Locating transition states by quadratic image gradient descent on potential energy surfaces

Sun, Jun-Qiang; Ruedenberg, Klaus
August 1994
Journal of Chemical Physics;8/1/1994, Vol. 101 Issue 3, p2157
Academic Journal
An analysis is given of the so-called ‘‘image function’’ approach to finding transition states. It is demonstrated that, in fact, such functions do not exist for general potential energy surfaces so that a plain minimum search is inappropriate. Nonconservative image gradient fields do exist, however, and their field lines, defined by Euler’s equation, can lead to transition states as exemplified by quantitative integrations of these equations for the Müller–Brown surface. As do gradient fields, image gradient fields contain streambeds and ridges, but their global structure is considerably more complex than that of gradient fields. In particular, they contain certain singular points where the image gradients change sign without passing through zero. They are the points where the two lowest eigenvalues of the Hessian are degenerate. Some of them can act as singular attractors for the image gradient descent and any algorithm must contain safeguards for avoiding them. (Such regions are equally troublesome for quasi-Newton-type transition-state searches.) Image gradient fields appear to have considerably larger catchment basins around transition states than do quasi-Newton-type or gradient-norm-type transition-state searches. A quantitative quadratic image-gradient-following algorithm is formulated and, through applications to the Müller–Brown surface, shown to be effective in finding transition states.


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