Transition states using the image method

Transition states are found as saddle points on the potential energy surface. The simplest way of locating a first-order transition state, which are the chemically most interesting ones, is to use the trust-region image minimization method described in Ref. [24]. Such geometry optimizations may be considered as a special case of walks on second-order surfaces, and can be done using either the *WALK module or the *OPTIMIZE module. Just like for minimization the former uses a pure second-order method (analytical Hessians calculated at every geometry), while the latter gives you a choice of first- and second-order methods or combinations of the two.

A second-order optimization of a transition state can be requested by either adding .SADDLE and .2NDORD in the *OPTIMIZE section:

**DALTON INPUT
.OPTIMIZE
*OPTIMIZE
.SADDLE
.2NDORD
**WAVE FUNCTIONS
.HF
**END OF DALTON INPUT

or by adding the keyword .IMAGE in the *WALK section:

**DALTON INPUT
.WALK
*WALK
.IMAGE
**WAVE FUNCTIONS
.HF
**END OF DALTON INPUT

Any property may of course be specified at all stages of the optimization in the same fashion as for geometry minimizations.

The principle behind the trust-region image minimization is simple. A first-order transition state is characterized by having one negative Hessian eigenvalue. By reversing the sign of this eigenvalue, we have taken the mirror image of our potential surface along the associated mode, thus turning our problem into an ordinary minimization problem. A global one-to-one correspondence between the image surface and our potential energy surface is only valid for a second-order surface, but in general the lack of a global one-to-one correspondence seldom gives any problems.

The advantage of the trust-region image optimization as compared to for instance following gradient extremals lies mainly in the fact that we may take advantage of well-known techniques for minimization. In addition, the method does not need to be started at a stationary point of the potential surface which is necessary when following a gradient extremal (in fact, when using *OPTIMIZE the starting geometry should not be a minimum). We have so far not experienced that the trust-region image optimization fails to locate a first order transition state, even though this is by no means globally guaranteed from the approach. Note that the first-order saddle-points normally obtained using the image method starting at a minimum, often corresponds to conformational transition states, and thus not necessarily to the chemically most interesting transition states.

There are two approaches for locating several first-order saddle points with the trust-region image optimization. One may take advantage of the fact the image method is not dependent upon starting at a stationary point, and thus start the image minimization from several different geometries, and thus hopefully ending up at different first-order saddle points, as the lowest eigenmode at different regions of the potential energy surface may lead to different transition states.

The other approach is to request that not the lowest mode, but some other eigenmode is to be inverted. This can be achieved by explicitly giving the mode which is to be inverted through the keyword .MODE. This keyword is the same in both the *WALK and the *OPTIMIZE module. However, one should keep in mind that there will be crossings where a given mode will switch from the chosen mode to a lower mode. However, what will happen in these crossing points cannot be predicted in advance. Thus, for such investigations, the gradient extremal approach may prove equally well suited. Let us give an example of an input for a trust-region image optimization where the third mode is inverted:

**DALTON INPUT
.WALK
*WALK
.IMAGE
.MODE
 3
**WAVE FUNCTIONS
.HF
**END OF DALTON INPUT