4. modeling 3d-periodic systems
DESCRIPTION
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…. - PowerPoint PPT PresentationTRANSCRIPT
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