I'm looking at Figure 1 on p. 2 of Jansen, "Effects of relativistic motion of electrons on the chemistry of gold and platinum" (2005) (should be a free download). From the bottom curve, there is such a dip around elements 78 and 79 that it seems like if the curve were "filled in" then one would see a mathematical singularity where the orbital radius goes to zero here.
As I understand it, the relativistic contraction of valence orbitals, specifically the 6s orbital in elements gold and platinum, contains some kind of numerical instability which could be explained in terms of positive feedback: The relativistic mass increase of these high-velocity electrons causes them to orbit closer to the nucleus. But when orbiting closer to the nucleus, these electrons experience less screening from other orbitals, so they "see" a greater nuclear charge and their kinetic energy increases, causing further contraction.
I'm still trying to understand the basics of relativistic quantum chemistry, see Do relativistically-contracted electron states have the same energy and angular momentum values? and Why does an electron's orbital contract as its relativistic speed increases?.
I'm wondering if my interpretation is correct, and for example, if it would be possible to have a situation where a 6s electron fully "collapses in" on the nucleus under certain circumstances:
if one could simulate a nucleus with fractional charge, say between 78 and 79 or between 79 and 80
if one could slightly perturb some physical constant such as the fine structure constant or the speed of light or planck's constant
if one could find a way to temporarily increase the energy of a valence electron, perhaps with a high energy photon
I've read on the extended periodic table Wikipedia article that inner (1s) orbitals are expected to "dive" into the "Dirac sea" for very heavy nuclei, $Z\approx 173$, but I wonder if something like this could happen with a lighter atom, for example if a 6s electron, which has more energy than a 1s electron, were to approach just as closely to the nucleus through some of the de-screening and relativistic effects I mentioned previously.
A related item of curiosity is how to generate the data appearing in Jansen's figure 1, for example if the referenced software (Grant, "An atomic multiconfigurational Dirac-Fock package", 1980) is still available in some form, and whether it is easy to modify it to explore the three scenarios I proposed.