where a superscript plus symbol indicates the Moore-Penrose pseudoinverse. As before, we treat singular values below 10 -6 as zero and assign the corresponding singular vectors to the unconstrained space. Using the projection matrices defined in equations 1 and 5 above, we define the m × p matrix P and the m × (3n -6 -p) matrix Q for projection directly from the redundant internal coordinate system to the constrained and unconstrained nonredundant internal coordinate spaces respectively as