9.1 Equilibrium Geometries and Transition-State Structures

9.1.3 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.873 For a geometry optimization, it solves for the lowest eigenvalue of the Hessian (λ1) and checks if λ1>0 (a negative λ1 indicates a saddle point); for a TS search, it solves for the lowest two eigenvalues, and λ1<0 and λ2>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×Niter total energy + gradient calculations, where Niter 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. 873, 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.

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

       Overide the built-in convergence criterion for the Davidson solver.
       0 (use the built-in default value 10-5)
       n Set the convergence criterion to 10-n.
       Use the default. If it fails to converge, consider loosening the criterion with caution.

Example 9.3  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.

   -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

   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_ENERGY        10
   GEOM_OPT_CHARAC            TRUE

Example 9.4  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.

   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

   JOBTYPE         freq
   EXCHANGE        B3LYP
   BASIS           6-31G
   SYMMETRY        false
   SYM_IGNORE      true



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