Transition State Optimization

The optimization of transition state geometries is best performed with the EF optimization algorithm. The TS keyword must be used in this case in order to tell the algorithm to look for a transition instead of a ground state. The following example uses the hydroboration reaction of ethylene with BH₃ to demonstrate the most important keywords. The algorithm used for ts optimization is rather similar to the one used in Gaussian and also locates the transition state in a local search after calculation of the Hessian matrix.

AM1 TS T=128H RECALC=5 GNORM=0.1 SCFCRT=0.000001
TS opt ethylene + BH3 (AM1)

 C    0.000000 0    0.000000 0    0.000000 0       0    0    0
 B    1.836364 1    0.000000 0    0.000000 0       1    0    0
 C    1.677200 1   47.717037 1    0.000000 0       2    1    0
 H    1.104000 1  120.462990 1  102.225822 1       1    3    2
 H    1.104000 1  120.463005 1 -102.226204 1       1    3    2
 H    1.104600 1  119.662224 1  106.660477 1       3    1    2
 H    1.104600 1  119.662292 1 -106.660301 1       3    1    2
 H    1.169800 1  118.564735 1  -87.511246 1       2    1    3
 H    1.243700 1   65.253059 1 -179.998581 1       2    1    3
 H    1.169800 1  118.563843 1   87.512947 1       2    1    3

In this particular case the TS keyword is augmented by RECALC=5 in order to recalculate the Hessian matrix every five geometry optimization steps. The accuracy of the optimization process is defined with GNORM=0.1 setting the average gradient norm convergence criterion to 0.1 (default = 1.0). In order to increase the accuracy of the energies, an increased SCF convergence criterion with SCFCRT=0.000001 is used. This defines a convergence criterion of 0.000001 kcal/mol between two consecutive SCF iterations (default = 0.0001 kcal/mol). This particular example optimizes the transition state within 25 geometry optimization cycles. The last cycles are described in the output files as follows:

 CYCLE:  21 TIME:    .01 TIME LEFT: 460799.2 GRAD.:      .146 HEAT: 49.08696    
 CYCLE:  22 TIME:    .01 TIME LEFT: 460799.2 GRAD.:      .179 HEAT: 49.08178    
 CYCLE:  23 TIME:    .01 TIME LEFT: 460799.2 GRAD.:      .195 HEAT: 49.07754    
 CYCLE:  24 TIME:    .00 TIME LEFT: 460799.2 GRAD.:      .329 HEAT: 49.07659    

          HESSIAN CALCULATED NUMERICALLY

 CYCLE:  25 TIME:    .14 TIME LEFT: 460799.0 GRAD.:      .040 HEAT: 49.07637    

     RMS GRADIENT =   .03970  IS LESS THAN CUTOFF =   .10000



 -------------------------------------------------------------------------------
 AM1 TS T=128H RECALC=5 GNORM=0.1 SCFCRT=0.000001
 TS opt ethylene + BH3 (AM1)
 


     GEOMETRY OPTIMISED USING EIGENVECTOR FOLLOWING (EF).   
     SCF FIELD WAS ACHIEVED

The HESSIAN CALCULATED NUMERICALLY statement indicates recalculation of the Hessian after every five optimization cycles. For each cycle, the gradient norm is given together with the heat of formation of the current structure. In cycle 25 the gradient norm becomes smaller than the convergence limit of 0.1 and the optimization is therefore terminated with the statement GEOMETRY OPTIMISED USING EIGENVECTOR FOLLOWING (EF). One of the problems that can occur during transition state optimization is that the Hessian has more than one negative eigenvalue. The optimization is then terminated with the WARNING! HESSIAN DOES NOT HAVE THE REQUIRED STRUCTURE statement and a new starting structure must be found in order to continue the search. Guessing a good starting structure is therefore an important condition for successful transition state optimizations.

Whether or not the optimized transition state structure is a true transition state must, as usual, be verified through explicit calculation of the second derivative matrix. The FORCE keyword is used for this purpose in MOPAC. The output files of these frequency calculations can be read with MOLDEN in order to obtain an animated picture of the vibrational modes.