Methods for finding transition states on reduced potential energy surfaces

Three new algorithms are presented for determining transition state (TS) structures on the reduced potential energy surface, that is, for problems in which a few important degrees of freedom can be isolated. All three methods use constrained optimization to rapidly find the TS without an initial Hessian evaluation. The algorithms highlight how efficiently the TS can be located on a reduced surface, where the rest of the degrees of freedom are minimized. The first method uses a nonpositive definite quasi-Newton update for the reduced degrees of freedom. The second uses Shepard interpolation to fit the Hessian and starts from a set of points that bound the TS. The third directly uses a finite difference scheme to calculate the reduced degrees of freedom of the Hessian of the entire system, and searches for the TS on the full potential energy surface. All three methods are tested on an epoxide hydrolase cluster, and the ring formations of cyclohexane and cyclobutenone. The results indicate that all the methods are able to converge quite rapidly to the correct TS, but that the finite difference approach is the most efficient.