Structural Optimization in Tensor LEED Using a Parameter Tree and $R$-Factor Gradients

TL;DR

Confronts the high cost of LEED I(V) structure optimization by reformulating the parameter space with a tree-like, affine-parameter mapping to an irreducible space and by enabling end-to-end differentiability through automatic differentiation in JAX. The core contributions are a on-the-fly tensor-LEED implementation (viperleed-jax) with gradients of the $R$-factor with respect to irreducible parameters and a two-stage search combining CMA-ES exploration with gradient-based refinement. Benchmarks on a-Fe$_2$O$_3$ and related surfaces show >10x reductions in wall time and convergence to consistent minimum $R$-factors compared to previous TensErLEED-based workflows. The framework is modular, open-source under GPLv3, and extensible to other surface diffraction methods.

Abstract

Quantitative low-energy electron diffraction [LEED $I(V)$] is a powerful method for surface-structure determination, based on a direct comparison of experimentally observed $I(V)$ data with computations for a structure model. As the diffraction intensities $I$ are highly sensitive to subtle structural changes, local structure optimization is essential for assessing the validity of a structure model and finding the best-fit structure. The calculation of diffraction intensities is well established, but the large number of evaluations required for reliable structural optimization renders it computationally demanding. The computational effort is mitigated by the tensor-LEED approximation, which accelerates optimization by applying a perturbative treatment of small deviations from a reference structure. Nevertheless, optimization of complex structures is a tedious process. Here, the problem of surface-structure optimization is reformulated using a tree-based data structure, which helps to avoid redundant function evaluations. In the new tensor-LEED implementation presented in this work, intensities are computed on the fly, eliminating limitations of previous algorithms that are limited to precomputed values at a grid of search parameters. It also enables the use of state-of-the-art optimization algorithms. Implemented in \textsc{Python} with the JAX library, the method provides access to gradients of the $R$ factor and supports execution on graphics processing units (GPUs). Based on these developments, the computing time can be reduced by more than an order of magnitude.

Figures (5)

• Figure 1: Schematic representation of the parameter tree. The input parameters of the tensor-LEED calculation are $\xi = \{\xi_i\} = \{ \mathbf{r}_i, v_i, c_i, \mu \}$post_processing_footnote. They are linked to the vector of irreducible parameters $\tilde{\xi} = \{\tilde{\xi}_j\}$ via system and user constraints. Dashed lines represent links due to user constraints, which depend on user input and may be changed. Solid lines represent links which are automatically created and do not depend on user inputs (symmetry restrictions and normalizations).
• Figure 2: Timing benchmarks for the calculation of Pendry's R factor and its gradient for the test systems: a-Fe2O3(1-102)-(1x1)R (described in \ref{['sec:benchmarks']}\ref{['sec:benchmarks']}), Ir(100)-(2x1)R O, and Pt25Rh75-(3x1)R O (described in Section S14 of the Supporting Information supporting). The number of free (irreducible) parameters for these systems are 33, 14 and 40 respectively. Timing benchmarks for the evaluation of Pendry's R factor are also shown for Pt(111)-(10x10)R Te10x10 (119 free parameters; no gradients calculated due to memory limitations, see Section S14 of the Supporting Information supporting for details). The computing times (with an Nvidia A100 GPU) are shown as a function of the angular momentum quantum number cutoff $\ell_{\mathrm{max}}$. $R_{\mathrm{P}}$-factor evaluations are shown with solid symbols, while gradient evaluations are shown with open symbols. The evaluation time scales almost exponentially with $\ell_{\mathrm{max}}$. Dashed lines indicate the exponential fit of the timing data against the cutoff $\ell_{\mathrm{max}}$. Arrows indicate the factor by which the gradient calculation is slower than the R-factor calculation at the respective $\ell_\text{max}$ value.
• Figure 3: Comparison of runtime and number of evaluations (top), as well as reliability (bottom) of local optimization algorithms near the global minimum. Algorithms were compared using the second segment (48 irreducible parameters) of the a-Fe2O3(1-102)-(1x1)R optimization described in \ref{['sec:hematite_oneby']}\ref{['sec:hematite_oneby']}, starting from 98 configurations (one per generation) selected from the CMA-ES run. The algorithms shown were found to be the most reliable: sequential least-squares quadratic programming (SLSQP) kraftSoftwarePackageSequential1988, Broyden--Fletcher--Goldfarb--Shanno (BFGS) nocedalNumericalOptimization2006, and Powell powellEfficientMethodFinding1964pressNumericalRecipesArt2007, as implemented in SciPy2020SciPy-NMeth. Gradient-based algorithms are shown to the left of the dashed line, while finite-difference methods are shown to the right. Algorithms labeled as "2-point" employ two-point finite-difference approximations for the gradients. Computational details are provided in Section S3 of the Supporting Information supporting. In each box plot, boxes extend from the first to the third quartile; whiskers extend to the furthest data point within $1.5 \times$ of the inter-quartile range. Small circles indicate single outliers beyond the range of the whiskers. Tests using an improved R factor Imre2025PendryRfactor show that outliers in the bottom panel are due to the noisiness of Pendry's R factor.
• Figure 4: Comparison of structure optimization for the a-Fe2O3(1-102)-(1x1)R surface using viperleed-jax and the TensErLEED backend in viperleed.calc. The running minimum of the R factor is shown as a function of execution time. Optimization proceeds in three segments of alternating reference calculations and tensor-LEED-based optimizations. The starting configuration for each segment corresponds to the final structure of the previous segment, with the first segment beginning from a bulk-truncated structure. R-factor values of the reference calculations are shown as filled circles. Broken horizontal lines indicate the time required for setup and just-in-time compilation in the new tensor-LEED implementation. Dotted vertical lines mark differences in the R factor values between the end of an optimization segment and the subsequent reference calculation, reflecting errors from the perturbative tensor-LEED approximation that are corrected by the reference calculations. A dashed gray line indicates the smallest R-factor value ($R_{\mathrm{P}}=0.165$) achieved by the new tensor-LEED implementation.
• Figure 5: Structure optimization for a-Fe2O3(1-102)-(1x1)R using a two-stage minimization approach. Exploratory and local stages used CMA-ES wanzenbockClinamen2FunctionalstyleEvolutionary2024 and SLSQP 2020SciPy-NMeth algorithms, respectively. (a) $R_{\mathrm{P}}$ factor as a function of time. The dashed line marks the switch from the exploratory to the local optimization stage. (b) Points evaluated by the CMA-ES algorithm, projected onto two dimensions using PCA. Colors indicate generation numbers with the same scheme as in panel (a). Contour lines display the R-factor in a plane spanned by the first two principal components. The mean and the individual with the smallest R factor for each generation are highlighted by orange and a black lines, respectively. (c) Trajectory of the local optimization, projected onto a 2D plane [different from that of panel (b)] to highlight convergence towards the minimum. Evaluated configurations and the running minimum are represented as individual points and as a black line, respectively. The blue star marks the final configuration with $R_{\mathrm{P}}=0.387$.