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Crystal Structure Prediction

The structure of the material is essential in order to carry out any materials modelling work. Very often, this information has to be obtained experimentally in the first place, or a reasonable guess must be performed. For example, if one want to compute the band structure of TiO2 using DFT, the crystal structure of that particular TiO2 polymorph (and there are many of them) is needed as the input. However, what if one want to model materials that has not been synthesised? Or perhaps looking for other polymorphs of the same composition that can potentailly have different properties?

Crystal structure prediction is the process of predicting the crystal structure of materials from no or limited experimental data. Quantum mechanics based first-principles calculations allow accurate energy/force/stress to be determined for any given atomic structure. Using these techniques, in theory, one can just type in the formula and the computer should be able to output the experimental crystal structure(s). This, however, turned out to be not so trivial and may involve significant amount of the computational work load. The root of the problem is the vast configurational spaces to be explored - for a periodic system with N atoms there are 3N + 1 dimensions, making locating the global minimum, or points having similar energy to the global minimum challenging.

Fortunately, many approaches have been developed to tackle this problem.

  • Simulated annealing
  • Basin hopping
  • Minimum hopping
  • Genetic algorithms
  • Particle swarm optimisation
  • Random search

The phrase crystal structure prediction is sometimes used to describe those based on subsituting species in known structures from databases, usually coupled with machine learning. While they offers essentially cost-free predictions for a given composition, they are often limited to systems that are well studied and have many experimental data available. While low energy structures for particular composition can be suggested, but there is no guarantee that the true ground state structure is among them, due to the lack of true exploration. Nevertheless, when apply to a large number of compositions, they can still provide valuable information of the chemical space.

Ab initio Random Structure Searching - brief introduction

Ab initio random structure searching (AIRSS)12 is an approach to search for low energy structure by simply generating random structures, followed by local optimisation. Local optimisation is a relatively simple and mathematically well-established, it found a nearby local minimum in the configuration space by simply going down-hill in energy. The potential energy surface (PES) can be divided into many basins-of-attraction. Performing a local optimisation inside a basin would lead us to bottom of this basin, e.g. the local minimum.

From first glance, such random sampling approach is deemed to fail for complex and high-dimensional configuration space due to the existence of many high energy local minima, making the chance of landing in the basin of attraction of the global minimum diminishing. However, one must not forget that the potential energy surface (PES) is a physical quantity, and hence their structure follow certain rules. While the number of high energy local minima may be overwhelming, the chance of landing into a basins-of-attraction is proportional to its hypervolume, and the lower the energy, the larger the hypervolume, favouring even a random sampler. In addition, we know that a large portion of the configuration space is not likely to contain any low energy minima - the atoms the bonds between them do have well-known finite sizes. The region of the PES including atoms very close to each other are unlikely to have any low energy minima, and hence can be excluded from sampling. Likewise, structures with atoms far apart from each other should also be excluded from sampling. With a few physical constraints, such as the species-wise minimum separations, including symmetry operations, a simple random search approach can be made highly efficient for locating the ground state structure and other low energy polymorphs. While the structures are randomly generated, the parameters controlling the generation process are not randomly chosen - they are motivated by physical reasons.

The phrase ab initio not only means that this approach can be used with first-principles methods for energy evaluation, but also that it should be used with them, since it is the physical nature of the PES that is been exploited here. While random search can be applied with parameterized interatomic potentials, the latter are unlikely to reproduce the true PES especially in the region far away from fitted phases. Hence, it is not a surprise if random search does not work well in those cases.

A consequence of random sampling is that the computational workload can be made fully parallel and distributed. There is no need to coordinate the search as a whole, each worker can work independently little or no inter-communication at all. In constrast, \"global optimisation\" methods such as basin hopping, requires each calculation to be performed in serial. Population based approaches such as genetic algorithms and particle swarm optimisation allow certain degree of parallelism within a single generation (tenth of structures), but each generation still have to be evaluated iteratively.

This means that for random searching:

  • The underlying DFT calculations can be parallelised over only a few CPUs each, maximising the parallel efficiently which otherwise can drop sharply with increasing core counts.
  • The elimination of iterative process means there is no dilemma of exploration or exploitation.
  • A further consequence is that the DFT calculations can be performed at relatively low basis set qulaity and accuracy to maximise the speed, usually at several times lower in the cost compared to normal calculations.
  • In addition, most structures include and perserv symmetry operations throughout the process, which can be used to speed-up DFT calculations by several folds.

  1. Pickard, C. J.; Needs, R. J. High-Pressure Phases of Silane. Phys. Rev. Lett. 2006, 97 (4), 045504. https://doi.org/10.1103/PhysRevLett.97.045504. ↩

  2. Pickard, C. J.; Needs, R. J. Ab Initio Random Structure Searching. Journal of physics. Condensed matter : an Institute of Physics journal 2011, 23 (5), 053201--053201. https://doi.org/10.1088/0953-8984/23/5/053201. ↩