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CALYPSO Method for Structure Prediction and Its Applications to Materials Discovery

  • Yanchao WangEmail author
  • Jian Lv
  • Quan Li
  • Hui Wang
  • Yanming Ma
Living reference work entry

Abstract

Atomic-level structure prediction for condensed matters, given only a chemical composition, is a major challenging issue in a broad range of science (e.g., physics, chemistry, materials, and planetary science, etc.). By combining the global swarm optimization algorithm with a number of specially designed structure-dealing techniques (e.g., symmetry constraints, structure fingerprints, etc.), we developed the CALYPSO (Crystal structure AnaLYsis by Particle Swarm Optimization) structure prediction method that is able to predict the structures of a wide range of materials including isolated clusters/nanoparticles, two-dimensional layers and reconstructed surfaces, and three-dimensional bulks and holds the promise for the functionality-driven design of materials (e.g., superhard, electride, and optical materials, etc.). It has been demonstrated in a wide range of applications that CALYPSO is highly efficient when searching for the structures of materials and becomes an invaluable tool for aiding materials discovery. In this chapter, we provide an overview of the basic theory and main features of the CALYPSO approach, as well as its versatile applicability to the design of superconductors and superhard materials. Finally, the conclusion and opportunities for further research on CALYPSO method are presented.

1 Introduction

The traditional way of discovering new materials is by Edisonian iterations of trial and error, which choose materials to synthesize and test depending on chemical intuition, experience, or theoretical knowledge until some combination of success and exhaustion is achieved. This approach is painstaking, long, and costly due to the extraordinary number of experimental attempts required. Advances in computing now allow new materials with target properties to be simulated and analyzed theoretically before their synthesis (Agrawal and Choudhary 2016; Jain et al. 2013), significantly reducing the necessary trials (and errors) and thus the expected cost of discovery (Jain et al. 2016; Oganov et al. 2010; Needs and Pickard 2016).

A material’s structure determines its properties and functionalities. Therefore, theoretical structure prediction is critical in aiding materials discovery (Wang and Ma 2014). The goal of crystal structure prediction is to develop an efficient computational scheme to explore the high-dimensional potential-energy surface (PES) in order to identify the global energy minimum that corresponds to the global stable ground-state structure observable in experiment. Empirical observations and heuristic estimates indicate that the number of local minima grows exponentially as the system size increases (Stillinger 1999; Gavezzotti 1994). Therefore, an efficient computational approach is urgently needed to find quickly the most stable structure of a large assembly of atoms in the vast configuration space of the PES.

Various strategies to generate hypothetical structures have been recently pursued in the field of crystal structure prediction, and several efficient structure prediction methods (e.g., data mining (Gavezzotti 1994), simulated annealing (Kirkpatrick et al. 1983; Schön and Jansen 1996), genetic algorithms (Deaven and Ho 1995; Oganov and Glass 2006; Abraham and Probert 2006; Trimarchi and Zunger 2007; Kolmogorov et al. 2010), minima hopping (Goedecker 2004; Amsler and Goedecker 2010), basin hopping (Wales and Doye 1997; Doye and Wales 1998), metadynamics (Laio and Parrinello 2002), random sampling (Pickard and Needs 2011; Shang and Liu 2013), and CALYPSO (Wang et al. 2010)) have been developed. Their applications to materials science have led to a number of breakthroughs, as exemplified by a series of theory-driven discoveries of novel functional materials (Ceder 2010; Ma et al. 2009; Oganov et al. 2009; Zhu et al. 2011; Pickard and Needs 2006; Drozdov et al. 2015; Ceder et al. 1998; Teter 1998; Li et al. 2014). Structure prediction is now becoming an irreplaceable tool to accelerate the discovery of materials. Several reviews consider structure prediction methods and their applications (Wang and Ma 2014; Woodley and Catlow 2008; Oganov 2011; Ciobanu et al. 2013; Wang et al. 2015; Zhang et al. 2017a), but this work focuses only on our CALYPSO method and its application to superconductive and superhard materials.

This chapter is organized as follows. Section 2 briefly introduces the basic theory and general features of the CALYPSO methodology, including swarm intelligence algorithm. Several recent applications of CALYPSO in discovering superconductive and superhard materials are provided in Sect. 3, followed by general conclusions and future prospects in Sect. 4.

2 CALYPSO Structure Prediction Method

2.1 General Theory of the CALYPSO Method

The CALYPSO method for structure prediction run comprises several main steps (Fig. 1). Initial structures are randomly generated with symmetry constraints. Once a new structure is generated, structure characterization is performed to examine its similarity to all the previous ones, where duplicate or similar structures are eliminated. After all the structures of each population have been generated, local structural optimizations are performed to reduce the noise of the energy surface and drive the systems to local minima. Structural evolution via the particle swarm optimization algorithm is then carried out to produce new structures for the next generation. These steps are iterated until a termination criterion (such as a prescribed threshold or a fixed number of iterations) is attained (Wang et al. 2012a). Below we give an introduction to the general theory of the CALYPSO method.
Fig. 1

The flow chart of CALYPSO

2.1.1 Generation of Random Structures with Symmetry Constraints

Most natural structures of materials are governed by symmetry rules; any crystal structure must belong to one of the 230 crystallographic space groups. For a given space group, only a subset of the optimized variables (lattice parameters and atomic coordinates) is independent, while the rest are determined via symmetry rules. It is found that the majority of randomly generated structures (particularly for large systems, e.g., >30 atoms per simulation cell) without symmetry constraints exhibits unreasonably low symmetries, while proper consideration of symmetry constraints can significantly reduce the optimization variables and enrich the diversity of structures during structure evolution. Symmetry constraints on random structure generation are therefore implemented in CALYPSO to improve the starting structures for the subsequently evolutionary structure searches. Different strategies for dealing with 0D-isolated molecule/clusters, 2D layers or surfaces, and 3D crystals are designed to generate random symmetric structures. In particular, the generation of random structures is constrained by ∼40 point groups for isolated systems (e.g., molecules, clusters, and nanoparticles), 17 planar space groups for 2D layers or surfaces, and 230 space groups for 3D crystals. For example, the space group of initial crystal structures is randomly selected among 230 ones for a crystal structure prediction. Once the space group is selected, the lattice parameters are generated within the chosen symmetry according to the defined volume, and the corresponding atomic coordinates are obtained by a combination of a set of symmetrically related coordinates (Wyckoff positions) in accordance to the number of atoms in the simulation cell. Notably, a backup database of space groups of all generated structures is established in order to compare them with the space groups of newly generated structures. The appearance of identical space group is forbidden with a certain probability (80%). This allows the initial samplings to cover different regions of the search space and generate the diverse structures, which are found to be crucial for a high efficiency of a global minimization.

To demonstrate the effects of symmetric constraints on structure generation, bulk crystals (Wang et al. 2012a) and isolated clusters (Lv et al. 2012) are chosen as two test systems, where crystalline TiO2 consists of 16 formula units per cell, while the cluster contains 100 atoms described by the Lennard–Jones potential. For each system, a large number of structures (3,250 for the crystal and 10,000 for the cluster) were randomly generated with and without symmetric constraints, respectively, and then locally optimized using the GULP code (Gale 1997). The obtained energy distributions of crystal and cluster are shown in Fig. 2a and b, respectively. The results (Fig. 2a) showed that the rutile structure, i.e., the global stable structure, cannot be generated without symmetry constraints. However, once symmetric constraints were employed in the generation of random structures, 203 (∼6.2% of the total) rutile structures were successfully produced. The generated cluster structures (Fig. 2b) without symmetric constraints have a Gaussian-like energy distribution with a well-defined sharp peak centered at a high energy of 0.3 ε/atom. Many of the cluster structures appeared disordered or liquid-like; their generation greatly reduced structural diversity. Instead, once point group symmetry was applied, a much broader energetic distribution emerged with the generation of a large number of lower-energy structures.
Fig. 2

Energetic distributions of randomly generated structures for (a) TiO2 and (b) LJ100 clusters with and without symmetries

2.1.2 Structural Characterization Techniques

During the structural evolution process, many trial structures are produced for each generation, and some may be very similar or even identical. The existence of similar structures decreases the diversity of the population, potentially leading to stagnation. Moreover, given that the most computationally expensive parts of the algorithm are the energy calculations and local structural optimizations, repetitive optimizations of similar structures will significantly increase the computational cost and impede the structure searching efficiency. Thus, efficient searching must employ techniques to fingerprint each structure, measure its similarities to other structures, and eliminate similar structures. Two effective methods, the bond characterization matrix (BCM) (Lv et al. 2012) and the coordination characterization function (CCF) (Su et al. 2017), have been developed based on the geometrical information of a structure.

The BCM is an advanced version of the bond-orientational order parameter technique originally introduced by Steinhardt et al. (Steinhardt et al. 1983). For a given structure, a bond vector rij between atoms i and j is defined if the interatomic distance is less than a given cutoff distance. This vector is associated with the spherical harmonics Ylm(θij,\( {\varphi_{ij}} \)), where θij and \( {\varphi_{ij}} \) are the polar angles. A weighted average over all bonds formed by, for instance, the type A and B atoms is then evaluated by Eq. 1:
$$ {\mathrm{Q}}_{lm}^{\updelta_{\mathrm{AB}}}=\frac{1}{{\mathrm{N}}_{\updelta_{\mathrm{AB}}}}{\sum}_{\mathrm{i}\in \mathrm{A},\mathrm{j}\in \mathrm{B}}{\mathrm{e}}^{-\upalpha \left({r}_{ij}-{\mathrm{b}}_{\mathrm{AB}}\right)}{\mathrm{Y}}_{lm}\left({\theta}_{ij},{\phi}_{ij}\right), $$
(1)
where δAB and \( {N}_{\delta_{AB}} \)denote the type and the number of a bond, respectively. To avoid dependence on the choice of reference frame, it is important to consider the rotationally invariant combinations:
$$ {Q}_l^{\delta_{AB}}=\sqrt{\frac{4\pi }{2l+1}{\sum}_{m=-l}^l{\left|{Q}_{lm}^{\delta_{AB}}\right|}^2}, $$
(2)
where each series of \( {Q}_l^{\delta_{AB}} \)for l = 0, 2, 4, 6, 8, and 10 can be used to represent a type of bond and is thus an element of the BCM. Only even-l spherical harmonics are used to guarantee invariant bond information with respect to the direction of the bonds. By including the complete structural information of bond lengths and angles, this technique unambiguously fingerprints the structures. As a result, the similarity of any two structures (u and v) can be quantitatively evaluated by the Euclidean distance between their BCMs:
$$ {\mathrm{D}}_{\mathrm{uv}}=\sqrt{\frac{1}{{\mathrm{N}}_{\mathrm{type}}}{\sum}_{\updelta_{\mathrm{AB}}}{\sum}_{\mathrm{l}}{\left({\mathrm{Q}}_{\mathrm{l}}^{\updelta_{\mathrm{AB}},\mathrm{u}}-{\mathrm{Q}}_{\mathrm{l}}^{\updelta_{\mathrm{AB}},\mathrm{v}}\right)}^2}, $$
(3)
where Ntype is the number of bond types.
Two LJ38 clusters with different structural motifs are chosen as examples to test the BCM. The first structure is the global stable structure, having a face-centered cubic structural motif (LJ38-Oh), while the second one is a metastable incomplete Mackay icosahedron (LJ38-C5v). Their BCMs, shown by histograms in Fig. 3a, are dramatically different. Except for Q2 and Q10, the LJ38-Oh structure has obviously larger Ql values than those in the LJ38-C5v structure. The resultant distance between these two structures is a very large value of 0.618, characterizing their large structural differences. The distances to the isotropic system are 0.728 for LJ38-Oh and 0.163 for LJ38-C5v, illustrating a high degree of order in LJ38-Oh. Randomly distorting LJ38-Oh, and plotting its distances to the unaltered structure as a function of distortion (Fig. 3b), illustrates that the distances increase linearly with the distortions at the initial stage, reflecting correctly the structural deviations. Eventually, the curves become flat at a distortion of about 0.4r0. At this stage, the distances are approaching those distances relative to the isotropic limits favorable for disordered systems.
Fig. 3

(a) Histograms of BCMs for LJ38-Oh and LJ38-C5v. (b) BCM distances with respect to the unaltered structures as a function of distortions for LJ38-Oh. Each data point was obtained by averaging over the distances of 100 randomly distorted structures, and the error bar denotes the standard deviation. (c) Comparisons of the CCFs of different structures of P4/mmm and Pnma for PtK2Cl4. For ease of comparison, the plots show -CCFs for the Pnma structure. (d) Calculated ΔE and structural distances between the optimized structure of CuInS2 and all intermediate structures in each ionic step. Inset shows the detailed optimization procedure approaching the equilibrium point

Although the BCM can provide a precise measurement of structures, it suffers from heavy computational costs for large systems (e.g., the number of atoms per simulation cell >30) since all the geometrical information (bond lengths and angles) is considered and evaluated.To reduce the computational cost, CCF, an alternative and fast method depending only on the two-body correlation function (interatomic distance), was developed to fingerprint large structures. The CCF method employs a matrix involving different atomic types:
$$ \mathrm{M}=\left[\begin{array}{ccc}{\mathrm{ccf}}_{11}& \cdots & {\mathrm{ccf}}_{1\mathrm{Nt}}\\ {}\vdots & \ddots & \vdots \\ {}{\mathrm{ccf}}_{\mathrm{Nt}1}& \cdots & {\mathrm{ccf}}_{\mathrm{Nt}\mathrm{Nt}}\end{array}\right]. $$
(4)
Here, each matrix element related to different pairs of atomic types i and j is calculated as follows:
$$ {\mathrm{ccf}}_{\mathrm{i}\mathrm{j}}\left(\mathrm{r}\right)=\left\{\begin{array}{c}\frac{1}{\mathrm{N}}{\sum}_{{\mathrm{n}}_{\mathrm{i}}}{\sum}_{{\mathrm{n}}_{\mathrm{j}}}\mathrm{f}\left({\mathrm{r}}_{{\mathrm{n}}_{\mathrm{i}}{\mathrm{n}}_{\mathrm{j}}}\right)\sqrt{\frac{{\mathrm{a}}_{\mathrm{pw}}}{\uppi}}\exp \left[-{\mathrm{a}}_{\mathrm{pw}}{\left(\mathrm{r}-{\mathrm{r}}_{{\mathrm{n}}_{\mathrm{i}}{\mathrm{n}}_{\mathrm{j}}}\right)}^2\right],\left(\mathrm{i}\ne \mathrm{j}\right),\\ {}\frac{1}{2\mathrm{N}}{\sum}_{{\mathrm{n}}_{\mathrm{i}}}{\sum}_{{\mathrm{n}}_{\mathrm{j}}}\mathrm{f}\left({\mathrm{r}}_{{\mathrm{n}}_{\mathrm{i}}{\mathrm{n}}_{\mathrm{j}}}\right)\sqrt{\frac{{\mathrm{a}}_{\mathrm{pw}}}{\uppi}}\exp \left[-{\mathrm{a}}_{\mathrm{pw}}{\left(\mathrm{r}-{\mathrm{r}}_{{\mathrm{n}}_{\mathrm{i}}{\mathrm{n}}_{\mathrm{j}}}\right)}^2\right],\left(\mathrm{i}=\mathrm{j}\right),\end{array}\right. $$
(5)
where ni runs over all the atoms of the i-th type within the cell, nj runs over all the atoms of the j-th type within the extended cell, \( {r}_{n_i{n}_j} \)is the interatomic distance less than the cutoff radius (usually 9.0 Å), \( f\left({r}_{n_i{n}_j}\right) \) is the weighting function for different interatomic distances, N is the number of atoms in the cell, apw is an empirical parameter that controls the peak width of the Gaussian function, and \( \sqrt{\frac{a_{pw}}{\pi }}\mathit{\exp}\left[-{a}_{pw}{\left(r-{r}_{n_i{n}_j}\right)}^2\right] \) is the normalized Gaussian function. As a result, the similarity of two structures can be estimated by the distance between their CCF matrices. The Pearson correlation coefficient was employed to measure the degree of similarity between two matrices, and the corresponding distance (d) is defined as d = 1 − R from their correlation coefficient (R). A detailed description of the CCF can be found in Su et al. (2017).

Two distinct crystal structures of PtK2Cl4 with space groups of P4/mmm and Pnma are chosen to test the CCF method. The CCFs calculated for the two structures (Fig 3c) are clearly rather different, with a large calculated distance of 0.655, giving a good measure on the degree of dissimilarity between the two structures. Note that our tests recommend a user-defined threshold of 0.075 as a good number to identify dissimilar structures. CCF is a continuous function with respect to the motion of the atoms. Through the optimization of a CuInS2 structure with Pbam symmetry, it is seen from the calculated ΔE and structural distances between the optimized structure and the intermediate structures of each ionic step (Fig. 3d) that the structure closest to the equilibrium point tends to have the smaller ΔE and distance values.

2.1.3 Structural Evolution via Particle Swarm Optimization Algorithm

The CALYPSO method adopts a “self-improving” strategy to locate the global minimum of the PES via particle swarm optimization (PSO) (Kennedy 2011). The PSO algorithm is a typical swarm-intelligence scheme inspired by natural biological systems (e.g., ants, bees, or birds) and has been applied to a variety of fields in engineering and chemistry (Ourique et al. 2002; Call et al. 2007). Its application to extended systems for structure prediction started only recently (Wang et al. 2010, 2012b). Within the CALYPSO method, the structures at the t + 1th generation (xt + 1) evolve in the energy landscape through velocity. In practice, the lattice parameters (unit cell) of new structures remain unaltered or are randomly generated, while the atomic positions are updated according to the following formula:
$$ {\mathrm{x}}^{\mathrm{t}+1}={\mathrm{x}}^{\mathrm{t}}+{\mathrm{v}}^{\mathrm{t}+1}. $$
(6)
It is noteworthy that the velocity plays a significant role in governing the speed and direction of particle movement. The updated velocity (vt + 1) is calculated based on its previous location (xt), previous velocity (vt), current location (pbestt) derived from geometrical optimization of this individual, and the global best location (gbestt) for the entire population as follows:
$$ {\mathrm{v}}^{\mathrm{t}+1}=\mathrm{w}{\mathrm{v}}^{\mathrm{t}}+{\mathrm{c}}_1{\mathrm{r}}_1\left({\mathrm{pbest}}^{\mathrm{t}}-{\mathrm{x}}^{\mathrm{t}}\right)+{\mathrm{c}}_2{\mathrm{r}}_2\left({\mathrm{gbest}}^{\mathrm{t}}-{\mathrm{x}}^{\mathrm{t}}\right), $$
(7)
where w denotes the inertia weight controlling the momentum of the particle, which dynamically varies and decreases linearly from 0.9 to 0.4 during the iteration in our methodology; the self-confidence factor (c1) and swarm confidence factor (c2) are equal to 2, which give the best overall performance; r1 and r2 are random numbers distributed in the range [0, 1]. The velocity formula includes random parameters (r1) and (r2) that ensure good coverage of the searching space and avoid entrapment in local optima. The initial velocity of each structure is generated randomly, while the updated one is calculated using both its individual properties and those of the entire population.
Two versions of the PSO algorithm (local and global) have been implemented in CALYPSO (Lv et al. 2012; Wang et al. 2011). The global PSO, outlined in Fig. 4a, has only one global best structure acting as the learning example or attractor for the entire structure population, and all particles seek new positions only in the regions close to the uniquely overall best position. This method converges quickly for small systems (i.e., those with fewer than ∼30 atoms in the simulation cell), but it may be less effective for larger systems because the PES becomes much more complex. The local PSO method is outlined in Fig. 4b. Each particle (i.e., a candidate structure) selects a set of other particles as its neighbors, and its velocity is adjusted according to both its position and the best position achieved so far in the community formed by its neighborhood. Thus, at each iteration, the particle will move toward its own best position and the best position of its local neighborhood, rather than the overall best position in the swarm. By maintaining multiple attractors corresponding to different regions of the PES, the local PSO allows for a finer exploration of the PES and can effectively avoid stagnation during the structure searches.
Fig. 4

The schematic diagram of global (a) and local (b) PSO

2.1.4 Local Structure Optimization

The PES of a material can be regarded as a multidimensional system of many peaks and valleys connected by saddle points. Given that the global minimum is one of the possibly many local minima, local structure searching is an inevitable part of the global search and therefore should be included in any efficient structure searching method. The CALYPSO method currently has interfaces with various ab initio and force-field-based total-energy packages (e.g., VASP (Kresse and Furthmüller 1996), SIESTA (Soler et al. 2002), Quantum ESPRESSO (Giannozzi et al. 2009), CASTEP (Segall et al. 2002), CP2K (Vandevondele et al. 2005), LAMMPS (Plimpton 1995), and GULP (Gale 1997)) for local structural optimization. Other external total-energy programs can also be easily interfaced with CALYPSO as required.

2.2 Features of the CALYPSO Method

CALYPSO can be used to perform unbiased searches of the energetically stable/metastable structures of a given chemical composition (Wang et al. 2015, 2016a). Several attractive features have been implemented in CALYPSO to resolve various structure prediction problems, including 3D solids (Wang et al. 2010), 2D layers (Wang et al. 2012b; Luo et al. 2011) and atomic adsorption (Gao et al. 2015), 2D surfaces (Lu et al. 2014), 0D nanoclusters or molecules (Lv et al. 2012), X-ray diffraction data-assisted structure searches (Gao et al. 2017a), and the inverse design of novel functional materials (e.g., superhard, electride, and optical materials) (Zhang et al. 2013, 2017b; Xiang et al. 2013).

2.2.1 3D Crystals

A crystallographic structure can be regarded as an infinitely repeating array of 3D boxes, termed unit cells, represented by lattice parameters and atomic coordinates. As shown in Fig. 5a, there are six lattice parameters: three lengths and three angles of the lattice vectors. Each atom has three coordinates coded as a fraction of each of the lattice vectors. Structure searches of 3D crystals evolve both types of variables simultaneously to achieve the global minimum on the PES with 3N + 3 dimensions, where 3N − 3 degrees of freedom are the atomic positions, and the remaining six dimensions are the lattice parameters. To reduce the search space, several hard constraints, including the minimum interatomic distances, the minimum lattice lengths, and constraints on the range of angles between the lattice vectors, are imposed in the CALYPSO method. For example, the distance between two atoms should not be smaller than a threshold or the sum of their covalent radii, and the angles between lattice vectors are free to vary between 30° and 160°. An advantage gained by using these constraints is to force attention onto regions containing physically reasonable minima. The CALYPSO method has been widely applied to the prediction of crystal structures for various systems (Wang and Ma 2014; Zhu et al. 2011; Lv et al. 2011; Nishio-Hamane et al. 2012; Li et al. 2016; Chen et al. 2013), some of which have already been experimentally confirmed (see, e.g., Guillaume et al. (2011), Bai et al. (2015), Ma et al. (2012), and Yang et al. (2017)). Section 3 describes its applications to the discovery of novel superconductive and superhard materials.
Fig. 5

The models used in CALYPSO for (a) 3D crystal structure, (b) single-layer structure, (c) multilayer structure, (d) 2D atomic adsorption structure, (e) surface reconstruction structure, and (f) cluster structure

2.2.2 2D Layers and Atomic Adsorption

A 2D material is defined as having a finite thickness in one dimension and an infinite extent in the other two. The slab model is employed to simulate 2D-layered structures in the CALYPSO method (Wang et al. 2012b). As illustrated in Fig. 5b and c, these models contain two regions: the layered material and vacuum. The vacuum ensures that the studied layers are isolated from their periodic images. Any 2D-layered structure can be characterized by 1 of 17 planar space groups. To ensure unbiased sampling of the energy landscape, the lattice parameters and atomic positions are randomly generated with 2D symmetry constraints. A distortion parameter perpendicular to the in-plane layer (z in Fig. 5b) can also be activated to search for buckled layers, and a van der Waals gap parameter (i.e., the distance between two adjacent layers, as in Fig. 5c) is introduced for multilayered systems. Structural evolution in 2D space is achieved by a constrained PSO algorithm. The CALYPSO method has been widely applied to various 2D materials with exotic structures, including Be5C2 (Wang et al. 2016b), FeB2 (Zhang et al. 2016), and Si (Luo et al. 2014). Because of their intriguing structures and properties, these materials are expected to be promising candidates for future applications in specific fields (Gu et al. 2017).

The adsorption of atoms can efficiently functionalize two-dimensional layer materials with desirable properties. Therefore, structural information about the atoms adsorbed on 2D layers is crucial for understanding their functional properties. A new method was developed to predict the structures of atoms adsorbed on 2D-layered materials based on the CALYPSO methodology. The structural model used is shown in Fig. 5d; it contains three regions: vacuum, adsorption region, and 2D substrate. Several specially designed techniques (e.g., fixed adsorption sites, symmetry constraints, and a constrained PSO algorithm) were employed to improve the search efficiency. Application of our method to investigate fully and partially hydrogenated graphene and graphene oxide successfully predicts their energetically most favorable structures (Gao et al. 2015). In general, our method is promising for the prediction and design of structures of atoms adsorbed on 2D-layered materials (Li et al. 2015a; Zhou et al. 2016).

2.2.3 2D Surfaces

Surfaces play a decisive role in determining the properties and processing of almost all engineering materials, especially at the nanoscale. However, surface structures are often elusive, impeding significantly the engineering of devices. To predict surface structures, we developed an efficient method based on the CALYPSO methodology (Lu et al. 2014). Structure searches for surfaces are conducted using a slab model comprising three regions (Fig. 5e): the bulk material region, the unreconstructed surface, and the reconstructed surface. Generally, the bulk region (6–8 layers) remains fixed to preserve the bulk nature of the material, and the bottom side of the slabs is passivated by hydrogen atoms with integer or partial charges. The atoms in the unreconstructed surface region (usually composed of 2–4 layers) are subject to local structural relaxation, but they are not involved in the structural evolution. In contrast, the atoms in the reconstructed surface region are fully evolved during structure searching, and the choice of thickness for this region depends strongly on the specific energetic situation of the system. The surface excess free energy is calculated as the fitness during structure evolution by\( \triangle \gamma \left(\mu \right)=\frac{1}{A}\left({E}_{surf}^{tot}-{E}_{ideal}^{tot}-{\sum}_i{n}_i\ast {\mu}_i\right) \), where A is the area of the studied surface (usually a 1 × 1 planar unit cell), \( {E}_{surf}^{tot} \)and \( {E}_{ideal}^{tot} \)are the total energies of the reconstructed and unreconstructed surfaces, respectively; and ni and μi denote the number and chemical potential of the ith species in the surface region. Additionally, several specially designed methods have been incorporated into the search procedure to improve its efficiency. For example, symmetry operations of various two-dimensional space groups are applied, while the initial surface structures and surfaces of semiconductors are constructed using the electron-counting rules (Lu et al. 2014). Our approach is evaluated via its application to various semiconductor surface reconstructions (Gao et al. 2017b; Xu et al. 2017), including C, Si, SiC, AlN, and ZnO. Experimentally observed surfaces are readily reproduced by the CALYPSO method from input knowledge of only the chemical compositions, thus validating our approach (Lu et al. 2014).

It is noteworthy that application of new method to a simple diamond (100) surface reveals an unexpected surface reconstruction featuring self-assembled carbon nanotubes arrays (Lu et al. 2014). Such a surface is energetically competitive with the known dimer structure under normal conditions, but it becomes more favorable under a small compressive strain or at high temperatures. The novel surface structure exhibits a unique feature of carrier kinetics (i.e., one dimensionality of hole states, while two dimensionality of electron states) that could lead to novel design of superior electronics. The finding of a previously unknown formation of self-assembled carbon nanotubes on the diamond (100) surface highlights the power of our intelligent surface structure searching method.

2.2.4 0D Nanoclusters or Molecules

Clusters or molecules belong to 0D nonperiodic materials systems. They usually exhibit geometrical frustration due to the competition between surface and bulk, leading to various structural motifs that do not exist in periodic system. The CALYPSO method has been generalized to perform structure searches for these 0D systems (Call et al. 2007). In contrast to periodic systems with translational symmetry, only the point group symmetries are utilized in generating candidate structures of isolated systems, and Cartesian coordinates are used straightforwardly to represent structures. To comply with usual total energy calculations requiring periodic boundary conditions, a big box (Fig. 5f) is built, and the cluster is located at its center. The vacuum surrounding the clusters should be large enough to avoid interactions between the cluster and its periodic images. The main algorithm of PSO is properly revised to allow the structural evolution of 0D systems. The module for cluster structure searching has been extensively benchmarked using Lennard–Jones cluster systems of various sizes (up to 150 atoms), and high search efficiency was achieved, demonstrating the reliability of the current methodology (Lv et al. 2012). Many cluster systems have been investigated using the CALYPSO method (Lv et al. 2014, 2015; Lu et al. 2016; Li et al. 2013a), leading to the discovery of several intriguing structures with interesting chemical bonding. For example, we have used CALYPSO to investigate the structures for neutral boron clusters containing 38 atoms. A symmetric cage-like configuration is found to be the global minimum structure, which can be seen as an all-boron fullerene. Furthermore, a quasi-planar structure with a double-hexagonal vacancy is also revealed with nearly degenerate energy. This quasi-planar structure is recently confirmed by experiments as the ground-state structure for the anionic B38 cluster (Chen et al. 2017).

2.2.5 X-Ray Diffraction Data-Assisted Structure Searches

X-ray diffraction (XRD) is the most powerful technique for determining crystal structure information at the atomic level. However, it remains challenging to determine the crystal structure from only experimental powder XRD data because estimated structural information (e.g., unit cell parameters and space group) is required. This is exemplified by about half of the 300,000 Powder Diffraction Files having some unrefined atomic coordinates (Meredig and Wolverton 2012). Traditional structure prediction uses only energy as the fitness function, and the development of a structure prediction method that can effectively use XRD data is highly desirable. This issue is recently addressed by a first-principles-assisted structure solution (FPASS) method (Meredig and Wolverton 2012), which will be given in detail in the other chapter of this book. We also propose a versatile global search method based on the CALYPSO method to determine crystal structures from experimental powder XRD data without guessed structural information (Gao et al. 2017a). This search method uses the degree of dissimilarity between the simulated and experimental XRD patterns as the fitness (instead of energy), and a weighted cross-correlation function is used to compare the dissimilarity of two powder diffraction patterns. The efficiency and robustness of the new method are demonstrated by exploring the high-pressure phase of a binary compound, CaLi2. The resultant (Fig. 6a) candidate structure of C2221 for CaLi2 at 54 GPa gave a simulated XRD pattern matching with the experimental data (Debessai et al. 2008).
Fig. 6

(a) CALYPSO structure search history for CaLi2. Inset shows experimental and simulated XRD patterns of predicted structure. (b) Inverse-design results for binary electrides. Stability map of A2B (top) and AB (bottom) electrides. A and B elements are shown in horizontal and vertical directions, respectively

2.2.6 Inverse Design of Functional Materials

Metastable structures can exist in nature even though they are not thermodynamically stable. However, traditional structure prediction methods that concentrate on finding the ground-state structure of the PES may miss these metastable structures. It is therefore highly desirable to develop an inverse-design approach to design functional materials directly from certain target properties (Franceschetti and Zunger 1999; Zhang et al. 2015a). The inverse design aims to establish an efficient computational scheme to explore the structure–functionality landscape. The fitness function here is the intrinsic material property such as hardness, bandgap, or interstitial electron localization rather than energy as used in traditional structure searching methods. Our CALYPSO method implements a number of computer-assisted inverse-design techniques to search for functional materials (Zhang et al. 2013, 2017b; Xiang et al. 2013) (e.g., superhard, optical, and electride materials). Here, we demonstrate the performance of the CALYPSO inverse-design method by applying it to the design of inorganic electride materials by screening 99 binary compounds (Zhang et al. 2017b). Before our work, the known inorganic electrides at ambient pressure are only limited to several of them, e.g., C12A7 (Patel 2003) and Ca2N and its variants (Lee et al. 2013; Inoshita et al. 2014; Tada et al. 2014). With the use of the advanced inverse-design method, we are able to identify 89 new electrides, among which 18 are existing compounds that have not been identified as electrides (solid squares, Fig. 6b), while the other 71 electrides (circles, Fig. 6b) are unknown compounds that have not been synthesized, of which 18 (solid circles, Fig. 6b) are thermodynamically stable in their lowest-energy states and dynamically stable with the absence of any imaginary frequency in phonon spectra; the other 53 (void circles, Fig. 6b) are metastable with positive formation energies. In reality, these metastable electrides might be synthesizable, as various metastable electrides have already been synthesized (void squares, Fig. 6b). Our work provides a useful tool for aiding the discovery of electride materials and reveals the rich abundance of inorganic electrides in nature.

3 Materials Discovery Using CALYPSO

3.1 Superconductors

Superconductivity is among the most exciting properties in condensed-matter physics, and considerable effort in the past century has been devoted to the discovery of new superconductors (Bardeen et al. 1957). A significant boost came with the establishment of the microscopic theory of superconductivity by Bardeen, Cooper, and Schrieffer (BCS) (Bardeen et al. 1957) in 1957, which provides important guidance for the design of superconductors with high Tc (critical temperature for superconductivity). According to BCS theory, Tc is intimately tied to one key parameter, the electron–phonon coupling (EPC) constant, which can be estimated by first-principles calculations based on the band structure and Eliashberg theory (Eliashberg 1960). This EPC constant is clearly the critical factor in the design of high-Tc superconductors and is strongly correlated with crystal structure. The determination of crystal structure is therefore a critical step in the design of novel superconductors.

Advances in crystal structure prediction in the past decades have greatly aided the design of superconductors; many have been predicted theoretically, and some confirmed experimentally. Here, we provide an overview of recent advancements in the discovery of superconductors aided by the CALYPSO method. As most of the discoveries are related to hydrides, the overview is divided into two parts: one concerning hydrogen-rich compounds and the other non-hydrogen compounds.

3.1.1 Hydrogen-Rich Compounds

It has been suggested that the lightest element, H, forms metallic solids with sufficiently strong EPC necessary for a high-Tc phonon-mediated superconductivity at high pressures (Ashcroft 1968). However, the metallization of hydrogen had not been observed at low temperature at up to 388 GPa (Dalladay-Simpson et al. 2016). Metallization of solid H at 495 GPa has recently been reported (Dias and Silvera 2017), but additional experimental measurements are required to verify this claim (Liu et al. 2017a, b; Silvera and Dias 2017). Hydrogen-rich compounds have been considered as an alternative because they are expected to metallize at considerably lower pressures owing to the chemical “precompression” caused by other elements (Ashcroft 2004). Combining the CALYPSO method with first-principles EPC calculation leads to predictions of many unexpected superconducting phases in H-rich compounds (Zhang et al. 2017a; Wang et al. 2017). Theoretically calculated Tc values for some hydrogen-rich compounds are summarized in Fig. 7a. These works suggest the possibility of superconductivity with estimated Tc values up to 303 K. Strikingly, a Tc of about 80 K predicted for dense H2S has stimulated experimental observation of hydrogen sulfide superconductors above megabar pressures with Tc in the range of 30–200K (Drozdov et al. 2015). Results for sulfur hydrides (Li et al. 2014), calcium polyhydrides (Wang et al. 2012c), and yttrium polyhydrides (Peng et al. 2017; Liu et al. 2017c) will be introduced here in detail.
Fig. 7

(a) Histogram of calculated Tc of some hydrogen-rich compounds predicted by CALYPSO. Crystal structures of (b) H2S, (c) CaH6, and (d) YH10

H2S is a typical molecular solid at ambient pressure. At high pressure, it transforms into three high-pressure phases, whose crystal structures are the subject of intense debate. H2S is not considered a superconductor as it has been proposed to dissociate into its constituent elements before metallization (Rousseau et al. 2000). Li et al. performed extensive structure searches on solid H2S at pressures of 10–200 GPa (Li et al. 2014). In addition to the identification of candidate structures for the nonmetallic phases IV and V, two metallic structures with P-1 (Fig. 7b) and Cmca symmetry were predicted to be stable above 80 GPa, thus contradicting with the traditional belief (Rousseau et al. 2000) on elemental dissociation at high pressure. The subsequent EPC calculations revealed Tc values of 80 K for the P-1 structure at 158 GPa and 82 K for the Cmca structure at 160 GPa. Motivated by this prediction, electrical measurements of compressed H2S observed the appearance of superconductivity with a Tc in a range of 30–150 K in a sample prepared at 100–150 K, in a good accordance with our prediction in the low Tc regime of 30–80 K; a higher Tc of ∼200 K was observed in another sample prepared at above 220 K (Drozdov et al. 2015; Duan et al. 2015), which is later on suggested as H3S through the decomposition of H2S (see, e.g., (Li et al. 2016; Duan et al. 2015; Einaga et al. 2016; Errea et al. 2015)).

Extensive structure searching on calcium polyhydrides (CaHx) at high pressure (Wang et al. 2012c) showed the CaH6 stoichiometry (Fig. 7c) to be stable at above 150 GPa: it had a body-centered cubic structure, with hydrogen forming unusual “sodalite” cages containing enclathrated Ca at the center of the cage. Its stability is derived from the acceptance by two H2 of electrons donated by Ca to form an “H4” unit as the building block for the construction of the three-dimensional sodalite cage. This unique structure has partially occupied degenerate orbitals at the zone center. The resultant dynamic Jahn–Teller effect helps to enhance the EPC, leading to the superconductivity of CaH6. A Tc of 220–235 K at 150 GPa obtained by solving the Eliashberg equation is the highest among all hydrides studied thus far. Similar sodalite-like structures of MH6 (M = Mg and Y) compounds (Li et al. 2015b; Feng et al. 2015) are also high-Tc superconductors at high pressure, with Tc reaching as high as 264 K. These findings suggest that metal polyhydrides with clathrate structures are potential high-Tc superconductors.

Two recent works conducted structure searches using CALYPSO for stable H-rich clathrate structures in rare earth hydrides at high pressures (Peng et al. 2017; Liu et al. 2017c). Hydrogen clathrate cage structures with stoichiometries of H24, H29, and H32, in which H atoms are weakly covalently bonded to one another, forming cages around rare earth atoms are predicted to be thermodynamically stable at high pressures, some of which exhibit high superconductivity. It is found that high-Tc superconductivity is closely associated with H clathrate structures, with large H-derived electronic densities of states at the Fermi level and strong electron–phonon coupling related to the stretching and rocking motions of H atoms within the cages. In particular, an yttrium H32 clathrate structure of stoichiometry YH10 (Fig. 7d) is predicted to be a potential room-temperature superconductor with an estimated Tc of up to 303 K at 400 GPa (Peng et al. 2017). In the same structure of LaH10, the estimated Tc reaches 288 K at 200 GPa (Peng et al. 2017; Liu et al. 2017c). But it is unstable with respect to LaH3 and LaH11 at 200 GPa. There might be a need of very high temperature to synthesize it (Geballe et al. 2018). Encouragingly, two recent independent experiments seem support the predicted superconductivity of LaH10 at high pressures, where the measured Tc can reach as high as 260-280 K at 190 GPa (Maddury et al. 2018) and 215 K at 150 GPa Drozdov et al.

3.1.2 Non-hydrogen Compounds

Despite a wealth of high-Tc hydrogen-rich hydrides theoretically predicted at high pressure, experimental exploration of these compounds remains a great challenge. The potentially broader ranges of stability make non-hydrogen compounds attractive. Combining CALYPSO with first-principles EPC calculations has predicted many unexpected superconducting compounds of light elements.

Tin telluride is an established high-pressure superconductor, but understanding the underlying mechanism of its transition from semiconductor has been impeded by unsettled issues concerning its structural identification and phase boundary at high pressure. Zhou et al. investigated the high-pressure phase transitions of SnTe using angle-dispersive synchrotron X-ray diffraction combined with the CALYPSO method (Zhou et al. 2013). Three coexisting intermediate phases of Pnma, Cmcm, and GeS-type structures were identified. The Pnma and Cmcm phases were predicted to be superconducting by first-principles calculations, with Tc values of 0.70–0.37 K and 0.01–0.03 K, respectively.

As a sister of CO2, CS2 is a transparent liquid under ambient conditions and found to transform a molecular solid with a Cmca structure at 1 GPa. Upon compression, it became a superconductor with a Tc of 6 K which remains almost constant from 60 to 170 GPa (Dias et al. 2013). However, the lack of an accurate structural determination impedes further understanding the origin of this superconductivity. The later theoretical work using the particle swarm optimization and genetic algorithm methods investigated the crystal structures of CS2 at high pressures (Zarifi et al. 2015). The predicted structure with P21/m symmetry was found to be the most stable from 60 to 120 GPa. The calculated pair distribution functions are in agreement with the experimental assignment. By calculating the electron–phonon coupling parameters, the estimated Tc at 60 GPa is 13 K that is slightly higher than observed 6 K in experiment. The computed Eliashberg spectral function reveals that the S-S vibrations play a critical role in determining the superconductivity of CS2.

The binary semiconductor boron phosphide (BP) with the zinc blende crystal structure has attracted tremendous attentions due to novel properties such as high hardness, high temperature stability, and high thermoelectric powers for direct energy conversion. In order to understand the pressure-induced structural behavior of BP, the CALYPSO method was employed to determine the crystal structures of boron phosphide at high pressure. A novel C2/m structure, which possesses zigzag phosphorus chain structure, was uncovered at 113 GPa, followed by another P42/mnm structure above 208 GPa (Zhang et al. 2015b). Theoretical calculations indicate that the C2/m phase was superconductor with Tc (9.4–11.5 K). This work reveals that pressure-induced zigzag phosphorus chain in BP exhibits high superconducting temperature, opening a new route to design superconductors with zigzag phosphorus chains.

3.2 Superhard Materials

To be superhard, a material should have a Vickers hardness exceeding 40 GPa. Superhard materials are widely used industrially in cutting, polishing, abrasives, protective coatings, etc. The origin of their excellent mechanical properties is strong directional covalent bonds with high electron density that resist both elastic and plastic deformation. Diamond has the highest Vickers hardness of 60–120 GPa but is incompatible with ferrous metals and high temperatures due to its low thermal and chemical stability (Haines et al. 2001). The theoretical design of new superhard materials has become a major research focus and is greatly desirable to assist experimental investigations (Kaner 2005; Zhao et al. 2016). Over the past several decades, covalent compounds formed by light elements, namely, boron (B), carbon (C), nitrogen (N), and oxygen (O), have been the preferred targets owing to their ability to form strong and densely packed 3D covalent bonding networks (Kaner 2005). A new family of materials formed by heavy transition metals and light elements has recently been proposed as potential superhard materials, given that heavy transition metals can introduce high valence electron density into the compounds to resist both elastic and plastic deformations. Our CALYPSO method has helped to design new superhard materials comprising light elements or heavy transition metals and light elements (Fig. 8a, Zhang et al. 2017a).
Fig. 8

(a) Histogram of calculated hardness of some compounds predicted by CALYPSO. Crystal structures of (b) BC3, (c) WB3, and (d) WN

3.2.1 Light-Element Compounds

Boron-doped diamond has been expected to show better oxidation and ferrous resistance than diamond, thus expanding its applicability in electronic devices (Zinin et al. 2012; Dubrovinskaia et al. 2007; Liu et al. 2015; Solozhenko et al. 2004). However, it remains a huge challenge to introduce a high amount of B into the cubic diamond cell. Encouragingly, diamond-like BC3 with one-quarter B content has been synthesized at high pressure and high temperature. Several structures have been proposed for this novel diamond-like BC3; however, none of the proposed sp3 structural models satisfy the experimentally observed cubic symmetry, thus severely impeding further exploration of its properties. The CALYPSO method has solved the crystal structure of recently synthesized cubic BC3 (Zhang et al. 2015c). In contrast to previously proposed tetragonal and orthorhombic structures, our predicted highly symmetric BC3 phase for the cubic diamond structure has a space group of I-43m symmetry and 64 atoms per unit cell, denoted as d-BC3 (Fig. 8b). It becomes stable above 41.3 GPa, in good agreement with the reported synthesis pressure of 39 GPa. Simulated X-ray diffraction and Raman spectra of the predicted d-BC3 phase agree well with the experimental data (Zinin et al. 2012). The calculated hardness (62 GPa) and ideal strength (52.5 GPa) demonstrate d-BC3 as an intrinsic superhard material with intriguing bond elongation and sequential bond-breaking processes that lead to remarkable extended ductility and elastic response. These results represent a significant advance in the understanding of a distinct type of superhard material that exhibits superior ductility to diamond or c-BN.

The prediction of several superhard materials with C3N4 stoichiometry has motivated intense experimental interest to synthesize and characterize them and also intense debate. To design new stoichiometric carbon nitrides with four-coordinated carbon and three-coordinated nitrogen atoms, a body-centered tetragonal CN2 with I-42d space group is predicted at high pressure using CALYPSO (Li et al. 2012). The current predicted structure is built up by strong covalent C–N bonds and N–N bonds, denoted bct-CN2, which resist decomposition into a mixture of diamond + N2 or 1/3(C3N4 + N2) above 45.4 GPa. The strong covalent C–N bonds, N–N bonds, and non-bonding lone-pair are together the driving force for its high bulk (407 GPa) and shear (386 GPa) modulus and simulated hardness (77 GPa) at equilibrium. Our results show that the lone-pair non-bonding states are much more flexible and mobile than the covalent bonds under large strain, and thus bct-CN2 possesses lower ideal strength (47 GPa).

We used CALYPSO method to design two potential superhard ternary materials, B2CO and B3NO, that are isoelectronic with diamond (Li et al. 2015c). For B2CO, two energetically competitive diamond-like structures with space groups of P-4m2 (tP4) and I-42d (tI16) with all sp3 bonding states are predicted. Our simulated results show that they exhibit excellent and similar mechanical properties: e.g., high bulk modulus (∼31 GPa), high shear modulus (∼260 GPa), high hardness (∼50 GPa), and low Poisson’s ratio (∼0.17). For B3NO, we further performed a superhard-driven search, finding a variety of new structures with short and strong 3D covalent bonds. Among them, two orthorhombic structures with space groups of Imm2 (oI20) and Pmn21 (oP20) were found to be dynamically stable. They are both superhard materials with simulated Vickers hardnesses above 45 GPa, exceeding the criterion for superhardness. Electronic results show that the oI20 and oP20 structures are semiconductors with optimal bandgaps of 0.87 and 0.12 eV, respectively, and thus have broad prospects for industrial application.

3.2.2 Transition-Metal–Light-Element Compounds

Tungsten borides are transition-metal–light-element compounds that exhibit diverse polymorphism, excellent functional properties, and potential industrial applicability; for example, they have strong mechanical properties, inexpensive components, and feasible synthesis conditions achievable in a cubic anvil apparatus at relatively low pressures. The conventional expectation is their high boron concentration forms a strong covalent bonding network to enhance their hardness/strength. However, the latest theoretical studies show the structural determination and even the chemical composition of these synthesized tungsten borides still to be open questions. Using CALYPSO, we identified the thermodynamically stable structures as well as many metastable structures over a wide range of boron concentrations for tungsten borides (Li et al. 2013b). Comparison of experimental and simulated X-ray diffraction patterns leads to the identification of previously synthesizedI4/m-4u W2B (γ-phase), I41/amd-8u WB (α-phase), P63/mmc-4u WB2 (ε-phase), and P63/mmc-4u WB3 (Fig. 8c). Based on the calculated convex hull, P63/mmc-2u WB2, R-3m-6u WB3, and P63/mmc-2u WB4 are thermodynamically stable and thus viable for experimental synthesis. Our first-principles calculations reveal that, contrary to common expectations, increasing the boron content does not raise the mechanical strength of boron-rich tungsten borides; instead, ideal strength show little improvement or even decreases with rising boron content in most cases. This intriguing behavior is fundamentally rooted in boron’s ability to form versatile bonding states, which produces distinct structural configurations (Li et al. 2015a). This work represents a comprehensive study of boron-rich tungsten borides based on a global structure search, and the results have important implications for a large class of transition-metal borides that share the same boron concentration range, along with likely similar bonding configurations and deformation mechanisms as revealed in the present study.

Transition-metal nitrides are another class of materials that can be tailored into a new generation of superhard solids. CALYPSO structural searching yields many crystal structures for tungsten nitrides (Lu et al. 2017). X-ray diffraction indicates the presence of hP4-WN in a recently synthesized specimen. Here we report findings from first-principles calculations for two tungsten nitrides, hP4-WN (Fig. 8d) and hP6-WN2, which exhibit extraordinary strain stiffening that remarkably enhances their indentation strengths above 40 GPa, raising exciting prospects for nontraditional superhard solids. Calculations show that hP4-WN is metallic both at equilibrium and under indentation, making it the first known intrinsic superhard metal.

4 Conclusions and Prospects

This chapter briefly introduced the basic theory and general features of our CALYPSO approach. Its validity has been extensively demonstrated via successful applications to various material systems, including 3D bulk crystals, 2D layers and surfaces, and 0D clusters. Its successful application to the discovery of superconductive and superhard materials was presented, demonstrating its great promise for designing functional materials.

Design-orientated experiments guided by structure prediction offered, for example, by CALYPSO have greatly expedited materials discoveries. However, they are sometimes hampered by the high computational costs needed for structural optimizations, especially for multicomponent or large systems. Machine learning as a data-driven method for making prediction, decision, or classification is now starting to enter the heart of computational physics, chemistry, and material sciences in a manifold way. Several machine learning potentials have been developed, which show accuracy comparable to those of quantum mechanical simulations but require less computational effort by many orders of magnitude (Behler and Parrinello 2007; Behler 2016). The replacement of the heavy computational cost of DFT calculations with a state-of-art machine learning potential is a viable route for structure prediction in large systems (Tong et al. 2018; Deringer et al. 2018).

The structural prediction at finite temperature remains challenging due to the requirement of accurate calculation of free energy. The harmonic/quasi-harmonic lattice dynamics models (Van de Walle and Ceder 2002) can be used to estimate the free energy. However, numerous trial structures are required to estimate free energy for structure prediction. It is computationally unaffordable to calculate phonons of all structures. In principle, metadynamics or molecular dynamics simulations (Martoňák et al. 2003; Parrinello and Rahman 1981) can be widely used to predict structures at finite temperatures. However, in reality, metadynamics or molecular dynamics simulations may suffer frequently from the constraint of short simulation run, especially when time-consuming first-principles simulations are performed. Short simulation runs naturally lead to inadequate sampling of free-energy landscape, limiting the wide use of these methods on prediction of high-temperature structures. Future development of cheaper and more efficient methods in calculations of free energy is essential to accomplish the goal of structure prediction at high temperature.

Knowledge of stable structures does not automatically provide detailed synthesis pathways. It is highly desirable to know whether the identified new materials can be eventually synthesized experimentally, but the theoretical prediction of synthesis conditions leading to a desired structure remains challenging. This requires us to compute the stability of a material with respect to alternative atomic configurations. For doing so, it is worth developing relevant methods to address the minimum-energy path of phase transition or chemical reaction after new materials are predicted.

Notes

Acknowledgments

The authors acknowledge funding support from the National Key Research and Development Program of China under Grant No. 2016YFB0201200, No. 2016YFB0201201, and No. 2017YFB0701503; NSAF (No. U1530124)? the National Natural Science Foundation of China under Grants No. 11774127, No. 11534003, No. 11622432 and No. 11722433); supported by Program for JLU Science and Technology Innovative Research Team (JLUSTIRT); and the Science Challenge Project, No. TZ2016001.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Yanchao Wang
    • 1
    Email author
  • Jian Lv
    • 1
  • Quan Li
    • 1
    • 2
  • Hui Wang
    • 1
  • Yanming Ma
    • 1
    • 2
  1. 1.State Key Laboratory of Superhard Materials & Innovation Center of Computational Physics Methods and Software, College of PhysicsJilin UniversityChangchunChina
  2. 2.International Center of Future ScienceJilin UniversityChangchunChina

Section editors and affiliations

  • Cai-Zhuang Wang
    • 1
  • Christopher M. Wolverton
    • 2
  1. 1.Ames Laboratory and Department of Physics and AstronomyIowa State UniversityAmesUSA
  2. 2.Northwestern UniversityEvanstonUSA

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