Smoothing of Gradients

In ADF 2004.01 a method is implemented which is designed to smooth the gradient for small-ish perturbations in molecular geometry. This should help convergence in the last stages of a geometry optimization, and frequency calculations. We anticipate, for example, that it will be possible to perform frequency calculations with accint 4 with this option, rather than 5 or 6.

The reason for the smoothing is as follows: ADF generates integration points by dividing the 3D space up into Voronoy cells, and a spherical region around each atom. Unfortunately, the topology of the Voronoy cells is not always stable. The result is that in virtually every step in a geometry optimization the number of integration points changes. This can cause 'noise' in the gradient: even though the error in the gradient may not be excessively large, its magnitude and sign varies randomly with each change in geometry. This can cause the hessian (second derivative matrix) to be of poor quality. The new smoothing method is designed to make the error in the gradient vary systematically.

The way the smoothing works is to freeze the Voronoy cells in place from one step to the next whenever possible. The atoms are allowed to move (with their spherical regions) within these cells. Obviously after the atoms have been perturbed, the cells are no longer Voronoy cells of the molecular geometry, but there is nothing in the integration scheme that requires this.

By fixing the cells, we are able to regenerate the same integration points and weights each step. These points are shifted, and the weights adjusted, according to the atom's position in the cell. If an atom gets too close to the side of a cell, the freezing is relaxed, and the Voronoy cells recalculated. Another attempt to freeze the cells is then made at the next step. The smoothing can be made more effective if the DISHUL parameter in the INTEGRATION block key is increased, for example, to dishul=5.

This smoothing technique has been tested and found to considerably improve geometry optimization and frequency calculation results due to reduction of the numerical noise. As of ADF2005 smoothing is switched on for frequency calculations. It is off in all other cases by default. You can turn it on for any geometry run (e.g. geometry optimization, TS search, linear transit, etc) by using the 'smooth' subkey. You can use this in several ways. The first option is less dramatic:

GEOMETRY
 smooth freezecells
end

This option attempts to freeze the Voronoy cells between geometry steps, but does not reuse the points from the previous step. The points are instead recalculated, using the standard test integrals. Because the topology of the cells does not change, it is thought this may help somewhat, whilst still providing a rigorously tested integration grid.

The second option is more severe, but also more effective.

GEOMETRY
 smooth conservepoints
end

This option not only freezes the cells between steps, but also reuses the integration points of the previous step. It is recommended for frequency runs, as it should result in better gradient smoothing. The only disadvantage of this method is that there is no guarantee that the integral tests that ADF uses would be passed by the perturbed grid.

The third option is the most aggressive, but can be also most effective.

GEOMETRY
 smooth aggressive
end

This option not only freezes the cells between steps, reuses the integration points of the previous step, but also ignores some checks, which might lead to extra cells. By default, this option is on during frequency calculations.

The smoothing should be particularly effective for frequency calculations of molecules with no symmetry. In theory one should be able to rerun old frequency calculations with lower accint, for example, and still accurately reproduce the frequencies.

The method should help in the last stages of geometry optimizations, where the geometry is almost converged. In theory, you should now be able to use much lower gradient tolerances than were previously possible. It should also be possible to converge optimizations with lower accint than previously possible. accint 4 should suffice, rather than accint 6.

Geometry optimizations in which the molecule almost reaches convergence, but then continuously takes small steps around the minimum, should benefit greatly from the gradient smoothing.