where q k and J k are the internal coordinate vector and internal-Cartesian Jacobian matrix corresponding to x k , respectively. While this approach is computationally facile and generally converges in only a few iterations, it does not account for the curvature of the space of valid internal coordinates, which results in poor quality displacements when using highly redundant coordinate systems, for example for molecules with rings. We have previously developed an alternate approach for realizing displacements in redundant internal coordinate systems based on geodesics of the curved internal coordinate manifold. This geodesic approach accounts for the correlation between internal coordinates not only at the endpoints of the displacement, but also throughout the entire displacement trajectory. We have shown that this geodesic displacement strategy dramatically reduces the number of steps required to reach convergence in a minimization algorithm as compared to the traditional Newton displacement method. Following each geometry optimization step, the approximate Hessian is updated using the geodesic secant update, which is also described in our previous work. As mentioned previously, the parameter α in equation 28 (for minimization) and equations 29 and 30 (for saddle point optimization) is chosen to satisfy a trust region condition. In our trust region algorithm, a displacement vector s k is accepted only if it satisfies the trust region condition,