Optimization for Atomistic Simulations

Nowadays, atomistic simulation is widely used to elucidate the mechanisms of experimentally known phenomena and to develop new materials by analyzing elemental combinations and structures. Building on ab initio or (semi-)empirical force fields, atomistic simulation models materials at the atomic level, seeking equilibrium states (including saddle points) and transition paths over the potential energy surface. One of the most fundamental tasks in this domain is structure relaxation (also known as structure/geometry optimization)¹.

Structure relaxation can be formulated as an optimization problem in terms of atomic positions (and lattice vectors, if considering crystals). Conventional methods include conjugate gradient methods (CG)² and quasi-Newton methods (QN)³, commonly found in off-the-shelf materials simulation software. Practically, CG offers better robustness but lacks efficiency, while QN converges quickly only near local minima. Theoretically, both lack global convergence guarantees, especially when physical constraints are present. These constraints are significant in reducing the dimensionality of the configurational space and are relevant in various downstream applications.

Regarding structure relaxation, my interests basically concern with developing computationally efficient and theoretically convergent optimization methods catering for various applications. Currently, my collaborators and I are primarily working on:

• Crystal structure relaxation under physical constraints.

Crystal structure relaxation under physical constraints

Collaborators (in the alphabetic order):

• Xingyu Gao (China Academy of Engineering Physics);
• Xin Liu (Chinese Academy of Sciences);
• Haifeng Song (China Academy of Engineering Physics);
• Zhen Yang (University of Science and Technology Beijing).
• Junlei Yin (Northwest Polytechnical University);
• Yafan Zhao (China Academy of Engineering Physics);

We address requirements from several contexts, encompassing relaxation

1. with fixed lattice vectors;
2. with a fixed unit cell volume;
3. under a given external pressure.

Notably, (2) is frequently seen in the calculations of the equations of state as well as material response functions, (3) finds itself in structure searches under high external pressures. Mathematically, (2) corresponds to a nonconvex determinant constrained problem, which has received little attention from the optimization community.

Our works significantly improve the execution of the line minimization process, where CG can easily get trapped. Our developed methods imitate the Newton method by exploiting the curvature of the potential energy surface, yet maintain single-step computational costs similar to CG. Additionally, to handle nonconvex constraints, we incorporate search direction projections onto the tangent cones of the feasible set. The overall convergence is ensured by an improved nonmonotone stopping rule for the line minimization process. Across a benchmark set containing over 200 structures with good universality, our methods achieve over 40% speedup compared to CG, without any failures.

We have developed a suite called “ProME-SuRe” for crystal structure relaxation with the above functionalities. The suite can interface with some off-the-shelf materials simulation packages. Static-link library files are available upon request!

References

• H., X. Gao, Y. Zhao, X. Liu, and H. Song. Force-based gradient descent method for ab initio atomic structure relaxation. Physical Review B, 2022, 106(10): 104101. (link)

• H., J. Yin, X. Gao, X. Liu, and H. Song. Projected gradient descent algorithm for ab initio crystal structure relaxation under a fixed unit cell volume. Physical Review B, 2024, 109(22): 224108. (link)

Patents and copyrights

• X. Gao, X. Liu, H., H. Song, X. Chen, Y. Wang, and L. Wang. 晶体结构弛豫软件包 ProMe-SuRe. CN Software Copyright, 2023, 2023SR1558824.

• X. Gao, X. Liu, H. Song, H., X. Chen, Y. Wang, J. Fang, L. Wang, and L. Zhang. 固定晶格体积晶体结构弛豫的计算方法及装置. CN Patent, 2023, ZL 202211210741.3.

• X. Gao, X. Liu, H. Song, H., J. Fang, Z. Yang, Y. Zhao, L. Wang, and H. Liu. 原子结构弛豫的非单调线搜索方法及装置. CN Patent, 2022, ZL 202111534901.5.

Ongoing works

• Optimization methods for crystal structure relaxation with spacegroup symmetry constraints.
• Error propagation analysis on structure relaxation.