A.7 Nearly singular basis sets: Strange results from small-unit-cell periodic calculation with many k-points

We have observed numerous times that periodic bulk calculations with small unit cells and many $k$-points are apparently much more prone to ill-conditioning of the basis set than any other type of calculation. The symptom is that, with the usual accuracy and grid settings, large but still affordable basis sets (e.g., tier 3) will show reasonable convergence behavior at the outset, but then suddenly show a large jump and unphysical total energies at some point in the s.c.f. cycle.

The underlying reason is that the basis sets used by FHI-aims are overlapping and non-orthogonal. As the basis functions located at each atom of the structure approach completeness, the basis set as a whole becomes overcomplete. The result may be that certain linear combinations of basis functions are approximately expressable as linear combinations of some others. The eigenvalue problem Eq. (3.38) becomes ill-conditioned, and small amounts of numerical noise in the Hamiltonian / overlap matrix elements can group together to produce large unphysical effects in the eigenvakue spectrum.

If this happens, a number of strategies are available to deal with this situation. These are summarized in the following. Note, however, that ill-conditioning does indicate that your chosen basis set is already closer to completeness than even your computer can handle, and a smaller basis set for production calculations should be equally sufficient (and much faster) for high-quality results.

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  Employ the basis_threshold keyword. This allows to identify the near-linear dependent components of the overlap matrix and eliminate them from the calculation. The successful threshold value depends on your chosen basis set and system, so test different choices (typically, 10^-4 or 10^-5). Note, however, that a large basis_threshold value may also impact the the total energy found at a level of a few meV/atom.

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  In addition, the keyword override_illconditioning must be set in order to run with a basis set that is reduced by basis_threshold. This should serve as an indicator that extra care is required in this situation—in particular, a detailed convergence analysis of the behaviour of the problem with increasing basis set size, and (separately!) with increasing cutoff radius, up to the value you are using. In most cases, it should turn out that either the basis set, or the cutoff radius, or both, were chosen to be far overconverged.

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  Increase the accuracy of the integration grids via radial_base and angular_grids. This is an expensive strategy (use for proof-of-principle only!), but it will serve to reduce the numerical noise in your calculations and thus increase the validity range of the eigenvalue solution, Eq. (3.38).

Again, note that we do not usually observe any ill-conditioning related problems for large periodic structures (e.g., surface slabs) or even very large molecules, even when employing very large basis sets.