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 4s<4p<3d in K, 4s<3d<4p in Ca and 3d<4s<4p for 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 K and Ca before 3d, but 3d before 4s in the transition elements."
He states the orbital energies ε are represented by the effective quantum numbers neff = 1/(√|ε|/13.606 eV)^3. This seems to imply the orbital energy of 4s is lower than 3d for K and Ca hence the 4s orbitals are occupied in preference to 3d in their atoms.
This contradicts various literature which state according to Hartree-Fock calculations the 4s orbital never has a lower energy than 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)
Criticisms of a similar graph used in textbooks is that it features out-dated 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 renconcile the contrasting information presented?