Once a guess structure to the transition state is obtained, standard
eigenvector-following methods such as Baker’s partitioned rational-function
optimization (P-RFO) algorithm^{46} 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.^{832} 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.^{506} 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^{45, 267} 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.

$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 -0.855603 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