1. Charge flipping in superspace, Aperiodic 2006, 21.9.2006 Structure solution of modulated structures by charge flipping in superspace Lukas Palatinus EPFL Lausanne Switzerland

2. Charge flipping in superspace, Aperiodic 2006, 21.9.2006 The principle of charge flipping Superspace Limitations and how to overcome them Implementation and demonstration

3. Charge flipping in superspace, Aperiodic 2006, 21.9.2006 Published by Oszlanyi & Sütö (2004), Acta Cryst A Iterative algorithm Requires only lattice parameters and reflection intensities The output is an approximate scattering density of the structure sampled on a discrete grid No use of atomicity, only of the “sparseness” of the electron density No use of symmetry apart from the input intensities Related to the LDE (low density elimination) method (Shiono & Woolfson (1992), Acta Cryst. A; Takakura et al. (2001), Phys.Rev.Lett.)

4. Charge flipping in superspace, Aperiodic 2006, 21.9.2006 Flow chart structure factors electron density “flipped” electron density “flipped” structure factors random phases + experimental amplitudes inverse FT flip all charge below a (small) threshold δ FT Combine phases of the flipped SF with amplitudes of the experi- mental SF

5. Charge flipping in superspace, Aperiodic 2006, 21.9.2006 structure factors electron density “flipped” electron density “flipped” structure factors random phases + experimental amplitudes inverse FT flip all charge below a (small) threshold δ FT Combine phases of the flipped SF with amplitudes of the experi- mental SF Flow chart

6. Charge flipping in superspace, Aperiodic 2006, 21.9.2006 Charge flipping reconstructs the density always in P1 Reason: in P1 the maxima can appear anywhere in the cell. In higher symmetry the choice is limited -> lower effectivity. Advantage: No need to know the symmetry, symmetry can be read out from the result Disadvantage: The structure is randomly shifted in the cell -> it is necessary to locate the origin

7. Charge flipping in superspace, Aperiodic 2006, 21.9.2006 Charge flipping does not use “atomicity” -> no problem to apply to superspace densities: The 3D density is replaced by a (3+d)D superspace density sampled using a (3+d)D grid The structure factors are indexed by (3+d) integer indices. They represent the coefficients of the Fourier transform of the superspace density. No need to know the average structure!

8. Charge flipping in superspace, Aperiodic 2006, 21.9.2006 All tested modulated structures could be solved by charge flipping: structuresymmetrycompositionV UC atoms tantalum germanium telluridePnma(00γ)s00TaGe 0.354 Te 2 347.316 lanthanum niobium sulphideF′m2m(α00)00s(LaS) 1.14 NbS 2 439.95.32 4,4 ’ -azoxyphenetole I2(α0γ)0C 16 H 18 N 2 O 3 1457.042 quininium (R)-mandelateP21(α0γ)0 C 20 H 25 N 2 O 2 + · C 8 H 7 O 3 - 1214.670 tetraphenylphosphonium hexabromotellurate- (IV) bis{dibromoselenate(I)} C2/m(α0γ)0s[(C 6 H 5 ) 4 P] 2 [TeBr 6 (Se 2 Br 2 ) 2 ] 2913.9130 hexamethylenetetramine sebacateP21(α0γ)0 N 4 (CH 2 ) 6 · (CH 2 ) 8 (COOH) 2 942.148 hexamethylenetetramine resorcinolI′mcm(0β0)s0s N 4 (CH 2 ) 6 · C 6 H 4 (OH) 2 1232.432 chromium(II) diphosphateC2/m(α β0)0sCr 2 P 2 O 7 258.724 Ce 13 Cd 58 Amma(00γ)s00Ce 13 Cd 58 6692.377 d-QC AlCoNi10 5 /mmcAl 70 Co 15 Ni 15 d-QC AlIrOs10 5 mcAl 70 Ir 14.5 Os 12.5 i-QC AlPdMnFm-3-5

9. Charge flipping in superspace, Aperiodic 2006, 21.9.2006 tetraphenylphosphonium hexabromotellurate(IV)bis{dibromoselenate(I)} <-CF Br1 Fourier-> <-CF C5 Fourier-> 4086 out of 4247 reflections correctly phased ( ~ 96%)

10. Charge flipping in superspace, Aperiodic 2006, 21.9.2006 published section final structure as obtained from charge flipping d-QC Al-Co-Ni, Steurer et al., Acta Cryst. B49, 1993

11. Charge flipping in superspace, Aperiodic 2006, 21.9.2006 Requirements on the data: Atomic resolution dmin=<1.0 A Small to medium-sized structure (below ca 1000 atoms in the unit cell) X-ray diffraction data Complete dataset Individual intensities are known (no powder, no twins)

12. Charge flipping in superspace, Aperiodic 2006, 21.9.2006 Atomic resolution dmin=<1.0 A Small to medium-sized structure (below ca 1000 atoms in the cell) X-ray diffraction data Complete dataset Individual intensities are known (no powder, no twins)

13. Charge flipping in superspace, Aperiodic 2006, 21.9.2006 Atomic resolution dmin=<1.0 A Small to medium-sized structure (below ca 1000 atoms in the cell) X-ray diffraction data Complete dataset Individual intensities are known (no powder, no twins) i-QC AlPdMn, unpublished neutron data provided by Marc de Boissieu Solution: flip everything between -  and +  (Oszlanyi & Sütö, ECM23)

14. Charge flipping in superspace, Aperiodic 2006, 21.9.2006 Atomic resolution dmin=<1.0 A Small to medium-sized structure (below ca 1000 atoms in the cell) X-ray diffraction data Complete dataset Individual intensities are known (no powder, no twins) from Palatinus & Steurer, in preparation Solution: extrapolate the missing reflections by MEM

15. Charge flipping in superspace, Aperiodic 2006, 21.9.2006 Atomic resolution dmin=<1.0 A Small to medium-sized structure (below ca 1000 atoms in the cell) X-ray diffraction data Complete dataset Individual intensities are known (no powder, no twins) Two techniques to overcome this problem: a)Repartitioning of the overlapping reflections according to the “flipped” structure factors (Wu et al. (2006), Nature Mater.) b)Repartitioning using histogram matching (Baerlocher, McCusker & Palatinus (2006), submitted)

16. Charge flipping in superspace, Aperiodic 2006, 21.9.2006 Atomic resolution dmin=<1.0 A Small to medium-sized structure (below ca 1000 atoms in the cell) X-ray diffraction data Complete dataset Individual intensities are known (no powder, no twins) Two techniques to overcome this problem: a)Repartitioning of the overlapping reflections according to the “flipped” structure factors (Wu et al. (2006), Nature Mater.) b)Repartitioning using histogram matching (Baerlocher, McCusker & Palatinus (2006), submitted)

17. Charge flipping in superspace, Aperiodic 2006, 21.9.2006 Superflip Superflip = charge FLIPping in SUPERspace Program for application of charge flipping in arbitrary dimension Some properties: Keyword driven free-format input file Automatic search for δ Automatic search for the origin of the (super)space group Support for the histogram-matching procedure and intensity repartitioning Continuous development Palatinus & Chapuis (2006), http://superspace.epfl.ch/superflip

18. Charge flipping in superspace, Aperiodic 2006, 21.9.2006 EDMA EDMA = Electron Density Map Analysis (part of the BayMEM suite) Program for analysis of discrete electron density maps: Originally developed for the MEM densities Analysis of periodic and incommensurately modulated structures Location of atoms and tentative assignment of chemical type based on a qualitative composition Export of the structure in Jana2000 format (SHELX and CIF formats in preparation) Writes out the modulation functions in a form of a x 4 -x i table Palatinus & van Smaalen, University of Bayreuth

19. Charge flipping in superspace, Aperiodic 2006, 21.9.2006 s d

20. s d d = (I-R).s

21. Charge flipping in superspace, Aperiodic 2006, 21.9.2006 d = (I-R).s How to find d? Patterson function: Symmetry correlation function: S will have the “origin peak” at d

22. Charge flipping in superspace, Aperiodic 2006, 21.9.2006 Example of a solution of a modulated structure:

23. Charge flipping in superspace, Aperiodic 2006, 21.9.2006 δ determines the amount of the flipped density. If δ is too small, the perturbation of the density is too small and the iteration does not converge. If δ is too large, too much of the density is flipped. In an extreme case all the density is flipped, which leads to no change of the amplitude of the structure factors. In practice δ can be determined easily by trial and error. Parameter δ