HERO toolkit to accelerate DFT structural optimization by tuning the Hessian information in the BFGS method

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 nonquadratic 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 (WO3−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).

Citation: Li, Mingzhe, Piao Ma, Limin Li, Weijie Yang, and Hao Li. "HERO (Hessian-Engineered Relaxation Optimizer): Suppressing “Hessian Pollution” for Accelerated First-Principles Structural Relaxation." Computers, Materials, & Continua 88, no. 1 (2026).