This section describes our redundant internal coordinate saddle point optimization algorithm, illustrated in figure . Section 2.1 describes the novel aspects of our algorithm for constructing internal coordinate system, in particular the addition of dummy atoms and how their associated degrees of freedom are handled. Section 2.2 discusses the construction and transformation of the various projected internal coordinate bases that are used in this work, including the nonredundant internal coordinates, the constrained internal coordinates, and the unconstrained nonredundant internal coordinates. Section 2.3 describes our iterative Hessian diagonalization algorithm, as well as the multi-secant Hessian update formula that is used to construct and update the approximate Hessian matrix. Section 2.4 describes how we have combined the null-space SQP constrained optimization formalism with the RS-PRFO saddle point optimization procedure.