The reaction path of a mechanically induced chemical transformation changes under stress. It is well established that the force-induced structural changes of minima and saddle points, i.e., the movement of the stationary points on the original or stress-free potential energy surface, can be described by a Newton Trajectory (NT). Given a reactive molecular system, a well-fitted pulling direction, and a sufficiently large value of the force, the minimum configuration of the reactant and the saddle point configuration of a transition state collapse at a point on the corresponding NT trajectory. This point is called barrier breakdown point or bond breaking point (BBP). The Hessian matrix at the BBP has a zero eigenvector which coincides with the gradient. It indicates which force (both in magnitude and direction) should be applied to the system to induce the reaction in a barrierless process. Within the manifold of BBPs, there exist optimal BBPs which indicate what is the optimal pulling direction and what is the minimal magnitude of the force to be applied for a given mechanochemical transformation. Since these special points are very important in the context of mechanochemistry and catalysis, it is crucial to develop efficient algorithms for their location. Here, we propose a Gauss-Newton algorithm that is based on the minimization of a positively defined function (the so-called -function). The behavior and efficiency of the new algorithm are shown for 2D test functions and for a real chemical example.