Precision¶

The quality of the calculation, given the selected model Hamiltonian - density-functional, relativistic features, spin-restricted/unrestricted… - is determined to a large extent by several technical precision parameters.

The most significant ones are:

Basis set

Obviously, the quality of the basis set may have a large impact on the results. As a general rule, minimum and almost-minimum basis sets (types SZ and DZ) may be used for pilot calculations, but polarization functions should be included (DZP, TZP) for more reliable results.

SCF convergence

The self-consistent-field (SCF) and geometry optimization procedures terminate when convergence criteria are satisfied. If these are set sloppy the results may carry large error bars. The default SCF convergence tolerance is tight enough to trust the results from that aspect. However, when the SCF procedure encounters severe problems an earlier abort may occur, namely if a secondary (less stringent) criterion has been satisfied (see the key SCF). Although this still implies a reasonable convergence, one should be aware that for instance the energy may be off by a few milli-Hartree (order of magnitude, may depend quite a bit on the molecule). It is recommended that in such cases you try to overcome the SCF problems in a secondary calculation, by whatever methods and tricks you can come up with, rather than simply accept the first outcomes. Note: in a geometry optimization the SCF convergence criteria are relaxed as long as the geometry optimization has not yet converged. This should generally not affect the final results: the SCF density and hence the energy gradients may be somewhat inaccurate at the intermediate geometries, but since these are not a goal in themselves the only concern is whether this might inhibit convergence to the correct final geometry. Our experiences so far indicate that the implemented procedure is reliable in this aspect.

Geometry convergence

This is a far more troublesome issue. Three different types of convergence criteria are monitored: energy, gradients and coordinates. The energy does not play a critical role. Usually the energy has converged well in advance of the other items. The coordinates are usually what one is interested in. However, the program-estimated uncertainty in the coordinates depends on the Hessian, which is not computed exactly but estimated from the gradients that are computed in the various trial geometries. Although this estimated Hessian is usually good enough to guide the optimization to the minimum - or transition state, as the case may be - it is by far not accurate enough to give a reasonable estimate of force constants, frequencies, and as a consequence, neither of the uncertainties in the coordinates. An aspect adding to the discrepancy between the Hessian-derived coordinate-errors and the true deviations of the coordinates from the minimum-energy geometry is that the true energy surface is not purely quadratic and using the Hessian neglects all higher order terms. The gradients provide a better criterion for convergence of the minimizer and therefore it is recommended to tighten the criterion on the gradients, rather than anything else, when stricter convergence than the default is required. The default convergence criteria, in particular for the gradients, are usually more than adequate to get a fair estimate of the minimum energy. Tighter convergence should only be demanded to get more reliable coordinate values (and in particular, when the equilibrium geometry needs to be determined as a preliminary for a Frequencies run).

Numerical integration accuracy

The key BeckeGrid key block (or alternatively the old INTEGRATION key block) determines the numerical precision of integrals that are evaluated in ADF by numerical integration, primarily the Fock matrix elements and most of the terms in the gradients. In addition the integration settings also determine several other computational parameters. The demands on numerical integration precision depend quite a bit on the type of application. The SCF convergence seems to suffer hardly from limited integration precision, but geometry convergence does, especially when tight convergence is required and also in transition state searches, which are generally more sensitive to the quality of the computed energy gradients. An extreme case is the computation of frequencies, since they depend on differences in gradients of almost-equal geometries. Frequency calculations on molecules with sloppy modes suggest that a BeckeGrid of “good” quality may be required. Note: a large integration value implies that a lot more points will be used in the numerical integrals, thereby increasing the computational effort (roughly linear in the number of points).