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ACR-USPEX-2011 Accounts o f c h e m i c a l r e s e a r c h March 2011 Volume 44 number 3 pubs.acs.org/accounts Published on the Web 03/01/2011 www.pubs.acs.org/accounts Vol. 44, No. 3 ’ 2011 ’ 227–237 ’ ACCOUNTS OF CHEMICAL RESEARCH ’ 227 10.1021/ar1001318 & 2011...

ACR-USPEX-2011
Accounts o f c h e m i c a l r e s e a r c h March 2011 Volume 44 number 3 pubs.acs.org/accounts Published on the Web 03/01/2011 www.pubs.acs.org/accounts Vol. 44, No. 3 ’ 2011 ’ 227–237 ’ ACCOUNTS OF CHEMICAL RESEARCH ’ 227 10.1021/ar1001318 & 2011 American Chemical Society How Evolutionary Crystal Structure Prediction Works;and Why ARTEM R. OGANOV,*, †, ‡ ANDRIY O. LYAKHOV,† AND MARIO VALLE§ †Department of Geosciences and Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-2100, United States, ‡Geology Department, Moscow State University, 119992 Moscow, Russia, and §Data Analysis and Visualization Group, Swiss National Supercomputing Centre (CSCS), via Cantonale, Galleria 2, 6928 Manno, Switzerland RECEIVED ON OCTOBER 2, 2010 CONS P EC TU S O nce the crystal structure of a chemical substance is known, many prop-erties can be predicted reliably and routinely. Therefore if researchers could predict the crystal structure of a material before it is synthesized, they could significantly accelerate the discovery of newmaterials. In addition, the ability to predict crystal structures at arbitrary conditions of pressure and temperature is invaluable for the study of matter at extreme conditions, where experiments are difficult. Crystal structure prediction (CSP), the problem of finding the most stable arrangement of atoms given only the chemical composition, has long remained amajor unsolved scientific problem. Two problems are entangled here: search, the efficient exploration of the multidimensional energy landscape, and ranking, the correct calculation of relative energies. For organic crystals, which contain a few molecules in the unit cell, search can be quite simple as long as a researcher does not need to include many possible isomers or conformations of the molecules; therefore ranking becomes the main challenge. For inorganic crystals, quantum mechanical methods often provide correct relative energies, making search the most critical problem. Recent developments provide useful practical methods for solving the search problem to a considerable extent. One can use simulated annealing, metadynamics, random sampling, basin hopping, minima hopping, and data mining. Genetic algorithms have been applied to crystals since 1995, but with limited success, which necessitated the development of a very different evolutionary algorithm. This Account reviews CSP using one of the major techniques, the hybrid evolutionary algorithm USPEX (Universal Structure Predictor: Evolutionary Xtallography). Using recent developments in the theory of energy landscapes, we unravel the reasons evolutionary techniques work for CSP and point out their limitations. We demonstrate that the energy landscapes of chemical systems have an overall shape and explore their intrinsic dimensionalities. Because of the inverse relationships between order and energy and between the dimensionality and diversity of an ensemble of crystal structures, the chances that a random search will find the ground state decrease exponentially with increasing system size. A well-designed evolutionary algorithm allows for much greater computational efficiency. We illustrate the power of evolutionary CSP through applications that examine matter at high pressure, where new, unexpected phenomena take place. Evolutionary CSP has allowed researchers to make unexpected discoveries such as a transparent phase of sodium, a partially ionic form of boron, complex superconducting forms of calcium, a novel superhard allotrope of carbon, polymeric modifications of nitrogen, and a new class of compounds, perhydrides. These methods have also led to the discovery of novel hydride superconductors including the “impossible” LiHn (n = 2, 6, 8) compounds, and CaLi2. We discuss extensions of the method to molecular crystals, systems of variable composition, and the targeted optimization of specific physical properties. 228 ’ ACCOUNTS OF CHEMICAL RESEARCH ’ 227–237 ’ 2011 ’ Vol. 44, No. 3 Evolutionary Crystal Structure Prediction Oganov et al. 1. Combinatorial Complexity of the Problem Following a simple combinatorial argument,1 the number of possible distinct structures can be evaluated as C ¼ V=δ 3 N � �Y i N ni � � (1) where N is the total number of atoms in the unit cell of volume V, δ is a relevant discretization parameter (for instance, 1 Å) and ni is the number of atoms of ith type in the unit cell. Already for small systems (N ≈ 10-20), C is astronomically large (roughly 10N if one uses δ = 1 Å and typical atomic volume of 10 Å3). It is useful to consider the dimensionality of the energy landscape: d ¼ 3Nþ3 (2) where 3N - 3 degrees of freedom are the atomic posi- tions, and the remaining six dimensions are lattice para- meters. For example, a systemwith 20 atoms/cell poses a 63-dimensional problem! We can rewrite eq 1 as C ∼ exp(Rd), where R is some system-specific constant. With such high-dimensional problems, simple exhaustive search strategies are clearly unfeasible. The global optimization problem can be greatly simpli- fied if combined with relaxation (local optimization). During relaxation, certain correlations between atomic positions set in: interatomic distances adjust to reasonable values and unfavorable interactions are avoided to some extent. The intrinsic dimensionality of this reduced energy landscape consisting only of local minima (Figure 1) is now d� ¼ 3Nþ3-K (3) where κ is the (noninteger) number of correlated dimen- sions. d* depends both on system size and on chemistry. We found2 d* = 10.9 (d=39) for Au8Pd4, d*=11.6 (d=99) for Mg16O16, and d* = 32.5 (d = 39) for Mg4N4H4. The number of local minima is then C�∼ exp(βd�) (4) with β < R, d* < d, and C* , C, implying that efficient search must include local optimization. Even simple ran- dom sampling, when combined with relaxation,3 can deliver correct solutions for systems with N< 8-10.With USPEX, the limit is much higher, but the exponential increase of C* with system size means that CSP is an NP-hard problem and for sufficiently large sizes CSP will always be intractable. In most cases, we are interested in systems with N < 20-200, and systems with N < 100 are tractable, while the range 100 < N < 200 may become accessible in the foreseeable future. 2. How the Method Works Evolutionary algorithms work best when the energy (or, more generally, fitness) landscape has an overall shape, as in Figure 1. Analysis2 suggests suchoverall shape in the energy landscapes of chemical systems and implies that evolu- tionary algorithms are highly appropriate for CSP. Such overall structure is also expected for landscapes of many physical properties. In evolutionary simulations, a population of struc- tures evolves, gradually “zooming in” on the most promising regionsof the landscape and leading to further reductionof d*. The evolutionary algorithm USPEX (Universal Structure Predictor: Evolutionary Xtallography1,4,5), unlike a previous FIGURE 1. Energy landscape:2 (a) 1D scheme showing the full landscape (solid line) and reduced landscape (dashed line joining local minima); (b) 2D projection of the reduced landscape of Au8Pd4, showing clustering of low-energy structures in one region. Vol. 44, No. 3 ’ 2011 ’ 227–237 ’ ACCOUNTS OF CHEMICAL RESEARCH ’ 229 Evolutionary Crystal Structure Prediction Oganov et al. genetic algorithm for crystals6 but similarly to an algorithm for clusters,7 includes local optimization and treats structural variables as physical numbers, instead of nonintuitive bin- ary “0/1” strings (the latter is the defining difference between “genetic” and more general “evolutionary” algorithms; the former use binary strings). Other important considerations are as follows: (1) The algorithm incorporates “learning from history” (i.e., offspring structures bear resemblance to the more successful of the previously sampled structures), which is done through selection of the low-energy structures to become parents of the new generation, survival of the fittest structures, and variation opera- tors (i.e., recipes for producing child structures from parents). Acting upon low-energy structures, variation operators lead, with high probability, to yet other low- energy structures. Four variationoperators are used in our method:1,4,5 (i) heredity (creating child structures from planar slabs cut from two parent structures8) (ii) lattice mutation (large random deformation ap- plied to the unit cell shape) (iii) permutation (swaps of chemical identity in pairs of chemically different atoms) (iv) special coordinate mutations (displacements of the atoms, but not in a fully random way, see below). For molecular crystals, where the structure is assembled from entiremolecules (of a particular isomer), rigid or flexible, the above variation op- erators act on molecular centers, and additional variation operators must act on orientation and conformation of the molecules. (2) The population should remain diverse, allowing very different solutions to be produced throughout the simulation. Diversity can be measured by the collec- tive quasientropy, Scoll: Scoll ¼ - (1-Dij)ln(1-Dij) � � (5) where Dij are abstract cosine distances between all pairs of structures (these distances measure struc- tural dissimilarity and can only take values be- tween 0 and 12). Figure 2 shows that in a good simulation quasientropy retains large values and can exceed quasientropy of the first random gen- eration, that is, evolutionary search not only is more efficient in finding low-energy structures but also can have more structural diversity than random search, thus depriving the latter of any potential advantages. Initialization of the first generation can be random for small systems (N < 20). For large systems, most of the structures produced by random samplingwill be very similar (Figure 3), disordered and with high energies.2 It will be hard to produce good structures from such a population. There is an inverse relationship between the intrinsic dimensionality and the mean μ of the distance distribution, μ - � (d�)-m (6a) and variance of this distribution, σ � (d�)-n (6b) where positivem and n depend on the distance measure used (cosine vs Euclidean distances). FIGURE 2. Evolutionary simulation of the binary Lennard-Jones A5B16 (the potential model used here is known to yield low-energy quasi- crystalline structures9). The insets show the lowest energy as a function of generation number, and the lowest-energy structure. FIGURE 3. Distribution of distances between randomly sampled local minima in a binary Lennard-Jones system AB2. 230 ’ ACCOUNTS OF CHEMICAL RESEARCH ’ 227–237 ’ 2011 ’ Vol. 44, No. 3 Evolutionary Crystal Structure Prediction Oganov et al. To obtain a diverse population, one should reduce the number of degrees of freedom in the first generation by (i) assembling initial structures from ready-made building blocks (molecules, coordination polyhedra, and low-energy seed structures) or (ii) generating the initial population using symmetry and/or pseudosymmetry. Since variation opera- tors break symmetry, structures with different symmetries will have a chance to emerge. Consider splitting the unit cell into S subcells. When all ni/S are integers, splitting is done into identical subcells, introducing additional translational symmetry.When ni/S is a noninteger, random vacancies are created to maintain the correct number of atoms (Figure 4), introducing pseudosymmetry and leading to nontrivial or- dered structures that are difficult to create otherwise. Such structures are well-known in nonstoichiometric compounds and even in the elements, for example, complex high-pressure phases Cs-III and Rb-III can be represented as supercells of thebody-centered cubic structurewithadditional atoms (which can be thought of as additional partially occupied sublattices). After a generation is completed, locally optimized struc- tures are compared using their fingerprints,10 and all non- identical structures are ranked in order of their free energies. The probability P of selecting a structure to be a parent is determined by its fitness rank i, e.g. in a linear scheme: P(i) ¼ P1- (i-1)P1c , Xc i¼1 P(i) ¼ 1 (7) where c is a selection cutoff. This scheme is superior to Boltzmann-type selection, because it is insensitive to peaks and gaps in energy distributions and does not require an additional parameter (“temperature”) needed for defining Boltzmann probabilities; a quadratic analo- gue of (7) often works even better. Niching (i.e., removal of identical structures using fingerprints5,11) allows a large number of lowest-energy structures to be carried over into the next generation, increasing the learning power, retaining diversity, and en- abling a more thorough exploration of low-energy meta- stable structures. The current algorithm is efficient for systems with <300 degrees of freedom and can be enhanced.5 Directed moves that have a higher likelihood of leading to lower-energy structures or new promising areas of the energy landscape will be essential. For instance, moving the atoms along the eigenvectors of lowest-frequency phonon modes typically leads to low-energy structures. For this, one has to construct and solve the dynamical matrix, which is computationally extremely demanding at the ab initio level. We have effi- ciently solved this problem5 by constructing an approximate dynamical matrix using the bond hardnesses computed from bond lengths and atomic covalent radii and electronegativities. Given the usually good correlation between the energy and degree of order,2 one could preferentially use the more ordered pieces of parent structures in the heredity operator. Figure 5 shows local order in a hypothetical defective structure of SiO2; clearly, defective regions correspond to low-order atoms. Giving low-order atoms larger displace- ments while preserving positions of high-order atoms leads to a very effective coordinate mutation operator;5 note that “blind” indiscriminate displacement of all atoms is much more likely to destroy than to create good structural motifs.4 Figure 6 shows different types of optimization. These examples show that with very minor adaptations, this method is powerful for solving a wide range of problems. The limits of applicability of this approach are not fully known. Using a partly successful reimplementation of the method,1,4 Trimarchi and Zunger14 failed to predict FIGURE 5. Illustration of the concept of local order for defective SiO2. Low-order atoms are blue; high-order atoms are red. Low-order regions correspond to the planar defect. FIGURE 4. Pseudo-subcells for composition A3B6 (atoms A, large black circles; atoms B, small filled circles; vacancies, empty circles). The true cell (thick lines) is split into four pseudo-subcells (thin lines). Vol. 44, No. 3 ’ 2011 ’ 227–237 ’ ACCOUNTS OF CHEMICAL RESEARCH ’ 231 Evolutionary Crystal Structure Prediction Oganov et al. fcc-ordered structures of the Au8Pd4 alloy that would be competitive with structures predicted by cluster expansion; however, this failure was probably due to inadequate Bril- louin zone sampling. With this in mind, we found12 a P21/m ordering, more stable than any structure found to be stable by cluster expansion. Thus, atomic ordering can be effi- ciently predicted. What are the real limitations then? First, like for any NP-hard problem, dimensionality is a great restriction. The current version of the method was found to work very well for dimensionalities below 100-200. Re- cently, we could determine, at DFT level of theory, a very complex high-pressure structure of methane with 105 atoms/cell (Q. Zhu, unpublished), but to enable this, we had to operate with molecular, rather than atomic, entities, which considerably lowers the dimensionality of the problem. Topology of the landscape is an important factor - single-funnel landscapes (as in Figure 1) are much more amenable for evolutionary algorithms than multifunnel or, even worse, featureless landscapes. Energy landscapes usually have an overall shape with a small number of funnels,2 but landscapes of some properties may be more erratic. Below we consider some this method's recent applica- tions to high-pressure chemistry; most of these findings could not be expected by chemical intuition and required a powerful CSP method, nonempirical, unbiased and cap- able of arriving at completely unexpected solutions. 3. Applications 3.1. The High-Pressure Chemistry of “Inert” Atomic Cores and of the Electron Gas. Highly compressible atoms of alkali and alkali earth metals enter a chemically interest- ing regime at strong compression when their cores begin to overlap:15 valence electrons get increasingly “trapped” in the interstitial space, and valence bandwidth decreases on compression. Vacant in the free atoms, p- and d-orbitals become dominant at strong compression, eventually mak- ing K, Rb, Cs, Ca, Sr, and Ba d-metals, while Li becomes a predominantly p-element, even adopting the diamond structure above 483 GPa.16 The most interesting picture occurs for Na: in this element at megabar pressures, valence s-, p-, and d-orbitals are populated nearly equally.17 Sodium has a deep minimum on the melting curve at ∼118 GPa and 300 K, belowwhich extremely complex (and not resolved) crystal structures were observed.18 We predicted17 that above 250 GPa an insulating hP4 structure becomes stable, and this prediction was experimentally FIGURE 6. Predictions of (a) stable crystal structure of MgSiO3 with 80 atoms in the unit cell,12 (b) stable compounds and their structures in a binary Lennard-Jones system,12 and (c) the hardest structure of TiO2. 13 232 ’ ACCOUNTS OF CHEMICAL RESEARCH ’ 227–237 ’ 2011 ’ Vol. 44, No. 3 Evolutionary Crystal Structure Prediction Oganov et al. verified,17 although at lower pressures (>200 GPa).19 The band gap, estimated using the state-of-the-art GW calculations,17 turned out to be remarkably wide, from ∼2 eV at 200 GPa to over 5 eV at 500 GPa. This implies that the hP4 phase should be optically transparent already at 200 GPa (this was experimentally confirmed17) and even color- less above ∼320 GPa. The band gap is due to strong interstitial localization of electron pairs. The hP4 structure (Figure 7) can be described in several equivalent ways as (i) NiAs-type structure where both sites are occupied by Na atoms. (ii) Ni-sublattice of the Ni2In structure, In sublattice being occupied by the interstitial electron pairs. The Ni2In structure is known to be remarkably dense, consid- ered to be the highest-pressure phase for AB2 com- pounds20 until a post-Ni2In transitionwas discovered. 21 (iii) Double hexagonal close packed structure. The stack- ing of close-packed layers ofNa atoms is CACBCACB... (underlined layers contain interstitial electron pairs, Figure 7) and is squeezed by a factor of >2 along the c-axis, while the interstitial electron pairs form a nearly ideal hcp ABAB-stacking (c/a ≈ 1.3-1.6). The hP4 structure of Naminimizes core-valence overlap and maximizes packing efficiency of the interstitial electron pairs. It can be called an “electride”,22 that is, a “compound” made of ionic cores and localized interstitial electron pairs; electride formshavealsobeenpredicted inLi underpressure,16 and Li was experimentally shown to become a semiconductor under pressure.23 There are surprisingly faint hints of electride behavior in K in anarrowpressure range,24while Rb andCs do not showelectridebehavior at all. This is due to thepresenceof shallow d-orbitals in heavy alkali and alkali earth elements; when d-states get populated under pressure, the atom be- comes more compact
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