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 Zn 13 Sb 10. 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: The lattice a1, a2, a3 mag., and α, β, γ. The atomic coordinates Fractional coordinates: (x,y,z) 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 = 10 72.
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. Mutations 4. 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 a H Pick an x. Create a slab from the mother. The mother slab contains everything from 0 a H to x*a H. Create a slab from the father. The father slab contains everything from x*a H to 1 a H 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 * a H 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: Strain tensor 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.
N permute variable Example: Zn 13 Sb 10. N permute =1 Permutation:
Volume scaling: V 0 is the nominal volume of the cell. During some genetic operations, the vulume will change. It is rescaled back to V 0, 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) Some detasils: Hard constraints: Offspring must be viable at birth. -- ds 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 dont understand their algorithm.