Aug. 19, 2010
Meeting 001
The “important” questions – A name, and logo.
Electron Poor Materials Research Group
RG
EPM
Evolutionary crystal structure prediction
(***) USPEX, C.W. Glass, A.R. Oganov, N. Hansen, Computer Physics
Communications 175 713-720 (2006)
(*) Crystal structrue prediction using ab-initio evolutionary technique:
Principles and applications, A.R. Oganov and C.W. Glass, JCP 124,
244705 (2006)
USPEX = Universal structure preictor evolutionary Xtallography.
Thoughts
Many minima
High barriers
Part of landscape recognizable as inaccessible (d=0.5Angstrom)he bus
Symmetry degenerate structures
Deepest minima has large target area (?)
Favorable regions are cluttered with multiple minima.
Multiple minima play on a common theme – similar bond lengths, angles.
Barriers in favorable region are relatively shallow.
Advantages of evolution.
Non-local, no order parameters, useful for ab-iniito, no initial structure
needed, no thinking need be done once bus takes off.
The population at one generation
Each of these is a crystal of a specific composition, say Zn13Sb10.
Each structure is “optimized” – atoms have zero is force (CG, power
quench…) and lattice parameters optimized (at constant P).
How to get started? Random picks or include good guesses, poor
guesses,…
The variables:
  
a1, a2 , a3
The lattice  a1, a2, a3 mag., and α, β, γ.
The atomic coordinates
    
r1, r2 , r3 , r4 ....rN
Fractional coordinates:
(x,y,z)




r3  x3a1  y3a2  z3a3
The list: a1,a2,a3,alpha,beta,gamma,
X1,y1,z1, x2,y2,z2, …. xN,yN,zN  3N+3 (remove 3 uniform
translations).
Suppose just 10 values for each; e..g
X1=0, 0.1 ,…., 0.9.
.
10**(3N+3)
Zn13Sb10 has N=23, Landscape has 72 (=3N+3) dimensions.
Number of configurations we should check = 1072.
Evolutionary dynamics ---
Survival of the fittest.
Fitness landscape --- total energy (or Free energy)
Lowest free energy is the fittest.
“Unfit” structures will be eliminated (die off) .
Steps in the
dynamics:
1. Death.
2. Heredity.
3. Mutations
4. Permutations.
Heredity -- 2 parent cells give an offspring which is part “mother” and
part “father.”
Crude mating scheme – “slice and splice.”
Offspring 
Heredity –
Pick randomly a1, a2, or a3. Call it aH
Pick an x.
Create a slab from the mother. The mother slab
contains everything from 0 aH to x*aH.
Create a slab from the father. The father slab
contains everything from x*aH to 1 aH
Offspring 
Heredity –
But wait. The lattice vectors of a1, a2,
and a3 are different for Mom and Dad.
1. They each have an aH (e.g. an a2 if a2 is
chosen).
2. The atomic coordinates are fractions
(x,y,z), so one knows if the coordinate is
less than x*aH or greater than x*aH.
3.The lattice constants are randomly
weighted.
f=random number [0,1).
a1(offspring)= f * a1(mother) + (1-f) * a1(father)
Similar for a2, a3.
Heredity –Problems
1. Adjust chemical composition if necessary. Drop
atoms in or randomly take them out (?? Not sure
what their algorithm is. )
2. Choosing 0-x * aH could introduce bias, in that the
origin is special. Solution – shift atoms random
before mating.
r1=x*a1+y*a2+z*a3, suppose aH is a2.
Let rg(q) be a random number from a gaussian
distribution centered at q.
x(new)=(1+rg(0.05 (5%)))*x(old)
z(new)=(1+rg(0.05))*z(old)
y(new)=(1+rg(1.0))*y(old). (y special because aH =
a2)
Mutation:
Random pick
Mutant
Mutation:
Strain tensor
e11 e12 e13
e21 e22 e23
e31 e32 e33
e=rg(0,sigma)
gaussian distributed
about zero.
a1(new)= (1+e)*a1(old);
a2(new), a3(new) similar transformation.
Atomic coordinates are NOT mutated (i.e. x,y,z
remain unaltered).
Volume(new)=Volume(old)*trace(e) (small e)
Rescale Volume to target V_target. (?? Not sure)
Permutation:
Random pick
Permuted –
Shuffling of
chemical
species.
Permutation:
Npermute variable
Example:
Zn13Sb10. Npermute =1
Some detasils:
Volume scaling: V0 is the “nominal” volume of the cell.
During some genetic operations, the vulume will change.
It is rescaled back to V0,
before minimization. After minimization (at constant pressure), the volume will
change.
V0 changes during the run:
V0(next generation)=
sum(best structures) weight(i)*Volume(I, previous generation)
Hard constraints:
“Offspring” must be viable at “birth.”
-- d’s not too small ** [d(Zn, Sb)>minimum1, d(Zn,Zn)>minimum2, etc.]
-- angles not too large or small (alpha, beta, gamma)
-- each lattice vector not too small.
**Tip: for large systems, it becomes difficult to get all distances above
minimum distance. Use a vdW potential to relax the system first.
Kpoints – must be rescaled and grid changed. I don’t
understand their algorithm.
Other papers on USPEX -1. Oganov A.R., Glass C.W., Ono S. (2006). High-pressure phases of CaCO3:
crystal structure prediction and experiment. Earth Planet. Sci. Lett. 241, 95-103
2. Oganov A.R., Glass C.W. (2006). Crystal structure prediction using
evolutionary algorithms: principles and applications. J. Chem. Phys. 124, art.
244704
3. Glass C.W., Oganov A.R., Hansen N. (2006). USPEX: evolutionary crystal
structure prediction. Comp. Phys. Comm. 175, 713-720
4. Glass C.W., Oganov A.R., Hansen N. (2005). Predicting crystal structures of
new high-pressure phases. (Invited lecture, 20th IUCr congress, 23-31 August
2005, Florence, Italy). Acta Cryst. A61, C71, abstract MS54.27.5. (pdf-file).
5. Martonak R., Oganov A.R., Glass C.W. (2007). Crystal structure prediction and
simulations of structural transformations: metadynamics and evolutionary
algorithms. Phase Transitions 80, 277-298 (pdf-file).
6. Oganov A.R., Ma Y., Glass C.W., Valle M. (2007). Evolutionary crystal structure
prediction: overview of the USPEX method and some of its applications. Psi-k
Newsletter, number 84, Highlight of the Month, 142-171
7. Oganov A.R., Glass C.W. (2008). Evolutionary crystal structure prediction as a
tool in materials design. J. Phys.: Cond. Mattter 20, art. 064210 (invited paper)