Construction of crystal structure prototype database: methods and applications

The first critical step in the construction of the CSPD is the development of an effective structure descriptor to identify and eliminate redundant structure entries. This can solve the data-overloading problem without the loss of important information. Several successful methods [12, 19–22] have been previously developed to fingerprint structures automatically and evaluate their similarity. For example, Willighagen et al developed a method using a radial distribution function (RDF) that combines atomic coordinates with partial atomic charges [20], Wang et al developed a bond characterization matrix based on structure bond information [19], and Zhu et al proposed a crystal fingerprint considering the local environment of all the atoms in the cell [22].

After investigating the literature on structure descriptors, we suggest six criteria for the structure descriptor: (i) it should be independent of translation, rotation, or the choice of equivalent cell for the structure; (ii) it reflects differences among atomic species; (iii) it is a continuous function with respect to the motion of the atoms in the structure, and can yield a quantitative measure of the degree of similarity between two structures; (iv) it can generally describe materials of varying dimensions (e.g. 0D, 1D, 2D, and 3D); (v) an explicit value should be provided as a threshold for judging structures as similar or dissimilar; and (vi) the algorithm to calculate the descriptor should be robust and efficient.

Our proposed new structure descriptor, the coordination characterization function (CCF), considers these criteria and identifies the fingerprint of a structure on the basis of its interatomic distances. It can reflect the coordination character of a structure within a given cutoff radius. In fact, interatomic distances are good structure identifiers because they naturally satisfy criterion (i) and can be easily calculated at low computational cost for different systems, thereby meeting criteria (iv) and (vi). Interatomic distances are established basic information for the structure descriptors of RDFs [20, 21, 23]. However, it should be emphasized that our proposed structure descriptor is significantly different from RDFs, in that it better considers the long-range character of structures owing to it employing the same weighting for a large range of interatomic distances.

The CCF can be obtained by equation (1), and the Gaussian kernel was used to smooth the discrete interatomic distances into continuous functions. To satisfy criterion (ii), a matrix involving different atomic types was constructed to describe the structure. Specifically, each matrix element is related to components of the CCF coming from different pairs of atomic types i–j, so that

$$\begin{eqnarray}&&cc{{f}_{ij}}(r)=\left\{\begin{array}{c} \frac{1}{N}\underset{{{n}_{i}}}{\sum}\underset{{{n}_{j}}}{\sum}\,f\left({{r}_{{{n}_{i}}{{n}_{j}}}}\right)\sqrt{\frac{{{a}_{\text{pw}}}}{\pi}}\exp [-{{a}_{\text{pw}}}{{\left(r-{{r}_{{{n}_{i}}{{n}_{j}}}}\right)}^{2}}],\quad \left(i\ne j\right), \\ \frac{1}{2N}\underset{{{n}_{i}}}{\sum}\underset{{{n}_{j}}}{\sum}\,f\left({{r}_{{{n}_{i}}{{n}_{j}}}}\right)\sqrt{\frac{{{a}_{\text{pw}}}}{\pi}}\exp [-{{a}_{\text{pw}}}{{\left(r-{{r}_{{{n}_{i}}{{n}_{j}}}}\right)}^{2}}],\quad (i=j)\text{,} \end{array}\right. \end{eqnarray} \tag{ 1 }$$

where n_i runs over all the atoms of the ith type within the cell, and n_j runs over all the atoms of the jth type within the extended cell, ${{r}_{{{n}_{i}}{{n}_{j}}}}$ is the interatomic distance less than the cutoff radius r_cut (usually 9.0 Å), $f({{r}_{{{n}_{i}}{{n}_{j}}}})$ is the weighting function for different interatomic distances, N is the number of atoms in the cell, a_pw (usually 60.0 Å⁻²) is an empirical parameter that controls the peak width of the Gaussian function, and $\sqrt{\frac{{{a}_{\text{pw}}}}{\pi}}\exp [-{{a}_{\text{pw}}}{{(r-{{r}_{{{n}_{i}}{{n}_{j}}}})}^{2}}]$ is the normalized Gaussian function that satisfies the condition

$$\begin{eqnarray}&&{\int}_{-\infty}^{\infty}\sqrt{\frac{{{a}_{\text{pw}}}}{\pi}}\exp [-{{a}_{\text{pw}}}{{(r-{{r}_{{{n}_{i}}{{n}_{j}}}})}^{2}}]\text{d}r=1.\end{eqnarray} \tag{ 2 }$$

In equation (1), when i  =  j, we introduce a factor of 1/2 to avoid over-counting interatomic distances of the same atomic type. Note that a sufficiently large extended cell should be chosen to include all the atom pairs within the cutoff radius r_cut. Furthermore, a weighting function, f (x), was introduced to avoid discontinuities in the CCFs when an atom enters or leaves the sphere,

$$\begin{eqnarray}f(x)=\left\{\begin{array}{*{35}{l}} 1, & \left(x<{{r}_{\text{cut}}}-l\right), \\ \exp \left[-\frac{\alpha \left(x-{{r}_{\text{cut}}}+l\right)}{l}\right], & \left(x\geqslant {{r}_{\text{cut}}}-l\right). \end{array}\right. \end{eqnarray} \tag{ 3 }$$

Note that the shape of f (x) depends entirely on the parameters α and l. Parameter α is a decay factor: a large value allows quick decay (figure 1(a)). Parameter l controls the point where f (x) starts to decay (figure 1(b)). In our studies, α and l are set to 3.0 and 0.5 Å, respectively.

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The matrix to describe a structure containing Nt atomic types can be given as

$$\begin{eqnarray}M=\left[\begin{array}{c} cc{{f}_{11}} & cc{{f}_{12}} & \cdots & cc{{f}_{1Nt}} \\ cc{{f}_{21}} & cc{{f}_{22}} & \cdots & cc{{f}_{2Nt}} \\ \vdots & \vdots & {} & \vdots \\ cc{{f}_{Nt1}} & cc{{f}_{Nt2}} & \cdots & cc{{f}_{NtNt}} \end{array}\right].\end{eqnarray} \tag{ 4 }$$

Given that ccf_ij(r)  =  ccf_ij(r) can be deduced from equation (1), only the upper triangular matrix is required to characterize the structure. Obviously, a comparison between structures within our scheme then reduces to a comparison of the upper triangular matrix. We employ the Pearson correlation coefficient to measure the degree of similarity between two matrices. It is calculated for discrete points r_k within the range of 0.0–10.0 Å, so that

$$\begin{eqnarray}&&{{R}_{ij}}=\frac{\underset{{{r}_{k}}}{\sum}[ccf_{ij}^{A}({{r}_{k}})-\overline{ccf_{ij}^{A}}][ccf_{ij}^{B}({{r}_{k}})-\overline{ccf_{ij}^{B}}]}{\sqrt{\underset{{{r}_{k}}}{\sum}{{[ccf_{ij}^{A}({{r}_{k}})-\overline{ccf_{ij}^{A}}]}^{2}}\cdot \underset{{{r}_{k}}}{\sum}{{[ccf_{ij}^{B}({{r}_{k}})-\overline{ccf_{ij}^{B}}]}^{2}}}},\end{eqnarray} \tag{ 5 }$$

where R_ij denotes the Pearson correlation coefficient; $ccf_{ij}^{A}({{r}_{k}})$ and $ccf_{ij}^{B}({{r}_{k}})$ represent CCFs from structures A and B, respectively; and $\overline{ccf_{ij}^{A}}$ and $\overline{ccf_{ij}^{B}}$ are the average values of the CCFs for each respective structure. Therefore, the distance for two CCFs can be defined from their correlation coefficient as

$$\begin{eqnarray}&&{{d}_{ij}}=1-{{R}_{ij}}.\end{eqnarray} \tag{ 6 }$$

The summation of d_ij with different weighting coefficients yields the distance between the structures, so that

$$\begin{eqnarray}&&d=\frac{\underset{i=1}{\overset{Nt}{\sum}}\underset{j=i}{\overset{Nt}{\sum}}{{c}_{ij}}{{d}_{ij}}}{\underset{i=1}{\overset{Nt}{\sum}}\underset{j=i}{\overset{Nt}{\sum}}{{c}_{ij}}}.\end{eqnarray} \tag{ 7 }$$

We adopt the following formula to calculate the weighting coefficient c_ij for d_ij related to atoms of type i and j:

$$\begin{eqnarray}&&{{c}_{ij}}=\frac{1}{2}\underset{{{r}_{k}}}{\sum}[ccf_{ij}^{A}({{r}_{k}})+ccf_{ij}^{B}({{r}_{k}})] \Delta,\end{eqnarray} \tag{ 8 }$$

where Δ (usually 0.025 Å) is the step length for r_k. Considering that the Pearson correlation coefficient takes values from  −1 to 1, the distance between two structures ranges from 0 to 2.

To evaluate the performance of our structure descriptor, it was applied to illustrative examples including TaPd₂ and PtK₂Cl₄. We calculated the CCFs for the conventional cell and the primitive cell of TaPd₂, which crystallizes in a MoPt₂-type structure (space group Immm) [24]. Figure 2(a) shows that both CCFs are the same, and the distance calculated by equation (7) is 0.000. Obviously, our structure descriptor satisfies criterion (i). As a further test we considered PtK₂Cl₄, which has P4/mmm (experimental) [25] and Pnma (hypothetical) structures, and calculated the CCFs for both structures. The Pnma structure is adjusted to the same atom number density as the P4/mmm structure. Figure 2(b) shows that the CCFs of the two structures differ greatly, with a distance of 0.655. The value of distance is an effective measure of the degree of dissimilarity between the two structures. Another test examined whether our structure descriptor satisfies criterion (iii). We optimized a metastable structure of CuInS₂ (Pbam) with four formula units (f.u.). The optimization starts from a structure that lies on the slope of the potential energy surface. After 25 ionic steps, the structure reached the equilibrium point. We calculated ΔE and structural distances between the optimized structure and all the intermediate structures for each ionic step (figure 3). The structure closest to the equilibrium point tends to have the smallest ΔE and distance values. This test shows that our structure descriptor can quantitatively describe the path taken during structural optimization.

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To determine the threshold for classifying structures as similar or dissimilar, we chose to examine the distances between the structures of LaC₂ [26] and CaF₂ as an example. Structure prediction runs found 1101 and 385 structures for each, respectively. In each system, two distinct structures were selected as reference structures to calculate the distance to all other structures. Figure 4 shows the calculated distances sorted in ascending order. We have checked the similarity between the reference structures and all the structures of the prediction results via manual inspection. In each test as shown by figure 4, all the structures below the red dashed line are similar with the reference structure, and all the structures above the red dashed line are distinct from the reference structure. Obvious gaps exist between the distances in all four tests. We have tested a larger data set to check whether such a gap in distances exist or not. We used our program to randomly select 3000 structure pairs in the predicted structures of CaF₂. Figure 5 shows the correlation between the energy difference and the structural distance. There is a clear gap separating similar from distinct structure pairs. We did not find any structure pairs that have a large energy difference with a small distance. This test further proves the effectiveness of our structure descriptor CCF. Given these results, we recommend 0.075 as the threshold to determine whether structures are similar. However, this threshold may fail to distinguish large cells with different stacking sequences. We have built two hypothetical structures of element Zn with stacking sequences of ABABCBC and ABABCAB in hexagonal lattice system. The distance calculated by our structure descriptor is 0.001 28. The difference of stacking sequence can be reflected by our structure descriptor, but the distance is much smaller than the threshold. Because these two structures are very similar with each other, it is reasonable for the CCF to yield a small distance.

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We constructed the CSPD by extracting the crystal structure prototypes from crystal structure database. The norms for assessing the crystal structure prototype were used to classify structures in the COD [8], an open-access collection of crystal structures of organic, inorganic, metal–organic compounds, and minerals, excluding biopolymers. Note that the framework of the CSPD can easily consider crystal structures from multiple databases. The COD has aggregated more than 360 000 entries. All registered users can deposit published or unpublished structures as personal communications or pre-publication depositions at www.crystallography.net/cod/index.php. Therefore, all of the structures in the CSPD are freely obtained from the COD.

The procedure of constructing the CSPD comprises four main steps:

• (1)  
  filtering structures in the database;
• (2)  
  classifying structures according to their composition type and total number of atoms in the conventional cell;
• (3)  
  discriminating between inorganic and organic structures;
• (4)  
  assessing structural similarity.

Step 1: Filtering structures in the database.

We downloaded the entirety of the crystallographic data of the COD in Crystallographic Interchange File/Framework (CIF) format [27]. Given that the POSCAR format adopted in Vienna ab initio simulation package (VASP) [28] is convenient for the program to read geometrical information, the CSPD also stores crystallographic data in POSCAR format. Here, the CIF2Cell program [29] was used to transform the crystallographic data of the structures to POSCAR format for the conventional cell. To ensure the CSPD contains only justified structures, all the structures should be filtered: those with incorrect chemical symbols or interatomic distances below 0.6 Å were discarded. Note that the structures in POSCAR and CIF format are stored separately in different directories in the CSPD.

Step 2: Classifying structures according to composition and total number of atoms in the conventional cell.

We selected one specific structure (www.crystallography.net/7153176.cif) to demonstrate the determination of composition type. The structure has 46 C, 44 H, 4 N, and 6 O atoms. These numbers, sorted in ascending order, should be denoted as (4, 6, 44, 46); their highest common factor should then be calculated to give the 'formula unit'. The composition type is then defined as the set of numbers resulting from dividing the original numbers of atoms by the formula unit. The formula unit and composition type can be denoted as, for example, 2 (2, 3, 22, 23). As shown in figure 6, these numbers are used to indicate the file path and file name of the structure in the CSPD. This framework provides convenient access to the structures of a desired composition type in the CSPD.

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Step 3: Discriminating between inorganic and organic structures.

We developed a module in our program to automatically distinguish inorganic and organic structures. Generally speaking, structures containing both C–H and C–C bonds were treated as organic structures. Structures containing C and H had the C–C and C–H interatomic distances calculated. If these distances are less than 1.65 and 1.20 Å, respectively, the structure is recognized as organic. The inorganic and organic structures are stored in different directories in the CSPD.

Step 4: Assessing structural similarity.

To remove multiple entries of similar structures from a database, structural similarity should be assessed and compared with the structures in the CSPD to determine a structure's prototype. We only compared the similarities between structures of the same composition, with the same total number of atoms in the conventional cell. Figure 7 shows a flow chart of the procedure used to compare structural similarity. Specifically, the space groups of the two structures are compared first; if they are the same, the distances between the structures are calculated by our structure descriptor. If the distance is less than 0.075, they are identified as belonging to the same cluster. The CSPD preserves only one structure per cluster.

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Based on this method, we developed a program named the structure prototype analysis package (SPAP) to analyze the results of CALYPSO structure prediction. CALYPSO is a state-of-the-art structure prediction method and eponymous computer software. It requires only the chemical composition of a given compound to predict its stable or metastable structures at given external conditions (e.g. pressure). It can thus be used to predict or determine the crystal structure [30–32]. CALYPSO calculations typically generate a large number of candidate structures. In particular, many different initial structures lying on the slope of the free-energy surfaces in the same valley reach the same local minimum after local structure optimizations, resulting in many similar structures in the structure searches. The SPAP is a versatile tool for automatic structure analysis; it can be used to evaluate the similarity of the optimized structures and to extract distinct structure prototypes. The SPAP can analyze CALYPSO prediction results for clusters, layered materials, and solids. Furthermore, it can classify the predicted structures and extract the low-energy structures to construct a theoretical structure database. This database can be applied to generate structures for structure prediction. By taking advantage of prior knowledge, this method may improve the efficiency of structure prediction.

After transformation of the raw database to a composition–crystal-structure prototype database, the number of candidate structures in the database can be reduced dramatically. Based on our big-data technique (CSPD), we developed an advanced method named the big data method (BDM) to generate structures for structure prediction. It comprises four main steps, as follows:

• (1)  
  selecting structure prototypes in the CSPD for a given targeted composition type and formula unit;
• (2)  
  substituting elements for the selected structure prototype;
• (3)  
  adjusting the lattice parameters for a given volume;
• (4)  
  checking the minimal interatomic distances.

To evaluate the performance of our approach, we generated a series of candidate structures for three typical systems (e.g. CaF₂, KN₃, and CuInS₂) by our proposed BDM and also the random sampling method implemented in the CALYPSO package [18, 19]. Structure relaxations and energy calculations were performed in the framework of density functional theory within the all-electron projector-augmented wave (PAW) [35, 36] method as implemented in the VASP code [28]. The BDM and the random method with symmetry constraints both generate 204, 94, and 144 structures with four formula units for CaF₂, KN₃, and CuInS₂, respectively. The energies relative to the lowest energy of all the distinct structures were sorted in ascending order, and are presented in figure 8. Note that the structure with the lowest energy for all the systems was generated by the BDM, and the structures generated with the BDM tend to be distributed at lower energy. The average computation times for the structure relaxation of these candidate structures are listed in table 7. The BDM appears clearly more efficient than the random method. For CuInS₂, the experimental I-42d structure was successfully reproduced by the BDM, and it also predicted two novel, energetically degenerate structures of P-42c and P-4m2. The calculated structural parameters for these three phases are listed in table 8. The energies of the new structures relative to the I-42d structure are both only 0.001 eV/atom. Therefore, we believe that the proposed metastable structures of CuInS₂ are likely to be seen experimentally.

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The BDM outperforms the random sampling method, because most of the structures in the CSPD were determined by experiment. These structures are more physically justified than those generated by the random method. The initial structures are crucial for efficient structure prediction by global optimization algorithms (e.g. simulated annealing [37, 38], minima hoping [39], and evolution algorithm [18, 40]). Therefore, the BDM is a powerful approach to generate the initial structures for global structure prediction. Moreover, it can also be used to generate structures for high-throughput calculations.

The CSPD is valuable for both experimental and theoretical researchers, because it can be used to determine whether a new proposed structure is similar to any known ones. Furthermore, the structure prototype for a given structure can be determined by comparing its structural similarity with the prototype structures in the CSPD. Thanks to our structure descriptor, the structure prototype can be automatically determined by our program SPAP. The procedure comprises two main steps:

• (1)  
  screening structures of the same composition type as the given structure in the database;
• (2)  
  assessing the distances between the given structure and screened structures in the CSPD.

We took the diamond structure as an example. The distances between the diamond structure and all the structures for other elemental solids with eight f.u. in the databases (i.e. CSPD and COD) were calculated by SPAP. Note that the structures were adjusted to the same atom number density before calculating the distances. The calculated distances were sorted in ascending order, and are plotted in figure 9. According to our norms for crystal structure prototypes, the distance between similar structures must be less than 0.075. In this example, 50 structures in the COD appear under this threshold, all of which are the same diamond structure for C, O, Si, Ge, and Sn. Only one structure in the CSPD appears under the threshold: the diamond structure. This result further verifies that our crystal structure prototype definition is effective.

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