Aug. 19, 2010

Meeting 001

The “important” questions – A name, and logo.

Electron Poor Materials Research Group

EPMRG

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:

3,2,1 aaa

The lattice a1, a2, a3 mag., and α, β, γ.

The atomic coordinates

Nrrrrr

....4,3,2,1Fractional coordinates: (x,y,z) 3323133 azayaxr

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) .

1. Death.2. Heredity.3. Mutations4. Permutations.

Steps in the dynamics:

Offspring

Heredity -- 2 parent cells give an offspring which is part “mother” and part “father.”

Crude mating scheme – “slice and splice.”

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

333231

232221

131211

eee

eee

eee

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)

e=rg(0,sigma) gaussian distributed about zero.

Permutation:

Random pick

Permuted – Shuffling of chemical species.

Npermute variable

Example:

Zn13Sb10. Npermute =1

Permutation:

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 willchange.

V0 changes during the run:

V0(next generation)=sum(best structures) weight(i)*Volume(I, previous generation)

Some detasils:

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, 95-1032. Oganov A.R., Glass C.W. (2006). Crystal structure prediction using evolutionary algorithms: principles and applications. J. Chem. Phys. 124, art. 2447043. Glass C.W., Oganov A.R., Hansen N. (2006). USPEX: evolutionary crystal structure prediction. Comp. Phys. Comm. 175, 713-7204. 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)