Step Scaling Criteria

For *SCALE* 0 the `step length' is taken to be the total step length in the multidimensional
space of Cartesian coordinates. For *SCALE* 1 we scale according to the largest displacement
in Cartesian coordinates. For *SCALE* 2 we scale according to the largest displacement
in the Hessian eigenvector basis. For *SCALE* 3 we scale according to the total step size
called for by the steepest-descent routine. The default for all other *SEARCH* types is now
*SCALE* 10 for which a different trust ratio is calculated for each eigendirection and used
to adjust the maximum allowed step for each eigenvector. This
is described in more detail below.[79]

*SCALE* 10 employs a trust radius whose default value is 4.0. This value can be changed
using the keyword *TRAD*.
*Appropriate values for the maximum initial step size and the trust radius
for different systems may need to be found by experimentation*. After the first step the
program compares the predicted value of the Hessian eigenvalue for a given eigendirection,
as calculated by finite difference of the present and previous component of the gradient in
this direction, with the actual eigenvalue. The trust ratio for a particular eigendirection
is the modulus of this predicted value minus the actual value, divided by the actual value.
If the trust ratio is less than the trust radius then the maximum step allowed in this
direction is increased by a factor of (up to a maximum of 0.5); if it is more
than the trust radius the maximum allowed step in this direction is divided by (down
to a minimum allowed value of ). The maximum allowed step for each direction is
initialised at the value read from the `odata` file.

The trust ratio is not used in a transition state search for the uphill direction if the
corresponding eigenvalue is positive; the step size is instead set to the value
of the *MAXMAX* parameter.

This trust ratio approach assumes that all the eigenvectors are in correspondence for the present and previous geometries when they are arranged according to the eigenvalues in ascending or descending order. When large steps are being taken some eigendirections are likely to cross over, and it might be better to use an overlap criterion for all the present and previous eigenvectors. However, this has not proved to be necessary to date.

Scaling of the above sort does not apply in :BFGS minimisations, although the
trust radius approach described above for *SCALE* 10 is used for the eigenvector-following
step in hybrid transition state searches.

For *SEARCH* 0, 2, 3 and 4 OPTIM detects convergence to stationary points with the
wrong Hessian index and steps away from them. The test is triggered when the RMS force
falls below the value specified by *PUSHCUT* (default )
in the prevailing units. The test is only applied at the first step and every fourth step.
If a transition state search converges to (or starts from) a minimum
a step is taken in the direction specified by *MODE* of magnitude *PUSHOFF*.
If *PUSHOFF* has not been set then the displacement is one tenth of the
current maximum step size allowed for the eigenmode in question.
If a transition state
search converges to (or starts from) a higher index saddle then the action depends upon the
value of *MODE*. For *MODE* 0 displacements are taken for all the eigenmodes
with negative eigenvalues except for the one with the most negative eigenvalue. The magnitude
of the displacement is determined in the same way as above.
For *MODE* a displacement is applied only along the specified value of *MODE*.
For search types 0 and 3 similar action is taken. If the search converges to (or starts from)
a stationary point that is not a minimum then if *MODE* 0 displacements are applied
for all the eigenmodes with negative eigenvalues. If *MODE* then a displacement
is only applied in the direction specified by *MODE*.

David Wales 2018-07-21