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

Modeling 3D-periodic systems 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 E VdW - Does E coul really converge? - Discontinuity at R c periodicity -- handling long-range interactions

Modeling 3D-periodic systems periodicity -- handling long-range interactions * cut-off * ‘smooth’ cut-off + No discontinuity at R c - Artifacts at R c, due to large forces (F=  E/  R) E= q i x q j 4  0  r ij E approx = E x C(r ij ) C(r ij ) r ij 1 0

Modeling 3D-periodic systems periodicity -- handling long-range interactions * cut-off * ‘smooth’ cut-off * cut-off with charge groups Electrostatic energy can strongly depend on the chosen cut-off or atomic position. Z tot ….

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

Modeling 3D-periodic systems periodicity -- 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

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

Growth units ; slices Morphology from E attachment : growth rate ~ interaction energy between slice and bulk crystal

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

Growth units ; bonds Bond strength: the sum of all interactions between growth units 12 x 12 x 2=288 interactions, summed into one bond; molecule treated as rigid body

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

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%

Predicting crystal structures Generating trial crystal packings in a MC-like way O b a c 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 E MM 5. Accept new packing if: e (-  E/kt) > r (  E=E new - E old ; r= random number, 0  r  1) 6. Vary cell/orientational angles, and GOTO 2 … and at the same time, vary T. Asymmetric unit generated from space-group symmetry

Predicting crystal structures Generating trial crystal packings: MC and T Accept new structure if: e (-  E/kt) > r  E  e (-  E/kt)   E: E new - E old 0.69kt Always accepted Maybe accepted

Predicting crystal structures Generating trial crystal packings: varying T during MC “Simulated Annealing” (SA) Temp  Time  1 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. 2

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

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

Crystal structure prediction results

Crystal structure prediction clustering Reduce computation time via a representative subset of all structures. Remove identical structures. SamplingClusteringRankingClustering 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; …..

Crystal structure prediction descriptors: Radial Distribution Functions

C O H  C--C C--H C--O O--O O--H H--H

Crystal structure prediction descriptors: 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) ….

Crystal structure prediction - accuracy Accuracy from e.g. calculated  E’s between observed polymorphs. Polymorph pair  E ACPRET BAFLID BRESTO CIYRIL ESTRON ESTRON GASFAH LABHAX MHNPRY PROGST ZZZNUK =2.0 kcal/mol

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