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4. Modeling 3D-periodic systems

4. Modeling 3D-periodic systems. Cut-off radii, charges groups Ewald summation Growth units, bonds, attachment energy Predicting crystal structures. Modeling 3D-periodic systems. periodicity -- handling long-range interactions. Modeling just the asymmetric unit should be sufficient, but….

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4. Modeling 3D-periodic systems

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  1. 4. Modeling 3D-periodic systems Cut-off radii, charges groups Ewald summation Growth units, bonds, attachment energy Predicting crystal structures

  2. Modeling 3D-periodic systems periodicity -- handling long-range interactions Modeling just the asymmetric unit should be sufficient, but…. the number of VdW/Coulomb interactions becomes infinitely large! solutions: * cut-off + Contributions from VdW/Coulomb get smaller at larger distance, and will finally converge. - Systematic error in EVdW - DoesEcoul really converge? - Discontinuity at Rc

  3. 1 C(rij) 0 rij Modeling 3D-periodic systemsperiodicity -- handling long-range interactions E= qixqj 40rij Eapprox = E x C(rij) * cut-off * ‘smooth’ cut-off + No discontinuity at Rc - Artifacts at Rc, due to large forces (F=E/R)

  4. Modeling 3D-periodic systemsperiodicity -- handling long-range interactions Ztot 0 -4 0 -4 0 …. Electrostatic energy can strongly depend on the chosen cut-off or atomic position. * cut-off * ‘smooth’ cut-off * cut-off with charge groups

  5. Modeling 3D-periodic systemsperiodicity -- handling long-range interactions Electrostatic energy can strongly depend on the chosen cut-off or atomic position. Solution: consider neutral groups instead of individual atoms/ions. * cut-off * ‘smooth’ cut-off * cut-off with charge groups + Avoids discontinuity at Rc - Need to define charge groups

  6. Modeling 3D-periodic systemsperiodicity -- handling long-range interactions * cut-off * ‘smooth’ cut-off * cut-off with charge groups * Ewald summation Mathematical trick which makes use of the periodicity of the system: * part of the system treated ‘normally’ (in direct space) * part of the system treated via it’s Fourier transform (in reciprocal space) Combines good accuracy with efficiency. Can be implemented in 3 and 2 (and 1) dimensions, corresponding to an infinite crystal or an infinite slice

  7. Growth units,bonds and attachment energy Growth units: basic building blocks in crystal growth. Organic crystals  usually molecules as growth units. Example: benzene

  8. Growth units ; slices Morphology from Eattachment: growth rate ~ interaction energy between slice and bulk crystal

  9. solution/vapour/melt surface crystal bulk Growth units,bonds and attachment energy Crystal growth occurs via the incorporation of growth units. This process is directed by the interactions between growth units, bonds.

  10. Growth units ; bonds Bond strength: the sum of all interactions between growth units 12x12x2=288 interactions, summed into one bond; molecule treated as rigid body

  11. Predicting crystal structures * what is the 3D structure of a given crystalline material? * which other crystal packings might be possible? * what will be the structure if we change some functional group(s)? Crystal structure prediction: - generate many hypothetical structures - determine which ones are reasonable (ranking) - remove similar/duplicate structures from your results

  12. spacegroup occurrence N S occurrence P21/c 35.9% 35.9% 4 P -1 13.7% 49.6% 2 P212121 11.6% 61.2% 4 P21 6.7% 67.9% 2 C2/c 6.6% 74.5% 8 Pbca 4.3% 78.8% 8 Pnma 1.9% 80.7% 8 Pna21 1.8% 82.5% 4 Pbcn 1.2% 83.7% 8 P1 1.0% 84.7% 1 Predicting crystal structures Step 1: generating trial crystal packings - use space-group statistics on organic solids for efficiency * non-chiral systems * any value for Z’ * top-17 covers 90%

  13. b a O c Predicting crystal structures Generating trial crystal packings in a MC-like way generated from space-group symmetry 0. Select the proper molecular conformation(s) 1. Choose cell angles (max. 3) and orientation of cell contents. 2. Set position of the molecule(s) in the asym. unit. 3. Apply symmetry, and ‘shrink to fit’ starting from long cell axes. 4. Calculate EMM 5. Accept new packing if: e(-E/kt) > r (E=Enew - Eold; r= random number, 0r1) 6. Vary cell/orientational angles, and GOTO2 … and at the same time, vary T. Asymmetric unit

  14. Predicting crystal structures Generating trial crystal packings: MC and T Accept new structure if: e (-E/kt) > r e (-E/kt)  E: Enew - Eold Always accepted 1.0 Maybe accepted 0.5 E  0.69kt 0.0

  15. Predicting crystal structures Generating trial crystal packings: varying T during MC “Simulated Annealing” (SA) 2 1 Temp Time  1: increase T until almost every ‘move’ gets accepted, so every configuration can be reached. 2: slowly cool down, to drive the system to low-energy regions.

  16. Predicting crystal structures Crystal structure prediction: - generate many hypothetical structures - determine which ones are reasonable (ranking) Relative MM energy as ranking criterion  optimize all hypothetical structures, using e.g. a MM energy function. Degrees of freedom: * cell parameters (up to 6) * all atomic coordinates in the asymmetric unit

  17. Predicting crystal structures Thermodynamically: structure(s) with lowest G can be found  relative MM energy as approximation.

  18. Crystal structure predictionresults

  19. Crystal structure predictionclustering Reduce computation time via a representative subset of all structures. Remove identical structures. Sampling Clustering Ranking Clustering 5000 structures 500 structures 50 structures Clustering works by comparing descriptors for different items (i.c. structures) in the set, and quantifying their difference. Descriptors for crystals: space group; cell parameters; density; atomic coordinates; orientation of dipoles in the cell; …..

  20. Crystal structure predictiondescriptors: Radial Distribution Functions

  21. Crystal structure predictiondescriptors: Radial Distribution Functions C O H  C--C C--H C--O O--O O--H H--H

  22. Crystal structure predictiondescriptors: Radial Distribution Functions RDF based on: atoms element force field type weight factors via: partial charge (c.f. Egon Willighagen) number of electrons (~powder diffraction pattern) ….

  23. Crystal structure prediction - accuracy Accuracy from e.g. calculated E’s between observed polymorphs. Polymorph pair E ACPRET00-03 1.30 BAFLID00-01 0.94 BRESTO11-10 4.32 CIYRIL00-01 2.29 ESTRON11-10 0.28 ESTRON12-10 -0.02 GASFAH00-02 4.14 LABHAX00-01 3.44 MHNPRY01-00 1.74 PROGST10-01 1.25 ZZZNUK11-12 2.51 <E>=2.0 kcal/mol

  24. Our ‘toolkit’ for polymorph prediction Database searching [guess at initial model; spgr statistics] Conformational search [good starting structures] Monte Carlo; simulated annealing [sampling] clustering [speed up] energy minimization [ranking] combination with experiments XRPD ssNMR IR phase diagrams

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