A Tutorial Review of Bayesian Optimization with Gaussian Processes to Accelerate Stationary Point Searches

Building local surrogates to accelerate stationary point searches on potential energy surfaces span decades of effort. Done correctly, surrogates can reduce the number of expensive electronic structure evaluations by factors of several, and in favorable regimes by roughly an order of magnitude, while preserving the accuracy of the underlying theory; the gain depends on oracle cost, search distance, and the availability of analytical forces. We present a unified Bayesian optimization view of minimization, single-point saddle searches, and double-ended path searches: all three share one six-step surrogate loop and differ only in the inner optimization target and the acquisition criterion. The framework uses Gaussian process regression with derivative observations, inverse-distance kernels, and active learning, and we develop optional extensions for production use, including farthest-point sampling with the Earth Mover's Distance, MAP regularization, an adaptive trust radius, and random Fourier features for scaling. An accompanying pedagogical Rust code demonstrates that all three applications use the same Bayesian optimization loop, bridging the gap between theoretical formulation and practical execution.