al. (1992); (8) Jouanneaux, Verbaere, Guyomard et al. (1991); (9) McCusker (1988); (10) PleÂvert et al. (1999); (11) McCusker et al. (1991). Code

Space group

Unit-cell content

Symmetry-independent atoms

CROX1 (X) EMT2 (S) GAPO3 (S) MCM4 (S) NBPO5 (S) NIZR6 (S) SAPO7 (S) SBPO8 (S) SGT9 (S) UTM110 (S) VFI11 (S)

P1 P63/mmc Pbca P6/mmm C2/c P21/n Pmmn P21/n I41/amd C2/m P63

Cr8O21 (Si,Al)96Na28O204 Ga32P32O128F8C56 Si72O144 Nb20O120P28 Ni2Zr8P4O48 Si32O64N2C48 Sb8P12O48 Si64O128C104 Si44O88 Al18P18O114

4Cr 11O 4(Si,Al) 13O 3Na 4Ga 4P 16O 1F 7C 8Si 13O 3Nb 4P 15O 2Zr 3P 1Ni 12O 4Si 10O 1N 6C 2Sb 3P 12O 4Si 7O 6C 6Si 13O 3Al 3P 19O

atoms in the chain is accepted; otherwise the process returns to step (iii). (vii) The polyhedron around Km2 is geometrically built by taking into account the positions of A22 and A21. (viii) The Rp value is calculated. (ix) The procedure returns to step (iii) and is repeated about 5000 times. (x) Among the feasible trials, that corresponding to the best pro®le ®t is selected. An extension of case 2 occurs when nm > 2. The cations Kmi, i > 2, and the corresponding anions are located via the steps of case 2 by considering the chain Ai2±Kmi±Ai1±Kp1±A11±Km1± A12±Kp2.

2.2.3. Case 3: np > 2, nm > 2, more than two cations must be located. The location of the ®rst missing cation and its

coordinated anions is carried out via the steps described for case 1. As soon as a new anion is located, its symmetryequivalent positions are calculated. Two situations may occur: (a) none of the symmetry-equivalent positions links the already located cations; the location of the other missing cations is executed by following the steps of case 1 or 2; (b) one or more of the symmetry-equivalent positions link the already located cations. Then a suitable atomic chain is hypothesized which takes into account the number of anions bonded to one or to two cations between which the missing cation should be positioned. The automation of the procedure is carried out in such a way that before locating each new cation, the connectivity status is examined and a convenient chain description is adopted. For the sake of brevity, the details of this situation will not be given. The above-described method has been integrated with the POLPO procedure in EXPO. The advantage is the following: as soon as the described process locates the missing cations and their coordination anions, the POLPO procedure will complete the structure. The structure completion is thus achieved in an easy and fast way. The full procedure proves more practical than the classical approach of combining difference Fourier map calculations with the Rietveld re®nement (Rietveld, 1969). J. Appl. Cryst. (2002). 35, 422±429

3. Applications The method has been applied to a number of test structures, the main crystallochemical information of which is given in Table 1. As for the original POLPO procedure, the following considerations hold. (a) The imperfect location of the cations provided by direct methods may lead to unreliable structure models (i.e. to remarkable errors in the positions of the missed cations and of the relative anions). (b) The strict geometrical conditions ®xed in the Z matrix for building the missing cations and the relative polyhedra do not take into account the distortion of the polyhedra in real structures and may lead to a ®nal model with a number of atoms that is larger than the true one. The ®rst problem can be reduced by a preliminary Rietveld re®nement, the second by considering a tolerance on the geometrical conditions. The new integrated POLPO procedure generates some feasible complete models which are stored and ranked according to the Rp value. Often the analysis of the bond distances immediately discards some trial models as chemically unacceptable. We will describe in detail the most relevant aspects and results of the new procedure for two of the test structures: SAPO (a zeolite) and SBPO (an antimony phosphate). For the other compounds, only the main results will be quoted.

3.1. SAPO

In SAPO, the only `heavy' atom is silicon. Direct methods in the standard EXPO program provide an incomplete structure model: 23 peaks are selected, 9 of them are close to the true positions of the atoms in the framework, with a distance Ê (the true number of atoms in the asymsmaller than 0.6 A metric unit is 14). Two atomic positions are automatically labelled as silicon: we selected and used them as pivots of the new procedure. Their distances from the true positions are Ê . The following directives are respectively 0.20 and 0.26 A given: Carmelo Giacovazzo et al.


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research papers tetrahedron Si 1 1:6 0:2 0:2 tetrahedron Si 2 1:6 0:2 0:2 missing 1 2 Si tetrahedron 1:6 0:2 0:2 missing 1 2 Si tetrahedron 1:6 0:2 0:2 The ®rst two directives state the coordination of the pivots: for each of them the ideal cation±anion distance is given, followed Ê units) and by the tolerance on by the tolerance on it (both in A the tetrahedral angle (see Altomare et al., 2000). A tolerance angle of 0.2 corresponds to the tolerance range 87.576± 131.364 . The directive `missing' activates the application of the new method. It is followed by the two order numbers of the cations between which the missing cation must be located, the missing cation label, its type of coordination, the expected cation±anion distance and the tolerances about the distance and the tetrahedron angle, respectively. For SAPO, two new Si atoms and their tetrahedrally coordinated anions should be located between the two already positioned Si atoms. Such a situation corresponds to case 2 (x2.2.2). The 15 feasible solutions (c.p.u. time 11 min 14 s on a Compaq Personal Workstation 500 SPECfp95:19.5; hereinafter all the c.p.u. time will refer to this workstation) provided by our procedure were ranked according to the Rp parameter. The model corresponding to the lowest Rp value (i.e. 1.23) was the most Figure 4

SBPO: (a) the two Sb cations, octahedrally coordinated, and the three P cations, tetrahedrally coordinated, in SBPO are shown; our procedure starts with the location of Sb(1) and Sb(2); (b) the model found by the new procedure is shown in the asymmetric unit; (c) the published structure is shown in the asymmetric unit.

informative (see Fig. 3): 2 more cations and 11 anions were found in the asymmetric unit. The average distance of the Ê , while for the cations from their true positions was 0.21 A Ê . In order to assess the reliability and the anions it was 0.38 A limits of the method, instead of using the two Si positions found by direct methods, their true positions were used. In this case, 18 feasible models were obtained and that corresponding to the lowest pro®le agreement parameter (1.18) was the best solution: 2 new cations and 12 anions were located in the asymmetric unit. The average distance of the cations from Ê , while it was 0.37 A Ê for the their true positions was 0.11 A anions. The reliability of the model obtained by using the true positions of the located cations is slightly better than for the model obtained by using the positions provided by direct methods.

3.2. SBPO

Figure 3

SAPO: (a) the four Si cations, tetrahedrally coordinated, in SAPO are shown; the starting point of our procedure is constituted by Si(1) and Si(2) cations; (b) the model found by the new procedure is shown in the asymmetric unit; (c) the published structure is shown in the asymmetric unit.

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In SBPO there are two types of heavy atoms, Sb and P. Direct methods in the standard EXPO program provide an incomplete structure model: among the 23 automatically selected peaks, 9 were close to the true atomic positions (the true number of atoms in the asymmetric unit is 17). Two atomic positions [Sb(1) and Sb(2) in the list below] labelled as antimony were selected and used as the pivots of our proceJ. Appl. Cryst. (2002). 35, 422±429

research papers Table 2

Results for each test structure. The code is followed (within parentheses) by the type of coordination (oct. if octahedral, tetr. if tetrahedral). np is the number of already positioned cations (with the atom type within parentheses). nm is the number of missing cations to be located (with the atom type within parentheses). nf is the number of atoms in the ®nal model provided by the procedure. nau is the true number of polyhedrally coordinated atoms in the asymmetric unit. nc is the number of atoms correctly located Ê ). hdi is the average distance between the correctly located atoms and the true ones. norder is the order (with a distance from the true position of less than 0.6 A number of the model corresponding to the largest number of correct positions (the models are ranked according to the Rp value). Time is the c.p.u. time spent by the procedure on a Compaq Personal Workstation 500 SPECfp95:19.5. The results in columns 5±8 were obtained by using the true cation positions to start our procedure; those in the last three columns were obtained when the trial coordinates supplied by EXPO were used. Code

np

nm

nf (nau)

nc

hdi

norder

Time

nc

hdi

norder

CROX (oct./tetr.) EMT (tetr.) GAPO (tetr.) MCM (tetr.) NBPO (oct./tetr.) NIZR (oct./tetr.) SAPO (tetr.) SBPO (oct./tetr.) SGT (tetr.) UTM1 (tetr.) VFI (oct./tetr.) VFI (oct./tetr.)

3 (Cr) 3 (Si) 5 (Ga) 5 (Si) 5 (Nb) 2 (Zr), 1 (P) 2 (Si) 2 (Sb) 3 (Si) 4 (Si) 2 (Al), 2 (P) 3 (Al)

1 (Cr) 1 (Si) 4 (P) 3 (Si) 2 (P) 1 (Ni), 2 (P) 2 (Si) 3 (P) 1 (Si) 2 (Si) 1 (Al), 1 (P) 3 (P)

16 (15) 16 (16) 27 (25) 25 (21) 28 (22) 21 (18) 16 (14) 22 (17) 12 (11) 20 (19) 21 (20) 21 (20)

15 16 23 20 21 16 14 17 11 19 20 19

0.20 0.15 0.22 0.24 0.24 0.30 0.30 0.28 0.18 0.16 0.26 0.38

1 1 1 5 5 2 1 3 1 1 3 3

2 min 5 s 11 min 22 s 35 min 37 s 25 min 26 s 25 min 26 s 21 min 38 s 11 min 57 s 35 min 10 s 12 min 7 s 7 min 15 s 8 min 4 s 13 min 59 s

15 16 20 17 21 15 14 17 11 19 20 17

0.32 0.16 0.25 0.21 0.27 0.29 0.32 0.27 0.14 0.27 0.31 0.31

2 1 1 2 3 2 1 1 1 1 1 1

dure: their distances from the true positions were about Ê . The following directives were given: 0.03 A octahedron Sb 1 2:0 0:15 0:15 octahedron Sb 2 1:9 0:15 0:15 missing 1 2 P tetrahedron 1:5 0:15 0:15 missing 1 2 P tetrahedron 1:5 0:15 0:15 missing 1 2 P tetrahedron 1:5 0:15 0:15 Three new cations (P) should be located, all of them bridging Sb(1) and Sb(2). Such a situation corresponds to the extension of case 2 (x2.2.2). The ®ve feasible solutions (c.p.u. time 30 min 12 s) provided by our procedure were ranked according to the Rp parameter. The model corresponding to the lowest Rp value (i.e. 0.80) was the most informative (see Fig. 4): 3 P cations and 16 anions were correctly located in the asymmetric unit. The average distance of the cations from their true Ê , while it was 0.32 A Ê for the anions. When positions was 0.15 A the true positions of the two located Sb atoms are used, four feasible models are obtained and the solution corresponding to the third model (as ranked according to the Rp pro®le agreement parameter; Rp = 0.80) was the best. Three new cations and 17 anions were located in the asymmetric unit. The average distance of the cations from their true positions was Ê , while for the anions it was 0.32 A Ê. 0.17 A 3.3. Test structures

In Table 2, the main results obtained by application of the new procedure to all the test structures are shown (see the caption to Table 2 for the meaning of the symbols). The results indicate that the method is ef®cient: almost all the missing cations and anions are well located. When the number of located atoms is larger than the number of atoms in the asymmetric unit (as for GAPO, MCM, NBPO, NIZR and VFI), the imperfectly positioned anions are misplaced, with respect to the true positions, by a distance value slightly larger J. Appl. Cryst. (2002). 35, 422±429

Ê . The structure completion is more straightforward, than 0.6 A on average, when no more than two cations have to be located: the best solution corresponds to the lowest Rp value and all the atoms are correctly located. The c.p.u. computing time required by our procedure is relatively modest and highly competitive with the classical approach (difference Fourier maps combined with Rietveld re®nements). The structure completion is often more accurate when the cation positions are used (the hdi values are smaller for seven out of twelve test structures; for ®ve cases the converse is true, which this is probably due to the aleatory character of our procedure). When the number of atoms in the ®nal model is larger than the true number, the Rietveld re®nement may improve the location of some atoms and discard some others. In some cases the best solution (that with the lowest hdi value) corresponds to the lowest Rp value, but it is always among the ®rst ®ve feasible models (in most of the cases all the feasible models are variants of the same solution).

4. Conclusions A method is described for locating one or more cations and their surrounding anions, provided that they bridge two or more located cations. It generalizes the POLPO procedure described by Altomare et al. (2000), which is able to generate only anion positions, provided that all the cations have been previously located by other techniques (e.g. by direct methods). The procedure has been implemented into the EXPO program so that EXPO is now able to complete a crystal structure starting from a few located cations, by using minimal additional information: the cation positions between which the missing cations have to be, the coordination of the missing and of the located cations (tetrahedral or octahedral), the expected cation±anion distances and the corresponding Ê in the location of the Kp cations tolerances. Errors up to 0.3 A do not hinder the success of the procedure; furthermore, tolerances up to 30% of the bond distances are not critical. So Carmelo Giacovazzo et al.


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research papers R21 xlKp2 R22 ylKp2 ycKp2 ;

far, the method has not been tested using unsolved structures; the success of the procedure suggests that the technique may have wide applications.

APPENDIX A Determination of the value of the angle a2 formed by A11, Kp1 and Kp2 in the chain A12±Km1±A11±Kp1±Kp2 The determination of the local Cartesian coordinates of the atoms in the chain A12±Km1±A11±Kp1±Kp2 may be carried out by following the paper by Andreev et al. (1997, Appendix A therein). The local Cartesian coordinates of Kp2 in a frame whose origin is in Kp1 are 0 l 1 2 3 xKp2 sin2 ÿ =2 ÿd B yl C 4 p2;p1 @ Kp2 A dp2;p1 cos2 ÿ =2 5 0 zlKp2 The local Cartesian coordinates of A12 xlA12 ; ylA12 ; zlA12 will depend on the stereochemical parameters of the chain. In accordance with the polyhedral coordination around Kp2, the distance of Kp2 from A12 must be equal to dKp2 . Therefore, it must be 2 ÿdp2;p1 sin2 ÿ =2 ÿ xlA12 2 dp2;p1 cos2 ÿ =2 ÿ ylA12 zlA212 dK2 p2 ; from which t2 2dp2;p1 ylA12 ÿ D 4dp2;p1 xlA12 t ÿ 2dp2;p1 ylA12 ÿ D 0; 1 2 ÿ xlA212 ÿ ylA212 ÿ where t = tg(2/2 ÿ /4) and D = dK2 p2 ÿ dp2;p1 l2 zA12 . Solving (1) with respect to t, gives t1=2 ÿ2dp2;p1 xlA12 2 4dp2;p1 xlA212 2dp2;p1 ylA12 ÿ D 2dp2;p1 ylA12 ÿ D1=2 2dp2;p1 ylA12 ÿ D:

The condition 2 xlA212 2dp2;p1 ylA12 ÿ D2dp2;p1 ylA12 ÿ D 0 4dp2;p1

2

must be ful®lled to obtain a real value for the unknown quantity t [and, consequently, for 2; 2 = 2(tanÿ1t + /4)].

APPENDIX B The Eulerian matrix for the chain A12±Km1±A11±Kp1±Kp2 In the atomic chain A12±Km1±A11±Kp1±Kp2, only two atoms are located (Kp1 and Kp2). The local Cartesian coordinates of Kp2 in a frame whose origin is in Kp1 are xlKp2 ; ylKp2 ; 0 (see Appendix A). If xcKp2 ; ycKp2 ; zcKp2 denote the coordinates of the located Kp2 in a general Cartesian frame and (Rij; i = 1, 3; j = 1, 3) is the Eulerian matrix, then R11 xlKp2 R12 ylKp2 xcKp2 ;

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R31 xlKp2 R32 ylKp2 zcKp2 : In addition, the Eulerian matrix properties require: R211 R212 R213 1:0; R221 R222 R223 1:0; R231 R232 R233 1:0; R211 R221 R231 1:0; R212 R222 R232 1:0; R213 R223 R233 1:0: The last equation derives from the previous ®ve and the system is not unequivocally determined. Therefore, instead of deducing some Rij values analytically (and of considering free the other values) random values are assigned to all three Eulerian angles from which the Rij matrix is obtained.

References Altomare, A., Burla, M. C., Camalli, M., Carrozzini, B., Cascarano, G., Giacovazzo, C., Guagliardi, A., Moliterni, A. G. G., Polidori, G. & Rizzi, R. (1999). J. Appl. Cryst. 32, 339±340. Altomare, A., Burla, M. C., Cascarano, G. L., Giacovazzo, C., Guagliardi, A., Moliterni, A. G. G. & Polidori, P. (1995). J. Appl. Cryst. 28, 842±846. Altomare, A., Giacovazzo, C., Guagliardi, A., Moliterni, A. G. G. & Rizzi, R. (2000). J. Appl. Cryst. 33, 1305±1310. Andreev, Y. G., MacGlashan, G. S. & Bruce, P. G. (1997). Phys. Rev. B, 55, 12011±12017. Arnott, S. & Wonacott, A. J. (1966). Polymer, 7, 157±166. Baerlocher, C. (1982). The X-ray Rietveld System. ZuÈrich: Institut fuÈr Kristallographie und Petrographie. Baerlocher, C., McCusker, L. B. & Chiappetta, R. (1994). Microporous Mater. 2, 269±280. Camblor, M. A., Corma, A., DõÂaz-CabanÄas, M.-J. & Baerlocher, C. (1998). J. Phys. Chem. B, 102, 44±51. Chernyshev, V. V. & Schenk, H. (1998). Z. Kristallogr. 213, 1±3. Estermann, M. A., McCusker, L. B. & Baerlocher, C. (1992). J. Appl. Cryst. 25, 539±543. Giacovazzo, C. (1996). Acta Cryst. A52, 331±339. Giacovazzo, C., Monaco, H. L., Viterbo, D., Scordari, F., Gilli, G., Zanotti, G. & Catti, M. (1992). Fundamentals of Crystallography. Oxford University Press. Harris, K. D. M., Johnston, R. L. & Kariuki, B. (1998). Acta Cryst. A54, 632±645. Harris, K. D. M., Tremaine, M., Lightfoot, P. & Bruce, P. G. (1994). J. Am. Chem. Soc. 116, 3543±3547. Jansen, I., Peschar, R. & Schenk, H. (1992). J. Appl. Cryst. 25, 231± 236. Jouanneaux, A., Verbaere, A., Guyomard, D., Piffard, Y., Oyetola, S. & Fitch, A. N. (1991). Eur. J. Solid State Inorg. Chem. 28, 755±765. Jouanneaux, A., Verbaere, A., Piffard, Y., Fitch, A. N. & Kinoshita, M. (1991). Eur. J. Solid State Inorg. Chem. 28, 683±699. Kirkpatrick, S., Gelatt, C. D. Jr & Vecchi, M. P. (1983). Science, 220, 671±680. McCusker, L. (1988). J. Appl. Cryst. 21, 305±310. J. Appl. Cryst. (2002). 35, 422±429

research papers McCusker, L. B., Baerlocher, Ch., Jahn, E. & BuÈlow, M. (1991). Zeolites, pp. 308±313. Meden, A., Grosse-Kunstleve, R. W., Baerlocher, C. & McCusker, L. B. (1997). Z. Kristallogr. 212, 801±807. Norby, P., Christensen, A. N., FjellvaÊg, H., Lehmann, M. S. & Nielsen, M. (1991). J. Solid State Chem. 94, 281±293. Pawley, G. S. (1981). J. Appl. Cryst. 14, 357±361. PleÂvert J., Yamamoto, K., Chiari, G. & Tatsumi, T. (1999). J. Phys. Chem. B, 103, 8647±8649. Rietveld, H. M. (1969). J. Appl. Cryst. 2, 65±71.

J. Appl. Cryst. (2002). 35, 422±429

Rodriguez-Carvajal, J. (1990). Abstracts of the Satellite Meeting on Powder Diffraction of the XV Congress of the IUCr, p. 127, July 1990, Toulouse, France. Sivia, D. S. & David, W. I. F. (1994). Acta Cryst. A50, 703± 714. Wessels, T., Baerlocher, Ch., McCusker, L. B. & Creyghton E. J. (1999). J. Am. Chem. Soc. 121, 6242±6247. Zah-Letho, J. J., Jouanneaux, A., Fitch, A. N., Verbaere, A. & Tournoux, M. (1992). Eur. J. Solid State Inorg. Chem. 29, 1309± 1320.



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