As mentioned in section 2.2, displacements in the internal coordinate space must be realized as changes to the Cartesian coordinates x, as these are the independent coordinate variables. In our algorithm, these displacements are realized using a geodesic internal coordinate stepping approach we have previously developed for minimization in a redundant internal coordinate basis. At each iteration of the diagonalization algorithm, the part of the Hessian which has already been evaluated Hi = YT i Ši is exactly diagonalized to find the leftmost eigenvalue θ i and eigenvector zi . This eigenvector, when projected into the larger nonredundant internal coordinate space through the relation ži = Ši zi , approximates the leftmost eigenvector of ȞL . The diagonalization procedure is repeated until the convergence criterion ři 2 < γ|θ i | is met, where ři = Yi zi -θ i ži is the residual vector and γ is a unitless convergence parameter which defaults to 0.1 in Sella. This procedure is a particular implementation of the Rayleigh-Ritz method. Implementations of the Rayleigh-Ritz method differ in how the matrix Ši is constructed. Our implementation uses the Olsen method, 21 a method related to the generalized Davidson and Jacobi-Davidson diagonalization algorithms. Olsen's method extends the matrix Ši with the vector obtained by orthonormalizing the vector ťi against Ši , where ťi is the solution to the equation