Note that equation 25 is the traditional quadratic approximation to the potential energy surface, but with the gradient g replaced by the extrapolated gradient gex and the approximate Hessian of the potential energy B replaced by the approximate Hessian of the Lagrangian BL . Therefore, the procedure for obtaining s Q described above is equivalent to standard quasi-Newton minimization with some modifications to the gradient and approximate Hessian. It is possible to perform a constrained minimization using an alternative approach such as the restricted step rational function optimization (RS-RFO) method by making these same substitutions. This is useful, as unlike quasi-Newton, RS-RFO guarantees a downhill step even when the approximate Hessian BL is indefinite. Furthermore, making these substitutions in the restricted step partitioned rational function optimization (RS-PRFO) method will result in a constrained saddle point optimization step. Our algorithm implements constrained saddle point optimization by combining null-space SQP with RS-PRFO in this way.