Gradient extremals and steepest descent lines on potential energy surfaces

Relationships between steepest descent lines and gradient extremals on potential surfaces are elucidated. It is shown that gradient extremals are the curves which connect those points where the steepest descent lines have zero curvature. This condition gives rise to a direct method for the global determination of gradient extremals which is illustrated on the Müller–Brown surface. Furthermore, explicit expressions are obtained for the derivatives of the steepest-descent-line curvatures and, from them, for the gradient extremal tangents. With the help of these formulas, a new gradient extremal following algorithm is formulated.