LOCATING TRANSITION-STATES BY QUADRATIC IMAGE GRADIENT DESCENT ON POTENTIAL-ENERGY SURFACES

An analysis is given of the so-called ''image function'' approach to f inding transition states; It is demonstrated that, in fact, such funct ions do not exist for general potential energy surfaces so that a plai n minimum search is inappropriate. Nonconservative image gradient fiel ds do exist, however, and their field lines, defined by Euler's equati on, can lead to transition states as exemplified by quantitative integ rations of these equations for the Muller-Brown surface. As do gradien t fields, image gradient fields contain streambeds and ridges, but the ir 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 th e points where the two lowest eigenvalues of the Hessian are degenerat e. 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 lar ger catchment basins around transition states than do quasi-Newton-typ e or gradient-norm-type transition-state searches. A quantitative quad ratic image-gradient-following algorithm is formulated and, through ap plications to the Muller-Brown surface, shown to be effective in findi ng transition states.