Cell-Filter NEB for Large-Scale Reaction Pathways

• CFNEB is a computational method that computes minimum-energy reaction pathways by treating atomic positions and dynamic lattice deformations as unified coordinates.
• It employs an adaptive cell-filtering scheme that dynamically refines the discrete path representation to capture both smooth and nucleation-driven transitions.
• Leveraging GPU acceleration, CFNEB scales to simulate up to 10^5 atoms, enabling detailed studies of solid-state phase transitions and interface migration.

The cell-filter nudged elastic band (CFNEB) method is a computational framework for determining minimum-energy reaction pathways (MEPs) and activation barriers between known initial and final states in condensed matter systems. Building on the traditional nudged elastic band (NEB) approach, CFNEB introduces rigorous treatment of both atomic configurations and lattice deformations as generalized coordinates, enabling the scalable modeling of transition mechanisms—including localized, nucleation-driven processes—in simulation cells containing up to $10^5$ atoms. CFNEB achieves high computational throughput by leveraging a GPU-accelerated engine and incorporates adaptive image management to refine the discrete path representation dynamically.

1. Foundation and Motivation

The NEB method is extensively utilized for exploring transition paths and energy barriers between metastable configurations in atomic-scale systems. Conventional NEB approaches parameterize the transition pathway as a discrete sequence (chain) of system "images" interpolating between initial and final states, with a spring force between images enforcing continuity. Standard NEB formulations typically optimize atomic positions under fixed cell conditions, restricting large-scale transformation and precluding the appearance of collective, cell-mediated mechanisms such as nucleation and spontaneous symmetry breaking in extended solids.

CFNEB addresses these limitations by explicitly incorporating the cell (lattice vectors) as dynamic variables alongside atomic coordinates. The method thereby enables simulation of phenomena where both local atomic rearrangement and macroscopic cell deformation are intertwined—crucial for modeling solid-solid phase transitions, interface migration, and mechanical response in large supercells where system size precludes concerted transitions.

2. Deformation-Based Cell Filtering Scheme

At the core of CFNEB is a generalized coordinate transformation that links atomic and lattice degrees of freedom on a unified scale. For a given reference simulation cell with lattice vectors $h_0$, the deformation gradient is defined as $D = h_0^{-1} h$, where $h$ are the current lattice vectors. Real-space atomic positions $R$ (row vectors) are mapped to "filtered" coordinates: $\tilde{R} = R \cdot D^{-1}$ The system's degrees of freedom are recast in the generalized coordinate vector: $$\mathcal{R} = (\tilde{R}_1,\, \ldots,\, \tilde{R}_n,\, D_1,\, D_2,\, D_3)^T$$ where $n$ is the number of atoms and $D_i$ are the columns of the deformation gradient.

To prevent unphysical rotational degrees of freedom that arise when lattice vectors evolve freely, CFNEB applies a rotation-elimination procedure. This switches to a symmetric strain formulation using the left Cauchy–Green tensor: $B = D D^T$ By optimizing the cell deformation through $B$, rotational contributions are excluded, resulting in numerically robust and physically meaningful cell dynamics during path optimization. The generalized force on the lattice is given by: $F_\text{lattice} = D^{-T} V D^{-1}$ where $V$ is the virial tensor.

3. Adaptive Image Insertion and Deletion

Capturing sharp, localized changes in the transition path—particularly those characteristic of nucleation mechanisms—necessitates dynamic refinement of the discrete chain of images. CFNEB implements an adaptive strategy that monitors image spacing in configuration space. Where the path exhibits large deviations or high curvature, additional images are inserted to enhance spatial resolution; conversely, images are removed from path segments displaying near-linear progression.

This adaptive management allows for accurate resolution of both smooth, concerted pathways and abrupt, spatially localized transitions, improving robustness when the initial guess is rough or the true MEP contains highly nonlinear segments.

4. Computational Scaling and Implementation

CFNEB is constructed for large-scale simulations by employing a GPU-accelerated implementation within the Graphics Processing Units Molecular Dynamics (GPUMD) engine. All major vector and matrix operations, including force evaluation and geometry transformations, are executed natively on the GPU. This approach achieves throughput on the order of $10^6$ atom$\cdot$steps per second on consumer GPUs such as the NVIDIA GeForce 4090.

The design allows simulation of systems with up to $10^5$ atoms and supports chains extending to 100 or more images, rendering scenarios with realistic nucleation and growth phenomena tractable. An implementation within the Atomic Simulation Environment (ASE) also exists, broadening compatibility with standard atomistic simulation workflows.

5. Representative Applications

CFNEB's capabilities are demonstrated on prototypical structural transitions inaccessible to small-scale NEB implementations:

• $\beta$–$\lambda$ Phase Transition in $Ti_3O_5$: Conventional NEB in small supercells reproduces a concerted sliding of layers. With large supercells and CFNEB, the pathway exhibits spontaneous symmetry breaking, producing a helical (spiral-like) layer displacement. This result indicates a nucleation-and-growth mechanism with localized shear–rotation coupling instead of a system-wide concerted event.
• Graphite-to-Diamond Transformation: In the graphite-to-cubic-diamond transition, CFNEB uncovers a nucleation-driven mechanism. The process begins with local bending of graphite layers and formation of interlayer bonds, nucleating a small diamond-like region. As the reaction proceeds, the diamond nucleus expands to a critical size at the transition state, showing faceting along specific crystallographic planes. As system size increases, this nucleated pathway supplants the concerted buckling mechanism commonly observed in smaller cells.

These examples establish that CFNEB not only reproduces known concerted mechanisms at small scales but also reveals emergent, spatially localized transition modes vital for understanding realistic transformation phenomena in solids.

6. Mathematical Formulation and Force Projection

The NEB force acting on each image is decomposed into a real component (energy gradient) and an artificial spring force along the chain: $$f^{(\mathrm{NEB})} = (f^{(\mathrm{real})})_\perp + f^{(\mathrm{spring})}$$ Strain evolution of the simulation cell is governed by: $h' = h (I + \epsilon)$ where $\epsilon$ is an infinitesimal strain tensor. The virial is expressed as: $$V = -\frac{1}{2} \sum_{i,j} [ r^{(ij)} \otimes f^{(ij)} ]$$ and the Cauchy stress as $\sigma = V/\Omega$ with $\Omega$ the cell volume.

By integrating deformation-based cell filtering into this framework, CFNEB achieves consistent scaling between atomic and cell degrees of freedom, ensuring stability and fidelity in path optimization for large systems.

7. Impact and Outlook

CFNEB substantially extends the domain of transition-path sampling by enabling atomistic investigation of processes involving both atomic rearrangement and cell deformation in large supercells. The method is suitable for the study of solid-state phase transitions, interfacial reactions, and mechanical deformation events where elastic and structural responses are coupled.

Its compatibility with machine-learning force fields and ab initio methodologies allows integration into multiscale modeling schemes. Anticipated extensions include incorporation of finite-temperature effects via free-energy corrections (e.g., thermodynamic integration or umbrella sampling) to better estimate kinetic viability, as well as further refinement of adaptive image management in regions exhibiting pronounced anharmonicity.

A plausible implication is that CFNEB will facilitate the discovery of previously obscured transition mechanisms—such as the onset of spontaneous symmetry breaking, critical nucleus formation, and growth—thus enabling more predictive modeling of complex materials transformations.