Newton Leaves on Potential Energy Surfaces

by
 MICHAEL HIRSCH &  WOLFGANG QUAPP 

 Mathematical Institute,  University of Leipzig, 
 Augustus-Platz,  D-04109 Leipzig,  Germany
 


abstract:
The reaction path (RP) is an important concept of theoretical
chemistry. We generalize the definition of the Newton trajectory (NT),
as RP, to Newton leaves in a higher dimensional subspace of the 
configuration space. 
Our standpoint is that of Bofill and Anglada (2001) TCA {105}:{436} 
who have 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
 3-dimensional test surface in R^4 
[W.Quapp et al.\,(1998) TCA {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 Newton trajectories.