Schematic representation of the distance constrain in the saddle interpolation. The dashed line represents the hypersphere surface around the higher energy structure , is the starting point (displaced lower energy structure) and origin of the local optimization, and is the new optimized structure.
Schematic plot of the function. Here are also shown the singularities at and which usually occur but are not necessary for the discussion in the text.
Schematic plot of the function. The linearization right from is essential for the efficient root finding, i.e., the determination of .
Schematic plot of the function in the hard case, i.e., without the singularity at .
Schematic plot of the function for the second case discussed in the text (Sec. III B) with (a) and with (b).
Schematic plot of the function for the third case discussed in the text (Sec. III C) where the Hessian matrix is positive definite.
Schematic plot of the function for the hard case, i.e., without the singularity at . The situation for is depicted in (a), whereas the situation for is depicted in (b).
Approximated (left) and refined (right) TS structures for the rearrangement (top) and the addition to ethylene (bottom).
Refined TS structures for the ring opening of cyclopropyl obtained from a hierarchical TS search employing a calculated (a) and an approximated (b) start Hessian for the local uphill optimization.
Zero point corrected relative energies (kcal/mol) of all isomer and TS structures found for the IMDA reaction of -8-(5-methylfuran-2-yl)oct-3-en-2-one. The depicted reaction paths sketch the calculated IRC paths.
Test suite of 28 reactions covering dissociation/addition, rearrangement, inner rotation, and ring opening reactions. This table lists the starting reaction coordinate distances between the reactants and products ( column, in bohr) as well as the number of saddle interpolation cycles needed to approximate the TS structure (Cycles column). For comparison, the data from the PBE and VWN calculations are listed.
Validation of the uphill trust region optimization for the PBE functional using the same test suite of 28 reactions as in Table I. The numbers of local uphill optimization steps for the calculated start Hessian matrix as well as for the start Hessian approximations from Fischer and Almlöf (Ref. 86) and Lindh et al. (Ref. 91) are listed. The uphill optimization itself is always a quasi-Newton optimization.
Starting reaction coordinate distances (in bohr) for the three TS searches in the IMDA reaction of -8-(5-methylfuran-2-yl)oct-3-en-2-one. Also given are the number of saddle interpolation cycles needed to approximate the TS structure (Saddle) and the number of uphill trust region optimization steps (Uphill). For the local uphill optimization, the start Hessian matrix was calculated.
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