# 10.1.3 Hessian-Free Characterization of Stationary Points

Q-Chem allows the user to characterize the stationary point found by a geometry optimization or transition state search without performing a full analytical Hessian calculation, which is sometimes unavailable or computationally unaffordable. This is achieved via a finite difference Davidson procedure developed by Sharada et al.832 For a geometry optimization, it solves for the lowest eigenvalue of the Hessian ($\lambda_{1}$) and checks if $\lambda_{1}>0$ (a negative $\lambda_{1}$ indicates a saddle point); for a TS search, it solves for the lowest two eigenvalues, and $\lambda_{1}<0$ and $\lambda_{2}>0$ indicate a transition state. The lowest eigenvectors of the updated P-RFO (approximate) Hessian at convergence are used as the initial guess for the Davidson solver.

The cost of this Hessian-free characterization method depends on the rate of convergence of the Davidson solver. For example, to characterize an energy minimum, it requires $2\times N_{\mathrm{iter}}$ total energy + gradient calculations, where $N_{\mathrm{iter}}$ is the number of iterations that the Davidson algorithm needs to converge, and “2" is for forward and backward displacements on each iteration. According to Ref. 832, this method can be much more efficient than exact Hessian calculation for substantially large systems.

Note:  At the moment, this method does not support QM/MM or systems with fixed atoms.

GEOM_OPT_CHARAC
Use the finite difference Davidson method to characterize the resulting energy minimum/transition state.
TYPE:
BOOLEAN
DEFAULT:
FALSE
OPTIONS:
FALSE do not characterize the resulting stationary point. TRUE perform a characterization of the stationary point.
RECOMMENDATION:
Set it to TRUE when the character of a stationary point needs to be verified, especially for a transition structure.

GEOM_OPT_CHARAC_CONV
Overide the built-in convergence criterion for the Davidson solver.
TYPE:
INTEGER
DEFAULT:
0 (use the built-in default value 10${}^{-5}$)
OPTIONS:
$n$ Set the convergence criterion to 10${}^{-n}$.
RECOMMENDATION:
Use the default. If it fails to converge, consider loosening the criterion with caution.

Example 10.3  Geometry optimization of a triflate anion that converges to an eclipsed conformation, which is a first order saddle point. This is verified via the finite difference Davidson method by setting GEOM_OPT_CHARAC to TRUE.

$molecule -1 1 C 0.00000 -0.00078 0.98436 F -1.09414 -0.63166 1.47859 S 0.00000 0.00008 -0.94745 O 1.25831 -0.72597 -1.28972 O -1.25831 -0.72597 -1.28972 O 0.00000 1.45286 -1.28958 F 1.09414 -0.63166 1.47859 F 0.00000 1.26313 1.47663$end

$rem JOBTYPE opt METHOD BP86 GEOM_OPT_DMAX 50 BASIS 6-311+G* SCF_CONVERGENCE 8 THRESH 14 SYMMETRY FALSE SYM_IGNORE TRUE GEOM_OPT_TOL_DISPLACEMENT 10 GEOM_OPT_TOL_ENERGY 10 GEOM_OPT_TOL_GRADIENT 10 GEOM_OPT_CHARAC TRUE$end


Example 10.4  TS search for alanine dipeptide rearrangement reaction beginning with a guess structure converges correctly. The resulting TS structure is verified using the finite difference Davidson method.

$molecule 0 1 C 3.21659 -1.41022 -0.26053 C 2.16708 -0.35258 -0.59607 N 1.21359 -0.16703 0.41640 C 0.11616 0.82394 0.50964 C -1.19613 0.03585 0.74226 N -2.18193 -0.02502 -0.18081 C -3.43891 -0.74663 0.01614 O 2.19596 0.25708 -1.63440 C 0.11486 1.96253 -0.53088 O -1.29658 -0.59392 1.85462 H 3.25195 -2.14283 -1.08721 H 3.06369 -1.95423 0.67666 H 4.20892 -0.93714 -0.22851 H 1.24786 -0.78278 1.21013 H 0.25990 1.31404 1.47973 H -2.02230 0.38818 -1.10143 H -3.60706 -1.48647 -0.76756 H -4.29549 -0.06423 0.04327 H -3.36801 -1.25875 0.98106 H -0.68664 2.66864 -0.27269 H 0.01029 1.65112 -1.56461 H 1.06461 2.50818 -0.45885$end

$rem JOBTYPE freq EXCHANGE B3LYP BASIS 6-31G SCF_MAX_CYCLES 250 SYMMETRY false SYM_IGNORE true$end

@@@

$molecule read$end

$rem JOBTYPE ts SCF_GUESS read GEOM_OPT_DMAX 100 GEOM_OPT_MAX_CYCLES 1500 EXCHANGE B3LYP BASIS 6-31G MAX_SCF_CYCLES 250 GEOM_OPT_HESSIAN read SYMMETRY false SYM_IGNORE true GEOM_OPT_CHARAC true$end