Liquid mercury, relativistic effects and Fermi velocity

Question

Reading this news article and some related references, I learned that mercury’s liquidity is due to relativistic effects in Hg, where "the electron approaches about 58% of the speed of light." I do not see how this can be reconciled with the Fermi velocity of Hg being $1.58 \times 10^6$ m/s, almost the same as Cu $1.57 \times 10^6$ m/s and in the same order of magnitude of virtually all other metals, making relativistic corrections barely significant. The relevant scientific article relies on numerical calculations and arguments which I am unable to relate to basic solid-state physics concepts (cf. this too). My question is: how do electrons with relativistic velocities (e.g. 0.58c as above) relate to the Fermi velocity of a metal?

PS I am aware that the articles cited above are chemistry publications, but I assume that an electron's velocity should be the same in physics, possibly modulus some convention that I am not aware of.

Answer 1

As it is stated in the Scientific American paper you cited,

for the 1s electron of mercury (atomic number 80) this effect becomes significant; the electron approaches about 58% of the speed of light, ...

(Bold is mine). As you see the most important relativistic effects have to do with the innermost orbital, whose energy is much lower than the energy of the valence states (and of the Fermi level). There is no point to compare them. Actually, what is reported in the paper, is that the presence of strong relativistic effects on the $1s$ state has some indirect consequence on all the higher energy orbitals, up to the valence states, even if these states do not show direct relativistic effects.

A final comment on your concern about the fact that the original research was published in a chemistry journal. It is not an issue. This kind of research is interdisciplinary and many of the computational techniques used in the field were introduced by physicists. I know many physicists working on related topics. Therefore your question is completely in-topic here.

A side remark about the language used in the Scientific American paper is in order. First of all, one should not take literally the word velocity with quantum particles. As it is well known, velocity is not directly appearing in the Schrödinger or Dirac equations. Moreover, the paper uses the obsolete concept of relativistic mass which is fading in the physics literature. However, this is an issue with the words. The equations are the real basis of the calculations.