6.1 Variable-Cell Nudged-Elastic-Band (VCNEB) method

Prediction of a phase transition mechanism can be considered as a double-ended problem, in which the algorithm has to locate the intermediate states. The nudged elastic band (NEB) 30; 31; 32 method is a widely used technique for solving double-ended problems, an efficient and robust approach for seeking the reaction paths and the saddle points along the “minimum energy path” (MEP) on the potential energy surface between the two endpoints. The NEB method has been successfully applied to molecular chemical reactions, surfaces, and defect migration, in particular it could provide the energy barrier between the given initial and final states of a phase transition process. Unfortunately, most of the problems treated by the NEB method are considered under the constraint of constant unit cell — which precludes it from being used for phase transitions (which involve the variation of the unit cell along the transition path).

Figure 14: The minimum energy path (line with gray circles) and initial path on a model 2D enthalpy surface. The forces in the VC-NEB method on Image $i$ are shown in the inset. $\mathbf{F}^{\nabla }_{i}$ is the potential force in the gradient direction. $\mathbf{F}^{\nabla \bot }_{i}$ and $\mathbf{F}^{s\parallel }_{i}$ are the transverse component of $\mathbf{F}^{\nabla }_{i}$ and the spring force, respectively.

The variable cell NEB (VC-NEB) method 14, which we have developed with somewhat different formulation, treats the cell and atomic coordinates on an equal footing and operates in an expanded configuration space under the condition of constant pressure. Our VC-NEB method within the first principles framework has been added to USPEX code as a new part. The VC-NEB method is a more general tool for exploring the activation paths between the two endpoints of a phase transition process within a larger configuration space. Every structure on the pathway in the VCNEB method is regarded as an “Image”.