Gradient extremals on N-dimensional energy hypersurfaces V=V(x1 ⋯ x n ) are curves defined by the condition that the gradient ∇V is an eigenvector of the hessian matrix ∇∇V. For variations which are restricted to any (N−1) dimensional hypersurface ∇V(x1⋯ x N ) = V0= constant, the absolute value of the gradient ∇V is an extremum at those points where a gradient extremal intersects this surface. In many, though not all, cases gradient extremals go along the bottom of a valley or along the crest of a ridge. The properties of gradient extremals are discussed through a detailed differential analysis and illustrated by an explicit example. Multidimensional generalizations of gradient extremals are defined and discussed.
Key wordsPotential energy surfaces Reaction paths
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