9.2.3 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. ¹⁰⁹⁰ Sharada S. M., Bell A. T., Head-Gordon M.
J. Chem. Phys.
(2014), 140, pp. 164115. Link 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 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.  1090 Sharada S. M., Bell A. T., Head-Gordon M.
J. Chem. Phys.
(2014), 140, pp. 164115. Link , 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.