How do Eigenfunctions of Douglas-Kroll Operators Behave in the Vicinity of Point-like Nuclei?

There is no consensus in the recent literature how singular the eigenfunctions of quasirelativistic operators are close to a point-like nucleus. This question has far-reaching implications, e.g. for the convergence properties when such eigenfunctions are expanded in a basis of regular functions. For this reason the spectrum of a Douglas-Kroll operator in a large basis of spherical waves has been investigated. Such a basis sets shows a very slow but regular convergence pattern from which information on the singularity at the origin can be extracted. This calculation involves multiplications and a diagonalization of very large dense matrices, which has been performed in parallel (with up to 256 CPU cores) using functions from the ScaLAPACK library. For first-order Douglas-Kroll operators the eigenfunctions were known to be more singular than for the Dirac operator, and this manifests itself in our results. Second-order Douglas-Kroll and beyond, however, behaves very similar to the Dirac case, and there is ample evidence that the Douglas-Kroll eigenfunctions beyond first-order Douglas-Kroll have the same Singularity close to the nucleus than the Dirac operator. It is thus likely that an expansion of Douglas-Kroll eigenfunctions in basis sets conventionally used in relativistic quantum chemistry will show essentially the same convergence rate as found e.g. for the Dirac operator.