Crystal structure prediction via particle-swarm optimization

We have developed a method for crystal structure prediction from “scratch” through particle-swarm optimization (PSO) algorithm within the evolutionary scheme. PSO technique is different with the genetic algorithm and has apparently avoided the use of evolution operators (e.g., crossover and mutation). The approach is based on an efficient global minimization of free-energy surfaces merging total-energy calculations via PSO technique and requires only chemical compositions for a given compound to predict stable or metastable structures at given external conditions (e.g., pressure). A particularly devised geometrical structure parameter which allows the elimination of similar structures during structure evolution was implemented to enhance the structure search efficiency. The application of designed variable unit-cell size technique has greatly reduced the computational cost. Moreover, the symmetry constraint imposed in the structure generation enables the realization of diverse structures, leads to significantly reduced search space and optimization variables, and thus fastens the global structure convergence. The PSO algorithm has been successfully applied to the prediction of many known systems (e.g., elemental, binary, and ternary compounds) with various chemical-bonding environments (e.g., metallic, ionic, and covalent bonding). The high success rate demonstrates the reliability of this methodology and illustrates the promise of PSO as a major technique on crystal structure determination.

Figure 1

The flow chart of CALYPSO.

Figure 2

(Color online) Depiction of the velocity and position updates in PSO. The black solid line approximates a typical energy landscape. Arrows represent either the velocity or the relative positions of a particle (here, a structure). An initial velocity $V^t$ develops to $V^{t+1}$ after one PSO step through Eq. (2). A high-energy structure $\left({\text{X}}^t\right)$ evolves to a much lower-energy structure $\left({\text{X}}^{t+1}\right)$ after one PSO operation through Eq. (1). This illustrates the efficiency of PSO in exploration of low-energy structures.

Figure 3

Main figures: the history of CALYPSO structure search on Li at 200 GPa. Insets: the predicted $Cmca\text{−}56$ structure at the fourth generation. For the $Cmca\text{−}56$ structure at 220 GPa, the lattice parameters are $a=15.687\text{Å}$, $b=3.855\text{Å}$, and $c=3.847\text{Å}$ with atomic positions at $16g$ (0.14302, 0.83356, 0.43863), (0.21482, 0.10157, 0.15158), (0.92849, 0.86370, 0.87551), and $8f$ (0.00000, 0.44292, 0.83303), respectively.

Figure 4

(Color online) The metastable structures of carbon predicted by CALYPSO. (a) $bc8$ structure predicted at 2000 GPa. (b) ${\text{C}}_6$ $Im\overline{3}m$ structure predicted at 0 GPa. (c) $\beta$-Sn structure predicted at 0 GPa. (d) Chiral framework $P{6}_{1}22$ or $P{6}_{5}22$ structure predicted at 0 GPa.

Figure 5

(Color online) Main figures: enthalpy history of CALYPSO structure search on ${\text{SiO}}_2$ (a) at 70 GPa and ${\text{CaCO}}_3$ (b) at 120 GPa. Insets: the predicted ${\text{CaCl}}_2$ $\left(Pnnm\right)$ structure of ${\text{SiO}}_2$ (a) at the fifth generation and perovskite structure of ${\text{CaCO}}_3$ (30 atoms/cell) (b) at the 13th generation.