Crystal structure prediction supported by incomplete experimental data

We propose an efficient theoretical scheme for structure prediction on the basis of the idea of combining methods, which optimize theoretical calculation and experimental data simultaneously. In this scheme, we formulate a cost function based on a weighted sum of interatomic potential energies and a penalty function which is defined with partial experimental data totally insufficient for conventional structure analysis. In particular, we define the cost function using “crystallinity” formulated with only peak positions within the small range of the x-ray-diffraction pattern. We apply this method to well-known polymorphs of ${\mathrm{SiO}}_2$ and C with up to 108 atoms in the simulation cell and show that it reproduces the correct structures efficiently with very limited information of diffraction peaks. This scheme opens a new avenue for determining and predicting structures that are difficult to determine by conventional methods.

Figure 1

Schematic of our scheme of utilizing available experimental data for crystal structure prediction. The aim of crystal structure prediction is to find the stable or metastable structures which are reachable experimentally (the orange sphere) of a potential-energy surface $E\left(\mathbf{R}\right)$ (the blue line), where $\mathbf{R}$ denotes atomic position. To destabilize the trivial local minima, a penalty function $D$, defined with available experimental data, is introduced. The cost function $F$ (the red line) is the sum of $E$ and $D$.

Figure 2

Crystal structures of three polymorphs of ${\mathrm{SiO}}_2$ and the corresponding powder x-ray-diffraction patterns. Crystallographic unit cells and diffraction patterns for (a) coesite, (b) low cristobalite, and (c) low quartz. Si and O atoms are represented by the blue and red spheres, respectively. Only information of the peak positions from ${20}^{\circ }$ to ${45}^{\circ }$ (the green squares) is used to define the penalty function $D$.

Figure 3

Finding the correct structure supported by insufficient x-ray-diffraction patterns. (a) The penalty function $D^{\mathrm{coesite}}$ is defined using the diffraction data for coesite. The green solid bold line represents the time evolution of crystallinity for coesite during optimization of the cost function, the red dashed bold line for low-cristobalite and the blue dotted bold line for low quartz. The green thin line shows the time evolution of crystallinity for coesite during optimization using only the potential energy. (b) and (c) The penalty function is defined using the diffraction data for low cristobalite and low quartz. The parameter $\alpha$ is set to $\frac{4\alpha }{9k_B}=20000\mathrm{K}$ in all cases.

Figure 4

$\alpha$ dependence of the potential-energy $E$ distributions for optimized structures. (a) The target material is the unit cell of coesite, which contains 48 atoms. Some 2500 (500) samples were optimized for $\frac{4\alpha }{9k_B}=0.0\mathrm{K}$ (other values) by simulated annealing at low temperatures (500 K). The most stable structure corresponds to the red bars. Simulated annealing at high temperatures (2500 K) was also performed (lower panel). (b) The target material is a $2\times 2\times 1$ supercell of low cristobalite, which contains 48 atoms. (c) The target material is a $2\times 2\times 1$ supercell of low quartz, which contains 36 atoms.

Figure 5

The change in $E$ and the cost function $F$ during successful optimization of $F$. Starting from the metastable structure shown in the upper left, the correct structure of coesite as shown in the upper right is found by optimization of the cost function as depicted by the red line. $\frac{4\alpha }{9k_B}=20000\mathrm{K}$ in this simulation.

Figure 6

Crystal structures of two polymorphs of carbon and the corresponding powder x-ray-diffraction patterns. Crystallographic unit cells and diffraction patterns for (a) diamond and (b) graphite. Only information of the peak positions from ${20}^{\circ }$ to ${45}^{\circ }$ (the green squares) is used to define the penalty function $D$.

Figure 7

Diffraction patterns with which the penalty function is formulated and $E$ distributions for optimized structures. (a) All seven peaks are correctly detected. (b) Two peaks are detected, but five small peaks are not detected. (c) The angle range is limited.