A periodic genetic algorithm with real-space representation for crystal structure and polymorph prediction

A genetic algorithm is described that is suitable for determining the global minimum energy configurations of crystal structures and which can also be used as a polymorph search technique. This algorithm requires no prior assumptions about unit cell size, shape, or symmetry, nor about the ionic configuration within the unit cell. This therefore enables true ab initio crystal structure and polymorph prediction. Our algorithm uses a real-space representation of the population members, and makes use of a periodic cut for the crossover operation. Results on large Lennard-Jones systems with fcc- and hcp-commensurate cells show robust convergence to the bulk structure from a random initial assignment and an ability to successfully discriminate between competing low enthalpy configurations. Results from an ab initio carbon polymorph search show the spontaneous emergence of both Lonsdaleite and graphite-like structures.

Figure 1

Diagram showing crossover in a fractional representation. For each $(\zeta ,\eta )=\text{either}(\mathbf{a},\mathbf{b})$, $(\mathbf{b},\mathbf{c})$, or $(\mathbf{c},\mathbf{a})$ then $\mathbf{X}=\text{either}\mathbf{c}$, $\mathbf{a}$, or $\mathbf{b}$.

Figure 2

Real-space representation of the periodic cuts in the crossover operation. Different wavelengths and amplitudes can be used for the cuts along the different cell directions. The cuts are calculated in fractional coordinates which allows crossover between parents with different cells. The dark gray sections represent one part of the cell, the light gray the other, and it is these parts that are swapped in crossover.

Figure 3

Summary of the convergence times (in number of generations) for each variation of the method presented. For each run, either periodic or planar cuts were used, using either the roulette or the simple update scheme, and the number of atoms could either be kept fixed, or be allowed to vary.

Figure 4

Typical results from a 150 atom (variable), 16 population member calculation starting from an initial random configuration with a mutation rate of 10% and mutation amplitude of $2.5\mathrm{\AA }$, using roulette wheel selection in the update procedure. Periodic cuts were used and the system converged in 29 generations to a structure with 150 atoms and a local minimum enthalpy $+0.024\%$ above the hcp minimum.

Figure 5

Side on view of minimized structure from Fig. 4, looking down the $\left[0\overline{1}1\right]$ direction. The colors show the mixture of fcc (gray) and hcp (black) stacking.

Figure 6

Top view of structure 1 of the carbon structures (graphite-like) showing the layer stacking. The top layer is colored black and the layer below gray for clarity.

Figure 7

Top view of structure 5 of the carbon structures (graphite-like) showing the mismatch in the layer stacking. The top layer is colored black and the layer below gray for clarity.

Figure 8

View of structure 7 of the carbon structures (Lonsdaleite) looking down the [110] direction.