Once a guess structure to the transition state is obtained, standard eigenvector-following methods such as the Partitioned-Rational Function Optimization (P-RFO) [432] can be employed to refine the guess to the exact transition state. The reliability of P-RFO depends on the quality of the Hessian input, which enables the method to distinguish between the reaction coordinate (characterized by a negative eigenvalue) and the remaining degrees of freedom. In routine calculations therefore, an exact Hessian is determined via frequency calculation prior to the P-RFO search. Since the cost of evaluating an exact Hessian typically scales one power of system size higher than the energy or the gradient, this step becomes impractical for systems containing large number of atoms.

The exact Hessian calculation can be avoided by constructing an approximate Hessian based on the output of FSM. The tangent direction at the transition state guess on the FSM string is a good approximation to the Hessian eigenvector corresponding to the reaction coordinate. The tangent is therefore used to calculate the correct eigenvalue and corresponding eigenvector by variationally minimizing the Rayleigh-Ritz ratio [433]. The reaction coordinate information is then incorporated into a guess matrix which, in turn, is obtained by transforming a diagonal matrix in delocalized internal coordinates [434] [435] to Cartesian coordinates. The resulting approximate Hessian, by design, has a single negative eigenvalue corresponding to the reaction coordinate. This matrix is then used in place of the exact Hessian as input to the P-RFO method.

An example of this one-shot, Hessian-free approach that combines the FSM and P-RFO methods in order to determine the exact transition state from reactant and product structures is shown below:

**Example 9.199**

$molecule 0 1 Si 1.028032 -0.131573 -0.779689 H 0.923921 -1.301934 0.201724 H 1.294874 0.900609 0.318888 H -1.713989 0.300876 -0.226231 H -1.532839 0.232021 0.485307 **** Si 0.000228 -0.000484 -0.000023 H 0.644754 -1.336958 -0.064865 H 1.047648 1.052717 0.062991 H -0.837028 0.205648 -1.211126 H -0.8556026 0.079077 1.213023 $end $rem jobtype fsm fsm_ngrad 3 fsm_nnode 18 fsm_mode 2 fsm_opt_mode 2 method b3lyp basis 6-31g symmetry false sym_ignore true $end @@@ $rem jobtype ts scf_guess read geom_opt_hessian read max_scf_cycles 250 geom_opt_dmax 50 geom_opt_max_cycles 100 method b3lyp basis 6-31g symmetry false sym_ignore true $end $molecule read $end