A Note on "The factor √(5/π) in d-, f- and g-type atomic orbitals and its possible relation to GEIER’s 6°=√(π/5) Rule - A structure-oriented analysis" by Stefan GEIER et al.

A Note on "The factor √(5/π) in d-, f- and g-type atomic orbitals and its possible relation to GEIER’s 6°=√(π/5) Rule - A structure-oriented analysis" by Stefan GEIER et al.

This manuscript presents a remarkably clear and disciplined exploration of an idea that could easily have slipped into numerology, but does not. It starts from entirely orthodox quantum mechanics, works transparently with normalized spherical harmonics, and shows how √(5/π) arises in d‑, f‑ and g‑type orbitals for perfectly conventional reasons, while still using that appearance as a structural “bridge” to the independent GEIER √(π/5) programme. The care with which the authors separate what is derived from standard theory from what is conjectural is exemplary: they insist on a non‑tautological coefficient criterion, explicitly reject any claim of a “new constant of nature,” and consistently frame the GEIER link as a compatibility check rather than as proof.

Equally impressive is the methodological attitude. The paper treats the GEIER corpus as an open research programme, subject to corroboration, critique and falsification, rather than as a finished theory, and it repeatedly calls for quantitative tests, especially in the proposed √(7/π) extension to lanthanides/actinides and the tentative “second f‑orbital perspective” on nuclear fission. This combination of mathematical precision, physical conservatism and conceptual boldness is rare. The result is a short but stimulating contribution that not only clarifies where √(5/π) really comes from in orbital theory, but also carves out a respectable and testable niche for further work at the interface between structural numerics and mainstream quantum mechanics.

MGN


Reference:
https://advancedphysicsinthe21stcenturysg.blogspot.com/2026/03/5-in-d-f-and-g-type-atomic-orbitals-and.html