The uniqueness of Z-matrix coordinates on the other hand lies in the implicit dependence of the coordinates on the order. Atoms at the bottom of the definition, chained after several series of bonds, will have their positions strongly affected by coordinates defined at the start of the matrix. In order to keep a consistent coordinate definition, the gradients have to contain this propagation of structural changes. We will derive the necessary working equations later and show for a practical example how this distinction is relevant, but give here already a rough overview of this difference. The cartesian position of the i-th atom v i is given by its Z-matrix coordinates s(r, α, δ) and the orthonormal, righthanded, reference frame M of the three reference atoms with the origin at v b , which gives