4. Modeling 3D-periodic systems

Cut-off radii, charges groupsEwald summationGrowth units, bonds, attachment energyPredicting crystal structures

Modeling 3D-periodic systems

Modeling just the asymmetric unit shouldbe sufficient, but….the number of VdW/Coulomb interactionsbecomes infinitely large!

solutions:* cut-off+ Contributions from VdW/Coulomb getsmaller at larger distance, and will finallyconverge.- Systematic error in EVdW

- Does Ecoul really converge?- Discontinuity at Rc

periodicity -- handling long-range interactions

Modeling 3D-periodic systemsperiodicity -- handling long-range interactions

* cut-off* ‘smooth’ cut-off+ No discontinuity at Rc

- Artifacts at Rc, due to large forces (F=E/R)

E= qixqj

40rij

Eapprox = E x C(rij)

C(rij)

rij

1

0

Modeling 3D-periodic systemsperiodicity -- handling long-range interactions

* cut-off* ‘smooth’ cut-off* cut-off with charge groups

Electrostatic energy can strongly dependon the chosen cut-off or atomic position.

Ztot

0-4 0-4 0….

Modeling 3D-periodic systemsperiodicity -- handling long-range interactions

* cut-off* ‘smooth’ cut-off* cut-off with charge groups+ Avoids discontinuity at Rc

- Need to define charge groups

Electrostatic energy can strongly dependon the chosen cut-off or atomic position. Solution: consider neutral groups instead of individual atoms/ions.

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 toan 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 ; slicesMorphology from Eattachment:growth rate ~ interaction energy between slice and bulk crystal

Growth units,bonds and attachment energy

surface

crystalbulk

solution/vapour/melt

Crystal growth occurs via the incorporation of growth units.This process is directed by the interactions between growthunits, bonds.

Growth units ; bondsBond strength: the sum of all interactions between growth units

12x12x2=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 structuresStep 1: generating trial crystal packings- use space-group statistics on organic solids for efficiency

spacegroup occurrence occurrence N35.9% 35.9% 4

P -1 13.7% 49.6% 211.6% 61.2% 46.7% 67.9% 2

C2/c 6.6% 74.5% 8Pbca 4.3% 78.8% 8Pnma 1.9% 80.7% 8

1.8% 82.5% 4Pbcn 1.2% 83.7% 8P1 1.0% 84.7% 1

P21/c

P212121

P21

Pna21

* non-chiral systems* any value for Z’* top-17 covers 90%

Predicting crystal structuresGenerating trial crystal packings in a MC-like way

O

b

a

c

0. Select the propermolecular conformation(s)1. Choose cell angles (max. 3)and orientation of cell contents.2. Set position of themolecule(s) in the asym. unit.3. Apply symmetry, and‘shrink to fit’ starting fromlong cell axes.4. Calculate EMM

5. Accept new packing if: e(-E/kt) > r(E=Enew - Eold; r= random number, 0r1) 6. Vary cell/orientational angles, and GOTO 2

… and at the same time, vary T.

Asymmetric unit

generated from space-group symmetry

Predicting crystal structuresGenerating trial crystal packings: MC and T

Accept new structure if: e (-E/kt) > r

E

e (-E/kt)

1.0

0.0

0.5

E: Enew - Eold

0.69kt

Always accepted

Maybe accepted

Predicting crystal structuresGenerating 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 structuresCrystal 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 structuresThermodynamically: structure(s) with lowest G can be found relative MM energy as approximation.

Crystal structure predictionresults

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; …..

Crystal structure predictiondescriptors: Radial Distribution Functions

Crystal structure predictiondescriptors: Radial Distribution Functions

C O H

C--CC--HC--OO--OO--HH--H

Crystal structure predictiondescriptors: Radial Distribution Functions

RDF based on:atomselementforce 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 EACPRET00-03 1.30BAFLID00-01 0.94BRESTO11-10 4.32CIYRIL00-01 2.29ESTRON11-10 0.28ESTRON12-10 -0.02GASFAH00-02 4.14LABHAX00-01 3.44MHNPRY01-00 1.74PROGST10-01 1.25ZZZNUK11-12 2.51

<E>=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 experimentsXRPDssNMRIRphase diagrams