Perspective: crystal structure prediction at high pressures.

Perspective: Crystal structure prediction at high pressures Yanchao Wang and Yanming Ma Citation: The Journal of Chemical Physics 140, 040901 (2014); doi: 10.1063/1.4861966 View online: http://dx.doi.org/10.1063/1.4861966 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Theoretical prediction of high pressure phase transition in ScC and YC: Ab initio calculations J. Appl. Phys. 114, 053516 (2013); 10.1063/1.4817504 Hexagonal high-pressure phase of tantalum mononitride predicted from first principles J. Appl. Phys. 113, 083502 (2013); 10.1063/1.4792731 Room-temperature structures of solid hydrogen at high pressures J. Chem. Phys. 137, 074501 (2012); 10.1063/1.4745186 Structural, electronic, and optical properties of crystalline iodoform under high pressure: A first-principles study J. Chem. Phys. 134, 034508 (2011); 10.1063/1.3528728 Theoretical prediction of the Cmca phase in Ge under high pressure J. Appl. Phys. 89, 2547 (2001); 10.1063/1.1341215

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THE JOURNAL OF CHEMICAL PHYSICS 140, 040901 (2014)

Perspective: Crystal structure prediction at high pressures Yanchao Wang and Yanming Maa) State Key Laboratory of Superhard Materials, Jilin University, Changchun 130012, China

(Received 4 November 2013; accepted 30 December 2013; published online 22 January 2014) Crystal structure prediction at high pressures unbiased by any prior known structure information has recently become a topic of considerable interest. We here present a short overview of recently developed structure prediction methods and propose current challenges for crystal structure prediction. We focus on first-principles crystal structure prediction at high pressures, paying particular attention to novel high pressure structures uncovered by efficient structure prediction methods. Finally, a brief perspective on the outstanding issues that remain to be solved and some directions for future structure prediction researches at high pressure are presented and discussed. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4861966] I. INTRODUCTION

Pressure is one of the fundamental thermodynamic variables, which can allow precisely tuning of the interatomic distance and alter the bonding patterns of materials, leading to phase transformations into high pressure (HP) structures with unusual chemical and physical properties. An old, but classic, example is the HP induced phase transformation of graphite (a soft material with a sp2 C–C bonding) into diamond (a hardest known superhard material with a sp3 C–C bonding).1 In recent years, advances in experimental and theoretical techniques have fostered remarkable breakthroughs in HP sciences, as exemplified from the discovery of HP superconductors,2 metal-to-semiconductor transitions,3, 4 superfluid,5 etc. HP has therefore become an irreplaceable tool in the syntheses of new materials via phase transformations. To note, only limited number of elements were known as superconductors at ambient pressure. However, the “superconducting periodic table” now extends to most elements under HP conditions.6 It has also been well recognized that HP can effectively lower the barrier of chemical reaction and thus makes the chemical reaction possible only at HP conditions. HP synthesis of functional materials with desirable chemical or physical properties via chemical reaction of distinct species has recently received great attentions. Experimentally, some of successful examples can be referred to the syntheses of mechanically important PtN2 7 and IrN2 ,8 and hydrogen-storage materials of methane-hydrogen mixtures,9 SiH4 (H2 )2 ,10, 11 Xe(H2 )8 ,12 etc. Theoretically, HP hydrogen-storage materials with high hydrogen contents in Li–H2 ,13 Na–H2 ,14 Mg–H2 ,15 and Ca–H2 16 systems were designed, some of which appear to be potential high temperature superconductors.16 Reactions of chemically inert rare gases (e.g., He, Kr, Xe) with various metals (e.g., Na,17 Mg,18 Fe, and Ni19 ) at HP were predicted to form hitherto unexpected compounds. Without HP, all these chemical reactions are not possible.

The chemical and physical properties depend on crystal structures, thus knowledge of crystallography is essential if the properties of materials are to be understood and exploited. Experimentally, diamond-anvil-cell devices20 combined with other techniques, such as X-ray diffraction and neutron diffraction, have been used to determine HP structures. However, due to the small size of the samples the diffracted X-ray beam is usually weak. It happens frequently that experiments fail to determine the HP structures. Moreover, the pressure range accessible to diamond anvil cells is limited to several millions atmospheric pressure. Extreme HP conditions relevant to the interiors of giant planets (e.g., pressures up to 40 Mbar in the core of Jupiter21 ) are beyond the current experimental reach. Unbiased theoretical prediction of HP structures with the only given information of chemical compositions of materials has to be relied on and become a topic of considerable interest in recent years. Breakthroughs on findings of HP structures were made through crystal structure prediction: HP phases with long-puzzling structures were frequently solved4, 22 and hitherto unexpected HP structures with novel properties were predicted,3, 23–27 some of which were already confirmed by experiments.3, 28 We emphasize that without relying on crystal structure prediction, these HP research works would not be possible. In this article, we review crystal structure prediction methods29–34 developed recently and outline our perspective on the outstanding issues that remain to be solved and some directions for future researches at HP. This article is organized as follows. In Sec. II structure prediction approaches including biased and unbiased methods and recent accomplishments of crystal structure predictions at HP in conjunction with firstprinciples total energy calculations are reviewed. In Sec. III, we discuss some issues on the challenges of current structural prediction methods. Some directions for future HP researches are presented in Sec. IV. Finally, a summary is provided in Sec. V. II. METHODS ON CRYSTAL STRUCTURE PREDICTION

a) Author to whom correspondence should be addressed. Electronic mail:

[email protected] 0021-9606/2014/140(4)/040901/11/$30.00

The aim of crystal structure prediction is to establish an efficient computational scheme to explore the vast

140, 040901-1

© 2014 AIP Publishing LLC


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FIG. 1. (a) One-dimensional model potential-energy surface. LM and EB are the abbreviations for local minimum and energy barrier, respectively. The illustration of the basin hopping scheme (b), minima hopping algorithm (c), and metadynamics algorithm (d).

configuration space of high-dimensional potential-energy surface (PES). The ultimate goal is to find the global energy minimum among a huge number of local minima on PES separated by high energy barriers (Fig. 1(a)). In principle, finding the global energy minimum structure or the so-called “most stable” structure needs a full visit of all local minima. However, according to empirical observations and heuristic estimates, the number of local minima grows exponentially with increasing system sizes.35 Thus, it is not feasible to adopt the exhaustive ergodic searching strategy.36 Generally, the approaches addressing the crystal structure prediction face a twofold challenge. On one hand, the PES needs to be accurately evaluated, thus necessitating the high cost quantummechanical methods that yield the total energy for a given atomic configuration. On the other hand, a major theoretical concern is related to the systematic and efficient exploration of the entire PES. Much effort has been devoted to solving the structure prediction problem by properly reducing the configuration space or effectively enhancing the sampling efficiency of the PES. Local structure optimization is able to dramatically eliminate the noise of energy surface, and drives the structure into its nearby local minima. Thus, structure searching methods rely heavily on the structure optimization or relaxation to reduce the configuration space. However, local structure optimization scans only local environment of the PES, an efficient sampling technique over the entire PES is a must for crystal structure prediction. Here we focus on the crystal structure prediction methods designed for inorganic systems and their applications to HP structures based on first-principles density functional calculations (e.g., see Refs. 29–32). Note that considerable progress has also been made in predicting crystal structures of organic compounds (e.g., see Refs. 37–39), where empirical potentials were primarily used for total-energy calculations, but these methods

are not within the scope of current review. According to the ways of sampling, structure prediction methods can be classified into two categories: biased or unbiased methods. Biased methods are possible to produce a faster global optimization by learning from prior known structural knowledge. The approaches (e.g., substitutional40 and data mining methods using databases of known structures41, 42 ) focus only on structural motifs provided by chemical intuition and hence can vastly reduce the configuration space. However, it suffers from a notable shortcoming, incapable of generating new structure types not listed in the structural database. To circumvent this problem, unbiased stochastic methods (e.g., see Refs. 29, 30, and 43) without relying on any prior experience have been developed and will be briefly discussed below. A. Methods designed for overcoming energy barriers

The simulated annealing approach44, 45 is based on the concept arising from physical annealing. There are two key factors on simulated annealing algorithm: Monte Carlo scheme46 via Metropolis sampling47 and an annealing schedule. A new configuration of ions is generated from an initial configuration through random ion move. The Metropolis criterion is employed to determine whether the random move is accepted or not. The simulation starts at high temperature where almost all moves are accepted and moves are attempted until system reaches thermal equilibrium; the temperature is then reduced and the process is repeated until the system becomes frozen. In principle, if this annealing process is continuous and carried out slowly enough, the resultant structure sits likely on the global minimum. The algorithm has been successfully applied into a variety of systems (e.g., see Refs. 48–51). The basin-hopping algorithm52 explores PES, which is transformed into a collection of interpenetrating staircases


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(Fig. 1(b)). The algorithm involves the use of Monte Carlo simulation with a constantly reduced temperature by 0.8. At each step, fractional coordinates are initially displaced by random numbers in the range [−1,1], and then minimized. If the energy of the new configuration is lowered, the initial move is accepted. Otherwise the move is only accepted with a probability of exp((Eold − Enew )/kB T). The reasons why the basinhopping method can be successful might stem from that the minimization converts the PES into the set of basins of attraction of all the local minima. This transformation removes the barriers between local minima without changing their energies and is thus feasible for the system to hop between basins. The use of this algorithm is able to detect the global energy minimum with a high probability even for a multiplefunnel surface.53 Basin hopping algorithm has been used to predict the global stable structures for Lennard-Jones clusters containing 1–110 particles.52 The minima hopping algorithm54 performs molecular dynamics simulation instead of the Monte Carlo simulation in basin hoping method to overcome the energy barrier between local minima. The acceptance rate of new configuration satisfies 50% of all attempted configurations through the adjustment of specially designed energy threshold (E). The illustration of this approach is shown in Fig. 1(c). New configuration will be accepted only if its energy difference relative to initial configuration is less than E, otherwise it is discarded. In addition, the minima hopping algorithm recognizes regions that have been already explored. The algorithm is also possible to explore broader regions of the configuration space through more violent configuration moves. The method has been successfully applied to study the HP structures of Si2 H6 ,55 LiAlH4 ,56 Zn(BH4 )2 ,57 etc. The metadynamics algorithm58 adds a positive Gaussian potential to the energy landscape of the system to lower and overcome the energy barrier (Fig. 1(d)). Gaussians are deposited by causing the underlying biased potential to grow with the increase of simulation time and high biased potential discourages the system to go back to its previous steps

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and forces it to explore the full energy landscape. Thus, metadynamics has a strong ability to overcome the large barrier in comparison to standard molecular dynamic simulation. Successful application of metadynamics algorithm on HP structures has been exemplified in SiO2 ,59 CO2 ,60 and Ca.61 All these methods described above need the input of initially guessed structure, where the simulation started. These simulations have advantages and disadvantages. If the initial structure on PES is best chosen in good close to the global minimum, the method has the advantage to converge fast into the global stable structure. However, if the initial structure is too far from the global minimum, the simulation faces to the disadvantage of an unpleasantly long simulation run. Below, we briefly discuss other unbiased methods, which are not relying on any initial guess of structures. B. Random sampling method

Random sampling methods30, 38, 62, 63 involve in the random generation of a large number of structures on PES (Fig. 2(a)), among which the energetically best structure is the natural choice of the structure searching (Fig. 2(b)). Recently, Pickard and Needs made substantial contribution on making the method more efficient by introducing several structural dealing techniques (e.g., choosing stoichiometry, symmetry constraints, and shaking) as implemented in the AIRSS (ab initio random structure searching) code.30 AIRSS has been widely applied to the prediction of HP structures.24, 26, 64–66 The predicted HP structures of hydrogen-rich SiH4 and AlH3 compounds were confirmed by experiments.67–69 Please refer to the review in Ref. 30 for a better understanding of the random sampling algorithm and its application. Below we discuss another example on the prediction of HP structures of solid hydrogen. Solid hydrogen was expected to be a good candidate for high-temperature superconductors at high pressure once it was metalized.70 Thus, compressed solid hydrogen has attracted considerable attention.71, 72 At high pressures,

FIG. 2. Illustration of random sampling scheme (a) and (b), the genetic algorithm (c), and CALYPSO method (d) in a one-dimensional model potential. (a) Random distributions of a large number of structures on PES and (b) distributions of locally optimized structures on PES.


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three nonmetallic molecular phases have been established experimentally,73–75 which are labeled as phases I, II, and III. The structure of phase I is well understood as a rotationally disordered structure. However, the structures of phases II and III remain unsolved. Recently, several molecular structures (e.g., P63 /m, C2/c, and Cmca-12) were predicted by AIRSS,64 among which C2/c structure is found to be the most stable structure known thus far in the pressure range 105–270 GPa and has thus been suggested as a good candidate for phase III. At higher pressures (>∼500 GPa), quasi-molecular and atomic phases (mC24, I41 /amd, R3m, oC12, and cI16)64, 76–78 were predicted by CALYPSO (Crystal structure AnaLYsis by Particle Swarm Optimization) and AIRSS methods, respectively. The quasi-molecular mC24 structure77 (space group C2/c) is interesting because it contains two different intramolecular bonds with distinct pressure variations, with which the understanding of pressure-induced molecular dissociation of solid hydrogen is possible. Recently, a new insulating phase IV of solid hydrogen was experimentally observed above 220 GPa and at room temperature.79–81 Remarkably, phase IV was interpreted as having an orthorhombic Pbcn structure (24 molecules per cell), which was previously predicted by AIRSS method in 2007.64 This Pbcn structure is rather peculiar and consists of unique alternate layers of graphene-like three-molecule rings and unbound H2 molecules; however, this structure is dynamically unstable with the existence of imaginary phonons. Subsequently, monoclinic Pc82 and Cc83 structures containing larger number of atoms in unit cell (96 atoms per primitive cell) were proposed. These structures are dynamically stable and are geometrically rather similar to Pbcn. The peculiar structure features of alternate layers of graphenelike three-molecule rings and unbound H2 molecules remain. Molecular dynamics simulations revealed83 that phase IV is a partially disordered structure: disorder in the unbounded H2 molecules layers but ordered in the strongly coupled graphene-like layers. More recently, a first-principles variable-cell molecular dynamics simulation84 suggested a novel hydrogen diffusive process in phase IV: intralayer hydrogen hoppings and collaborative rotation of three hydrogen molecules in a ring fashion in the graphene-like layers.84 This finding provides a direct explanation on the observed abrupt increase of Raman linewidth at the formation of phase IV and its large increase with pressure.79, 80 Other ab initio molecular dynamics simulations85, 86 on phase IV found similar results on collaborative rotation of three hydrogen molecules ring.

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ated by applying two major operators: “crossover” to pairs of parents (Fig. 2(c)), so combining parents structural features into new individuals, and “mutation” to individual candidate structures (Fig. 2(c)), so introducing new structural features to the population. The work by Deaven and Ho in 1995 was important since it bridged for the first time the real-space representation of structure to genetic algorithm.89 Very recently, several efficient software packages (e.g., USPEX,32 XtalOpt,33 MAISE,90 GSGO,91 EVO,92 and GASP93 ) on structure prediction94 based on genetic algorithm or its variation were developed. These methods have received wide application into the prediction of HP structures,3, 90, 93, 95–99 some of them including insulating phase of sodium,3 metallic structure of oxygen98 and superconductive Fe–B compounds90 were confirmed by experiments.100–102 We address below two examples. The light alkali element sodium is often considered to be a “simple” metal as its electronic properties are well described by the nearly free-electron model at ambient conditions. Under pressure, metals exhibit increasingly shorter inter-atomic distances. Intuitively, this response is expected to be accompanied by an increase in the widths of the valence and conduction bands and hence a more pronounced freeelectron-like behavior. However, earlier theoretical calculations suggested that Na might be demetalized by adopting a pairing dimer structure.103 To explore the possible existence of stable yet hitherto unknown high-pressure structures in Na, an extensive structure prediction study on Na up to 300 GPa97 was performed and an unexpected six-coordinated simple double-hexagonal close-packed hP4 structure (Fig. 3(a)) was predicted, rather than the earlier proposed paired structure.103 The hP4 structure is predicted to be stable above 260 GPa with an unexpected wide bandgap (bandgap at 3.2 eV at 300 GPa). The predicted hP4 structure was then confirmed by HP experiments,3 whose stability is lowered to 200 GPa in the experimental observation. The insulating state in Na-hP4 arises from the strong localization of valence electrons in the interstices of the lattice. The case of dense Na provides an

C. Methods based on genetic algorithm

Genetic algorithm (GA)87, 88 was proposed in the mid1970s and it works by mimicking Darwinian evolution, the basic tenets of which are that offspring resembles their parents and procreation is rewarded for success. When GA is applied to structure prediction, a population of candidate structures is randomly generated and structurally relaxed to the local minimum. Using the optimal energies as the criteria of fitness, a fraction of the candidate structures with lower energies is selected as parents. The new candidate structures are gener-

FIG. 3. High pressure hP4 structure of Na (a), Aba2-40 structure of Li (b), N10 -cage structure of N2 (c), and cI14 structure of CaH6 (d).


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unexpectedly extreme example of a wide-gap insulator created by compression of a “simple” metal. Boron element is situated right in between metals and insulators in the periodic table, and has only three valence electrons. This electron deficiency compared with carbon for a nonmetallic element determines its chemical complexity. As a result, known structures of boron contain typical B12 icosahedra with multi-centered chemical bonds. HP structures of boron have been the subject of considerable interests. A new form of boron104 was earlier synthesized at pressures above 10 GPa and temperatures above 1500 ◦ C. However, the structure remains unknown until crystal structure prediction simulations23 were performed with the constraints of fixed cell parameters as derived from experimental data. This disclosed form of boron (named as γ -B28 ) consists of icosahedral B12 clusters and B2 dimers and is theoretically stable between 19 and 89 GPa. The γ -B28 resembles a NaCltype structure, with the B12 icosahedra and B2 dimers playing the roles of “anions” and “cations,” respectively. The charge transfer between B12 and B2 indicates the partially ionic nature of this new phase. Note that an independent experimental work105 claimed also on the finding of the γ -B28 structure. D. Methods based on particle swarm optimization algorithm

The particle swarm optimization (PSO) algorithm was proposed in the mid-1990s.106, 107 As a stochastic global optimization algorithm, PSO is inspired by the social behavior of birds flocking or fish schooling and designed to solve problems related to multidimensional optimization. PSO belongs to the family of swarm intelligence and is fundamentally different with GA since it directly avoids the use of two major operators: “crossover” and “mutation” employed in GA. The PSO algorithm has been successfully applied to many optimization problems.108–111 We have recently applied PSO algorithm into the structure prediction of extended systems (e.g., crystals, and two-dimensional layers and surfaces) as implemented in the CALYPSO method31 and its same-name code.112 Within the CALYPSO method, structures are evolved in the energy landscape through velocity (Fig. 2(d)) according to the formula (1). The new velocity of each structure v t+1 is calculated based on its previous location (xt ) before optimization, previous velocity (v t ), current location (pbestt ) with an achieved best fitness, i.e., lowest enthalpy, of this individual, and the population global location (gbestt ) with the best fitness value for the entire population according to Eq. (2): x t+1 = x t + v t+1 , v t+1 = ωv t + c1 r1 (pbest t − x t ) + c2 r2 (gbest t − x t ),

(1) (2)

where ω denotes the inertia weight, which is dynamically varied and decreases linearly from 0.9 to 0.4 during the iteration. c1 and c2 are self-confidence factor and swarm confidence factor, respectively. r1 and r2 are random numbers, which distributed in the range [0, 1]. The velocity update formula includes random parameters r1 and r2 to ensure good

coverage of the searching space and avoid entrapment in local minimum. As shown in Fig. 2(d), it is quite obvious that the velocity plays an important role on determination of the speed and direction of structural movement and it is dynamically influenced by individual past experience (pbestt ,v t ) and the successful experience of the whole swarm (gbestt ). We found that PSO has superior ability to overcome large barriers of energy landscapes by making use of the swarm intelligence. Several important and efficient techniques31, 112 not considered earlier111 including symmetry constraints on structural generation, the bond characterization matrix on the elimination of similar structures, partial random structures per generation on enhancing structural diversity, and the penalty function on selection of low-energy structures, were implemented in CALYPSO method.112 We found that the use of these techniques can effectively prevent structure premature (i.e., entrapment in local minimum), a known shortcoming of the evolutionary algorithm (PSO and GA), and also can accelerate the structure searching convergence. We emphasize that the search space metric in our method is “structure,” rather than the cell parameters and atomic positions as misinterpreted by Lyakhov and co-authors.113 Evolution of structures on PES in CALYPSO method takes the advantage of swarm smart learning of both the individual and global best structures by manipulating unit cell and atomic positions. The so-called corrected-PSO in Ref. 113 is not a PSO algorithm, but rather a variation of GA algorithm since their major structure operators employed are “crossover” and “mutation” developed well in GA. The use of fingerprint distances in energy landscape as search space metric113 is less productive since an identical fingerprint distance would correspond to infinite number of structures.114, 115 Currently, CALYPSO has many attractive features including predictions of the energetically stable/metastable structures at given chemical compositions for isolated nanoparticles/clusters or molecules,115 2D layers (single/ multi layers and buckled layers),116, 117 2D surfaces, and 3D crystals;112 design of novel functional materials, e.g., superhard materials;118 structure searching with automatic variation of chemical compositions; structure predictions with fixed unit cell parameters, or fixed spacegroups, or fixed molecules. CALYPSO method has been widely applied to the prediction of HP structures for various systems including Li,25 N2 ,119 O2 ,120 C,121 Si,122 CaH6 ,16 Cl2 ,123 Bi2 Te3 ,27 H2 O,124 CO2 ,125 FeTi3 O7 ,126 etc., some of which25, 27 have already received the experimental confirmation. Below, we discuss three examples on the successful applications of CALYPSO into HP structures. The light alkali element Li is a good metal at ambient conditions. However, above 70 GPa, experiments demonstrated that Li experiences an unexpected metalsemiconductor transition into a broken-symmetry semiconducting phase.4 Crystal structure of this semiconducting phase has attracted numerous attentions.4, 25, 28, 127–131 The observed insulation seems in accordance with earlier prediction of pairing mechanism,128 which however is unlikely in view of the proposed energetically much preferred metallic Cmca24 (24 atoms/cell) structure.127 Using CALYPSO method on structure prediction, a complex orthorhombic Aba2-40


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structure (Fig. 3(b), 40 atom/cell, Pearson symbol, oC40) was predicted to be energetically more stable than all earlier proposed structures127–130 in the pressure range 60–80 GPa.25 Band structure calculation of Aba2-40 structure reveals indeed a semiconducting behavior with a band gap >0.8 eV at 70 GPa. Similar to that in the insulating Na, the semiconducting state in Aba2-40 structure of Li arises from the strong localization of valence electrons in the interstices of the lattice. The predicted Aba2-40 structure has soon received experimental and theoretical confirmation.28, 131 Under HP, triply bonded molecular nitrogen dissociates into singly bonded polymeric nitrogen, a potential high-energy-density material. The exploration of stable high-pressure forms of polymeric nitrogen is of great interests (see, e.g., Refs. 65 and 132–134). Extensive structure searching through CALYPSO method on solid nitrogen at HP conditions was performed and a hitherto unexpected cage-like diamondoid structure of polymeric nitrogen (stable above 263 GPa)119 was predicted. The diamondoid structure of polymeric nitrogen adopts a highly symmetric body-centered cubic structure with lattice sites occupied by diamondoids, each of which consists of ten nitrogen atoms, forming a N10 tetracyclic cage (Fig. 3(c)). A blind prediction by AIRSS method30 confirmed the diamondoid structure. The prediction of diamondoid structure of polymeric nitrogen provides an unexpected example created by compression of a molecular solid and represents a significant step toward the understanding of the behavior of solid nitrogen and other nitrogen-related materials at extreme conditions. Once hydrogen-rich compounds are metalized under HP, they hold the promise as high-temperature superconductors. The searching of hydrogen-rich compounds requested the examination of the chemical reaction of CaH2 and H2 under HP.16 Through CALYPSO structure prediction calculations, it was predicted that HP syntheses of hydrogen-rich stoichiometries CaH4 , CaH6 , and CaH12 are possible. In particular, a novel body-centered cubic structure for CaH6 (Fig. 3(d)) was predicted to be stable above 150 GPa, where hydrogen forms unusual “sodalite” cages with enclathrated Ca.16 The stability of this structure is derived from the acceptance by two H2 of electrons donated by Ca forming an “H4” unit as the building block in the construction of the three-dimensional sodalite cage. A superconducting critical temperature of 220–235 K at 150 GPa was estimated from the solution of the Eliashberg equations, the highest among all hydrides studied thus far. Though crystal structure prediction at HP has received tremendous successes via various global optimization algorithms, challenges of crystal structure prediction remain, in particular for large systems, and the prediction at finite temperature. In Sec. III, we address these challenges.

III. CHALLENGES: CRYSTAL STRUCTURE PREDICTION A. Crystal structure prediction for large systems

Most practical systems (e.g., materials with defects, nanomaterials, and biomaterials) contain usually a large number of atoms in the simulation cells, reaching several hun-

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dreds or thousands of atoms. Knowledge of structures for these large systems is more practical for engineering applications. Theoretical structure prediction of these large systems is challenging since the number of energy minima on PES grows exponentially with increasing number of atoms in the systems. A known problem of structure prediction is the lack of structure diversity, especially for large systems. One way to overcome this problem is to choose randomly trial solutions on the search space. However, entire random sampling for large systems leads to predominantly high-energy disordered (fluid-like) structures, and thus a low structural diversity.135 Note that biased structural dealing techniques (such as symmetric constraints and the use of structural database) can generate ordered structures. It is thus desirable to implement some particularly biased techniques for enhancing the structure diversity. One of the key issues related to the structure prediction of large systems is the so-called “adequate sampling.” The sampling needs an accurate description of PES. For this purpose, first-principles calculations on total energies and structural optimizations are often relied on since the ab initio potentials are rather accurate to describe the PES. However, the first-principles structural relaxation is computationally so expensive that structure predictions for large systems can fail in short of studied structures, leading to an inadequate sampling. In order to solve aforementioned problem, much work remains to be done with regard to the development of fast methods on first-principles calculations (e.g., orbitalfree density functional theory136 ) or more reliable force field methods.137 B. Crystal structure prediction at finite temperature

Temperature as a fundamentally thermodynamical variable is of vital importance for both scientific research and technological applications. The structural prediction at finite temperature becomes a great necessity. However, most of the crystal structure prediction methods are based on standard lattice energy minimization algorithms, which are restricted to 0 K calculations. These studies are unable to deal with structures whose stability order is a function of temperatures. To address this issue, the accurate calculations of free energy at finite temperature, rather than the static lattice energy, is essential. The harmonic/quasi-harmonic lattice dynamics models138, 139 can be used to estimate the free energy. On one hand, it is problematic for some systems (particular at very high temperature) due to ignoring of higher order term. On the other hand, structure prediction deals with numerous trial structures. It is computationally unaffordable to calculate phonons of all structures. The relative free energy differences may be calculated using statistical simulations via thermodynamic integration140 or umbrella sampling141 based on molecular dynamics simulations. In principle, metadynamics142 or molecular dynamics simulations143 can be widely used to predict structures at finite temperatures. To ensure an adequate sampling of free energy landscape, enough long simulation runs are required. However, in reality, metadynamics or molecular dynamics


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simulations 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. By all appearance, the bottleneck in structure prediction at finite temperature is the high computational cost of the free energy calculations of the 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.

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quent pseudopotential151 constructed with a smaller core cutoff radius (∼0.42 Å)152 amended the core overlapping problem and found that P41 32 structure is unstable with respect to Cmca-8, in good consistence with the results of all-electron full-potential method.153 The current example warns us that the transferability and reliability of pseudopotentials (though well established at ambient pressure) must be carefully examined for applications at HP.

IV. PERSPECTIVE ON STRUCTURE PREDICTION AT HP A. Earth and planetary materials

C. Crystal structure prediction at high pressure

Accurate description of the PES is central to the structure prediction. Thus, the quantum-mechanical methods, that yield accurate total energy for a given atomic configuration and are able to deal with various inter-atomic bondings independent of structures, are often used on structure prediction. The density functional theory (DFT)144, 145 has been proved to be a reliable and computationally tractable tool in condensed matter physics, albeit it has many limitations (e.g., failures for molecular systems with van der Waals interactions).146, 147 First-principles pseudopotential method within the framework of DFT has been widely applied into structure predictions at HP because of its relatively lower computational cost compared to the full-potential DFT method. Caution must be taken by using pseudopotential method at HP conditions since inter-atomic distances of materials can be significantly shortened compared with those at ambient conditions. The appropriate valence electrons explicitly treated and the choices of rigid core radii of elements in the pseudopotentials have significant effects on the accuracy of total energies. At HP conditions, the repulsive interaction between core and valence orbitals becomes more important because of the broadening of valence and core bandwidth under compression, and therefore, core potential will have a greater impact on the electronic properties. The constructed pseudopotentials suitable for HP studies should accommodate more electrons than as usual at ambient conditions. For example, pseudopotentials of sodium for studies at ambient pressure148 only treated 3s electrons as valence states with nonlinear partial core correction.149 However, for HP studies, it should be generated by treating more electrons (2s, 2p, and 3s electrons) as valence states.3 Moreover, sufficiently small core radii in pseudopotentials are necessary to avoid any core overlap at HP. We discuss a failure example on prediction of HP structures of elemental lithium resulting from the use of inappropriate pseudopotential. A cubic P41 32 structure was predicted to be energetically more favorable than Cmca-8 structure at 300 GPa by using a pseudopotential with a large core radius (∼0.82 Å).150 This pseudopotential works well at ambient and moderate pressures (3000 K) and high pressures.165 According to these analyses, Uranus and Neptune might contain large amounts of diamond. Recent structure prediction research did reveal the decomposition of CH4 into diamond and hydrogen166 and the idea of diamond formation in the interiors of giant planets such as Neptune was supported. Future crystal structure prediction is needed to understand the HP structures of CH4 since at room temperature CH4 was observed to remain in molecular form up to ∼200 GPa.167 B. Functional materials under extreme conditions

Two different cases of chemical reaction at HP conditions can be described: (i) the chemical compounds are pre-existing and in such a case, HP leads to structural phase transformations, (ii) the chemical compounds do not exist and HP is able to help on the synthesis of these materials. The first case has been investigated by theories and experiments for years and significant progresses have been made. A brief perspective on the first case will be presented below and we address the second case in Sec. IV C. In general, it is expected that material experiences a number of phase transitions under pressures. This opens the possibility to find hitherto unexpected functional materials in a large variety at HP conditions. These functional materials cover superhard materials,168, 169 superconductors,170–172 high-energy-density materials,119, 132 hydrogen storage materials,10, 16 thermoelectric materials,173, 174 etc. However, experimental syntheses of materials at HP conditions are exhaustive and challenging, especially when deals with the materials having only narrow pressure-temperature stability. Crystal structure prediction is helpful to the HP synthesis. Such effort has been devoted to design on superhard materials and potential HP superconductors.175, 176 It is foreseeing that with the guidance of structure prediction, the experimental cost can be reduced and the materials synthesis can be accelerated.176–178 We address below two perspective directions on crystal structure prediction of superhard materials and potential superconductors. The widely used superhard materials are diamond and cubic boron nitride. However, diamond has a major drawback in that it reacts with iron and cannot be used for machining steel, whereas the synthesis of bulk crystals of cubic boron nitride is challenging.179 Searching for new superhard materials has remained as a general interest for long. The compounds formed by light elements (B–C–N–O or Si–B–C–N) are potential superhard materials since these elements are able

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to form strong covalent bonds. However, combination of these elements leads to a large number of potential compounds. The structure prediction calculation can be adopted to determine the phase diagrams of potential superhard systems to offer a possible route for experimental synthesis. In addition, it is also worthwhile to search for superhard materials that are not composed entirely of light elements, such as the compounds formed by transition metals and light elements with partially covalent bonds.168, 169, 180 These researches are largely unexplored and may be important for searching of new superhard materials at HP. For decades, scientists have been ongoing to great effort to design high-temperature superconductor. Superconductivity at HP has been intensively studied and various new superconductors were discovered at HP conditions (e.g., Si, O, and S).6 For elemental superconductor Ca, the superconducting transition temperature increases from 2 K at 44 GPa to 29 K at 161 GPa,181, 182 which is the highest among all the elements. Two classes of compounds may be important for searching new superconductors. The first class can be traced to various metal borides, silicides, and aluminides with the planar or puckered-sheet structures similar to MgB2 .183 Some of these compounds (e.g., BaSi2 ) were already found to exhibit superconductivity at HP.184 Another class is related to hydrogen-rich compounds. Under HP conditions, hydrogenrich compounds can be metalized and have the high potential to be superconductive because of their high Debye temperatures.70, 185 Along this line, HP metallic structures of hydrogen-rich compounds (e.g., IVB hydrides, IIIB hydrides, and alkaline earth hydrides15, 16, 55 ) have been extensively investigated by crystal structure predictions. Experimental evidences68 seem to support these ideas, but intensive debates exist.67 Future theoretical and experimental studies of HP superconductivity on hydrogen-rich compounds are greatly needed. C. Chemical reactions at HP

Pressure can induce chemical reaction not taking place at normal conditions. Therefore, HP chemical reaction often leads to the formation of hitherto unknown compounds. Recent crystal structure prediction studies toward chemical reaction at HP created a number of unexpected novelties related to various disciplines, such as physical, chemical, and geological science. Below, we particularly emphasize examples related to creation of high oxidation states and the formation of stable compounds of chemically inert noble gases as predicted by crystal structure predictions. The oxidation state is an indicator of the degree of oxidation of an atom in a compound and is the hypothetical charge that an atom would have if all bonds to atoms of different elements are fully ionic. The goal of physics and chemistry is to prepare unusual states of matter beyond the naturally occurring forms. With the help of crystal structure prediction, chemical reactions of Hg + F2 and Cs + F2 at HP conditions were predicted with the formation of novel compounds with unusual stoichiometries.186, 187 Hg in HgF4 compound was found in an unusual high oxidation state of 5d8 , while


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Cs in CsF4 compound was even predicted to be a 5p element by adopting a high oxidation state of 5p4 . These novel results suggested that HP chemical reaction is able to generate oxidation states of elements unusual to our chemical intuitions. Noble gases are chemically very inert due to their closed electron shells. At ambient conditions, they hardly form stable compounds with other elements.188–190 At HP conditions, however, situation has been dramatically changed. Ar and Xe were found to form stable compounds with H2 .191, 192 Recently, Xe was even observed to react with water and predicted to react with Fe19 and Mg18 through crystal structure predictions. The formation of stable Fe-Xe and Mg-Xe compounds at HP might contribute to the understanding of missing Xe paradox. Under HP, chemical reactions often lead to compounds with unusual stoichiometries, which are clearly against on our conventional wisdom. For examples, at ambient conditions, the only known Ca hydride is CaH2 , but theoretical simulations predicted chemical reaction of CaH2 + H2 at HP, leading to the formation of stoichiometries CaH4 , CaH6 , and CaH12 .16 Some of other examples are seen in Li–B,193, 194 Na–Cl,195 and Si–C196 systems at HP conditions. These studies described above are just a corner of entire HP researches on chemical reactions. It is highly expected that more fascinating phenomena benefited from HP chemical reactions will be disclosed soon or after. Crystal structure prediction will inevitably play a critical role on uncovering these exciting new chemical compounds since it is essential to know the structure of the products of chemical reactions. In a wider scope, chemical reaction will be a general phenomenon when the mixtures are suffering at sufficient HP. V. SUMMARY

Crystal structure prediction has recently received great successes and attracted numerous attentions. We presented a short review on efficient structure prediction methods developed recently and their applications into a number of novel HP structures, some of which have been confirmed by experiments. We point out that further methodological developments on enhancing the structure searching efficiency are necessary and issues related to the structure predictions of large systems and at the conditions of finite temperatures are still challenging. A brief perspective on crystal structure prediction at HP conditions was proposed and discussed. We paid particular attention to HP research directions toward Earth and planetary materials, functional materials, and chemical reactions, where a large number of open questions remain unaddressed. It is concluded that crystal structure prediction will play a critical role on solving these HP problems in the near future. ACKNOWLEDGMENTS

The authors acknowledge funding support from the National Natural Science Foundation of China (under Grant Nos. 91022029, 11274136, and 11025418), 2012 Changjiang Scholar of Ministry of Education, the open project of key lab of computational physics in Beijing Computing Physics and

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Applied Mathematics, and Changjiang Scholar and Innovative Research Team in University (Grant No. IRT1132), and the China 973 Program under Grant No. 2011CB808204. 1 F.

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Crystal structure and prediction.

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Data mining approaches to high-throughput crystal structure and compound prediction.

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Ultralow viscosity of carbonate melts at high pressures.

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Crystal structure of allyl-ammonium hydrogen succinate at 100 K.

Crystal structure of thermitase at 1.4 A resolution.

Crystal structure of hevein at 2.8 A resolution.