where w i are the trapezoidal rule weights. This objective function is minimized subject to path boundary conditions and constraints arising from parameterization invariance and translational/rotational symmetry. For this minimization, the nonlinear optimization problem solver IPOPT and its Python wrapper cyipopt 35 are used. For computationally expensive energy calculations, such as in first-principles methods, we adaptively update the energy evaluation points {t i } to focus the search around the TS using fewer points. The Python implementation of the direct MaxFlux method, which interfaces with energy and gradient calculations from various software through ASE, is available on GitHub. The direct MaxFlux method has a limitation in accurately describing the low-energy region when β is large, as the contribution of this region falls below the numerical precision in the objective function defined by eqs 16 and 18. However, this is not critical since the main interest is typically in the high-energy region, including TSs.