Each element of q corresponds to one of the internal coordinates determined by the algorithm described in appendix A, and m is the total number of internal coordinates that have been determined. In practice, only the Cartesian coordinate vector x behaves as an independent variable for the optimization algorithm, and the internal coordinate vector q is calculated from x. Therefore, all changes to the internal coordinate vector q are realized as changes to x; this has important implications for the iterative diagonalization algorithm described in section 2.3 and the constrained saddle point optimization algorithm described in section 2.4 below. In this work, all undecorated vector and matrix quantities are defined to be in a basis of redundant internal coordinates with the exception of the Cartesian coordinate vector x. Quantities in other coordinate systems will be decorated with an accent character, as described below.