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The geometric blueprint of perovskitesMarina R. Filipa,1 and Feliciano Giustinoa,b,1,2

aDepartment of Materials, University of Oxford, Oxford OX1 3PH, United Kingdom; and bDepartment of Materials Science and Engineering, CornellUniversity, Ithaca, NY 14853

Edited by Roberto Car, Princeton University, Princeton, NJ, and approved April 13, 2018 (received for review November 3, 2017)

Perovskite minerals form an essential component of the Earth’smantle, and synthetic crystals are ubiquitous in electronics, pho-tonics, and energy technology. The extraordinary chemical diver-sity of these crystals raises the question of how many and whichperovskites are yet to be discovered. Here we show that the“no-rattling” principle postulated by Goldschmidt in 1926, describ-ing the geometric conditions under which a perovskite can form,is much more effective than previously thought and allows usto predict perovskites with a fidelity of 80%. By supplementingthis principle with inferential statistics and internet data min-ing we establish that currently known perovskites are only thetip of the iceberg, and we enumerate 90,000 hitherto-unknowncompounds awaiting to be studied. Our results suggest thatgeometric blueprints may enable the systematic screening of mil-lions of compounds and offer untapped opportunities in structureprediction and materials design.

perovskites | structure prediction | Goldschmidt | data mining |materials design

Crystals of the perovskite family rank among the most com-mon ternary and quaternary compounds and are central to

many areas of current research (1). For example, silicate per-ovskites constitute the most abundant minerals on Earth (2), andsynthetic oxide perovskites find applications as ferroelectrics (3),ferromagnets (4), multiferroics (5), high-temperature supercon-ductors (6), magnetoresistive sensors (7), spin filters (8), superi-onic conductors (9), and catalysts (10). Halide perovskites arepromising for high-efficiency solar cells, light-emitting diodes,and lasers (11–13); their double perovskite counterparts are effi-cient scintillators for radiation detection (14). The unique ver-satility of the perovskite crystal structure stems from its unusualability to accommodate a staggering variety of elemental combi-nations. This unparalleled diversity raises the questions of howmany new perovskites are yet to be discovered and which oneswill exhibit improved or novel functionalities. In an attempt toanswer these questions, we here begin by mapping the entirecompositional landscape of these crystals.

Fig. 1A shows the structure of a cubic ABX3 perovskite. In thisstructure the A and B elements are cations and X is an anion.B-site cations are sixfold coordinated by anions to form BX6

octahedra. The octahedra are arranged in a 3D corner-sharingnetwork, and each cavity of this network is occupied by oneA-site cation (15). All perovskites share the same network topol-ogy, but can differ in the degree of tilting and distortions of theoctahedra (15–19). The quaternary counterpart of the perovskitecrystal is the double perovskite A2BB′X6. When the B and B′

cations alternate in a rock-salt sublattice, the crystal is calledelpasolite (14) (SI Appendix, Fig. S1A). In the following we usethe term “perovskite” to indicate both ternary and quaternarycompounds.

ResultsHow many perovskites do currently exist? If we search for thekeyword “perovskite” in the inorganic crystal structure database(ICSD; www2.fiz-karlsruhe.de), we find 8,866 entries, but afterremoving duplicates the headcount decreases to 335 distinctABX3 compounds. Similarly, the keyword “elpasolite” yields224 distinct A2BB′X6 compounds. By including also the exten-

sive compilations of refs. 20–26, we obtain a grand total of1,622 distinct crystals that are reliably identified as perovskites(Dataset S1).

How many perovskites are left to discover? Direct inspec-tion of the elemental composition in Dataset S1 indicates that,with the exception of hydrogen, boron, carbon, phosphorous, andsome radioactive elements, these crystals can host every atomin the Periodic Table. Therefore an upper bound for the num-ber of possible perovskites is given by all of the combinationsof three cations and one anion. By considering only ions withknown ionic radii (27) we count 3,658,527 hypothetical com-pounds. Our goal is to establish which of these compounds canform perovskite crystals. Ideally we should resort to ab initiocomputational screening (28), but these techniques are not yetscalable to millions of compounds. For example, by making theoptimistic assumption that calculating a phase diagram requiredonly 1 h of supercomputing time per compound, it would take160 y to complete this task.

An empirical approach to investigate the formability of ABX3

perovskites was proposed by Goldschmidt almost a century ago(29). In this approach the perovskite structure is described asa collection of rigid spheres, with sizes given by the ionic radiirA, rB, and rX. These radii can be combined into two dimen-sionless descriptors, the tolerance factor t = (rA + rX)/

√2(rB +

rX) and the octahedral factor µ= rB/rX. Goldschmidt postu-lated that perovskites arrange so that “the number of anionssurrounding a cation tends to be as large as possible, sub-ject to the condition that all anions touch the cation” (ref.15, p. 34). This statement constitutes the “no-rattling” princi-ple and limits the range of values that t and µ can take for aperovskite.

Significance

Perovskites constitute one of the most versatile and chem-ically diverse families of crystals. In perovskites the samestructural template supports a staggering variety of proper-ties, from metallic, insulating, and semiconducting behaviorto superconducting, ferroelectric, ferromagnetic, and multi-ferroic phases. Approximately 2,000 perovskites are currentlyknown. In this work we revisit the century-old model pro-posed by Goldschmidt to predict the formability of perovskitesin the key of modern inferential statistics and internet datamining. We demonstrate that the nonrattling rule postulatedby Goldschmidt can predict the stability of perovskites with asuccess rate of 80%. Using this tool we predict the existenceof 90,000 hitherto unknown perovskites.

Author contributions: M.R.F. and F.G. designed research, performed research, analyzeddata, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Published under the PNAS license.1 M.R.F. and F.G. contributed equally to this work.2 To whom correspondence should be addressed. Email: feliciano.giustino@materials.ox.ac.uk.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1719179115/-/DCSupplemental.

Published online May 7, 2018.

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Fig. 1. No-rattling principle for ternary perovskites. (A) Rigid-sphere representation of the conventional cell of a cubic ABX3 perovskite, with A in gray,B in blue, and X in red. (B) Cross-sectional view of the stretch limit. In this configuration the A cation sits at the center of the cavity and touches 12nearest-neighbor X anions. (C) Cross-section of the perovskite structure in the octahedral limit. Here the nearest-neighbor X anions belonging to the sameoctahedron touch. (D) Schematic representation of the tilt limit. This configuration can be thought of as obtained from B by reducing the size of the Acation (gray), increasing the size of the X anion (red), and tilting the octahedra so that anions of adjacent octahedra touch. In this case the A cation movesaway from the center of the cavity to optimally fill the available space and touches five X anions. (E) Stability area of ternary perovskites (blue), derived fromthe no-rattling principle and from additional chemical limits (dashed lines): chemical limits (CL1 and CL2), octahedral limit (OL), stretch limit (SL), secondarystretch limits (SSL1 and SSL2), and tilt limits (TL1 and TL2). These boundaries are derived in SI Appendix, section 3.

Goldschmidt’s principle was recently tested on larger datasetsthan those available in 1926. By analyzing a few hundred ternaryoxides and halides, it was found that in a 2D t vs. µ mapperovskites and nonperovskites tend to cluster in distinct regions(22, 23, 30). Based on this observation, much work has beendedicated to identifying a “stability range,” either by postulat-ing boundaries for t and µ (22, 23, 31) or by using machinelearning to draw t vs. µ curves enclosing the data points (30).The merit of these efforts is that they addressed the predictivepower of the no-rattling principle in qualitative terms. However,these approaches suffer from relying too heavily on empiricalionic radii. It is well known that the definition of ionic radii isnonunique and that even within the same definition there arevariations reflecting the coordination and local chemistry (1, 15,27, 29, 32). As these uncertainties transfer to the octahedraland tolerance factors, the stability ranges proposed so far aredescriptive rather than predictive. To overcome these limita-tions, instead of defining a map starting from empirical data, ourstrategy is to construct a stability range from first principles, byrelying uniquely on Goldschmidt’s hypothesis. This choice allowsus to also derive a structure map for quaternary compounds, astep that has thus far remained elusive.

We describe our strategy starting from ternary perovskites,and then we generalize our findings to quaternary perovskites.Fig. 1B shows that, for the A cation to fit in the cavity, theradii must satisfy the condition rA + rX≤

√2(rB + rX) or equiva-

lently t ≤ 1 (SI Appendix, section 3). Similarly, Fig. 1C shows thatthe octahedral coordination of the B cation by six X anions isnot possible when

√2(rB + rX)< 2rX, and therefore µ≥

√2− 1.

We refer to these conditions as the “octahedral” limit and the“stretch” limit, respectively, as shown in Fig. 1E. When theseconditions are not fulfilled, the lattice tends to distort towarda layered geometry, with edge-sharing or face-sharing octahe-dra or lower B-site coordination (15, 31). These two bounds arewell known (15, 29) but are insufficient for quantitative structureprediction.

When t is smaller than 1, the corner-sharing octahedra exhibitan increased degree of tilting, and the A-site cation is dis-

placed from the central position in the cuboctahedral cavity,as shown Fig. 1D and SI Appendix, Fig. S2 (17, 18). Previouswork has shown that the displacement of the A-site cation canbe determined by optimizing Coulomb interactions between theperovskite ions, taking into account the bond-valence configu-ration of each ion (19). Here we explore this scenario usinga purely geometric approach, by identifying the ionic positionswhich achieve the tightest packing of ions in a tilted perovskiteconfiguration.

Fig. 1D shows that the octahedra can tilt only until twoanions belonging to adjacent octahedra come into contact. Inthis extremal configuration, to satisfy the no-rattling principlethe A-site cation must exceed a critical size. Using the geomet-ric construction described in SI Appendix, section 3 and shown inSI Appendix, Fig. S2, this condition translates into a lower limitfor the tolerance factor, t ≥ ρµ/

√2(µ+ 1), where ρµ is a simple

piecewise linear function of µ (SI Appendix, Fig. S3). We referto this condition as the “tilt” limit (Fig. 1E). When this crite-rion is not met, the perovskite network tends to collapse intostructures with edge-sharing or face-sharing octahedra. Alongthe same lines we must consider the limit of two neighboring A-site cations coming into contact (Fig. 1E and SI Appendix, Fig.S4A) and of the A and B cations touching (SI Appendix, Fig.S4B). Besides these geometric constraints, we also should takeinto account that the rigid spheres of the model represent chem-ical elements, and therefore we have additional bounds on thesize of the ionic radii: The largest tolerance factor correspondsto the combination of Cs and F, and the oxidation number of Acannot exceed that of B according to Pauling’s valency rule (32).By considering all of the (µ, t) points that satisfy these conditionssimultaneously, we obtain the perovskite stability area shown inblue in Fig. 1E.

These considerations are readily generalized to double perov-skites. In this case we have two different cations B and B′ (SIAppendix, Fig. S1A), and therefore we must consider two octa-hedral parameters: the average octahedral factor, µ= (rB +rB′)/2 rX, and the octahedral mismatch, ∆µ= |rB− rB′ |/2 rX. InSI Appendix, section 4 we derive the generalized tolerance factor,

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Fig. 2. Stability range of ternary and quaternary perovskites. (A) The blue volume represents the stability range of perovskites in the (µ, t, ∆µ) space,derived from Goldschmidt’s no-rattling principle. The blue and red markers correspond to perovskites (Datasets S1 and S3) and nonperovskites (DatasetS2), respectively, calculated for all of the compounds in Dataset S1. The location of each marker is the center of the rectangular cuboid definedin SI Appendix. (B–E) Slices of the stability volume shown in A, reporting all perovskites with octahedral mismatch ∆µ in the range indicated atthe top of each panel. In these 2D representations the cuboids appear as rectangles, with blue and red indicating perovskites and nonperovskites,respectively.

which takes the form t = (rA/rX + 1)/[2(µ+ 1)2 + ∆µ2]1/2

. Aswe now have three structure descriptors, the generalized stabilityregion is a closed volume in the (µ, t , ∆µ) space, as seen in Fig.2A. If we slice this volume through the plane ∆µ= 0, we revertto the perovskite area of Fig. 1D. We emphasize that our presentresults derive exclusively from Goldschmidt’s principle (barringthe chemical limits which are not essential) and make no refer-ence to the definition and values of the ionic radii. The bounds ofthe perovskite regions are easy to evaluate for any structure andare described by six inequalities for µ, t , and ∆µ in SI Appendix,Table S1.

Can the inequalities in SI Appendix, Table S1 be used for struc-ture prediction? To answer this question we analyzed a record of2,291 ternary and quaternary compounds that we collected fromthe ICSD and from refs. 20–26, as described in SI Appendix, sec-tion 2. Datasets S1–S3 include 1,622 perovskites (Dataset S1),592 nonperovskites (Dataset S2), and 77 compounds which cancrystallize either as a perovskite or as another structure (DatasetS3). Fig. 2A shows the distribution of all these compounds inthe (µ, t , ∆µ) space. We see that our perovskite volume (blue)delimits remarkably well the regions occupied by perovskites(blue markers) and nonperovskites (red markers). A detailedview of these data is provided in Fig. 2 B–E, where we show slicesof the perovskite volumes at fixed octahedral mismatch, and in SIAppendix, Fig. S5, where data for perovskites and nonperovskitesare presented separately. In Fig. 2 B–E and SI Appendix, Fig.S5 we see that, as ∆µ increases, the stability region decreasesin size and moves toward higher octahedral factors. Remark-ably, most datapoints from Dataset S1 closely follow this trend.Case-by-case inspection reveals several outliers, i.e., perovskitemarkers falling outside of the perovskite region or vice versa.The existence of outliers is to be expected given the simplic-

ity of Goldschmidt’s model, but intriguingly we find many caseswhere the presence of outliers signals the occurrence of polymor-phism. An important example is BaTiO3: While this compoundis mostly known as a ferroelectric perovskite, it is also stable ina hexagonal structure under the same pressure and temperatureconditions (33).

We now assess the predictive power of the model on quanti-tative grounds. The simplest way to proceed would be to classifycompounds based on whether the corresponding (µ, t , ∆µ) pointfalls inside or outside the stability region. However, this proce-dure is unreliable as it is very sensitive to small variations in theionic radii. A better strategy is to replace each point by a rect-angular cuboid, with dimensions representing the uncertaintyin the ionic radii. The uncertainty calculation is detailed in SIAppendix, section 5. With this choice we define the “formability”as the fraction of the cuboid volume falling within the perovskiteregion, and we classify the compound as a perovskite if this frac-tion exceeds a critical value (SI Appendix, Fig. S6A). To quantifythe accuracy of this classification procedure and the associateduncertainty, we determine the classification of large subsets ofcompounds, randomly selected from Datasets S1 and S2, and werepeat this operation several thousand times. By the central limittheorem, the average success rates tend to a normal distribution(SI Appendix, Fig. S7B); the center of this distribution gives themost probable success rate, and the standard deviation yields thestatistical uncertainty (SI Appendix, Fig. S6C).

Our main result is that, for sample sizes of 100 compounds ormore, the geometric model correctly classifies 79.7 ± 4.0% of allcompounds with a 95% confidence level. This predictive poweris unprecedented among structure prediction algorithms.

For completeness we also assess how our model compareswith previous models. To this end, we calculate how many of the

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known compounds in Datasets S1 and S2 are correctly classifiedwithin the original model of Goldschmidt (which considers onlythe stretching and octahedral limits); within three other empiri-cal models reported in refs. 22, 31, and 34; and within our presentmodel (SI Appendix, Fig. S7). SI Appendix, Fig. S7E shows thatthe stretching and octahedral limits correctly categorize nearlyall perovskites in Dataset S1, but fail to discriminate againstmore than half of nonperovskites in Dataset S2. The empiricalregions in SI Appendix, Fig. S7 B–D clearly demonstrate that bysetting a lower bound to the tolerance factor, the accuracy ofthe model improves significantly for both perovskites and non-perovskites. However, up to now, this bound has been describedvia empirical data fitting. Our model predicts the tilt limit fromfirst principles, while retaining a very good accuracy in distin-guishing perovskites from nonperovskites. By comparison, theperovskite regions reported in refs. 22, 31, and 34 can be under-stood as zeroth- and first-order approximations to our bounds,respectively.

By applying our classification algorithm to all possible3,658,527 quaternary combinations, we generated a library of94,232 hitherto-unknown perovskites and double perovskitesthat are expected to form with a probability of 80% (SI Appendix,Fig. S8). The complete library of predicted perovskites is pro-vided in Datasets S4–S6. Our library of future perovskitesdwarves the set of all perovskites currently known and is com-parable in size to the ICSD database of all known inorganiccrystals, which contains approximately 193,000 structures (www2.fiz-karlsruhe.de).

How many of our predicted perovskites are genuinely un-known compounds, i.e., have never been synthesized? To answerthis question we performed a large-scale web data extractionoperation by querying an internet search engine about each andevery one of the nearly 100,000 compounds in Datasets S4–S6(SI Appendix). This procedure revealed that the overwhelm-ing majority of these compounds have never been reported ormentioned before (Dataset S4) and that fewer than 1% of thestructures were already known, namely 786 of 94,232 compounds(Datasets S5 and S6).

The 786 previously known compounds reported in DatasetsS5 and S6 were not included in our initial Datasets S1–S3 ofknown crystals. We use this additional set of known compoundsto perform a second blind test of our predictions. According toour inferential analysis we expect 626 compounds of DatasetsS5 and S6 (79.7% of 786) to be perovskites. By carrying outa manual literature search we confirmed that 555 crystals areindeed perovskites (Dataset S5). This result is remarkably con-sistent with our prediction. This blind test replaces a validationbased on resource-intensive experimental synthesis of hundredsof new compounds with faster and inexpensive data analytics.The success of the blind test clearly demonstrates that, despiteits simplicity, Goldschmidt’s principle has a considerable pre-dictive power. Naturally, by combining our structure map withexperiments and ab initio calculations on selected subfamilies,the predictive accuracy of the model is bound to improve evenfurther.

What is the topography of our geometric structure map? Fig.3A and SI Appendix, Fig. S9B show that the majority of predictedperovskites tend to cluster toward the region with the lowestoctahedral, tolerance, and mismatch factors. This high densityof compounds stems from the occurrence of a large number oflanthanide oxide and actinide oxide perovskites, which tend tohave similar geometric descriptors due to the lanthanide con-traction (35). We also note that the concentration of compoundsnear the bottom of the map shows that the geometric tilt limitderived in this work (Fig. 1D) is essential to accurately predictthe formability of perovskites.

Fig. 3B shows the relative abundances of predicted perov-skites. The majority of compounds are oxides (68%), followed

Fig. 3. Topography of the perovskite landscape. (A) Density of predictedternary and quaternary perovskites from Datasets S1 and S3–S6. For clar-ity the density of perovskites has been integrated over all possible valuesof the octahedral mismatch ∆µ, to obtain a 2D map in the (µ, t) plane.The two plots show the same quantity as a 2D color map and a 3D surface,respectively. (B) Crystallographic site preference in Datasets S1 and S3–S6.The horizontal bar illustrates the relative abundance of perovskites with agiven anion X. For each anion, the rings illustrate the relative abundancesof the A-site cation (inner ring) and of the B-site cation (outer ring). Cationsare grouped using the standard classification: actinides (AC), alkaline earthmetals (AE), alkali metals (AM), halogens (HA), lanthanides (LA), metals andsemiconductors (MS: Al, Ga, In, Sn, Tl, Pb, Bi, B, Si, Ge, As, Sb, Te, Po),and transition metals (TM). Datasets S4–S6 contain 59 binary, 2,834 ternary,and 90,606 quaternary perovskites.

by halides (16%), chalcogenides (12%), and nitrides (4%). Whyis the perovskite landscape dominated by oxides, and nitrides areso rare instead? To answer this question we observe that the −2oxidation state of O admits as many as 10 inequivalent charge-neutral combinations of the oxidation states of the cations (SI

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Appendix, Table S2). Furthermore, the oxygen anion has a smallradius (1.3 A), which is compatible with most transition metals,lanthanides, and actinides; and these elements form the mostnumerous groups in the Periodic Table. A similar argumentcould be made for chalcogens, which share the same oxidationstate as oxygen. However, the chalcogen radii are too large (1.8–2.2 A) to accommodate most transition metals and actinides, andhence chalcogenide perovskites constitute a much smaller family.Our finding is consistent with recent ab initio calculations (37).Halide perovskites are even less numerous than chalcogenides,mostly owing to their more restrictive −1 oxidation number:In fact, this oxidation state admits only +1 A-site cations and+1/+3 or +2/+2 B-site cations (SI Appendix, Table S2). Nitridesconstitute an interesting exception to these trends. Indeed, whilethe ionic radius of N in the −3 oxidation state (1.5 A) is very simi-lar to that of O and its oxidation number admits as many as seveninequivalent combinations of oxidation states for the cations, inall such combinations at least one B-site cation must have theunusually high +5 oxidation state (SI Appendix, Table S2). Asthe ionic radii tend to decrease with the oxidation number (32),most B-site cations turn out to be too small to be coordinatedby six nitrogen anions in an octahedral environment. As a result,if we exclude radioactive elements, we find fewer than 80 nitrideperovskites across the entire Periodic Table. We note that, owingto the lower electronegativity of nitrogen, the ionic character ofthe chemical bonds in nitride perovskites is reduced, and ourgeometric model is reaching its limits of applicability (31). How-ever, the scarcity of nitride perovskites predicted by our modelis fully consistent with recent ab initio calculations (38) and withexperimental observations (only one ternary nitride perovskitecan be found in ICSD), indicating that the rigid sphere approx-imation can still provide meaningful predictions for nitrides.Among our predicted compounds we also identified many unex-pected binary compounds of the type A2X3. One such exampleis iron oxide, Fe2O3. While this oxide is primarily known in theform of hematite (corundum structure), it was recently foundthat the crystal undergoes a phase transition to a perovskite athigh pressure and temperature (39). The stabilization of Fe2O3

as a perovskite under high pressure is in agreement with our abinitio calculations (discussion in SI Appendix, section 8 and SIAppendix, Fig. S10) and can be associated with the well-knownphase transition of ilmenites (ternary ordered corundum) intoperovskites, observed, for example, for FeTiO3 (34). This find-ing suggests that several other binary compounds may hide aperovskite phase in their phase diagram, an intriguing possibilitythat is open to investigation.

To demonstrate the applicability of our model, we take theexample of ternary oxide perovskites. Fig. 4 shows a compari-son between the combinatorial screening of ternary oxides per-formed using our model, ab initio calculations reported in ref. 36,and experimental data collected from Datasets S1, S3, and S5.The compositions classified as perovskites by our model include92% of the experimentally observed oxide perovskites. In par-ticular, when A is a lanthanide and B is a first-row transitionmetal, our model predicts that most compositions can form asa perovskite, in excellent agreement with DFT predictions andexperiment. This can be explained by the similar ionic sizes oftransition metals and rare earths, respectively. The same predic-tion is made for the case when both A and B are rare earths;however, fewer perovskites are found from DFT and experiment.The reason for this discrepancy is that the nonrattling princi-ple effectively probes the dynamical stability of a given chemicalcomposition in the perovskite structure. However, it does notcontain information on its stability against decomposition (ther-modynamic stability). Of the two criteria, the thermodynamicstability requirement is more stringent, and this explains whygenerally geometric blueprints tend to predict more perovskites

Fig. 4. Combinatorial screening of ternary oxide perovskites. ABO3 ternarycompounds classified as perovskites by our geometric model (blue squares),ab initio calculations within the generalized gradient approximation to den-sity functional theory (DFT/GGA) (red circles) reported by Emery et al. (36),and experimental data from Datasets S1, S3, and S5 (black crosses) areshown. The gray shading highlights the region of the A/B map that is nottaken into account in the ab initio screening. The curly bracket marks theempty valence shells corresponding to an interval of atomic numbers for Aand B; e.g., the 3d interval corresponds to Z = 21–30 (Sn-Zn). Emery et al.(36) reported the study of the thermodynamic stability of 5,329 candidateternary compounds and found that 382 compositions are stable perovskites.There are 383 ternary oxides in Datasets S1, S3, and S5, and DFT/GGA cal-culations predict 225 (59%) of them to be stable while our model correctlyclassifies 354 of them (92%).

than have actually been made or that are predicted from abinitio calculations. Therefore, further theoretical and experimen-tal studies are required to ascertain whether these proposedcompositions are also thermodynamically stable. Despite thislimitation, Fig. 4 demonstrates that the Goldschmidt principlecan be used as an efficient and reliable prescreening tool for thehigh-throughput combinatorial design of perovskites. In fact, inFig. 4 we show that our model can reduce the number of cal-culations by more than 70% in the combinatorial screening ofternary oxides. Importantly, the Golschmidt no-rattling principlebecomes increasingly useful in the context of screening all possi-ble perovskites beyond oxides, reducing the total number of 3.6million possible compositions by 97%, to fewer than 100,000 can-didates. Therefore, by leveraging the complementary strengths ofGoldschmidt’s empirical no-rattling principle and ab initio com-putational modeling it will be possible to explore the completechemical landscape of all possible perovskites.

ConclusionWe charted the complete landscape of all existing and futureperovskites. By combining inferential statistics with large-scaleweb data extraction, we validated Goldschmidt’s no-rattling prin-ciple on quantitative grounds and developed a structure map topredict the stability of perovskites with a fidelity of 80%. Ourmodel completes the general theory that Goldschmidt proposedalmost a century ago and formalizes the nonrattling hypothesisinto a mathematically rigorous set of criteria that can be usedin the design and discovery of perovskites. As an outcome ofour study, we were able to generate a library of almost 100,000

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hitherto-unknown perovskites awaiting discovery (Dataset S4).By releasing this library in full, we hope that this work will stim-ulate much future experimental and computational research onthese fascinating crystals. More generally, our findings suggestthat geometric blueprints could serve as a powerful tool to helptackle the exponential complexity of combinatorial materialsdesign.

Materials and MethodsA full description of the methods, data provenance, and statistical analysisused in this paper can be found in SI Appendix.

ACKNOWLEDGMENTS. This work was supported by the Leverhulme Trust(Grant RL-2012-001), the Graphene Flagship (Horizon 2020 Grant 696656-GrapheneCorel), and the UK Engineering and Physical Sciences ResearchCouncil (Grant EP/M020517/1).

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5402 | www.pnas.org/cgi/doi/10.1073/pnas.1719179115 Filip and Giustino

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