HERO (Hessian-Engineered Relaxation Optimizer): Suppressing “Hessian Pollution” for Accelerated First-Principles Structural Relaxation
Mingzhe Li1,2,3,4, Piao Ma2,4, Limin Li4, Weijie Yang1,*, Hao Li3,*
1 Department of Power Engineering, North China Electric Power University, Baoding, China
2 Gusu Laboratory of Materials, Suzhou, China
3 Advanced Institute for Materials Research (WPI-AIMR), Tohoku University, Sendai, Miyagi, Japan
4 Suzhou MatSource Technology Co., Ltd., Suzhou, China
* Corresponding Author: Weijie Yang. Email: 
; Hao Li. Email: 
Computers, Materials & Continua https://doi.org/10.32604/cmc.2026.079131
Received 15 January 2026; Accepted 02 April 2026; Published online 22 April 2026
Abstract
Structural optimization is a fundamental step in density functional theory (DFT) calculations, typically driven by the Broyden–Fletcher–Goldfarb–Shanno (BFGS) optimizer. However, the standard BFGS algorithm relies on a local quadratic approximation of the potential energy surface (PES), which frequently breaks down in highly non-quadratic regimes typical of complex surface adsorption systems and defective bulk materials. This breakdown leads to “Hessian pollution”, a phenomenon where higher-order anharmonicities introduce spurious off-diagonal inter-atomic couplings that distort curvature estimates and significantly stall convergence. Herein, we propose a physics-inspired algorithmic intervention to the BFGS method that systematically suppresses this pollution. Once the maximum residual force drops below a specific activation threshold (e.g., 0.5 or 0.1 eV/Å), our approach conditionally resets all off-diagonal Hessian blocks, and introduces an isotropic background stiffness strategy where these blocks can be repopulated with a small positive constant rather than zeroed completely. This balances the robust stability of diagonal dominance with accelerated convergence speed. Implemented as an add-on to the Atomic Simulation Environment (ASE) Library, the method is lightweight, transferable, and compatible with standard DFT codes. Tests across diverse chemical systems, including atomic and molecular adsorbates (O*, H*, CO*) on Pt(111) surfaces and defective bulk oxides (WO
3–x), demonstrate substantial reductions in the number of required force calls without biasing the final optimized geometry. It offers a practical tool for high-throughput DFT workflows that eliminates the need for domain-specific training. This method is available via our open-source package, Hessian-Engineered Relaxation Optimizer (HERO).
Keywords
Hessian-engineered relaxation optimizer; density functional theory (DFT); structural optimization; BFGS algorithm; hessian pollution; structural relaxation
Open Access
Login
Register
Submit a Paper
Propose a Special lssue
Download PDF
Downloads
Citation Tools
