3. Crystals

1. 3. Crystals What defines a crystal? Atoms, lattice points, symmetry, space groups Diffraction B-factors R-factors Resolution Refinement Modeling!

2. Crystals What defines a crystal? 3D periodicity: anything (atom/molecule/void) present at some point in space, repeats at regular intervals, in three dimensions. X-rays ‘see’ electrons  (r) = (r+X) (r): electron density at position r X: n1a + n2b + n3c n1, n2,n3: integers a, b, c: vectors

3. Crystals What defines a crystal? primary building block: the unit cell lattice: set of points with identical environment crystal

4. Crystals Which is the unit cell? primitive vs. centered lattice primitive cell: smallest possible volume  1 lattice point

5. Crystalsorganic versus inorganic * lattice points need not coincide with atoms * symmetry can be ‘low’ * unit cell dimensions: ca. 5-50Å, 200-5000Å3 NB: 1 Å = 10-10 m = 0.1 nm

6. Crystalssome terminology * solvates: crystalline mixtures of a compound plus solvent c a b - hydrate: solvent = aq - hemi-hydrate: 0.5 aq per molecule * polymorphs: different crystal packings of the same compound * lattice planes (h,k,l): series of planes that cut a, b, c into h, k, l parts respectively, e.g (0 2 0), (0 1 2), (0 1 –2)

7. Crystalscoordinate systems Coordinates: positions of the atoms in the unit cell ‘carthesian: using Ångstrøms, and an ortho-normal system of axes. Practical e.g. when calculating distances. example: (5.02, 9.21, 3.89) = the middle of the unit cell of estrone ‘fractional’: in fractions of the unit cell axes. Practical e.g. when calculating symmetry-related positions. examples: (½, ½, ½) = the middle of any unit cell (0.1, 0.2, 0.3) and (-0.2, 0.1, 0.3): symmetry related positions via axis of rotation along z-axis.

8. Crystalssymmetry • Why use it? • efficiency (fewer numbers, faster computation etc.) • less ‘noise’ (averaging) • finite objects: crystals • rotation axes () rotation axes (1,2,3,4,6) • mirror planes mirror planes • inversion centers inversion centers • rotation-inversion axes rotation-inversion axes • ----------------------------- + screw axes • point groups glide planes • translations • --------------------------- + • space groups

9. Crystalssymmetry and space groups symmetry elements * translation vector * rotation axis * screw axis * mirror plane * glide plane * inversion center

10. Crystalssymmetry and space groups symmetry elements * translation vector * rotation axis * screw axis * mirror plane * glide plane * inversion center examples (x, y, z)  (x+½, y+½, z) (x, y, z)  (-y, x, z) (x, y, z)  (-y, x, z+½) (x, y, z)  (x, y, -z) (x, y, z)  (x+½, y, -z) (x, y, z)  (-x, -y, -z) equivalent positions Set of symmetry elements present in a crystal: space group examples: P1; P1; P21; P21/c; C2/c Asymmetric unit: smallest part of the unit cell from which the whole crystal can be constructed, given the space group. -

11. CrystalsX-ray diffraction diffraction: scattering of X-rays by periodic electron density diffraction ~ reflection against lattice planes, if: 2dhklsin = n X ~ 0.5--2.0Å Cu: 1.54Å dhkl  Data set: list of intensities I and angles  path: 2dhklsin

12. Crystalsinformation contained in diffraction data • * lattice parameters (a, b, c, , , ) obtained from the directions of • the diffracted X-ray beams. • *electron densities in the unit cell, obtained from the intensitiesof the • diffracted X-ray beams. • Electron densities  atomic coordinates (x, y, z) • Average over time and space • • Influence of movement due to temperature: atoms appear ‘smeared out’ • compared to the static model  ADP’s (‘B-factors’). • Some atoms (e.g. solvent) not present in all cells  occupancy factors. • Molecular conformation/orientation may differ between cells •  disorder information.

13. Crystalsinformation contained in diffraction data * How well does the proposed structure correspond to the experimental data?  R-factor consider all (typically 5000) reflections, and compare calculated structure factors to observed ones. R =  | Fhklobserved - Fhklcalculated | Fhkl =  Ihkl  Fhklobserved OK if 0.02 < R < 0.06 (small molecules)

14. Crystals - doing calculations on a structure from the CSD We can search on e.g. compound name

15. Crystals - doing calculations on a structure from the CSD We can specify filters!

16. Crystals - doing calculations on a structure from the CSD • ‘refcodes’ • re-determinations • polymorphs • *anthraquinone*

17. Crystals - doing calculations on a structure from the CSD

18. Crystals - doing calculations on a structure from the CSD

19. Crystals - doing calculations on a structure from the CSD Z: molecules per cell Z’: molecules per asymmetric unit

20. Crystals - doing calculations on a structure from the CSD

21. Crystals - doing calculations on a structure from the CSD

22. Crystals - doing calculations on a structure from the CSDexporting from ConQuest/importing into Cerius Cerius2 CSD cif cssr fdat pdb Not all bond (-type) information in CSD data  add that first!

23. Crystals - doing calculations on a structure from the CSDChecking for close contacts and voids how close is ‘too close’ default: ~0.9xRVdW minimal ‘void size’

24. Crystals - doing calculations on a structure from the CSDOptimizing the geometry CSD optimized*) a 7.86 7.76 b 3.94 4.36 c 15.75 15.12  90 90  102.6 107.4  90 90 ! * space-group symmetry imposed

25. Crystals - doing calculations on a structure from the CSDOptimizing the geometry CSD opt/spgr opt*) a 7.86 7.76 7.69 b 3.94 4.36 4.66 c 15.75 15.12 15.93  90 90 90  102.6 107.4 106.8  90 90 90 * space-group symmetry not imposed; Is it retained?

26. Crystals - doing calculations on a structure from the CSDOptimizing the geometry • Application of constraints during optimization: • space-group symmetry -- if assumed to be known • cell angles and/or axes -- e.g. from powder diffraction • positions of individual atoms -- e.g non-H, from diffraction • rigid bodies -- if molecule is rigid, or if it is too flexible...

27. Crystalssingle crystal versus powder diffraction Powder: large collection of small single crystals, in many orientations Single crystal  all reflections (h,k,l) can be observed individually, leading to thousands of data points. Powder  all reflections with the same  overlap, leading to tens of data points. Diffraction data can easily be computed  verification of proposed model, or refinement (Rietveld refinement)

28. Next week…. Modeling crystals: how does it differ from small systems? Applications: predicting morphology predicting crystal packing