Newton leaves on potential energy surfaces

The reaction path (RP) is an important concept of theoretical chemistry. We generalize the definition of the Newton trajectory (NT), as an RP, to Newton leaves in a higher dimensional subspace of the configuration space. Our standpoint is that of Bofill and Anglada [(2001) Theor. Chem. Acc. 105:436], who used a reduced potential energy surface for finding an RP. An NT follows an RP curve where the gradient is always a pointer to a fixed direction. More generally, a Newton leaf is a subspace of coordinates where the gradient can move in a subspace of directions. We report some known mathematical properties of Newton leaves. We explain the construction of Newton leaves with the example of a 3D test surface in ⁴ [W.Quapp et al. (1998) Theor. Chem. Acc. 100:285], because three coordinate dimensions are the smallest number of dimensions one needs at least to understand a Newton leaf in contrast to the known NTs.Acknowledgement The work was made possible through financial support of the Deutsche Forschungsgemeinschaft. The authors thank D. Heidrich for stimulating discussions.