On the implementation of the Runge–Kutta–Fehlberg algorithm to integrate intrinsic reaction coordinate paths

A new algorithm for the integration of the intrinsic reaction coordinate path on a potential energy surface is proposed, based in solving the set of differential equations associated to the corresponding characteristic curve, rather than the differential equation of the curve itself. The integration of this set of differential equations is carried out by employing a Runge–Kutta–Fehlberg with τ-stage and p-algebraic order technique. In addition, an update Hessian matrix formula is reported for this specific algorithm. The examples show that the proposed algorithm is numerically stable, with no extra computational effort with respect to the standard algorithms.