When I was perusing the works of Schwarz on atomic structure, I came across the unfamiliar term of d-orbital collapse.

He describes it as a variation in energetic sequence from group 1 to 3 elements such that the order changes from $\ce{4s < 4p < 3d}$ in $\ce{K}$, $\ce{4s < 3d < 4p}$ in $\ce{Ca}$ and $\ce{3d < 4s < 4p}$ for $\ce{Sc}$ and subsequent elements.
He continues to say "It is the result of the interplay of nuclear attraction, of imperfect shielding by the inner-core electrons, and of angular-momentum dependent centrifugal forces. It explains why 4s becomes occupied in $\ce{K}$ and $\ce{Ca}$ before $\ce{3d}$, but $\ce{3d}$ before $\ce{4s}$ in the transition elements."

Graph taken from "The Full Story of the Electron Configurations of the Transition Elements" J. Chem. Educ., 2010, 87 (4), pp 444–448

He states the orbital energies $ε$ are represented by the effective quantum numbers $$n_\mathrm{eff} = \frac{1}{\left(\frac{\sqrt{|ε|}}{\pu{13.606 eV}}\right)^3}.$$ This seems to imply the orbital energy of $\ce{4s}$ is lower than $\ce{3d}$ for $\ce{K}$ and $\ce{Ca}$ hence the $\ce{4s}$ orbitals are occupied in preference to $\ce{3d}$ in their atoms.

This contradicts various literature which state according to Hartree-Fock calculations the 4s orbital never has a lower energy than $\ce{3d}$, regardless of which atom is being considered. ("Transition Metals and the Aufbau Principle" VPD, J. Chem. Educ. 1994 71 (6), 469; "Why Is the 4s Orbital Filled before the 3d" MS, J. Chem. Educ. 74: 498–503, 1996; "4s is always above 3d! Or, how to tell the orbitals from the wavefunctions" Pilar, J. Chem. Educ. 1978 55 (1), 2; "Transition metal configurations and limitations of the orbital approximation" Scerri, J. Chem. Educ. 1989 66 (6), 481)

Graph taken from Shriver & Atkins Inorganic Chemistry (5th Edition)

Criticisms of a similar graph used in textbooks is that it features outdated calculations made by R. Latter in the 1950s that seemed to indicate the energy of the 4s orbital was lower than that of 3d. The graph which shows the crossing-over of the energies of the 4s and 3d orbital energies, "is now known to be erroneous". ("A Critique of Atkins' Periodic Kingdom and Some Writings on Electronic Structure" Scerri, Foundations of Chemistry 1: 297–305, 1999) Schwarz also alludes to this in his article: "The graphic is quantitatively correct, in contrast to qualitatively incorrect artwork in many chemical and physico-chemical textbooks.", yet his graph reproduces the same phenomenon.

Does Schwarz's d-orbital collapse actually imply 4s orbitals having lower energy than 3d? Is there any way to reconcile the contrasting information presented?

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    $\begingroup$ Prof. Schwarz is apparently active. Scerri is more of a historian and no need to rely too much on J.Chem.Ed. It may be worth contacting the author tc.chemie.uni-siegen.de/schwarz. I was lately looking for a historical reference and contacted the author directly. He was kind enough to scan original works and send them to me along with some details. $\endgroup$
    – ACR
    Commented Feb 29, 2020 at 19:20
  • $\begingroup$ Would this explain why 4s orbitals are occupied before the 3d orbitals in potassium and calcium whereas the reverse is true once you get to the transition elements? $\endgroup$ Commented Apr 5 at 12:02
  • $\begingroup$ It is worth pointed out that the entire atomic orbital picture becomes less and less meaningful as the electron correlation increases. The orbital picture is "just" a model, an extrapolation of the Hydrogen-like picture to multi-electronic systems, "augmented" with phenomenological parameters such as the shielding parameter. When the electron correlation becomes important, the HF atomic orbitals become mixed and the hydrogen like orbital picture losses its explanation power. $\endgroup$
    – Mils
    Commented Apr 26 at 9:08