# Transition State¶

A transition state (TS) search is very much like a minimization: the purpose is to find a stationary point on the energy surface, primarily by monitoring the energy gradients, which should vanish. The difference between a transition state and a (local) minimum is that at the transition state the Hessian has a negative eigenvalue.

Because of the similarities between a minimization and a TS search most subkeys in geometry are applicable in both cases, see the Geometry Optimization section. However, practice shows that transition states are much harder to compute than a minimum. For a large part this is due to the much stronger anharmonicities that usually occur near the ts, which threaten to invalidate the quasi-Newton methods to find the stationary point. For this reason it is good advice to be more cautious in the optimization strategy when approaching a Transition State and for some subkeys the default settings are indeed different from those for a simple optimization. In addition, certain additional aspects have to be addressed.

GEOMETRY
TransitionState
{Mode Mode}
{NegHess=NegHess}
End
End

NegHess
The number of negative eigenvalues that the Hessian should have at the saddle point. In the current release it is a rather meaningless key, which should retain its default value (1).
Mode

Controls the first step from the starting geometry towards the saddle point: it specifies in which direction the energy is to be maximized while the optimization coordinates will otherwise be varied so as to minimize the energy. A positive value means that the eigenvector #mode of the (initial) Hessian will be taken for the maximization direction. This means: put all Hessian eigenvalues in ascending order, ignoring those that correspond to impossible movements (rigid rotations and translations, symmetry breaking) and then take the eigenvector of #mode in the remaining list.

Default: mode=1. Generally the program performs best with this default: it will simply concentrates on the mode with the lowest eigenvalue, which should of course finally be the path over the transition state (negative eigenvalue).

After the first geometry step, the subsequent steps will attempt to maximize along the eigenvector that resembles most (by overlap) the previous maximization direction.

As mentioned before, the other subkeys have the same functionality as for minimizations, but different defaults or options may apply:

Hessupd
HessianUpdate

Different (fewer) options apply now. The methods available depend on the optimization branch being used. For both the old and new branches, the following options are available:

1. Powell: Powell
2. BFGS: Broyden-Fletcher-Goldfarb-Shanno
3. BOFILL : Bofill, Eq. (13) and (14) of Ref. [139]
4. MS : Murtagh-Sargent In the old branch, the following extra option is available: (i) DFP : Davidon-Fletcher-Powell The default is Powell for transition state searches.
Step
MaxRadStep
Default: 0.2 angstrom for Z-matrix optimization, 0.1 angstrom for Cartesian optimization. Not used in the new branch.
MaxAngleStep
Default: 5 degrees. Not used in the new branch.

Note: in Transition State searches precision is often much more critical than in minimizations. One might consider using NumericalQuality Good or better.

## Transition State Reaction Coordinate (TSRC)¶

A reaction coordinate for transition state search can be specified via the TSRC input block, similar to QUILD. The principle is very similar to the eigenvector-following (EF) method. In the EF method, the specified Hessian eigenvector is stored on the 1st iteration. On later iterations it is compared with eigenvectors of the updated Hessian. The eigenvector that matches the stored one the best is selected as the reaction coordinate and the step is made in the up-hill direction along this vector and down-hill along all other vectors. With TSRC, the specified TSRC vector is used instead of a Hessian eigenvector.

This feature is especially useful when the accurate Hessian is not available. In such a case ADF uses an approximate Hessian that can be poor when weak interactions and/or transition metals are involved. What then happens is that the mode with the lowest Hessian eigenvalue does not correspond to the reaction coordinate along which transition state is searched for, thus leading the optimization in the wrong direction.

This problem can now be solved by specifying a reaction coordinate along which the transition state should be searched for. Such a reaction coordinate can consist of one or more distance, valence or dihedral angle, or just a combination of vectors on certain atoms.

TSRC
{ATOM i x y z {fac}}
{DIST i j {fac}}
{ANGLE i j k {fac}}
{DIHED i j k l {fac}}
end

i, j, k, l, x, y, z, fac
Here, i, j, k, and l are atom indices, x, y, and z are corresponding components of a TSRC vector for atom i, and fac is the factor (and thus also the sign) of a particular coordinate in the TSRC. By default fac = 1.0.

Restrictions and notes

The TSRC feature does not work in combination with the old optimization branch. In general, the old branch is no longer developed so all new features related to geometry optimization work with the new branch only. The DIST, ANGLE and DIHED specifications should be used in combination with optimization in delocalized coordinates only (i.e. not with Cartesian).

Only one type of the keyword is allowed in a TSRC block. That is, the keys must be either all ATOM or all DIST, etc. Thus, mixing of different keywords is not allowed.

One should be careful when specifying more than one bond or angle as a TSRC. For example, suppose atom 2 is located between atoms 1 and 3. Then the following TSRC block:

TSRC
DIST 1 2  1.0
DIST 3 2 -1.0
END


means that the TSRC consists of two distances: R(1-2) and R(2-3). The positive direction of the TSRC is defined as increase of the distance R(1-2) and decrease of the distance R(2-3), which, in turn, means that this TSRC corresponds to atom 2 is moving along the R(1-3) axis.