Hessian Matrix Update Scheme for Transition State Search Based on Gaussian Process Regression

We show how Gaussian process regression can be used to update Hessian matrices using gradient-based information in the course of an optimization procedure. This is done by building a Gaussian process with at least one initial Hessian and some further energies and gradients from electronic structure calculations and evaluating the desired second derivative of the resulting Gaussian process. To a certain extent, we can overcome the significant scaling problems that occur when training a Gaussian process with Hessian information. We demonstrate in benchmark runs using the partitioned rational function optimization (P-RFO) that this new update method can outperform classical Hessian update methods for small systems.