Facing Your Evil Twin

1. Diagnosis and Treatment of Problem Structures: A Bruker Workshop on Single Crystal X-Ray Diffraction May 30, 2008Chemistry Department University of Tennessee Knoxville, TN Facing Your Evil Twin Victor G. Young, Jr. Department of Chemistry University of Minnesota 207 Pleasant St. S.E. Minneapolis, MN 55455 USA vyoung@umn.edu http://www.chem.umn.edu/services/xraylab

2. Outline • Introduction of Twinning • Merohedral Example • Pseudo-merohedral Example • Non-merohedral Example • Workshop Instruction • Final Thoughts MT PMT NMT

3. Twinning TermsGeneral • Twin: Aggregates consisting of regular crystals of the same species joined together in some definite mutual orientation. • Growth Twins: Those joined together in some random orientation as crystals radiate from their initial nucleation sites. • Deformation Twins: Those that form to relieve strain induced by some applied stress. • Transformation Twins: Those are a result of a polymorphic transformation from (usually) a higher symmetry, single crystal to a lower symmetry, twinned crystal.

4. Twinning TermsOperations • Rotation Twin: Twin components related by definite mπ/n rotation operations. • Twin Axis: The axis by which the twin components rotate. This often is a zone axis. Rotations in both direct- and reciprocal space are often referenced in publication. • Reflection Twin: Twin components related by a mirror operation. These are most often found in twinning by merohedry. • Twin Plane: The mirror plane relating two twin components.

5. Twinning TermsReference Frame • Composition plane: This is the plane which joins two or more twin components in the crystalline specimen. • Twin lattice: This is the reciprocal space representation of two or more twin components. If the twin operation is merohedral, then all reciprocal lattice points will be coincident with another. The other forms of twins produce fewer coincident reflections. • The concept of twin lattice is a useful construct to classify the various type of twins, however it does little to assist crystallographers in the task of unraveling the individual reciprocal lattices of twin components for accurate refinement.

6. Twin Lattice Symmetry Twin Lattice Symmetry (TLS) and Twin Lattice Quasi-Symmetry (TLQS) twins are classified into major groups based upon (a) whether the twin lattice superimposes over the true crystal lattice and (b) whether twinning promotes an apparent superlattice. ω stands for obliquity in angular deviation. n stands for twin lattice volume with respect to the individual. Reticular merohedral twins Pseudo-merohedral twins Non-merohedral twins Merohedral Inversion twins Merohedral Rotation and Reflection twins

7. TLS / TLQS Twins Pseudo-merohedral Non-merohedral Merohedral Reticular-merohedral

8. Examples of Twin Specimens Orthoclase twins Staurolite twins Fundamentals of Crystallography, Second Edition, Ed. Giacovazzo, pp. 280-281, Figs. 4.A.6-7

9. Twin Refinement in SHELX • Twinned crystals are refined by the method of a) Pratt, Coyle, and Ibers, J. Chem. Soc. (1971) 2146-2151 and b) Jameson, Acta Cryst., A38 (1982) 818-820. • The sum of the Fc2 values of the individual twin domains, each multiplied bt its fractional contribution, is fitted to the observed Fo2. • Since the n fractional contributions must sum to unity, only n-1 contributions can be refined independently. • (Fc2)* = osf2 [k1*Fc12 + k2*Fc22 + k3*Fc32 + …], where the values for k2, k3, … are given by the BASF instruction and k1 is defined such that Σ[km] = 1; (k1 = 1 - k2 - k3 - …). • MT and PMT use HKLF 4 data format. NMT use HKLF 5 data format. Interfering reflections are explicitly stated by format. Source: SHELX reference manual

10. Warning Signs for Merohedral and Pseudo-merohedral Twins 1) Laue symmetry does not fit for metric symmetry. 2) Rint for a higher symmetry Laue class is slightly higher than a lower symmetry Laue class. Rint often is lower than that for a normal crystal. 3) The reflection statistic |E2-1| value is shifted to a lower value: centrosymmetric. ~ 0.8, noncentrosymmetric ~ 0.6. 4) The space group may be in either trigonal or hexagonal: 2 crystal systems with same metrics, 5 Laue classes with 15 point groups. 5) The space group may be tetragonal: 2 Laue classes with 7 point groups. 6) The reflection conditions are not consistent with any known space group. 7) The structure cannot be solved. 8) The Patterson function is physically impossible. 9) The reciprocal lattice planes have a smooth variation in intensity from low to high resolution instead of a normal statistical variation. 10) SHELXL K-values are anomalously high – weak data fit is suspect. Source: R. Herbst-Irmer and G Sheldrick, Acta Cryst. B54, 443 (1998)

11. Example 1: Merohedral Twinning in a Trigonal Space Group - [Al(OC6H4(C3H7)2)]- . [NC7H8]+ • Laue statistics indicated P31/P32 or P3121/P3221 as the correct space groups with Rint = 0.041 and 0.045 respectively. The solution yielded an R = 0.16 result in P3221 with an unresolved cation. • The structure was re-solved in P32 which provided all atoms including the cation. • Inclusion of the twin law [01 0, 1 00, 001] improved the result to R = 0.043. • Some bond length restraints were included for the cation. • Twin ratio was 1:1.

12. Logic for the Determination of a Twin Law in the Example 1 1) Noting that two Laue classes, both 3 and 3m1 fit the data equally well and no acceptable solution was found in the enantiomorphic pair of space groups P3121 and P3221 from higher Laue class, we tested for a merohedral twin in a space group in the lower Laue class. 2) Table 1.3.4.2 from I.T. for C., Vol. C, suggests a possible “simulation” from P3121/P3221 to P31/P32. 3) The structure must be solved in one of the enantiomorphic pair of space groups P31 or P32 regardless of the quality. 4) Table 1.3.4.1 suggests inversion, a mirror or a two-fold rotation as possible twinning operations. The twinning is more severe than just a simple inversion. Since the heaviest atom is Al anomalous dispersion cannot distinguish enantiomorphs. 5) Provide the refinement program with possible applicable twin laws directly from Table 11.3 from I.T. for C., Vol. A. 6) Twin law was determined to be 2 x,x,0. The R improved dramatically.

13. Space Group Determination in the Trigonal Crystal System from the International Tables for Crystallography, Vol. A.

14. Possible Space Groups and Merohedral Twin Operations from the International Tables for Crystallography, Vol C, pg. 12: Tables 1.3.4.2 and 1.3.4.1 Step 1: Simulated space group of P3221 can be twinned P32. Step 2: Point group 3 can have inversion, mirror or two-fold twin operations. Step 3: Find the twin law and test it.

15. Symmetry Operations in the Hexagonal Coordinate System from the International Tables for Crystallography, Vol. A, pg. 798

16. Comparative Determination of the Twin Law: Equivalent Positions of P32 vs. P3221 • These are the listings of general positions for P32 (top) and P3221 (bottom). • Since a poorly-refining, simulated result is obtained in P3221, we can obtain the twin law directly from its set of general positions. • The twin law can be any general position (stripped of translational symmetry) in the simulated cell not appearing in the correct space group. • The y, x,-z(rotation) operation is the likely twin operation. • But, inversion twinning is also possible!!

17. Twinning in Enantiomorphic Space groups • If anomalous scattering significantly contributes to data, then enantiomorphs can be distinguished. For example 1, aluminum is the heavy atom for MoKα data, so this is ambiguous in refinement (high s.u. for Flack parameter). • The enantiomorphic pairs of space groups P31/P32 and P3121/P3221 introduce a complication when inversion twinning is indicated. To convert P31 to P32, for instance, reflect z-coordinates and change equivalent positions to P32. Remember the symmetry of P31 is a mirror image of P32! • The second example from the SHELX website is not ambiguous in anomalous scattering so both enantiomorphic and inversion twinning can be treated quantitatively.

18. Now, is that Twin Law Correct? In enantiomorphic space groups one needs to determine which of the two possible space groups is correct, because inversion twinning is also possible. Flack parameter may indicate conversion to the enantiomorph. Expansion of the twin law to: BASF 0.1 0.1 0.1 TWIN 0 1 0 1 0 0 0 0 -1 -4 To test all four possibilities: Two-fold [0 1 0 1 0 0 0 0 1 ] Two-fold + inversion [0 1 0 1 0 0 0 0 1 ] Mirror [0 1 0 1 0 0 0 0 1 ] Mirror + inversion [0 1 0 1 0 0 0 0 1 ]

19. Example 2: Merohedral Twinning in a Hexagonal Space Group – CsCp* . NC5H5 • Truly P61 or P65 but data appears as P6122 or P6522 in XPREP • The Cs atom in MoKα radiation provides sufficient anomalous scattering to refine racemic as well as 2 or m twin operations. • Structure is not interpretable until the 2 or m twin operation is refined. • Final R1 when completed is 0.02 From the SHELX website by Regine Herbst-Irmer: http://shelx.uni-ac.gwdg.de/~rherbst/merohedral.html

20. Common Pseudo-Merohedral Twins 1) A pseudo-merohedral twin can occur if the cell constants from a specimen imitate a higher crystal system. 2) Common examples are: a) monoclinic with b near 90° which could imitate orthorhombic (Laue class mmm). b) monoclinic with a and c near equal length and b near 120° which could imitate trigonal or hexagonal (several possible Laue classes). c) monoclinic with a and c near equal length and b near 90° which could imitate tetragonal (4/m Laue class). d) monoclinic with a and c near equal length and which could imitate C-orthorhombic n=2 (mmm Laue class). 3) The reduction in symmetry between the “simulated” Laue class and true Laue class leads to a possible twin law. 4) If the twin components are not equal in mass, then the “simulated” Laue class will be apparent when viewing the reciprocal lattice.

21. Example 3: Pseudo-merohedral Twinning in a Monoclinic Space Group – [CuC16H32N4]+[SbF6]- • The cell constants are metrically orthorhombic. • No solution could be determined in any orthorhombic space group. • Contents and unit cell volume suggested Z = 4. • P1 solution was generated: A possible c-glide plane suggested P21/c was true space group. • R = 0.24 before twin law was modeled. • R = 0.036 after [ 1 0 0, 0 -1 0, 0 0 1] twin law: ratio of twins was 0.55:0.45.

22. It looks like a c-glide, but… • Reciprocal lattice section h0l. • c-glide - h0l: l=2n . • h0l exhibits mm2 symmetry for orthorhombic instead of the expected 2 symmetry for monoclinic. • 0kl and hk0 sections also exhibit mm2 symmetry. • No space group fits! h l

23. Pseudo-Merohedral Twinning in a Monoclinic Space Group – Determination of Space Group Vol. indicates Z = 4 SPACE GROUP DETERMINATION Lattice exceptions: P A B C I F Obv Rev All N (total) = 0 6071 6055 6036 6064 9081 8052 8064 12090 N (int>3sigma) = 0 4989 5011 4880 5127 7440 6588 6610 9895 Mean intensity = 0.0 132.1 132.2 129.3 114.4 131.2 129.3 131.1 130.8 Mean int/sigma = 0.0 14.6 14.9 14.6 15.1 14.7 14.7 14.8 14.8 Crystal system M and Lattice type P selected Mean |E*E-1| = 0.771 [expected .968 centrosym and .736 non-centrosym] Chiral flag NOT set Systematic absence exceptions: -21- -a- -c- -n- N 18120 145 137 N I>3s 68 85 83 <I> 3.1 1.8 215.0 227.6 <I/s> 2.91.0 15.1 16.0 Option Space Group No. Type Axes CSD R(int) N(eq) Syst. Abs. CFOM [A] P2(1)/c # 14 centro 4 19410 0.079 7849 2.9 / 14.8 4.70 Option [A] chosen P-lattice unambiguous <|E2-1|> too low for centrosymmetric P21/c indicated Rint = 0.079 – high!

24. Pseudo-Merohedral Twinning in a Monoclinic Space Group – Analysis of Variance Indications Analysis of variance for reflections employed in refinement K = Mean[Fo^2] / Mean[Fc^2] for group Fc/Fc(max) 0.000 0.012 0.022 0.032 0.044 0.055 0.071 0.092 0.118 0.168 1.000 Number in group 444. 406. 402. 424. 395. 424. 413. 413. 407. 415. GooF 7.342 8.360 7.763 7.128 7.169 7.148 6.767 6.924 7.442 6.268 K 32.834 7.982 3.467 2.225 1.760 1.452 1.242 1.161 1.109 0.946 Resolution(A) 0.84 0.87 0.91 0.95 1.00 1.06 1.14 1.25 1.44 1.82 inf Number in group 425. 423. 410. 401. 414. 411. 417. 419. 408. 415. GooF 3.320 3.944 3.983 4.834 5.491 6.316 8.245 9.610 10.634 10.799 K 1.240 1.199 1.188 1.197 1.203 1.147 1.139 1.132 1.097 1.058 R1 0.279 0.283 0.259 0.254 0.267 0.254 0.270 0.247 0.227 0.192 Recommended weighting scheme: WGHT 0.1720 595.3703 Note that in most cases convergence will be faster if fixed weights (e.g. the default WGHT 0.1) are retained until the refinement is virtually complete, and only then should the above recommended values be used. Fc/Fc(max) and resolution: high Goof and K values

25. Pseudo-Merohedral Twinning in a Monoclinic Space Group – Determination of Twin Law • The apparent Laue class is mmm while we know the correct space group is P21/c (Laue class 2/m). • Compare the operations for point groups 2/m (1, m┴b, 2║b, and 1 ) vs. mmm (1, m┴a, 2║a, m┴b, 2║b, m┴c, 2║c, and 1 ). • Possible twin operations are those that exist in point group mmm that do not in point group 2/m (compare P2/m and Pmmm). • Thus m and 2 operations with respect to either the a or c axes can be used. Since it is arbitrary, we’ll select a 180º about (001) – [ -1 0 0 0 -1 0 0 0 1]. • Also, if the correct solution is an achiral (P21/P2) or a non-centrosymmetric (Pc/Pn) space group, then racemic twinning must also be tested.

26. Warning Signs of a Reticular Pseudo-Merohedral Twin 1) The specimen usually diffracts well. There is no indication of split reflections. 2) The specimen will index, but based on chemical knowledge it appears to have a unit cell with an unexpectedly large volume. Expected Z is too big. Often one cell constant is questionably long. 3) The space group choices are for unexpectedly low symmetry. 4) Lattice centering is present, but there are additional voids in the reciprocal lattice unaccounted by this reflection condition. 5) Visual inspection of reciprocal lattice indicates possible reflection conditions not found in the initial data reduction. 6) The reflection statistic <|E2-1|> value is an unreliable indicator of centrosymmetricity: False centering often raises this above 1.0. 7) Structure solution success varies: Usually, correct structural features related by the twin operation are observed in the model. 8) The twins in the correct, lower-symmetry crystal system must be determined with alternate indexing tools like Cell Now.

27. Example 4 : Reticular Pseudo-Merohedral Twinning: A True Triclinic Structure Twinned Through a Simulated C-centered Monoclinic • The triclinic cell was a = 7.336Å, b = 7.798Å, c = 8.396Å, a = 69.99°, b = 64.15,° g = 90.06°. • Data reduction indicated the cell could be transformed to C-centered monoclinic with a =15.112Å, b = 7.337Å, c = 7.797Å, and b = 112.35°. • <|E2-1|> was suspiciously low at 0.488. • Rint for triclinic =0.046 vs. monoclinic = 0.146. • Structure was solved in P1 with difficulty. • Twin law [ -1 0 0, 0 1 0, -1 0 1] was derived and applied. • R1 = 0.062 and wR2 = 0.132 for 87 parameters and 1386 reflections. M. Bolte and M. Kettner, Acta Cryst., C54, 963-964 (1998).

28. Example 4 : Reticular Pseudo-Merohedral Twinning: Derivation of the Twin Law • Three matrices are multiplied sequentially arranged right to left. • The central matrix is the twin symmetry element to be tested. • The transformation matrix from XPREP and its inverse surround the twin symmetry element. Twin symmetry element not allowed in Triclinic C-Monoclinicback to P-Triclinic P-Triclinic to C-Monoclinic Twin Law -1 x x =

29. Warning Signs of a Non-Merohedral Twin • The specimen will not index with the standard software. If it does index, then many reflections do not fit the cell. The reflections that do not index may have 1 or 2 non-integer h, k, l indices. • The specimen may index, but based on chemical knowledge it appears to have a unit cell with an unexpectedly large volume. The expected Z is likely too large based on the maximum number for the crystal system. • The unit cell has a sensible volume, but Laue symmetry is not confirmed in the initial analysis; twinning can make the Rint much greater than the final R1. • The reflection statistic <|E2-1|> value is shifted to a lower value when twins have frequent overlap: centrosymmetric ~ 0.8 and non-centrosymmetric ~0.6. • Reflections with Fo2>>Fc2 in the “50 Worst” list may have a systematic trend in some or combined indices of h, k, and/or l.

30. What should you consider before working on a non-merohedral twin? • Have you examined a representative number of specimens under a good polarizing microscope? Do you know what features to look for? • Have you attempted to index more than one specimen on the diffractometer? • If you have placed the specimen on the diffractometer with a cryostat and twinning is indicated, then you should consider repeating the indexing step at the same temperature the crystals were grown. • Given you have considered these questions you should proceed by collecting data on best twin specimen available. • A twin refinement using the method of Pratt, Coyle and Ibers can provide acceptable results especially if the data is treated with advanced software.

31. The Problem of Partially Overlapped Reflections • The reflection at h=0 is the sum of red and green twin comps. • Both h=1,2 are severely overlapped. The integration program has problems yielding accurate intensities for these. • There is no problem with the integration of h=3,4,5. • Proper integration and scaling/absorption correction are required. h=0 h=1 h=2 h=3 h=4 h=5 Regular and increasing separation in reciprocal space

32. The Better Solution for the Use of Partially Overlapped Reflections • The better method of processing data from frames is to integrate all exactly and partially overlapped reflections in similar ways. • This method profiles reflections and looks for any overlap. • The output data is essentially ready for refinement after an absorption correction is applied. h=0 h=1 h=2 (1/2) h=3 (1/2) h=4 (1/2) h=5 (1) (1) (1) Open circles represent the integration program applying a different algorithm to partially overlapping reflections.

33. Example 5: Non-Merohedral Twin – Twinning Discovered at the Point of Completion for [Nb(C9H12)2(CO)][V(CO)6] • The structure refines well and is acceptable for publication as is but… • R1 = 0.032, WR2 = 0.092 and the largest difference Fourier peak = 1.6 electrons Å-3. • 50 most disagreeable reflections indicate Fo2>>Fc2 for intensity. • Once twinning is properly accounted R1 = 0.022, WR2 = 0.0542, the largest difference Fourier peak = 0.26 electrons Å-3. B. Kucera and J. Ellis, unpublished results

34. Non-Merohedral Twin – Review 50 Most Disagreeable Listing ANOVA listing indicates nothing serious. The trend is Fo2 >> Fc2 with no systematic hkl variation.

35. Non-Merohedral Twin – Checking for a Minor Twin Component with Cell_Now #9 Matches original indexing

36. Non-Merohedral Twin – Checking for a Minor Twin Component with Cell_Now

37. Non-Merohedral Twin – Checking for a Minor Twin Component with Cell_Now

38. Non-Merohedral Twin – Cell_Now Summary • All reflections fit two components. • 132/157 fit the major component exclusively. • 25/157 fit the minor component exclusively. • 79 reflections are common to both components. • Twin law is [-1 0 0 -0.244 1 -0.113 0 0 -1], but it cannot be simply inserted into the SHELX ins file as for the MT and PMT examples. • Proper treatment of data begins in with integration with SAINT followed by TWINABS. • The SHELX refinement will be with HKLF5 data.

39. Non-Merohedral Twin – Integration Each twin component is integrated independently.

40. Non-Merohedral Twin – TWINABS Rint values are very good!

41. Non-Merohedral Twin – XL Refinement R improved 1.1%. Maximum residual electron density improved 5X!! Cost 1 additional parameter Too good to be true? Nah!

42. The 220 K structure in C2/c is in the high spin state for Fe+2. • Below 204 K the C-centered monoclinic cell transforms to a non-merohedral twin in P1. • This distortion is due to equal populations of both spin states. Example 6: FeN6 core at 220 K for{[HC(3,5-Me2pz)3]2Fe}[BF4]2 FE1 N1 2.172 FE1 N11 2.146 84.7 FE1 N21 2.166 84.6 84.0 FE1 N1A 2.172 180.0 95.3 95.4 FE1 N11A 2.146 95.3 180.0 96.0 84.7 FE1 N21A 2.166 95.4 96.0 180.0 84.6 84.0 N1 N11 N21 N1A N11A D.L. Reger, C.A. Little, V.G. Young, and M. Pink, Inorg. Chem. 40, 2870 (2001).

43. Low spin Comparison of FeN6 cores at 84 K for {[HC(3,5-Me2pz)3]2Fe}[BF4]2 High spin FE1 N1 1.977 FE1 N11 1.979 88.2 FE1 N21 1.976 88.6 88.4 FE1 N1A 1.977 180.0 91.8 91.4 FE1 N11A 1.979 91.8 180.0 91.6 88.2 FE1 N21A 1.976 91.4 91.6 180.0 88.6 88.4 N1 N11 N21 N1A N11A FE2 N31 2.171 FE2 N41 2.184 84.6 FE2 N51 2.160 83.9 85.1 FE2 N31A 2.171 180.0 95.4 96.1 FE2 N41A 2.184 95.4 180.0 94.9 84.6 FE2 N51A 2.160 96.1 94.9 180.0 83.9 85.1 N31 N41 N51 N31A N41A

44. Causative Factors in Reversible Phase Transitions Accompanied by Non-Merohedral Twinning • Bonds forming or breaking as temperature is lowered. These can be either covalent or hydrogen bonds. • Dynamic disorder freezes in to become static ordering. • It is quite certain that similar phase transitions involving non-merohedral twinning occur by varying magnetic fields, electro-magnetic fields, pressure, as well as temperature. • Our understanding of the structures of new materials will be enhanced by accurately accounting for non-merohedral twinning.

45. Phase Transition with Onset of Non-Merohedral Twinning: An Example You Can Use in Your Own Lab • 18-Crown-6 bis-(acetonitrile) clathrate is a simple crystal to grow for this demonstration. Dissolve 18-crown-6 in acetonitrile and chill in refrigerator. The specimens must be handled quickly or they will desolvate. Sub-ambient specimen transfer is recommended! • We have confirmed that the structure is a non-merohedral twin below ~168 K. The phase transition is reversible going between monoclinic and triclinic phases. • The phase transition is reversible without doing any damage to the specimen. • The twin law corresponds to a 180º rotation of [100].

46. Phase Transition with onset of Non-Merohedral Twinning: An Example You Can Use in Your Own Lab Formula 18-Crown-6 bis-(acetonitrile) clathrate Phase 1(GEFREO01) Phase 2(GEFREO) Space group P21/n P1 Temp, K 295 123 a, Å 9.123(3) 8.982(4) b, Å 8.524(3) 8.343(7) c, Å 13.676(4) 13.576(4) α, º 90.0 91.80(8) β, º 104.68(3) 105.50(3) γ, º 90.0 91.28(9) R, % 6.6 10.3 Phase 1, R.L. Garnell, J.C. Smyth, F.R. Fronczek, R.D. Gandour, J. Inclusion Phenom. Macrocyclic Chem., (1988)6, 73. Phase 2, R.D. Rogers, P.D. Richards, E.J. Voss, J. Inclusion Phenom. Macrocyclic Chem.,(1988) 6, 65.

47. Phase Transition with Onset of Non-Merohedral Twinning: hk0 Zone Image Showing the Phase Transition 163 K 173 K

48. What is the Origin of the Twinning? P21/n > 168 K – Disordered One-half molecule unique P1 < 168 K- Ordered by “Gearing” Two-half molecules unique Both drawn at 50% probability, viewing (101) axis of monoclinic setting The acetonitrile solvent was removed for clarity N. Brooks and V. G. Young, Jr., unpublished results

49. Questions After seeing these examples, do you now realize you may have a twin? Perhaps you know you have a twin, but you did not know how to treat the data. Do you have a better idea how to do this now? Is there anything else that is not clear? One thing I now realize after working on hundreds of twins is that there is a linkage between twinning, pseudo-symmetry, and the space groups linking phase transitions.

50. Example 1: Merohedral Twinning in a Trigonal Space Group - [Al(OC6H4(C3H7)2)]- . [NC7H8]+ • The SHELXL INS and HKL files are provided for both P32 and P3221 solutions. • View the P3221 solution first. While the anion appears OK, the cation is a total mess. The cation cannot be modeled. • As shown earlier, inclusion of the BASF and TWIN instructions fix the problem. Both anion and cation are well behaved. • Twin ratio was 1:1 with the Inclusion of the twin law [010, 1 00, 001].