Improved real-space genetic algorithm for crystal structure and polymorph prediction

Existing genetic algorithms for crystal structure and polymorph prediction can suffer from stagnation during evolution, with a consequent loss of efficiency and accuracy. An improved genetic algorithm is introduced herein which penalizes similar structures and so enhances structural diversity in the population at each generation. This is shown to improve the quality of results found for the theoretical prediction of simple model crystal structures. In particular, this method is demonstrated to find three new zero-temperature phases of the Dzugutov potential that have not been previously reported.

Figure 1

(Color online) Comparison of the Lennard-Jones and Dzugutov pair potentials.

Figure 2

(Color online) Summary of the enthalpies of the different Lennard-Jones structures found for different fitness weights, which controls how much the comparison factor is considered during selection for update and crossover. The values for $w=0.0$ are those from Abraham and Probert (Ref. 1). All points are averaged over 15 independent calculations.

Figure 3

(Color online) Summary of the convergence times for the results shown in Fig. 2. The values for $w=0.0$ are those from Abraham and Probert (Ref. 1). All points are averaged over 15 independent calculations.

Figure 4

(Color online) Plot showing convergence to hcp minimum structure for a Lennard-Jones calculation with $w=0.75$. The stacking patterns of the minimum-enthalpy solutions are shown next to their appearance during the course of the simulation. The system converged to a hcp structure in 55 generations, and by the 127th generation, all members were the same.

Figure 5

The unit cell of the Dzugutov potential (Ref. 7) $\sigma$ phase looking down the $\left[00\overline{1}\right]$ direction.

Figure 6

(Color online) Convergence plot of a variable-atom, variable-cell calculation, starting from 62 atoms. This gives rise to a previously unknown phase (labeled as phase $a$ in Fig. 8). The inset shows the complete calculation. The minimum-enthalpy structure found has 65 atoms and is shown in Fig. 7.

Figure 7

The structure of the new phase $a$, a 65-atom phase found in generation 64 of the calculation shown in Fig. 6.

Figure 8

(Color online) Comparison of the radial distribution function $g\left(r\right)$ for the distinct lower-enthalpy structures found with bcc, fcc, hcp, and the $\sigma$ phase. The Dzugutov potential (Ref. 7) is also shown.

Figure 9

(Color online) Energy-volume curve for the Dzugutov potential (Ref. 7) showing the three new phases calculated at zero pressure. The curves for the $\sigma$ phase and structures $a$, $b$, and $c$ were calculated assuming an isotropic expansion.