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Shry: Application of Canonical Augmentation to the Atomic Substitution Problem

Cite this: J. Chem. Inf. Model. 2022, 62, 12, 2909–2915
Publication Date (Web):June 9, 2022
https://doi.org/10.1021/acs.jcim.2c00389

Copyright © 2022 The Authors. Published by American Chemical Society. This publication is licensed under

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Abstract

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A common approach for studying a solid solution or disordered system within a periodic ab initio framework is to create a supercell in which certain amounts of target elements are substituted with other elements. The key to generating supercells is determining how to eliminate symmetry-equivalent structures from many substitution patterns. Although the total number of substitutions is on the order of trillions, only symmetry-inequivalent atomic substitution patterns need to be identified, and their number is far smaller than the total. Our developed Python software package, which is called Shry (Suite for High-throughput generation of models with atomic substitutions implemented by Python), allows the selection of only symmetry-inequivalent structures from the vast number of candidates based on the canonical augmentation algorithm. Shry is implemented in Python 3 and uses the CIF format as the standard for both reading and writing the reference and generated sets of substituted structures. Shry can be integrated into another Python program as a module or can be used as a stand-alone program. The implementation was verified through a comparison with other codes with the same functionality, based on the total numbers of symmetry-inequivalent structures, and also on the equivalencies of the output structures themselves. The provided crystal structure data used for the verification are expected to be useful for benchmarking other codes and also developing new algorithms in the future.

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Introduction

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Atomic substitution in solids is the most common strategy in materials science to tune material properties for applications such as alloying and cation/anion/vacancy doping, which often demonstrate unprecedented functionalities superior to nonsubstituted materials. (1) Within the scientific and technological communities, there is a considerable demand for the ability to predict the extent to which substitutions affect material properties using ab initio simulations, as well as the ability to evaluate whether substitutions are useful for achieving the desired properties. For instance, if we consider vacancy-type defects as a form of substitution (by a vacancy), this demand covers another industrially important problem: understanding how defects affect material properties. (2,3) This problem arises during the evaluation of sample qualities, damages, and degradation. Ab initio calculations based on density functional theory (DFT) are the most promising framework for such an assessment; they utilize the microscopic structure models of substitutions.
There are many strategies for studying disordered crystals or solid solutions within the DFT framework. If one considers disorder effects at the DFT level with the preservation of periodicity of a target crystal, virtual crystal approximation (VCA) (4) or coherent potential approximation (CPA) (5) combined with the Korringa–Kohn–Rostoker (KKR) method (6,7) are good choices. The former simulates disordered crystals or solid solutions by mixing (all-electron or pseudo) potentials of the constituent elements based on the composition, whereas the latter utilizes an effective potential constructed from the Green’s function of the effective medium. However, because both methods require special implementations, not all ab initio codes can perform them. The so-called supercell method is a considerably simpler yet very powerful strategy. In this method, substitutions are performed within the supercell of the original (i.e., unsubstituted) unit cell. While the supercell approach is simple and applicable for any ab initio framework, an intrinsic problem is that the approach introduces an artificial periodic boundary condition that does not exist in real materials. A direct approach to alleviate this artificial periodicity is to increase the size of the supercell. Although a larger supercell inevitably increases computational cost, recent advances in high-performance computing allow us to handle it even at the phonon analysis level. (8,9) Further, a larger supercell affords a finer resolution of substitution concentration compared to an unsubstituted unit cell because only discrete substitutions are allowed in the supercell approach. (10) In addition, more sophisticated methods to alleviate the artificial periodicity have been devised. One such successful method is the method of quasi-random structures (SQS), (11,12) which finds the closest periodic supercell to a genuine disordered state based on correlation functions. If the thermodynamical properties are in focus, the cluster expansion method with Monte Carlo sampling (13) is another very effective technique for studying disordered crystals. (14) Several schemes have been developed (15−18) to correct spurious interactions between supercell images for studying charged defects in crystals more quantitatively. These methods ensure that the supercell approach is effective and reliable. In this paper, we focus on the supercell approach.
Within the supercell approach, a solid solution or disordered system is modeled with a supercell in which certain amounts of target elements are substituted with other elements. The number of possible atomic substitution patterns grows combinatorically with the substitution concentration and size of the supercell; this readily causes technical issues in the ab initio framework. A simple but powerful solution is to randomly select several configurations from the vast number of the possible combinations. (19,20) With an efficient implementation, this random sampling method is not limited by the number of configurations. (20) Further, the method is easy to implement. In this paper, we focus on an exhaustive approach of exploring all the symmetry-inequivalent structures. Indeed, many configurations in the redundant structure set are related to crystallographic symmetry operations; however, ab initio simulations require only symmetry-inequivalent structures. This implies that these redundant configurations containing many symmetry-equivalent configurations can be further categorized into those composed of only symmetry-inequivalent configurations. If the number of symmetry-inequivalent structures is too large, one can also select several feasible configurations after enumerating the symmetry-inequivalent structures according to a criterion such as the Ewald energy.
Several software packages have been developed to address such vast combinations of atomic substitution in a solid; examples include Supercell, (21)Materials Studio (i.e., Disorder tool), (22)enumlib (23) (also combined with Pymatgen (Python Materials Genomics) (24)), Crystal, (25,26) and Sod. (27) Recently, we developed a Python package called Shry (a Suite for High-throughput generation of models with atomic substitutions implemented in Python). Shry allows one to select only symmetry-inequivalent structures from a set of redundant structures based on the canonical augmentation algorithm. (28)Shry is implemented in Python 3 and uses the CIF format as the standard for both reading and writing the reference and generated sets of substituted structures. Shry can be integrated into another Python program as a module or can be used as a stand-alone program. In the following, we describe the package specification, underlying method, implementation, and several benchmark and validation tests.

Summary of Technical Details

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Shry is implemented in Python 3 and uses the CIF format as the standard for both reading and writing the reference and generated sets of substituted structures. Structure abstractions, manipulations, and symmetry analysis rely on Pymatgen (24) and spglib. (29) The key point for the implementation is identifying and avoiding, among all possible permutations, multiple instances of symmetrically identical structures. The implementation of a variant of the canonical augmentation (30) algorithm is unique to Shry, which allows symmetrically unique structures corresponding to the specified substitute concentration to be recursively generated from the unique structures. As a command line tool, a CIF format file containing site occupation information can be used as an input structure. Shry writes all the identified symmetry-inequivalent structures as separate CIF files. A user can also specify the maximum number of outputted CIF files if the total number of structures is too large to write all of them. The total number of symmetry-inequivalent structures can be estimated a priori by Polya’s enumeration theorem, (31,32) which provides the expected number of irreducible structures. To enable a user to include Shry as a module in their own Python scripts, we provide several examples of Python scripts in the distribution, which also include interfaces for Shry with Pymatgen.
Since our formulation described in the Supporting Information is quite formal (but entirely self-contained), we here present several hints on the properties of canonical augmentation. The objective of Shry is to eliminate symmetry-equivalent structures from the possible atomic substitution patterns. A straightforward solution, classifying symmetry-inequivalent structures after determining all possible configurations, is redundant and practically unfeasible because the number of total combinations increases exponentially (i.e., combinatorial explosion). The key observation to alleviate the combinatorial explosion problem is that the atomic substitution problem can be mapped into a vertex coloring problem (Figures 1 and 2). Here, we consider a very simple case: the problem of coloring the eight vertices of a cube with two colors, as shown in Figure 1. We color white vertices with red in a stepwise manner until three of them become red. This is essentially equivalent to the substitution in which three out of eight atoms on a Wyckoff site are substituted with another element (e.g., see Figure 2; Ce8Pd24Sb → (Ce5,La3)Pd24Sb, where Ce and La atoms are on the 8g Wyckoff position and the space group is 221-Pm3m). Then, the question of how many symmetry-inequivalent structures exist in all the possible substitution patterns essentially becomes equivalent to the question of how many symmetrically unique coloring patterns exist in a given graph.

Figure 1

Figure 1. Example of the search tree, where three vertices out of eight are colored in red in a cube. p(X) denotes the parent node of X, whereas C(X) denotes the set of children of X.

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Figure 2

Figure 2. Atomic substitutions corresponding to Figure 1. Ce8Pd24Sb → (Ce5,La3)Pd24Sb, where Ce and La atoms are on the 8g Wyckoff site. The crystal structure (34) was obtained from the ICSD database (35,36) (CollCode: 83378). The space group is 221-Pm3m. The crystal structures are depicted using VESTA. (37)

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A search tree to find the unique coloring patterns can be constructed as shown in Figure 1, where the search tree is truncated at the depth corresponding to the number of colored vertices. The tree is traversed by a proper algorithm such as the depth-first traversal procedure. The mapping itself does not solve the combinational explosion problem, because one visits all the nodes in the worst-case scenario. If one can truncate large parts of the search tree at the upper level without visiting lower nodes, the computational cost can be drastically decreased. For instance, in Figure 1, since p(X) and p(Y) are symmetrically equivalent, all the children of p(Y) such as Y and W can be disregarded without visiting these nodes, if all the children associated with p(X) such as X and Z have been already visited. This so-called isomorphic rejection (28) procedure is one of the key points in the present study. Such truncations should be performed using only local information. Indeed, if one needs to store all the information on the visited nodes for the truncation (i.e., nonlocal information is needed), the combinatorial explosion problem still remains. The orderly generation and canonical augmentation exploit the so-called canonicity for a node representation to truncate a search tree using only local information (see the Supporting Information for the mathematical definition). They are very powerful algorithms for the classification of combinatorial structures, the main idea of which is to construct only nonequivalent objects (namely, symmetry-inequivalent structures in the atomic substitution problem) and thereby classify such objects. The orderly generation and canonical augmentation were first introduced by Read (33) and McKay, (30) respectively. (28) They are similar algorithms in the sense that both exploit the canonicity. However, they also show some significant differences. The orderly generation requires that all nodes should themselves be canonical. (28) In contrast, the canonical augmentation does not require the canonicity of nodes themselves; rather, it requires a search tree to be generated in a canonical manner. (28) This feature of the canonical augmentation allows us to occasionally avoid the computation of time-consuming canonicity tests in the tree search. In the Supporting Information, we outline the mathematical details of the algorithms implemented in Shry.

Comparison with Existing Software Packages

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Since the atomic substitution problem is very general and important, several software packages have already been developed. To our knowledge, the site-occupancy disorder (Sod) package is the pioneering work in this field. (27) The software package has been applied for many disordered systems since 2007. Another open-source package with similar functionalities is the enumlib package. (23) The code enables us to generate derivative superstructures of a parent lattice in a considerably efficient manner, and it has been integrated into Pymatgen. (24)Supercell (21) is the most recent and most sophisticated software package in the field. The package implements considerably efficient algorithms for the atomic substitution problem. The atomic substitution problem can be solved using several commercial software packages. The Disorder tool implemented in Materials Studio (22) can generate solid solutions with a very user-friendly graphical user interface (GUI). Similar functionalities are implemented in a well-known DFT code, Crystal program, (26,38) to treat disorder systems within the periodic ab initio framework. Within the existing solutions, Okhotonikov et al. (21) reported that only the Supercell package could handle cases with a large number of permutations because the other programs crash on such computationally demanding test cases. Therefore, for the benchmark and validation tests, the relevant comparison is mostly between Shry ver. 1.1.0 and Supercell ver. 2.0.2.

Benchmark Test

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AxPb1–xTe is a typical benchmark system used for testing the performance of an atomic substitution program. Table 1 summarizes the results for this benchmark test. Since Shry is implemented in Python (i.e., an interpreted language), it is intrinsically slower than the other tools implemented in compiled programming languages such as C++. Note that the actual computational time of Supercell is much smaller than its CPU time because Supercell is multithreaded very well. In contrast, Shry is not multithreaded at present.
Table 1. Total Number of Possible Atom Combinations for Different Supercell Sizes of the A0.5Pb0.5Te Systema
Total number of atoms in simulation cell816243264
Supercell size a × b × c1 × 1 × 11 × 1 × 21 × 1 × 31 × 2 × 22 × 2 × 2
Symmetry operations1921281922561536
Total combinations (symmetry-inequivalent structures)6(1)70(8)924 (34)12,870 (153)601,080,390 (404,582)
Performance of Shry (s)2.85(1.21)1.70(1.17)1.78(1.27)1.85(1.39)161.91(160.99)
Performance of Supercell (s)0.03(0.01)0.03(0.02)0.04(0.02)0.06(0.04)8.40(200.72)
a

The computational times of Shry and Supercell were measured on an Intel Xeon G-6242 (2.80 GHz, 16 cores, 32 threads) processor using the time command, where the time-consuming I/O operations were disabled. Both real and user (in parentheses) times are shown in the Shry and Supercell rows. The distributed binary Supercell program was used for the test.

We benchmarked Shry and Supercell for a variety of systems. For the benchmark data set, we randomly chose several CIF files from the Crystallography Open Database (COD) (39) for each space group, as listed in Table S2 (Supporting Information), and we generated CIF files containing up to three atomic substitution sites. The supercell size and atomic substitution sites were randomly and repeatedly selected for each compound until the total number of symmetry-inequivalent structures becomes less than 109. The number of compounds used for the benchmark test is 500. Table S2 lists the compounds, unique IDs in the database, and other relevant information used for the benchmark test. Further, we report the sizes of supercells, total numbers of substituted structures, and numbers of symmetry-inequivalent structures. The computational times required to find all symmetry-inequivalent structures are summarized in the rightmost columns of Table S2. We confirmed that both the number of substitution patterns including symmetry-equivalent structures and the number of symmetry-inequivalent structures are consistent between Shry and Supercell for all compounds listed in Table S2 with the given substitution patterns. We show the computational times of Shry on the benchmark set in Figure 3. The computational time is almost constant up to N ∼ 104 in the small N region, where N refers to the number of symmetrically distinct configurations. This is because preconditioning steps, such as finding the symmetry operations of an unsubstituted structure, are the rate-determining steps of the algorithms (e.g., the actual computational times are less than 3 s for N ≤ 104). Instead, the computational time in the large N region increases as N increases; however, it scales linearly up to N ∼ 109. Since Shry is purely implemented in Python and not yet parallelized, the actual computational time is longer than that of other packages realizing the same functionality, such as Supercell shown in Figure S3. There are several possible strategies for speeding up the computation in Shry, such as parallelizing the code or rewriting the core parts with Cython or C++, which would be promising future works.

Figure 3

Figure 3. Computational times of Shry on a benchmark data set. On the horizontal axis, we place the number of symmetrically distinct structures, whereas the vertical axis indicates the absolute computational time.

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Validation Test

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We compared not only computational times and the numbers of structures but also the redundancies and equivalencies of the generated structures among the software packages. We performed both (a) intra-software and (b) inter-software tests. (a) The intra-software code test checked whether each program generates zero redundant structures. Thus, for each program, we investigated whether each generated structure is symmetry inequivalent to all the others (Figure 4(a)). Brute-force comparisons were performed for this test (i.e., all the possible combinations were investigated). (b) The inter-software code test checked whether the symmetry-inequivalent structure sets generated by Shry and Supercell (or Disorder tool) are identical. Thus, we built pairs of Shry and Supercell (or Disorder tool) structures and confirmed that there is no excess or deficiency (Figure 4(b)).

Figure 4

Figure 4. Schematics of (a) intra-software and (b) inter-software tests. The intra-software test checked whether each program provides zero redundant structures, where brute-force comparisons were performed (i.e., all the possible combinations were considered). The inter-software test checked whether the symmetry-inequivalent structure sets generated by Shry and Supercell (Disorder tool) are identical; i.e., there is no excess or deficiency among them. The crystal structures are depicted using VESTA. (37)

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For the validation test, we used a total of 600 compounds, which were taken from COD. (39)Table S3 lists the compounds, unique IDs in the database, and other relevant information used for the validation test. No discrepancy was found among the three codes (Shry, Supercell, and Disorder tool). Both the intra-software and inter-software tests were successful for all the 600 compounds with Shry and Supercell (Disorder tool was tested only for some cases; see the Supporting Information). All the used CIF files are uploaded in our git repository.

Conclusion

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We developed a Python software package, Shry, that enables the selection of only symmetry-inequivalent structures from a vast number of candidates. This is very useful when employing the supercell approach to study a solid solution or a disordered system in an ab initio calculation. Although we showed several examples in this paper where some elements are substituted by other elements, the application of Shry is not limited to them. For instance, Shry can also be applied for studying vacancies and charge disproportionation in crystals within the supercell approach as far as symmetry-inequivalent patterns are concerned. Notice that the magnetic structures (e.g., configurations of up and down spins) cannot be studied with Shry at present because the current implementation does not support Shubnikov groups. There are several perspectives for the future development of Shry. One major concern is the long absolute computational time. Since Shry is purely implemented in Python and is not multithreaded at present, the absolute computational time is not compatible with other codes for large N. A possible solution is parallelizing the code, which should be very efficient because canonical augmentation can be performed independently. Another possibility is rewriting the core parts of Shry with much a faster language, such as C++, like other scientific packages. From the algorithmic point of view, a great advantage of the canonical augmentation algorithm used in Shry is the use of the subobject invariant (Supporting Information), which enables us to avoid computing the time-consuming canonicity test. The present implementation, i.e., the sum of the distance matrix, is sometimes not capable of distinguishing objects belonging to different orbits on a subobject. Finding a more distinguishable subject invariant is an interesting direction for future work that must be pursued to fully exploit the advantage of the canonical augmentation algorithm, realizing a more efficient exhaustive search.

Data and Software Availability

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The Python software package, Shry, is distributed on our GitHub repository [https://github.com/giprayogo/SHRY] under the MIT license. The package is archived in the Zenodo repository [10.5281/zenodo.5652360]. All the CIF files used for the benchmark and validation tests are also available from our GitHub repository [https://github.com/giprayogo/SHRY], which is also archived in the Zenodo repository [10.5281/zenodo.5652360], and those of unsubstituted structures are available from the Crystallography Open Database (COD) (39) [https://www.crystallography.net/cod/].

Supporting Information

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The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jcim.2c00389.

  • Mathematical details of the algorithms implemented in Shry. Details of crystal structures (CIF files) used in the benchmark and validations tests. Tables S2 and S3: Unique IDs of compounds in database, compositions, and substituted Wyckoff position(s). Measured computational times plotted in Figure 3 summarized in rightmost columns of Table S2. (PDF)

Terms & Conditions

Most electronic Supporting Information files are available without a subscription to ACS Web Editions. Such files may be downloaded by article for research use (if there is a public use license linked to the relevant article, that license may permit other uses). Permission may be obtained from ACS for other uses through requests via the RightsLink permission system: http://pubs.acs.org/page/copyright/permissions.html.

Author Information

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  • Corresponding Authors
  • Authors
  • Author Contributions

    R.M. and K.H. initiated the idea, and K.U. implemented the preliminary code of Shry. G.I.P. and K.N. were involved in the implementation of the production code of Shry. A.T. contributed to the theoretical foundations of the work. K.N. and A.T. prepared the initial draft of the manuscript, and then, all authors critically revised it, provided comments, and approved the final version.

  • Notes
    The authors declare no competing financial interest.

Acknowledgments

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The authors acknowledge the facilities of the Research Center for Advanced Computing Infrastructure at JAIST. G.I.P. gratefully acknowledges financial support from JST SPRING (Grant Number JPMJSP2102). A.T. acknowledges financial support from MIUR Progetti di Ricerca di Rilevante Interesse Nazionale (PRIN) Bando 2017 (Grant Number 2017BZPKSZ). K.H. is grateful for financial support from the HPCI System Research Project (Project IDs: hp210019, hp210131, and jh210045), MEXT-KAKENHI (JP16H06439, JP17K17762, JP19K05029, JP19H05169, JP21K03400, JP21H01998, and JP22H02170), and the U.S. Air Force Office of Scientific Research (Award Number FA2386-20-1-4036). R.M. is grateful for financial support from MEXT-KAKENHI (JP19H04692 and JP21K03400), the U.S. Air Force Office of Scientific Research (AFOSR-AOARD/FA2386-17-1-4049; FA2386-19-1-4015), and JSPS Bilateral Joint Projects (JPJSBP120197714). K.N. acknowledges support from JSPS Overseas Research Fellowships, a Grant-in-Aid for Early-Career Scientists (Grant Number JP21K17752), and a Grant-in-Aid for Scientific Research (C) (Grant Number JP21K03400). The authors thank Dr. Kirill Okhotnikov for help with the benchmark test using Supercell.

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    A model of a substitutional alloy is introduced based on the concept of an effective or coherent potential which, when placed on every site of the alloy lattice, will simulate the electronic properties of the actual alloy. The coherent potential is necessarily a complex, energydependent quantity. The model is evaluated for the simple case of a 1-dimensional alloy of δ-function potentials. To provide a basis for comparison, as well as to see if a simpler scheme will suffice. The spectrum of the same alloy was calcd. by using the av. t-matrix approxn. introduced by Beeby. Thus, the av. t-matrix approxn. is not adequate for the description of an actual transition-metal alloy, while the coherent-potential picture will provide a more reasonable facsimile of the d. of states in such an alloy.
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    Korringa, J. On the calculation of the energy of a Bloch wave in a metal. Physica 1947, 13, 392– 400,  DOI: 10.1016/0031-8914(47)90013-X
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    Kohn, W.; Rostoker, N. Solution of the Schrödinger equation in periodic lattices with an application to metallic lithium. Phys. Rev. 1954, 94, 1111,  DOI: 10.1103/PhysRev.94.1111
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    Solution of the Schroedinger equation in periodic lattices with an application to metallic lithium
    Kohn, W.; Rostoker, N.
    Physical Review (1954), 94 (), 1111-20CODEN: PHRVAO; ISSN:0031-899X.
    cf. ibid. 91, 234(1953); 92, 847; C.A. 45, 1403g; 46, 9374i. The problem of solving the Schroedinger equation in a periodic lattice is studied from the point of view of the variation-iteration method. This approach leads to a very compact scheme if the potential V(r) is spherically sym. within the inscribed spheres of the at. polyhedra and const. in the space between them. The band structure of the lattice is then detd. by (1) geometrical structure consts., characteristic of the type of lattice and (2) the logarithmic derivs., at the surface of the inscribed sphere, of the s, p, d, .... functions corresponding to V(r). By far the greater part of the labor is involved in the calcn. of (1), which needs to be done only once for each type of lattice; (2) can be obtained by numerical integration or directly from the at. spectra. Although derived from a different point of view, this scheme is essentially equiv. to one proposed by Korringa (Physica 13, 392(1947)) on the basis of the theory of lattice interferences. An application is made to the conduction band of metallic Li.
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    Nakano, K.; Hongo, K.; Maezono, R. Phonon dispersions and Fermi surfaces nesting explaining the variety of charge ordering in titanium-oxypnictides superconductors. Sci. Rep. 2016, 6, 29661,  DOI: 10.1038/srep29661
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    Phonon dispersions and Fermi surfaces nesting explaining the variety of charge ordering in titanium-oxypnictides superconductors
    Nakano, Kousuke; Hongo, Kenta; Maezono, Ryo
    Scientific Reports (2016), 6 (), 29661CODEN: SRCEC3; ISSN:2045-2322. (Nature Publishing Group)
    There has been a puzzle between expts. and theor. predictions on the charge ordering of layered titanium-oxypnictides superconductors. Unconventional mechanisms to explain this discrepancy have been argued so far, even affecting the understanding of supercond. on the compd. We provide a new theor. prediction, by which the discrepancy itself is resolved without any complicated unconventional explanation. Phonon dispersions and changes of nesting vectors in Fermi surfaces are clarified to lead to the variety of superlattice structures even for the common crystal structures when without CDW, including orthorhombic 2 × 2 × 1 one for BaTi2As2O, which has not yet been explained successfully so far, being different from tetragonal √2 × √2 × 1 for BaTi2Sb2O and BaTi2Bi2O. The electronic structure anal. can naturally explain exptl. observations about CDW including most latest ones without any cramped unconventional mechanisms.
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    Nakano, K.; Hongo, K.; Maezono, R. Investigation into Structural Phase Transitions in Layered Titanium-Oxypnictides by a Computational Phonon Analysis. Inorg. Chem. 2017, 56, 13732– 13740,  DOI: 10.1021/acs.inorgchem.7b01709
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    9
    Investigation into Structural Phase Transitions in Layered Titanium-Oxypnictides by a Computational Phonon Analysis
    Nakano, Kousuke; Hongo, Kenta; Maezono, Ryo
    Inorganic Chemistry (2017), 56 (22), 13732-13740CODEN: INOCAJ; ISSN:0020-1669. (American Chemical Society)
    We applied ab initio phonon anal. to layered titanium-oxypnictides, Na2Ti2Pn2O (Pn = As and Sb), and found a clear contrast between the cases with lighter/heavier pnictogen in comparison with expts. Na2Ti2As2O Na2Ti2Sb2O. The result completely explains the exptl. structure at low temp., C2/m for Pn = As, within the conventional charge d. wave, while there arise discrepancies when the pnictogen gets heavier. Our phonon calcn. using the GGA-PBE functional predicts that a Cmce polymorph is more stable than the exptl. obsd. one (Cmcm) for Pn = Sb. On the basis of further quant. anal., we suggest the possibility that the GGA-PBE functional does not properly reproduce the electron correlation effects for Pn = Sb, and this could be the reason for the present discrepancy.
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  10. 10
    Utimula, K.; Hunkao, R.; Yano, M.; Kimoto, H.; Hongo, K.; Kawaguchi, S.; Suwanna, S.; Maezono, R. Machine-Learning Clustering Technique Applied to Powder X-Ray Diffraction Patterns to Distinguish Compositions of ThMn12-Type Alloys. Adv. Theory Simul. 2020, 3, 2000039,  DOI: 10.1002/adts.202000039
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    Machine-Learning Clustering Technique Applied to Powder X-Ray Diffraction Patterns to Distinguish Compositions of thorium manganese-Type Alloys
    Utimula, Keishu; Hunkao, Rutchapon; Yano, Masao; Kimoto, Hiroyuki; Hongo, Kenta; Kawaguchi, Shogo; Suwanna, Sujin; Maezono, Ryo
    Advanced Theory and Simulations (2020), 3 (7), 2000039CODEN: ATSDCW; ISSN:2513-0390. (Wiley-VCH Verlag GmbH & Co. KGaA)
    A clustering technique is applied using dynamic-time-wrapping (DTW) anal. to X-ray diffraction (XRD) spectrum patterns in order to identify the microscopic structures of substituents introduced into the main phase of magnetic alloys. The clustering technique is found to perform well, identifying the concns. of the substituents with success rates of ≈;90%. This level of performance is attributed to the capability of DTW processing to filter out irrelevant information such as the peak intensities (due to the uncontrollability of diffraction conditions in polycryst. samples) and the uniform shift of peak positions (due to the thermal expansion of lattices). The established framework is not limited to the system treated in this work, but is widely applicable to systems the properties of which are to be tuned by at. substitutions within a phase. The framework has a broader potential to predict properties such as magnetic moments, optical spectra, etc. from obsd. XRD patterns, by predicting such properties evaluated from predicted microscopic local structure.
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    Wei, S.-H.; Ferreira, L. G.; Bernard, J. E.; Zunger, A. Electronic properties of random alloys: Special quasirandom structures. Phys. Rev. B 1990, 42, 9622– 9649,  DOI: 10.1103/PhysRevB.42.9622
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    Electronic properties of random alloys: special quasirandom structures
    Wei, S. H.; Ferreira, L. G.; Bernard, James E.; Zunger, Alex
    Physical Review B: Condensed Matter and Materials Physics (1990), 42 (15), 9622-49CODEN: PRBMDO; ISSN:0163-1829.
    Structural models needed in calcns. of properties of substitutionally random A1-xBx alloys are usually constructed by randomly occupying each of the N sites of a periodic cell by A or B. It is possible to design "special quasirandom structures" (SQS's) that mimic for small N (even N = 8) the first few, phys. most relevant radial correlation functions of an infinite, perfectly random structure far better than the std. technique does. These SQS's are shown to be short-period superlattices of 4-16 atoms/cell whose layers are stacked in rather nonstandard orientations (e.g., [113], [331], and [115]). Since these SQS's mimic well the local at. structure of the random alloy, their electronic properties, calculable via first-principles techniques, provide a representation of the electronic structure of the alloy. Authors demonstrate the usefulness of these SQS's by applying them to semiconductors alloys. They calc. their electronic structure, total energy, and equil. geometry, and compare the results to exptl. data.
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    Zunger, A.; Wei, S.-H.; Ferreira, L. G.; Bernard, J. E. Special quasirandom structures. Phys. Rev. Lett. 1990, 65, 353– 356,  DOI: 10.1103/PhysRevLett.65.353
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    Special quasirandom structures
    Zunger, Alex; Wei, S. H.; Ferreira, L. G.; Bernard, James E.
    Physical Review Letters (1990), 65 (3), 353-6CODEN: PRLTAO; ISSN:0031-9007.
    Structural models used in calcns. of properties of substitutionally random A1-xBx alloys are usually constructed by randomly occupying each of the N sites of a periodic cell by A or B. It is possible to design "special quasirandom structures" (SQS's) that mimic for small N (even N = 8) the first few, phys. most relevant radial correlation functions of a perfectly random structure far better than the std. technique does. The usefulness is demonstrated of these SQS's by calcg. optical and thermodn. properties of a no. of semiconductor alloys in the local-d. formalism.
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    Seko, A.; Koyama, Y.; Tanaka, I. Cluster expansion method for multicomponent systems based on optimal selection of structures for density-functional theory calculations. Phys. Rev. B 2009, 80, 165122,  DOI: 10.1103/PhysRevB.80.165122
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    Cluster expansion method for multicomponent systems based on optimal selection of structures for density-functional theory calculations
    Seko, Atsuto; Koyama, Yukinori; Tanaka, Isao
    Physical Review B: Condensed Matter and Materials Physics (2009), 80 (16), 165122/1-165122/7CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
    The cluster expansion (CE) method has been used to evaluate configurational properties in multicomponent systems based on the d.-functional theory (DFT) calcns. Appropriate selections of not only clusters but also structures for DFT calcns. (DFT structures) are crucial for the accuracy and the efficiency of the CE. In a conventional procedure to construct the CE, the CE error is reduced mainly through an appropriate selection of clusters. In the present paper, we propose an improved procedure that systematically leads to optimal selections of both clusters and DFT structures. DFT structures are chosen to cover as much of the configurational space as possible. During the iterative process, the predictive power of the out-of-sample structures can be increased up to the accuracy that is required to describe alloy thermodn. We apply the procedure to configurational behaviors in a simple MgO-ZnO pseudobinary system and in a complex MgAl2O4 system. The CE error is reduced in both systems, in particular, in the complex system, thereby significantly improving configurational properties at high temps. compared with the conventional CE procedure.
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    Yoshio, S.; Hongo, K.; Nakano, K.; Maezono, R. High-Throughput Evaluation of Discharge Profiles of Nickel Substitution in LiNiO2 by Ab Initio Calculations. J. Phys. Chem. C 2021, 125, 14517– 14524,  DOI: 10.1021/acs.jpcc.0c11589
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    14
    High-Throughput Evaluation of Discharge Profiles of Nickel Substitution in LiNiO2 by Ab Initio Calculations
    Yoshio, Satoshi; Hongo, Kenta; Nakano, Kousuke; Maezono, Ryo
    Journal of Physical Chemistry C (2021), 125 (27), 14517-14524CODEN: JPCCCK; ISSN:1932-7447. (American Chemical Society)
    Substitution of a Ni cation in a LiNiO2 cathode is a common strategy for improving the lifetime and thermal stability of lithium-ion batteries. However, this strategy does not improve the discharging capacity more than that of pristine LiNiO2. We investigated whether the capacity of a cation-substituted battery can be improved by an exhaustive search based on ab initio calcns. To ensure a feasible search at a practical computational cost, many data points obtained by expensive ab initio calcns. were bridged by interpolation using the cluster expansion method. Then, the candidates screened by the search were analyzed by more reliable calcns. based on d. functional theory with the strongly constrained and appropriately normed (SCAN) exchange-correlation functional, which dets. the discharging voltage profiles. The screening predicted that partial substitution of Ni with Pt and Pd can improve the discharging capacity of a lithium-ion battery.
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    Freysoldt, C.; Neugebauer, J.; Van de Walle, C. G. Fully Ab Initio Finite-Size Corrections for Charged-Defect Supercell Calculations. Phys. Rev. Lett. 2009, 102, 016402,  DOI: 10.1103/PhysRevLett.102.016402
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    15
    Fully Ab Initio Finite-Size Corrections for Charged-Defect Supercell Calculations
    Freysoldt, Christoph; Neugebauer, Jorg; Van de Walle, Chris G.
    Physical Review Letters (2009), 102 (1), 016402/1-016402/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
    In ab initio theory, defects are routinely modeled by supercells with periodic boundary conditions. Unfortunately, the supercell approxn. introduces artificial interactions between charged defects. Despite numerous attempts, a general scheme to correct for these is not yet available. We propose a new and computationally efficient method that overcomes limitations of previous schemes and is based on a rigorous anal. of electrostatics in dielec. media. Its reliability and rapid convergence with respect to cell size is demonstrated for charged vacancies in diamond and GaAs.
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    Rurali, R.; Cartoixà, X. Theory of defects in one-dimensional systems: application to Al-catalyzed Si nanowires. Nano Lett. 2009, 9, 975– 979,  DOI: 10.1021/nl802847p
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    16
    Theory of defects in one-dimensional systems. Application to Al-catalyzed Si nanowires
    Rurali, Riccardo; Cartoixa, Xavier
    Nano Letters (2009), 9 (3), 975-979CODEN: NALEFD; ISSN:1530-6984. (American Chemical Society)
    The energetic cost of creating a defect within a host material is given by the formation energy. Here we present a formulation allowing the calcn. of formation energies in one-dimensional nanostructures which overcomes the difficulties involved in applying the bulk formalism and the possible passivation of the surface. We also develop a formula for the Madelung correction for general dielec. tensors. We apply this formalism to the technol. important case of Al-nanoparticle-catalyzed Si nanowires, obtaining Al concns. significantly larger than in their bulk counterparts and predicting the fast consumption of the nanoparticles when the wires are grown on n-type substrates.
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    Murphy, S. T.; Hine, N. D. M. Anisotropic charge screening and supercell size convergence of defect formation energies. Phys. Rev. B 2013, 87, 094111,  DOI: 10.1103/PhysRevB.87.094111
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    Anisotropic charge screening and supercell size convergence of defect formation energies
    Murphy, Samuel T.; Hine, Nicholas D. M.
    Physical Review B: Condensed Matter and Materials Physics (2013), 87 (9), 094111/1-094111/6CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
    One of the main sources of error assocd. with the calcn. of defect formation energies using plane-wave d. functional theory (DFT) is finite size error resulting from the use of relatively small simulation cells and periodic boundary conditions. Most widely used methods for correcting this error, such as that of Makov and Payne, assume that the dielec. response of the material is isotropic and can be described using a scalar dielec. const. ε. However, this is strictly only valid for cubic crystals, and cannot work in highly anisotropic cases. Here we introduce a variation of the technique of extrapolation based on the Madelung potential that allows the calcn. of well-converged dil. limit defect formation energies in noncubic systems with highly anisotropic dielec. properties. As an example of the implementation of this technique we study a selection of defects in the ceramic oxide Li2TiO3 which is currently being considered as a lithium battery material and a breeder material for fusion reactors.
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    Kumagai, Y.; Oba, F. Electrostatics-based finite-size corrections for first-principles point defect calculations. Phys. Rev. B 2014, 89, 195205,  DOI: 10.1103/PhysRevB.89.195205
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    Electrostatics-based finite-size corrections for first-principles point defect calculations
    Kumagai, Yu; Oba, Fumiyasu
    Physical Review B: Condensed Matter and Materials Physics (2014), 89 (19), 195205/1-195205/15, 15 pp.CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
    Finite-size corrections for charged defect supercell calcns. typically consist of image-charge and potential alignment corrections. Regarding the image-charge correction, Freysoldt, Neugebauer, and Van de Walle (FNV) recently proposed a scheme that constructs the correction energy a posteriori through alignment of the defect-induced potential to a model charge potential. This, however, still has two shortcomings in practice. First, it uses a planar-averaged electrostatic potential for detg. the potential offset, which can not be readily applied to defects with large at. relaxation. Second, Coulomb interaction is screened by a macroscopic scalar dielec. const., which can bring forth large errors for defects in layered and low-dimensional structures. In this study, we use the at. site potential as a potential marker, and extend the FNV scheme by estg. long-range Coulomb interactions with a point charge model in an anisotropic medium. The authors also revisit the conventional potential alignment and show that it is unnecessary for correcting defect formation energies after the image-charge correction is properly applied. A systematic assessment of the accuracy of the extended FNV scheme is performed for defects and impurities in diverse materials: β-Li2TiO3, ZnO, MgO, Al2O3, HfO2, cubic and hexagonal BN, Si, GaAs, and diamond. Defect formation energies with -6 to +3 charges calcd. using supercells contg. around 100 atoms are successfully cor. even after at. relaxation within 0.2 eV compared to those in the dil. limit.
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    D’Arco, P.; Mustapha, S.; Ferrabone, M.; Noël, Y.; Pierre, M. D. L.; Dovesi, R. Symmetry and random sampling of symmetry independent configurations for the simulation of disordered solids. J. Phys.: Condens. Matter 2013, 25, 355401,  DOI: 10.1088/0953-8984/25/35/355401
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    19
    Symmetry and random sampling of symmetry independent configurations for the simulation of disordered solids
    D'Arco, Philippe; Mustapha, Sami; Ferrabone, Matteo; Noel, Yves; De La Pierre, Marco; Dovesi, Roberto
    Journal of Physics: Condensed Matter (2013), 25 (35), 355401, 13 pp.CODEN: JCOMEL; ISSN:0953-8984. (IOP Publishing Ltd.)
    A symmetry-adapted algorithm producing uniformly at random the set of symmetry independent configurations (SICs) in disordered cryst. systems or solid solns. is presented here. Starting from Polya's formula, the role of the conjugacy classes of the symmetry group in uniform random sampling is shown. SICs can be obtained for all the possible compns. or for a chosen one and symmetry constraints can be applied. The approach yields the multiplicity of the SICs and allows us to operate configurational statistics in the reduced space of the SICs. The present low-memory demanding implementation is briefly sketched. The probability of finding a given SIC or a subset of SICs is discussed as a function of the no. of draws and their precise est. is given. The method is illustrated by application to a binary series of carbonates and to the binary spinel solid soln. Mg(Al,Fe)2O4.
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    Okhotnikov, K. Comment on “Symmetry and random sampling of symmetry independent configurations for the simulation of disordered solids”. arXiv Preprint , arXiv:1606.08062, 2016.
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    Okhotnikov, K.; Charpentier, T.; Cadars, S. Supercell program: a combinatorial structure-generation approach for the local-level modeling of atomic substitutions and partial occupancies in crystals. J. Cheminformatics 2016, 8, 1– 15,  DOI: 10.1186/s13321-016-0129-3
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    Materials Studio, Version 18.1.0.2017; Dassault Systèmes BIOVIA, San Diego, 2018.
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    Hart, G. L. W.; Forcade, R. W. Algorithm for generating derivative structures. Phys. Rev. B 2008, 77, 224115,  DOI: 10.1103/PhysRevB.77.224115
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    Algorithm for generating derivative structures
    Hart, Gus L. W.; Forcade, Rodney W.
    Physical Review B: Condensed Matter and Materials Physics (2008), 77 (22), 224115/1-224115/12CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
    We present an algorithm for generating all deriv. superstructures-for arbitrary parent structures and for any no. of atom types. This algorithm enumerates superlattices and at. configurations in a geometry-independent way. The key concept is to use the quotient group assocd. with each superlattice to det. all unique at. configurations. The run time of the algorithm scales linearly with the no. of unique structures found.
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    Ong, S. P.; Richards, W. D.; Jain, A.; Hautier, G.; Kocher, M.; Cholia, S.; Gunter, D.; Chevrier, V. L.; Persson, K. A.; Ceder, G. Python Materials Genomics (pymatgen): A robust, open-source python library for materials analysis. Comput. Mater. Sci. 2013, 68, 314– 319,  DOI: 10.1016/j.commatsci.2012.10.028
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    Python Materials Genomics (pymatgen): A robust, open-source python library for materials analysis
    Ong, Shyue Ping; Richards, William Davidson; Jain, Anubhav; Hautier, Geoffroy; Kocher, Michael; Cholia, Shreyas; Gunter, Dan; Chevrier, Vincent L.; Persson, Kristin A.; Ceder, Gerbrand
    Computational Materials Science (2013), 68 (), 314-319CODEN: CMMSEM; ISSN:0927-0256. (Elsevier B.V.)
    We present the Python Materials Genomics (pymatgen) library, a robust, open-source Python library for materials anal. A key enabler in high-throughput computational materials science efforts is a robust set of software tools to perform initial setup for the calcns. (e.g., generation of structures and necessary input files) and post-calcn. anal. to derive useful material properties from raw calcd. data. The pymatgen library aims to meet these needs by (1) defining core Python objects for materials data representation, (2) providing a well-tested set of structure and thermodn. analyses relevant to many applications, and (3) establishing an open platform for researchers to collaboratively develop sophisticated analyses of materials data obtained both from first principles calcns. and expts. The pymatgen library also provides convenient tools to obtain useful materials data via the Materials Project's REpresentational State Transfer (REST) Application Programming Interface (API). As an example, using pymatgen's interface to the Materials Project's RESTful API and phase diagram package, we demonstrate how the phase and electrochem. stability of a recently synthesized material, Li4SnS4, can be analyzed using a min. of computing resources. We find that Li4SnS4 is a stable phase in the Li-Sn-S phase diagram (consistent with the fact that it can be synthesized), but the narrow range of lithium chem. potentials for which it is predicted to be stable would suggest that it is not intrinsically stable against typical electrodes used in lithium-ion batteries.
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    Mustapha, S.; D’Arco, P.; De La Pierre, M.; Noël, Y.; Ferrabone, M.; Dovesi, R. On the use of symmetry in configurational analysis for the simulation of disordered solids. J. Condens. Matter Phys. 2013, 25, 105401,  DOI: 10.1088/0953-8984/25/10/105401
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    25
    On the use of symmetry in configurational analysis for the simulation of disordered solids
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    Journal of Physics: Condensed Matter (2013), 25 (10), 105401, 16 pp.CODEN: JCOMEL; ISSN:0953-8984. (IOP Publishing Ltd.)
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Cited By

This article is cited by 2 publications.

  1. Joo Yeon D. Roh, David E. Sommer, Tyler J. Milstein, Scott T. Dunham, Daniel R. Gamelin. Evolution of Yb3+ Speciation in Cl–/Br–- and Yb3+/Gd3+-Alloyed Quantum-Cutting Lead-Halide Perovskite Nanocrystals. Chemistry of Materials 2023, Article ASAP.
  2. Keishu Utimula, Masao Yano, Hiroyuki Kimoto, Kenta Hongo, Kousuke Nakano, Ryo Maezono. Feature Space of XRD Patterns Constructed by an Autoencoder. Advanced Theory and Simulations 2023, 6 (2) , 2200613. https://doi.org/10.1002/adts.202200613
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  • Abstract

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    Figure 1

    Figure 1. Example of the search tree, where three vertices out of eight are colored in red in a cube. p(X) denotes the parent node of X, whereas C(X) denotes the set of children of X.

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    Figure 2

    Figure 2. Atomic substitutions corresponding to Figure 1. Ce8Pd24Sb → (Ce5,La3)Pd24Sb, where Ce and La atoms are on the 8g Wyckoff site. The crystal structure (34) was obtained from the ICSD database (35,36) (CollCode: 83378). The space group is 221-Pm3m. The crystal structures are depicted using VESTA. (37)

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    Figure 3

    Figure 3. Computational times of Shry on a benchmark data set. On the horizontal axis, we place the number of symmetrically distinct structures, whereas the vertical axis indicates the absolute computational time.

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    Figure 4

    Figure 4. Schematics of (a) intra-software and (b) inter-software tests. The intra-software test checked whether each program provides zero redundant structures, where brute-force comparisons were performed (i.e., all the possible combinations were considered). The inter-software test checked whether the symmetry-inequivalent structure sets generated by Shry and Supercell (Disorder tool) are identical; i.e., there is no excess or deficiency among them. The crystal structures are depicted using VESTA. (37)

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  • References

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      cf. ibid. 91, 234(1953); 92, 847; C.A. 45, 1403g; 46, 9374i. The problem of solving the Schroedinger equation in a periodic lattice is studied from the point of view of the variation-iteration method. This approach leads to a very compact scheme if the potential V(r) is spherically sym. within the inscribed spheres of the at. polyhedra and const. in the space between them. The band structure of the lattice is then detd. by (1) geometrical structure consts., characteristic of the type of lattice and (2) the logarithmic derivs., at the surface of the inscribed sphere, of the s, p, d, .... functions corresponding to V(r). By far the greater part of the labor is involved in the calcn. of (1), which needs to be done only once for each type of lattice; (2) can be obtained by numerical integration or directly from the at. spectra. Although derived from a different point of view, this scheme is essentially equiv. to one proposed by Korringa (Physica 13, 392(1947)) on the basis of the theory of lattice interferences. An application is made to the conduction band of metallic Li.
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    8. 8
      Nakano, K.; Hongo, K.; Maezono, R. Phonon dispersions and Fermi surfaces nesting explaining the variety of charge ordering in titanium-oxypnictides superconductors. Sci. Rep. 2016, 6, 29661,  DOI: 10.1038/srep29661
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      8
      Phonon dispersions and Fermi surfaces nesting explaining the variety of charge ordering in titanium-oxypnictides superconductors
      Nakano, Kousuke; Hongo, Kenta; Maezono, Ryo
      Scientific Reports (2016), 6 (), 29661CODEN: SRCEC3; ISSN:2045-2322. (Nature Publishing Group)
      There has been a puzzle between expts. and theor. predictions on the charge ordering of layered titanium-oxypnictides superconductors. Unconventional mechanisms to explain this discrepancy have been argued so far, even affecting the understanding of supercond. on the compd. We provide a new theor. prediction, by which the discrepancy itself is resolved without any complicated unconventional explanation. Phonon dispersions and changes of nesting vectors in Fermi surfaces are clarified to lead to the variety of superlattice structures even for the common crystal structures when without CDW, including orthorhombic 2 × 2 × 1 one for BaTi2As2O, which has not yet been explained successfully so far, being different from tetragonal √2 × √2 × 1 for BaTi2Sb2O and BaTi2Bi2O. The electronic structure anal. can naturally explain exptl. observations about CDW including most latest ones without any cramped unconventional mechanisms.
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    9. 9
      Nakano, K.; Hongo, K.; Maezono, R. Investigation into Structural Phase Transitions in Layered Titanium-Oxypnictides by a Computational Phonon Analysis. Inorg. Chem. 2017, 56, 13732– 13740,  DOI: 10.1021/acs.inorgchem.7b01709
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      9
      Investigation into Structural Phase Transitions in Layered Titanium-Oxypnictides by a Computational Phonon Analysis
      Nakano, Kousuke; Hongo, Kenta; Maezono, Ryo
      Inorganic Chemistry (2017), 56 (22), 13732-13740CODEN: INOCAJ; ISSN:0020-1669. (American Chemical Society)
      We applied ab initio phonon anal. to layered titanium-oxypnictides, Na2Ti2Pn2O (Pn = As and Sb), and found a clear contrast between the cases with lighter/heavier pnictogen in comparison with expts. Na2Ti2As2O Na2Ti2Sb2O. The result completely explains the exptl. structure at low temp., C2/m for Pn = As, within the conventional charge d. wave, while there arise discrepancies when the pnictogen gets heavier. Our phonon calcn. using the GGA-PBE functional predicts that a Cmce polymorph is more stable than the exptl. obsd. one (Cmcm) for Pn = Sb. On the basis of further quant. anal., we suggest the possibility that the GGA-PBE functional does not properly reproduce the electron correlation effects for Pn = Sb, and this could be the reason for the present discrepancy.
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    10. 10
      Utimula, K.; Hunkao, R.; Yano, M.; Kimoto, H.; Hongo, K.; Kawaguchi, S.; Suwanna, S.; Maezono, R. Machine-Learning Clustering Technique Applied to Powder X-Ray Diffraction Patterns to Distinguish Compositions of ThMn12-Type Alloys. Adv. Theory Simul. 2020, 3, 2000039,  DOI: 10.1002/adts.202000039
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      10
      Machine-Learning Clustering Technique Applied to Powder X-Ray Diffraction Patterns to Distinguish Compositions of thorium manganese-Type Alloys
      Utimula, Keishu; Hunkao, Rutchapon; Yano, Masao; Kimoto, Hiroyuki; Hongo, Kenta; Kawaguchi, Shogo; Suwanna, Sujin; Maezono, Ryo
      Advanced Theory and Simulations (2020), 3 (7), 2000039CODEN: ATSDCW; ISSN:2513-0390. (Wiley-VCH Verlag GmbH & Co. KGaA)
      A clustering technique is applied using dynamic-time-wrapping (DTW) anal. to X-ray diffraction (XRD) spectrum patterns in order to identify the microscopic structures of substituents introduced into the main phase of magnetic alloys. The clustering technique is found to perform well, identifying the concns. of the substituents with success rates of ≈;90%. This level of performance is attributed to the capability of DTW processing to filter out irrelevant information such as the peak intensities (due to the uncontrollability of diffraction conditions in polycryst. samples) and the uniform shift of peak positions (due to the thermal expansion of lattices). The established framework is not limited to the system treated in this work, but is widely applicable to systems the properties of which are to be tuned by at. substitutions within a phase. The framework has a broader potential to predict properties such as magnetic moments, optical spectra, etc. from obsd. XRD patterns, by predicting such properties evaluated from predicted microscopic local structure.
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    11. 11
      Wei, S.-H.; Ferreira, L. G.; Bernard, J. E.; Zunger, A. Electronic properties of random alloys: Special quasirandom structures. Phys. Rev. B 1990, 42, 9622– 9649,  DOI: 10.1103/PhysRevB.42.9622
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      Electronic properties of random alloys: special quasirandom structures
      Wei, S. H.; Ferreira, L. G.; Bernard, James E.; Zunger, Alex
      Physical Review B: Condensed Matter and Materials Physics (1990), 42 (15), 9622-49CODEN: PRBMDO; ISSN:0163-1829.
      Structural models needed in calcns. of properties of substitutionally random A1-xBx alloys are usually constructed by randomly occupying each of the N sites of a periodic cell by A or B. It is possible to design "special quasirandom structures" (SQS's) that mimic for small N (even N = 8) the first few, phys. most relevant radial correlation functions of an infinite, perfectly random structure far better than the std. technique does. These SQS's are shown to be short-period superlattices of 4-16 atoms/cell whose layers are stacked in rather nonstandard orientations (e.g., [113], [331], and [115]). Since these SQS's mimic well the local at. structure of the random alloy, their electronic properties, calculable via first-principles techniques, provide a representation of the electronic structure of the alloy. Authors demonstrate the usefulness of these SQS's by applying them to semiconductors alloys. They calc. their electronic structure, total energy, and equil. geometry, and compare the results to exptl. data.
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    12. 12
      Zunger, A.; Wei, S.-H.; Ferreira, L. G.; Bernard, J. E. Special quasirandom structures. Phys. Rev. Lett. 1990, 65, 353– 356,  DOI: 10.1103/PhysRevLett.65.353
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      Special quasirandom structures
      Zunger, Alex; Wei, S. H.; Ferreira, L. G.; Bernard, James E.
      Physical Review Letters (1990), 65 (3), 353-6CODEN: PRLTAO; ISSN:0031-9007.
      Structural models used in calcns. of properties of substitutionally random A1-xBx alloys are usually constructed by randomly occupying each of the N sites of a periodic cell by A or B. It is possible to design "special quasirandom structures" (SQS's) that mimic for small N (even N = 8) the first few, phys. most relevant radial correlation functions of a perfectly random structure far better than the std. technique does. The usefulness is demonstrated of these SQS's by calcg. optical and thermodn. properties of a no. of semiconductor alloys in the local-d. formalism.
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    13. 13
      Seko, A.; Koyama, Y.; Tanaka, I. Cluster expansion method for multicomponent systems based on optimal selection of structures for density-functional theory calculations. Phys. Rev. B 2009, 80, 165122,  DOI: 10.1103/PhysRevB.80.165122
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      13
      Cluster expansion method for multicomponent systems based on optimal selection of structures for density-functional theory calculations
      Seko, Atsuto; Koyama, Yukinori; Tanaka, Isao
      Physical Review B: Condensed Matter and Materials Physics (2009), 80 (16), 165122/1-165122/7CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
      The cluster expansion (CE) method has been used to evaluate configurational properties in multicomponent systems based on the d.-functional theory (DFT) calcns. Appropriate selections of not only clusters but also structures for DFT calcns. (DFT structures) are crucial for the accuracy and the efficiency of the CE. In a conventional procedure to construct the CE, the CE error is reduced mainly through an appropriate selection of clusters. In the present paper, we propose an improved procedure that systematically leads to optimal selections of both clusters and DFT structures. DFT structures are chosen to cover as much of the configurational space as possible. During the iterative process, the predictive power of the out-of-sample structures can be increased up to the accuracy that is required to describe alloy thermodn. We apply the procedure to configurational behaviors in a simple MgO-ZnO pseudobinary system and in a complex MgAl2O4 system. The CE error is reduced in both systems, in particular, in the complex system, thereby significantly improving configurational properties at high temps. compared with the conventional CE procedure.
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    14. 14
      Yoshio, S.; Hongo, K.; Nakano, K.; Maezono, R. High-Throughput Evaluation of Discharge Profiles of Nickel Substitution in LiNiO2 by Ab Initio Calculations. J. Phys. Chem. C 2021, 125, 14517– 14524,  DOI: 10.1021/acs.jpcc.0c11589
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      14
      High-Throughput Evaluation of Discharge Profiles of Nickel Substitution in LiNiO2 by Ab Initio Calculations
      Yoshio, Satoshi; Hongo, Kenta; Nakano, Kousuke; Maezono, Ryo
      Journal of Physical Chemistry C (2021), 125 (27), 14517-14524CODEN: JPCCCK; ISSN:1932-7447. (American Chemical Society)
      Substitution of a Ni cation in a LiNiO2 cathode is a common strategy for improving the lifetime and thermal stability of lithium-ion batteries. However, this strategy does not improve the discharging capacity more than that of pristine LiNiO2. We investigated whether the capacity of a cation-substituted battery can be improved by an exhaustive search based on ab initio calcns. To ensure a feasible search at a practical computational cost, many data points obtained by expensive ab initio calcns. were bridged by interpolation using the cluster expansion method. Then, the candidates screened by the search were analyzed by more reliable calcns. based on d. functional theory with the strongly constrained and appropriately normed (SCAN) exchange-correlation functional, which dets. the discharging voltage profiles. The screening predicted that partial substitution of Ni with Pt and Pd can improve the discharging capacity of a lithium-ion battery.
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    15. 15
      Freysoldt, C.; Neugebauer, J.; Van de Walle, C. G. Fully Ab Initio Finite-Size Corrections for Charged-Defect Supercell Calculations. Phys. Rev. Lett. 2009, 102, 016402,  DOI: 10.1103/PhysRevLett.102.016402
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      15
      Fully Ab Initio Finite-Size Corrections for Charged-Defect Supercell Calculations
      Freysoldt, Christoph; Neugebauer, Jorg; Van de Walle, Chris G.
      Physical Review Letters (2009), 102 (1), 016402/1-016402/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      In ab initio theory, defects are routinely modeled by supercells with periodic boundary conditions. Unfortunately, the supercell approxn. introduces artificial interactions between charged defects. Despite numerous attempts, a general scheme to correct for these is not yet available. We propose a new and computationally efficient method that overcomes limitations of previous schemes and is based on a rigorous anal. of electrostatics in dielec. media. Its reliability and rapid convergence with respect to cell size is demonstrated for charged vacancies in diamond and GaAs.
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    16. 16
      Rurali, R.; Cartoixà, X. Theory of defects in one-dimensional systems: application to Al-catalyzed Si nanowires. Nano Lett. 2009, 9, 975– 979,  DOI: 10.1021/nl802847p
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      16
      Theory of defects in one-dimensional systems. Application to Al-catalyzed Si nanowires
      Rurali, Riccardo; Cartoixa, Xavier
      Nano Letters (2009), 9 (3), 975-979CODEN: NALEFD; ISSN:1530-6984. (American Chemical Society)
      The energetic cost of creating a defect within a host material is given by the formation energy. Here we present a formulation allowing the calcn. of formation energies in one-dimensional nanostructures which overcomes the difficulties involved in applying the bulk formalism and the possible passivation of the surface. We also develop a formula for the Madelung correction for general dielec. tensors. We apply this formalism to the technol. important case of Al-nanoparticle-catalyzed Si nanowires, obtaining Al concns. significantly larger than in their bulk counterparts and predicting the fast consumption of the nanoparticles when the wires are grown on n-type substrates.
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    17. 17
      Murphy, S. T.; Hine, N. D. M. Anisotropic charge screening and supercell size convergence of defect formation energies. Phys. Rev. B 2013, 87, 094111,  DOI: 10.1103/PhysRevB.87.094111
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      Anisotropic charge screening and supercell size convergence of defect formation energies
      Murphy, Samuel T.; Hine, Nicholas D. M.
      Physical Review B: Condensed Matter and Materials Physics (2013), 87 (9), 094111/1-094111/6CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
      One of the main sources of error assocd. with the calcn. of defect formation energies using plane-wave d. functional theory (DFT) is finite size error resulting from the use of relatively small simulation cells and periodic boundary conditions. Most widely used methods for correcting this error, such as that of Makov and Payne, assume that the dielec. response of the material is isotropic and can be described using a scalar dielec. const. ε. However, this is strictly only valid for cubic crystals, and cannot work in highly anisotropic cases. Here we introduce a variation of the technique of extrapolation based on the Madelung potential that allows the calcn. of well-converged dil. limit defect formation energies in noncubic systems with highly anisotropic dielec. properties. As an example of the implementation of this technique we study a selection of defects in the ceramic oxide Li2TiO3 which is currently being considered as a lithium battery material and a breeder material for fusion reactors.
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    18. 18
      Kumagai, Y.; Oba, F. Electrostatics-based finite-size corrections for first-principles point defect calculations. Phys. Rev. B 2014, 89, 195205,  DOI: 10.1103/PhysRevB.89.195205
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      Electrostatics-based finite-size corrections for first-principles point defect calculations
      Kumagai, Yu; Oba, Fumiyasu
      Physical Review B: Condensed Matter and Materials Physics (2014), 89 (19), 195205/1-195205/15, 15 pp.CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
      Finite-size corrections for charged defect supercell calcns. typically consist of image-charge and potential alignment corrections. Regarding the image-charge correction, Freysoldt, Neugebauer, and Van de Walle (FNV) recently proposed a scheme that constructs the correction energy a posteriori through alignment of the defect-induced potential to a model charge potential. This, however, still has two shortcomings in practice. First, it uses a planar-averaged electrostatic potential for detg. the potential offset, which can not be readily applied to defects with large at. relaxation. Second, Coulomb interaction is screened by a macroscopic scalar dielec. const., which can bring forth large errors for defects in layered and low-dimensional structures. In this study, we use the at. site potential as a potential marker, and extend the FNV scheme by estg. long-range Coulomb interactions with a point charge model in an anisotropic medium. The authors also revisit the conventional potential alignment and show that it is unnecessary for correcting defect formation energies after the image-charge correction is properly applied. A systematic assessment of the accuracy of the extended FNV scheme is performed for defects and impurities in diverse materials: β-Li2TiO3, ZnO, MgO, Al2O3, HfO2, cubic and hexagonal BN, Si, GaAs, and diamond. Defect formation energies with -6 to +3 charges calcd. using supercells contg. around 100 atoms are successfully cor. even after at. relaxation within 0.2 eV compared to those in the dil. limit.
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    19. 19
      D’Arco, P.; Mustapha, S.; Ferrabone, M.; Noël, Y.; Pierre, M. D. L.; Dovesi, R. Symmetry and random sampling of symmetry independent configurations for the simulation of disordered solids. J. Phys.: Condens. Matter 2013, 25, 355401,  DOI: 10.1088/0953-8984/25/35/355401
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      19
      Symmetry and random sampling of symmetry independent configurations for the simulation of disordered solids
      D'Arco, Philippe; Mustapha, Sami; Ferrabone, Matteo; Noel, Yves; De La Pierre, Marco; Dovesi, Roberto
      Journal of Physics: Condensed Matter (2013), 25 (35), 355401, 13 pp.CODEN: JCOMEL; ISSN:0953-8984. (IOP Publishing Ltd.)
      A symmetry-adapted algorithm producing uniformly at random the set of symmetry independent configurations (SICs) in disordered cryst. systems or solid solns. is presented here. Starting from Polya's formula, the role of the conjugacy classes of the symmetry group in uniform random sampling is shown. SICs can be obtained for all the possible compns. or for a chosen one and symmetry constraints can be applied. The approach yields the multiplicity of the SICs and allows us to operate configurational statistics in the reduced space of the SICs. The present low-memory demanding implementation is briefly sketched. The probability of finding a given SIC or a subset of SICs is discussed as a function of the no. of draws and their precise est. is given. The method is illustrated by application to a binary series of carbonates and to the binary spinel solid soln. Mg(Al,Fe)2O4.
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    20. 20
      Okhotnikov, K. Comment on “Symmetry and random sampling of symmetry independent configurations for the simulation of disordered solids”. arXiv Preprint , arXiv:1606.08062, 2016.
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      Okhotnikov, K.; Charpentier, T.; Cadars, S. Supercell program: a combinatorial structure-generation approach for the local-level modeling of atomic substitutions and partial occupancies in crystals. J. Cheminformatics 2016, 8, 1– 15,  DOI: 10.1186/s13321-016-0129-3
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      Materials Studio, Version 18.1.0.2017; Dassault Systèmes BIOVIA, San Diego, 2018.
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      Hart, G. L. W.; Forcade, R. W. Algorithm for generating derivative structures. Phys. Rev. B 2008, 77, 224115,  DOI: 10.1103/PhysRevB.77.224115
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      Algorithm for generating derivative structures
      Hart, Gus L. W.; Forcade, Rodney W.
      Physical Review B: Condensed Matter and Materials Physics (2008), 77 (22), 224115/1-224115/12CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
      We present an algorithm for generating all deriv. superstructures-for arbitrary parent structures and for any no. of atom types. This algorithm enumerates superlattices and at. configurations in a geometry-independent way. The key concept is to use the quotient group assocd. with each superlattice to det. all unique at. configurations. The run time of the algorithm scales linearly with the no. of unique structures found.
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      Ong, S. P.; Richards, W. D.; Jain, A.; Hautier, G.; Kocher, M.; Cholia, S.; Gunter, D.; Chevrier, V. L.; Persson, K. A.; Ceder, G. Python Materials Genomics (pymatgen): A robust, open-source python library for materials analysis. Comput. Mater. Sci. 2013, 68, 314– 319,  DOI: 10.1016/j.commatsci.2012.10.028
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      Python Materials Genomics (pymatgen): A robust, open-source python library for materials analysis
      Ong, Shyue Ping; Richards, William Davidson; Jain, Anubhav; Hautier, Geoffroy; Kocher, Michael; Cholia, Shreyas; Gunter, Dan; Chevrier, Vincent L.; Persson, Kristin A.; Ceder, Gerbrand
      Computational Materials Science (2013), 68 (), 314-319CODEN: CMMSEM; ISSN:0927-0256. (Elsevier B.V.)
      We present the Python Materials Genomics (pymatgen) library, a robust, open-source Python library for materials anal. A key enabler in high-throughput computational materials science efforts is a robust set of software tools to perform initial setup for the calcns. (e.g., generation of structures and necessary input files) and post-calcn. anal. to derive useful material properties from raw calcd. data. The pymatgen library aims to meet these needs by (1) defining core Python objects for materials data representation, (2) providing a well-tested set of structure and thermodn. analyses relevant to many applications, and (3) establishing an open platform for researchers to collaboratively develop sophisticated analyses of materials data obtained both from first principles calcns. and expts. The pymatgen library also provides convenient tools to obtain useful materials data via the Materials Project's REpresentational State Transfer (REST) Application Programming Interface (API). As an example, using pymatgen's interface to the Materials Project's RESTful API and phase diagram package, we demonstrate how the phase and electrochem. stability of a recently synthesized material, Li4SnS4, can be analyzed using a min. of computing resources. We find that Li4SnS4 is a stable phase in the Li-Sn-S phase diagram (consistent with the fact that it can be synthesized), but the narrow range of lithium chem. potentials for which it is predicted to be stable would suggest that it is not intrinsically stable against typical electrodes used in lithium-ion batteries.
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    25. 25
      Mustapha, S.; D’Arco, P.; De La Pierre, M.; Noël, Y.; Ferrabone, M.; Dovesi, R. On the use of symmetry in configurational analysis for the simulation of disordered solids. J. Condens. Matter Phys. 2013, 25, 105401,  DOI: 10.1088/0953-8984/25/10/105401
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      25
      On the use of symmetry in configurational analysis for the simulation of disordered solids
      Mustapha, Sami; D'Arco, Philippe; De La Pierre, Marco; Noel, Yves; Ferrabone, Matteo; Dovesi, Roberto
      Journal of Physics: Condensed Matter (2013), 25 (10), 105401, 16 pp.CODEN: JCOMEL; ISSN:0953-8984. (IOP Publishing Ltd.)
      The starting point for a quantum mech. investigation of disordered systems usually implies calcns. on a limited subset of configurations, generated by defining either the compn. of interest or a set of compns. ranging from one end member to another, within an appropriate supercell of the primitive cell of the pure compd. The way in which symmetry can be used in the identification of symmetry independent configurations (SICs) is discussed here. First, Polya's enumeration theory is adopted to det. the no. of SICs, in the case of both varying and fixed compn., for colors numbering two or higher. Then, De Bruijn's generalization is presented, which allows anal. of the case where the colors are symmetry related, e.g. spin up and down in magnetic systems. In spite of their efficiency in counting SICs, neither Polya's nor De Bruijn's theory helps in solving the difficult problem of identifying the complete list of SICs. Representative SICs are obtained by adopting an orderly generation approach, based on lexicog. ordering, which offers the advantage of avoiding the (computationally expensive) anal. and storage of all the possible configurations. When the no. of colors increases, this strategy can be combined with the surjective resoln. principle, which permits the efficient generation of SICs of a problem in |R| colors starting from the ones obtained for the (|R| - 1)-colors case. The whole scheme is documented by means of three examples: the abstr. case of a square with C4v symmetry and the real cases of the garnet and olivine mineral families.
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    26. 26
      Dovesi, R.; Erba, A.; Orlando, R.; Zicovich-Wilson, C. M.; Civalleri, B.; Maschio, L.; Rerat, M.; Casassa, S.; Baima, J.; Salustro, S.; Kirtman, B. Quantum-mechanical condensed matter simulations with CRYSTAL. Wiley Interdiscip. Rev. Comput. Mol. Sci. 2018, 8, e1360  DOI: 10.1002/wcms.1360
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      A methodol. is presented, which reduces the no. of site-occupancy configurations to be calcd. when modeling site disorder in solids, by taking advantage of the crystal symmetry of the lattice. Within this approach, two configurations are considered equiv. when they are related by an isometric operation; a trial list of possible isometric transformations is provided by the group of symmetry operators in the parent structure, which is used to generate all configurations via at. substitutions. We have adapted the equations for configurational statistics to operate in the reduced configurational space of the independent configurations. Each configuration in this space is characterized by its reduced energy, which includes not only its energy but also a contribution from its degeneracy in the complete configurational space, via an entropic term. The new computer program SOD (site-occupancy disorder) is presented, which performs this anal. in systems with arbitrary symmetry and any size of supercell. As a case study we use the distribution of cations in iron antimony oxide FeSbO4, where we also introduce some general considerations for the modeling of site-occupancy disorder in paramagnetic systems.
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      The ternary compd. Ce8Pd24Sb, prepd. by arc-melting the elements, is very close in compn. to the intermediate valent binary CePd3. A single crystal study yielded a cubic cell with Pm‾3m symmetry and a = 8.461(1) Å, V = 605.71(2) Å3, Z = 1, ρc = 10.408 g/cm3, μ(MoKα) = 33.03 mm-1, F(000) = 1619, with R = 0.0170 and wR2 = 0.0412 based on 1453 reflections (222 unique) and 16 parameters. This new structure type is composed of distorted perovskite and Cu3Au subcells arranged with the perovskite-like units centered on the corners of the cube. Fitting the magnetic susceptibility data above 100 K to a Curie-Weiss expression yielded a Weiss const. of -15(3)K (antiferromagnetic) and an effective high-temp. moment per Ce of 2.45(4) μB indicating trivalent Ce. No ordering was obsd. above 3 K.
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      VESTA is a 3D visualization system for crystallog. studies and electronic state calcns. It was upgraded to the latest version, VESTA 3, implementing new features including drawing the external morphpol. of crysals; superimposing multiple structural models, volumetric data and crystal faces; calcn. of electron and nuclear densities from structure parameters; calcn. of Patterson functions from the structure parameters or volumetric data; integration of electron and nuclear densities by Voronoi tessellation; visualization of isosurfaces with multiple levels, detn. of the best plane for selected atoms; an extended bond-search algorithm to enable more sophisticated searches in complex mols. and cage-like structures; undo and redo is graphical user interface operations; and significant performance improvements in rendering isosurfaces and calcg. slices.
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      Dovesi, R.; Pascale, F.; Civalleri, B.; Doll, K.; Harrison, N. M.; Bush, I.; D’Arco, P.; Noël, Y.; Rérat, M.; Carbonnière, P.; Causà, M.; Salustro, S.; Lacivita, V.; Kirtman, B.; Ferrari, A. M.; Gentile, F. S.; Baima, J.; Ferrero, M.; Demichelis, R.; De La Pierre, M. The CRYSTAL code, 1976–2020 and beyond, a long story. J. Chem. Phys. 2020, 152, 204111,  DOI: 10.1063/5.0004892
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      CRYSTAL is a periodic ab initio code that uses a Gaussian-type basis set to express cryst. orbitals (i.e., Bloch functions). The use of atom-centered basis functions allows treating 3D (crystals), 2D (slabs), 1D (polymers), and 0D (mols.) systems on the same grounds. In turn, all-electron calcns. are inherently permitted along with pseudopotential strategies. A variety of d. functionals are implemented, including global and range-sepd. hybrids of various natures and, as an extreme case, Hartree-Fock (HF). The cost for HF or hybrids is only about 3-5 times higher than when using the local d. approxn. or the generalized gradient approxn. Symmetry is fully exploited at all steps of the calcn. Many tools are available to modify the structure as given in input and simplify the construction of complicated objects, such as slabs, nanotubes, mols., and clusters. Many tensorial properties can be evaluated by using a single input keyword: elastic, piezoelec., photoelastic, dielec., first and second hyperpolarizabilities, etc. The calcn. of IR and Raman spectra is available, and the intensities are computed anal. Automated tools are available for the generation of the relevant configurations of solid solns. and/or disordered systems. Three versions of the code exist: serial, parallel, and massive-parallel. In the second one, the most relevant matrixes are duplicated on each core, whereas in the third one, the Fock matrix is distributed for diagonalization. All the relevant vectors are dynamically allocated and deallocated after use, making the code very agile. CRYSTAL can be used efficiently on high performance computing machines up to thousands of cores. (c) 2020 American Institute of Physics.
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      Gražulis, S.; Chateigner, D.; Downs, R. T.; Yokochi, A. F. T.; Quirós, M.; Lutterotti, L.; Manakova, E.; Butkus, J.; Moeck, P.; Le Bail, A. Crystallography Open Database – an open-access collection of crystal structures. J. Appl. Crystallogr. 2009, 42, 726– 729,  DOI: 10.1107/S0021889809016690
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      Crystallography Open Database - an open-access collection of crystal structures
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      The Crystallog. Open Database (COD), which is a project that aims to gather all available inorg., metal-org. and small org. mol. structural data in one database, is described. The database adopts an open-access model. The COD currently contains ∼80,000 entries in crystallog. information file format, with nearly full coverage of the International Union of Crystallog. publications, and is growing in size and quality.
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    The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jcim.2c00389.

    • Mathematical details of the algorithms implemented in Shry. Details of crystal structures (CIF files) used in the benchmark and validations tests. Tables S2 and S3: Unique IDs of compounds in database, compositions, and substituted Wyckoff position(s). Measured computational times plotted in Figure 3 summarized in rightmost columns of Table S2. (PDF)


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