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CALYPSO: a method for crystal structure prediction

Yanchao Wang, Jian Lv, Li Zhu, Yanming Ma

arXiv:1205.2264v1cond-mat.mtrl-sci

TL;DR

Crystal structure prediction from composition and external conditions remains difficult, particularly because PSO can stagnate around an early sub-optimal solution. The paper describes CALYPSO’s implementation, combining PSO with structural constraints and diversity-control techniques; the authors report reliable and efficient structural searches and present it as a promising tool for crystal structure determination.

  • Problem

    Predicting crystal structures from chemical composition is extremely difficult, and rapid PSO convergence can cause stagnation around an early sub-optimal solution.

  • Method

    CALYPSO combines PSO with symmetry constraints, bond-characterization-based elimination of similar structures, partial random structures, and penalty functions.

  • Results

    The authors report that CALYPSO’s efficiency in structural searches demonstrates its reliability and promise as a major tool for crystal structure determination.

  • Takeaways & Limitations

    CALYPSO can be used to predict crystal structures of materials at given chemical compositions and external conditions.

  • Takeaways & Limitations

    If an early solution is sub-optimal, PSO can stagnate around it without pressure to continue exploration.

Abstract

from arXiv · show

We have developed a software package CALYPSO (Crystal structure AnaLYsis by Particle Swarm Optimization) to predict the energetically stable/metastable crystal structures of materials at given chemical compositions and external conditions (e.g., pressure). The CALYPSO method is based on several major techniques (e.g. particle-swarm optimization algorithm, symmetry constraints on structural generation, bond characterization matrix on elimination of similar structures, partial random structures per generation on enhancing structural diversity, and penalty function, etc) for global structural minimization from scratch. All of these techniques have been demonstrated to be critical to the prediction of global stable structure. We have implemented these techniques into the CALYPSO code. Testing of the code on many known and unknown systems shows high efficiency and high successful rate of this CALYPSO method [Wang et al., Phys. Rev. B 82 (2010) 094116][1]. In this paper, we focus on descriptions of the implementation of CALYPSO code and why it works.

1. Introduction

Crystal structure prediction from chemical composition and external conditions is difficult because the energy landscape contains many minima, while existing approaches have limited success generating new structures. CALYPSO addresses these challenges by combining PSO with structural-generation, diversity, and similarity-control techniques, whose implementation and rationale are the paper’s focus.

  • Motivation: Predicting crystal structures from chemical composition alone is extremely difficult and involves classifying many energy minima on the lattice energy surface.The paper frames this as a major challenge in chemistry and condensed matter physics.
  • Existing approaches: Database-based approaches have limited success and cannot generate new crystal structure types.The introduction contrasts this limitation with systematic search methods based on composition and external conditions.
  • PSO foundation: PSO performs multidimensional global search by allowing individuals to learn from local or global best solutions and their own past experience.The swarm adjusts flying speed and direction while searching through hyperspace.
  • PSO limitation: Rapid PSO convergence can cause premature stagnation when an early solution is sub-optimal.The swarm may remain near that solution without continuing exploration.
  • CALYPSO approach: CALYPSO combines PSO with symmetry constraints, bond characterization for eliminating similar structures, partial random structures, and penalty functions.These techniques were implemented to improve structural searches and global minimization from scratch.
  • Reported contribution: The authors report that the added techniques are critical for avoiding premature PSO behavior and significantly accelerating structure convergence.The paper presents detailed implementation descriptions and explains why the method works.

2. Implementation and discussions

CALYPSO combines symmetry-constrained random generation, local optimization, structural uniqueness checks, and population updates to search crystal structures globally. Symmetry constraints diversify sampling, reduce the search space, and improve search efficiency, including in test systems.

  • Workflow: CALYPSO generates random structures under symmetry constraints, locally optimizes them, identifies unique minima, and generates new structures.The workflow comprises four main steps: constrained generation, local optimization, bond-characterization post-processing, and new-structure generation.
  • Symmetry-constrained generation: Space-group selection determines lattice parameters and Wyckoff-position combinations for atomic coordinates.The lattice is generated within the selected symmetry and the coordinates are assembled according to the atom count.
  • Symmetry-constrained generation: 80% exclusion of previously used symmetries helps initial sampling cover different search-space regions and maintain population diversity.The generated-structure symmetry list is used to probabilistically forbid identical symmetric structures.
  • Symmetry-constrained generation: Symmetric constraints reduce the optimization search space and accelerate global structural convergence.They constrain lattice and atomic-coordinate variables while preserving unbiased random sampling of the energy landscape.
  • Symmetry-constrained generation: 11 generations with symmetry constraints versus 25.4 without were needed on average to find the global stable structure.The tests show that symmetry constraints greatly improve search efficiency, especially for larger systems.

2.2 Structural optimization

CALYPSO locally optimizes generated structures and compares them using a bond characterization matrix. This reduces redundant structures and supports faster convergence toward stable structures.

  • Local optimization: CALYPSO interfaces with ab initio and force-field programs to locally optimize structures and evaluate their free-energy fitness.At T = 0 K, free energy reduces to enthalpy; local optimization reduces energy-landscape noise despite increasing individual evaluation cost.
  • Local optimization: Local optimization provides locally optimal structures and improves comparability between different structures.The optimized structures are used for subsequent structural comparison and search.
  • Bond characterization matrix: The bond characterization matrix combines bond-orientational metrics Q_l with an exponential bond-length descriptor for all bond information.Bond vectors and related angular information are included when interatomic distances fall below the cutoff.
  • Bond characterization matrix: Euclidean distances between bond characterization matrices distinguish structures and increase monotonically with distortion magnitude.Graphite and diamond show different Q_l histograms, while distorted versions produce progressively larger distances.
  • Bond characterization matrix: Including the bond characterization matrix required much fewer optimization steps to find stable TiO2 structures.The technique avoids very similar or identical structures and accelerates global structure convergence.

2.4 Generation of new structures by PSO

CALYPSO uses particle-swarm optimization to update structural coordinates using individual and population-wide experience. A linearly decreasing inertia weight shifts the search from global exploration toward local search.

  • PSO representation: Each PSO individual is a crystal structure, and the population-wide best structure supplies a global reference for updates.The scheme tracks each structure’s previous location, optimized location, velocity, and population global best.
  • PSO updates: New coordinates are obtained by adding the updated velocity to the previous coordinates.The atomic positions follow the evolutionary coordinate-update equation after velocity calculation.
  • PSO updates: PSO velocity combines inertia, attraction toward each structure’s best location, and attraction toward the population’s best location.The random coefficients scale the self-confidence and swarm-confidence terms in the velocity update.
  • PSO parameters: The inertia weight decreases linearly from 0.9 to 0.4 during iterations, moving from global exploration toward faster local search.The authors keep c1 and c2 constant at 2 and generate r1 and r2 randomly between 0 and 1.
  • PSO outcome: PSO movement is intended to direct structures toward the global minimum and accelerate convergence.The update uses successful individual and population experiences to influence search direction and speed.

2.5 Penalty function

CALYPSO uses penalty-based selection and random-structure injection to focus evolution on promising low-energy regions while preserving structural diversity. Tests report faster convergence and successful recovery of global stable structures.

  • Penalty function: Penalty-function inclusion significantly accelerates convergence toward the global minimum in CALYPSO runs.The reported TiO2 evolution demonstrates the effect of concentrating structures in lower-energy regions.
  • Structural diversity: Structural diversity is maintained by including a certain percentage of random structures in every generation.The technique was designed because loss of diversity can cause stagnation, particularly for large systems.
  • Structural diversity: Structures with symmetries distinct from previously generated ones are crucial for convergence to the global minimum.Random structures provide diverse candidates, while only a few stable structures are generated randomly, especially in smaller systems.
  • Structural diversity: For most tested cases, structural evolution rather than random generation produces the global stable structures.The authors report that only a few stable structures are generated randomly, particularly for smaller systems.
  • Halting criterion: For systems with no more than 10 atoms per simulation cell, stable crystal structures can usually be found at approximately 10 generations.The default stopping criterion adds 10 further generations when no better structures are found.

3. Optimization of parameters

The CALYPSO parameters were calibrated using TiO2 benchmarks, adopting established PSO settings and testing repeated successful searches.

  • The TiO2 system with 16 formula units per simulation cell was used to benchmark CALYPSO parameters.Structural optimization and total-energy calculations used GULP.
  • c1 = c2 = 2 and a linear decrease of ω from 0.9 to 0.4 were adopted for PSO simulations.These settings were reported to provide the best overall performance in earlier work.
  • Population size, PSO-generated structure proportion, and maximum velocity were determined using the TiO2 benchmark.
  • Five successful CALYPSO runs were repeated, and the resulting parameter values were reported in Table 4.

4. Input and output files

The CALYPSO interface uses input.dat to define the chemical system, search settings, distance constraints, optimization command, and restart behavior; outputs summarize structures and iteration results.

  • Input file: input.dat contains the necessary CALYPSO parameters, including system identity, species, atom counts, formula-unit range, volume, and distance constraints.Chemical species and structural-generation settings are specified explicitly, with no defaults for several required fields.
  • Input file: Volume is specified per formula unit in Å3, while minimum interspecies distances are given in angstroms through a species-dependent matrix.A zero volume triggers automatic estimation from ionic radii.
  • Search controls: PsoRatio controls the fraction of PSO-generated structures, with the remainder generated randomly.PopSize sets the population size.
  • Optimization controls: Kgrid controls local-optimization k-point precision, with smaller values normally giving finer optimization results.The first and second values govern early and final optimizations, respectively; the default is 0.12 0.06.
  • Restart controls: MaxStep sets the maximum number of PSO iterations, while PickUp and PickStep control continuation from a previous calculation.PickUp=True requires a supplied PickStep because no default is provided.
  • Output files: The results folder stores logs, similarity matrices, initial structures, locally optimized structures, and enthalpy-sorted iteration data.CALYPSO.log records space groups, volumes, and atom counts; similar.dat records bond characterization matrices.

5. Applications.

Applications demonstrate CALYPSO searches across elemental, binary, and ternary systems, including high-pressure phases and previously unknown structures, with rapid convergence to global minima.

  • Lithium: CALYPSO reproduced experimentally observed high-pressure Li structures and predicted new Aba2-40 and Cmca-56 phases at 80 and 200 GPa.Aba2-40 was later verified by an independent experiment.
  • Bismuth telluride: CALYPSO predicted β-Bi2Te3 and γ-Bi2Te3 monoclinic phases at 12 and 14 GPa, respectively, later verified by Rietveld refinement.
  • Unknown structures: CALYPSO also discovered unknown high-pressure structures in Mg, BC3, and BC7.
  • Search efficiency: All tested structures rapidly converged to the global minimum in fewer than 150 local optimizations.The results were presented as evidence of CALYPSO’s efficiency in crystal structure determination.
  • Method rationale: The method combines PSO, symmetry-constrained generation, similarity elimination, high-energy rejection, structural diversity, and local optimization.These techniques are described as contributing to global structural convergence and reduction of landscape noise.

6. Conclusions

The paper documents CALYPSO as a crystal-structure prediction method for specified compositions and external conditions. It combines several search and diversity techniques, benchmarks parameter choices on TiO2, and reports high efficiency and success across known and unknown systems.

  • CALYPSO predicts crystal structures at given chemical compositions and external conditions.
  • The implementation combines PSO, symmetry constraints, bond characterization matrices, partial random structures, and penalty functions.These techniques are described as crucial for predicting global stable structures.
  • TiO2 benchmarks provide suggested values for various CALYPSO parameters.
  • The paper reports high success rate and high efficiency, supporting CALYPSO’s reliability and promise for crystal structure determination.

Program availability

CALYPSO is distributed as free software for non-profit organizations, with source code, installation documentation, a user manual, and examples included.

  • CALYPSO is available online for non-profit use at the listed project website.
  • The package is delivered with Fortran source code.
  • Installation instructions and a PDF user manual are included.
  • Examples are included in the package.

Table and Figure captions

The implementation uses symmetry-constrained generation, local optimization, bond characterization matrices, PSO, and random structures to search crystal configurations efficiently. Tests examine convergence, structural diversity, and parameter effects in TiO2 and related systems.

  • Implementation and parameters: Symmetry constraints are used during structural generation, and larger systems generally use larger populations and require fewer generations.The TiO2 illustration uses population size 20; the typical Vmax and PSO-generated fractions are 0.1 and 0.6.
  • Results: The TiO2 tests report finding the global stable rutile structure, while the calculations also discovered new structures.
  • Figure comparisons: Energy distributions compare TiO2 and A2B structures generated with and without symmetric constraints.
  • Figure comparisons: The reported figures examine distortion distances for graphite and diamond and lattice-energy evolution during structural iterations with and without a penalty function.The comparison also includes runs where 0.6 of structures are selected for PSO and the remainder generated randomly versus using all structures for PSO.
  • Parameter studies: 0.6 PPSO required 12, 15, 85, and 110 generations across the reported TiO2 cases, while 1.0 required 11, 23, 124, and 229 generations.One 1.0 PPSO run failed to find the global stable structure within 100 generations one time out of ten.
  • Implementation and parameters: The workflow evaluates structures by local optimization, defines pbest and gbest, updates bond characterization matrices, and generates new structures by PSO or symmetry-constrained random sampling.
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