Minimum energy paths for transitions such as atomic and/or spin rearrangements in thermalized systems are the transition paths of largest statistical weight. Such paths are frequently calculated using the nudged elastic band method, where an initial path is iteratively shifted to the nearest minimum energy path. The computational effort can be large, especially when ab initio or electron density functional calculations are used to evaluate the energy and atomic forces. Here, we show how the number of such evaluations can be reduced by an order of magnitude using a Gaussian process regression approach where an approximate energy surface is generated and refined in each iteration. When the goal is to evaluate the transition rate within harmonic transition state theory, the evaluation of the Hessian matrix at the initial and final state minima can be carried out beforehand and used as input in the minimum energy path calculation, thereby improving stability and reducing the number of iterations needed for convergence. A Gaussian process model also provides an uncertainty estimate for the approximate energy surface, and this can be used to focus the calculations on the lesser-known part of the path, thereby reducing the number of needed energy and force evaluations to a half in the present calculations. The methodology is illustrated using the two-dimensional Müller-Brown potential surface and performance assessed on an established benchmark involving 13 rearrangement transitions of a heptamer island on a solid surface.

The climbing image nudged elastic band method (CI-NEB) is used to identify reaction coordinates and to find saddle points representing transition states of reactions. It can make efficient use of parallel computing as the calculations of the discretization points, the so-called images, can be carried out simultaneously. In typical implementations, the images are distributed evenly along the path by connecting adjacent images with equally stiff springs. However, for systems with a high degree of flexibility, this can lead to poor resolution near the saddle point. By making the spring constants increase with energy, the resolution near the saddle point is improved. To assess the performance of this energy-weighted CI-NEB method, calculations are carried out for a benchmark set of 121 molecular reactions. The performance of the method is analyzed with respect to the input parameters. Energy-weighted springs are found to greatly improve performance and result in successful location of the saddle points in less than a thousand energy and force evaluations on average (about a hundred per image) using the same set of parameter values for all of the reactions. Even better performance is obtained by stopping the calculation before full convergence and complete the saddle point search using an eigenvector following method starting from the location of the climbing image. This combination of methods, referred to as NEB-TS, turns out to be robust and highly efficient as it reduces the average number of energy and force evaluations down to a third, to 305. An efficient and flexible implementation of these methods has been made available in the ORCA software.

The implementation of a novel tight-binding Hamiltonian within the QCEIMS program allows the first-principles based computation of EI mass spectra within a few hours for systems containing elements up to Z = 86.

Calculations of minimum energy paths for atomic rearrangements using the nudged elastic band method can be accelerated with Gaussian process regression to reduce the number of energy and atomic force evaluations needed for convergence. Problems can arise, however, when configurations with large forces due to short distance between atoms are included in the data set. Here, a significant improvement to the Gaussian process regression approach is obtained by basing the difference measure between two atomic configurations in the covariance function on the inverted interatomic distances and by adding a new early stopping criterion for the path relaxation phase. This greatly improves the performance of the method in two applications where the original formulation does not work well: a dissociative adsorption of an H 2 molecule on a Cu(110) surface and a diffusion hop of an H 2 O molecule on an ice Ih(0001) surface. Also, the revised method works better in the previously analyzed benchmark application to rearrangement transitions of a heptamer island on a surface, requiring fewer energy and force evaluations for convergence to the minimum energy path.