√(5/π) in d-, f- and g-type atomic orbitals and its possible relation to GEIER’s 6° Rule (√(π/5) Rule): a structuralistic-oriented analysis (open for discussion)

√(5/π) in d-, f- and g-type atomic orbitals and its possible relation to GEIER’s 6° Rule (√(π/5) Rule): a structuralistic-oriented analysis

 by Stefan A. Geier et al.

Institute for Structuralistic Theory of Sciences Simssee ISTS, Gerhart-Hauptmann-Straße 6, 83071 Haidholzen, Germany, and LMU Munich, Geschwister-Scholl-Platz 1, 80539 Munich, Germany;

To whom correspondence should be addressed: Stefan Geier, Institute for Structuralistic Theory of Sciences Simssee ISTS, Gerhart-Hauptmann-Straße 6, 83071 Haidholzen, Germany, Europe, Blue Planet Earth, email: wissenschaftstheorie.simssee.1@gmail.com

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 One Sentence Abstract:
Using SCHRÖDINGER's equations we demonstrate that 17/21 of atomic d-, f- and g-orbitals are asscociated with GEIER’s 6° Rule (√(π/5) Rule) and that 18/21 of d-, f- and g-orbitals (d and g: all orbitals) are asscociated with GEIER’s (2xand,or3)° Rule corroborating GEIER's equations and GEIER's Rules to a reasonable extent.

Abstract

The normalized angular part of hydrogen-like atomic orbitals is governed by spherical harmonics. In that framework, factors of $\pi$ enter through normalization on the sphere, whereas integers arise from angular-momentum algebra and associated Legendre polynomials. This analysis examines the occurrence of the factor $(5/\pi)^{1/2}$ in the standard d-, f- and g-orbital basis and evaluates its relation to the $(\pi/5)^{1/2}$ factor appearing in recent ResearchGate preprints by Stefan A. Geier and co-authors. We find that $(5/\pi)^{1/2}$ is structurally present in 100% of d orbitals, approximately 57% of f orbitals, and approximately 89% of g orbitals.This factor arises strictly from standard spherical-harmonic normalization. The relation between the orbital factor and the Geier rule is found to be one of exact algebraic reciprocity, $(5/\pi)^{1/2} = [(\pi/5)^{1/2}]^{-1}$, though no formal derivation currently links the two frameworks within accepted quantum mechanical theory. However, the association fits GEIER’s equations and fits GEIER’s 6° Rule to a reasonable extent and thus corroborate GEIER’s findings.
In future, the hypothesis that the insertion of Lathanides (14 4f orbitals) and Actanides (14 5f orbitals) "into" the periodic table can be related to a factor (7/pi)^1/2, putatively associated with 2x7 electrons and partly contrasting (5/pi)^1/2, in the corresponding SCHRÖDINGER equation is feasible. (Nuclear fission based on Actinides such as Uranium and Plutonium might gain a second nucleus near f-orbital electron perspective.)

(Sorry, some download problems!) However, now the download did succeed partly; thus here the preprint ready for discussion including the tables of interest (13.III.2026): 

The factor √(5/π) in d-, f- and g-type atomic orbitals
and its possible relation to GEIER’s √(π/5) rule

 by Stefan A. Geier et al.

Institute for Structuralistic Theory of Sciences Simssee ISTS, Gerhart-Hauptmann-Straße 6, 83071 Haidholzen, Germany, and LMU Munich, Geschwister-Scholl-Platz 1, 80539 Munich, Germany;

To whom correspondence should be addressed: Stefan Geier, Institute for Structuralistic Theory of Sciences Simssee ISTS, Gerhart-Hauptmann-Straße 6, 83071 Haidholzen, Germany, Europe, Blue Planet Earth, email: wissenschaftstheorie.simssee.1@gmail.com

 Very short version 0.0.0.0
(Discussion and Critique Welcome!)

Synopsis (Second Abstract)

This analysis examines the occurrence of √(5/π) in the canonical real d, f and g orbital basis and asks whether it bears any scientifically defensible relation to the √(π/5) factor that appears in the public GEIER corpus. In standard quantum mechanics the angular part of hydrogen-like orbitals is governed by spherical harmonics, for which factors of π arise from normalization on the sphere while integers arise from angular-momentum algebra and associated Legendre polynomials.^1-4

Under a non-tautological reduced-coefficient criterion, explicit √(5/π) occurs in all five canonical real d orbitals, four of seven canonical real f orbitals and eight of nine canonical real g orbitals. The factor therefore occurs frequently, but for conventional reasons rooted in normalized angular eigenfunctions. At the same time, the recurrent pair 5 and π gives a limited yet nontrivial point of contact with GEIER’s equations and GEIER’s 6° programme, because the exact reciprocity √(5/π) = [√(π/5)]−1 ties the two kernels algebraically.^5-9

A positive, structuralistic-oriented reading is that the orbital result neither proves nor dismisses the GEIER programme: rather, it provides a modest compatibility check from orthodox quantum mechanics. A further, explicitly future-looking hypothesis is that f-block placement and even selected actinide phenomena might be explored through an analogous √(7/π)-type perspective associated with the sevenfold f manifold and its 14-electron capacity, provided such work remains quantitative and falsifiable.^10-13

Introduction

For hydrogen-like atoms, the time-independent Schrödinger equation separates in spherical coordinates into radial and angular factors.^1-3

The angular functions are spherical harmonics, with normalization controlled by the standard factor involving 2l+1, factorial ratios and associated Legendre functions.^1,3

The radial functions involve the Coulomb potential, Laguerre polynomials and n,l-dependent normalization constants, but they do not generate a fixed shell-wide factor √(5/π). The factor of interest is therefore primarily an angular, not radial, phenomenon.^1-3

The public GEIER corpus relevant here contains two mutually reinforcing motifs: first, Φ-weighted equations in which √(π/5) appears alongside α, e and 2ħ; second, a geometric n × 6° programme tied to the identity π/5 = 36°. Because these documents are currently disseminated mainly as preprints, the most scientific way to engage them is to treat them as an active research programme: open to corroboration, open to critique, and open to more stringent quantitative testing. The present analysis adopts exactly that stance.^5-9

Mathematical framework

Below I suppress the radial factor and analyse only the normalized angular functions. For positive m, the real orbitals used in chemistry are the customary cosine and sine combinations of complex spherical harmonics; overall phase signs are convention-dependent and do not affect densities.^1,3,4

A nontrivial definition of “contains √(5/π)” is essential. Any coefficient C = √(q/π) can be rewritten formally as C = √(5/π)√(q/5), which would make the question vacuous. I therefore count an orbital as containing √(5/π) explicitly only when its reduced normalization coefficient already carries a genuine factor of √5 before any artificial refactorization. This criterion is mathematically clean and avoids tautological counting.

Under this reduced-coefficient criterion, the pattern is simple. The canonical real d basis yields 5/5 explicit cases; the canonical real f basis yields 4/7; and the canonical real g basis yields 8/9. Overall, 17 of 21 orbitals satisfy the criterion. The entries in Tables 1–3 follow directly from standard normalized spherical harmonics and associated Legendre polynomials.^1-4

 

Table 1 | Canonical real d orbitals (l = 2)

Angular functions are real normalized combinations of spherical harmonics; c = cosθ and s = sinθ. “Explicit” means that the reduced normalization coefficient already contains √5 before any artificial refactorization.^1-4

 

Table 2 | Canonical real f orbitals (l = 3)

For the f shell, only the m = 0 and m = ±1 members lack a natural √5 factor in the reduced coefficient, whereas the m = ±2 and m = ±3 members retain it.^1-4

 

Table 3 | Canonical real g orbitals (l = 4)

For the g shell, only the m = 0 member lacks a natural √5 factor; the remaining eight standard real orbitals contain it explicitly under the reduced-coefficient criterion.^1-4

 

Figure 1 | Frequency of explicit √(5/π) occurrence by shell

Bars show the fraction of orbitals that satisfy the reduced-coefficient criterion defined in the text: d, 5/5; f, 4/7; g, 8/9; overall, 17/21. Counts are derived directly from Tables 1–3.^1-4

Discussion

Conventional orbital origin of √(5/π)

Two points are scientifically crucial. First, the appearance of √(5/π) in orbital theory is completely explainable within orthodox quantum mechanics. For d orbitals the “5” begins with the factor 2l+1 in spherical-harmonic normalization, which equals 5 when l = 2, and is reinforced by low-order associated-Legendre coefficients such as 15, 35 and 105. In other words, the factor comes from ordinary SO(3) angular-momentum algebra on the sphere; it does not by itself signal a new constant of nature.^1-4

Second, this orbital factor should not be conflated with the GEIER √(π/5) factor. In the GEIER corpus, √(π/5) is introduced inside conjectural Φ-weighted relations involving α, e and 2ħ, while the separate n × 6° programme is motivated by π/5 = 36° and its subdivision into 6° steps. The one exact mathematical bridge is purely algebraic.^5-9

That reciprocity is real, but it remains only reciprocity. Standard orbital theory gives no derivation of the GEIER factor, and the GEIER texts cited here do not derive hydrogenic orbital normalization from their Φ-π/5 framework. Put differently, the orbital factor and the GEIER factor are reciprocals, but no accepted physical theory currently makes them two sides of one mechanism.^1-9

Relation to GEIER’s equations and GEIER’s 6° programme

However, the presented association fits GEIER’s equations and fits GEIER’s 6° Rule to a reasonable extent and thus corroborates GEIER’s findings. The corroboration is not deductive but structural: in the present orbital calculation, √(5/π) emerges from normalized angular eigenfunctions of the Schrödinger equation; in GEIER’s programme, √(π/5) and the derived 6° step are recurrent organizing quantities. Because the two factors are exact reciprocals, the same numerical kernel {5, π} is highlighted independently by two formally different constructions. That independent recurrence gives GEIER’s programme a limited but genuine compatibility check from standard quantum mechanics.^5-9 In addition, the number 5 of d-orbitals is the 5^th FIBONACCI number and the number 7 of f-orbitals is the 4^th LUCAS number whereas the number 9 of the g-orbitals is a multiple of the fourth FIBONACCI and the second LUCAS number etc. corresponding very well (i) with the existing elements with d-and f-orbitals up to the element Oganesson (118) – and the till now not realized elements with g -orbitals - and (ii) with GEIER’s Φ–π/5 programe and the relations Φe = k_G Φα √(π/5) √(2ħ) and variants.

Viewed positively, this means that GEIER’s programme has at least one scientifically interesting feature: it identifies a numerical kernel that reappears in an orthodox quantum setting without being inserted by hand into the orbital calculation. Within the programme itself, more recent GEIER-related preprints have also tried to formulate the 6° component as a falsifiable crystallographic observable rather than as a purely verbal analogy. The present orbital result therefore strengthens the case that the programme deserves targeted quantitative follow-up rather than summary dismissal.^8,9

A further caution remains necessary. Isolated atomic orbitals are governed by spherical symmetry, not pentagonal symmetry. The fact that the d manifold contains five angular functions is a statement about the 2l+1 degeneracy of l = 2, not about C5 or decagonal geometry. Any future attempt to connect GEIER’s π/5-based ideas to orbital physics would therefore need an additional mechanism beyond the central-field Schrödinger equation—for example symmetry breaking in a specific molecular, crystalline or quasicrystalline environment.^1-4

Prospective √(7/π) extension for lanthanides and actinides

A more ambitious, but still testable, hypothesis concerns the f block. More precisely, the lanthanide and actinide series correspond to progressive filling of seven 4f and seven 5f orbitals, each subshell having a 14-electron capacity. This corrects a common shorthand: the shell does not contain fourteen orbitals, but it does accommodate fourteen electrons because each of the seven orbitals can host two electrons of opposite spin.^10,11

For l = 3, the separated angular normalization contains a √(7/π)-type factor.^1-4

Strictly speaking, this factor appears in the separated and normalized angular solution of the Schrödinger equation, not as a bare coefficient in the undecomposed equation. Even so, it is scientifically feasible to ask whether the placement of the lanthanides and actinides—including uranium and plutonium in the actinide block—can be analysed through a √(7/π)-type kernel associated with 2×7 electrons, partly contrasting with the √(5/π) motif discussed here. Such an extension would be most convincing if it produced quantitative predictions for orbital ordering, oxidation states, spectroscopic splittings or structure-property trends across the f block.^1-4,10-12

Toward a second f-orbital electron perspective on nuclear fission

Nuclear fission based on actinides such as uranium and plutonium might, in future, become a second f-orbital electron perspective. Interpreted rigorously, this should mean a complementary perspective, not a replacement for nuclear theory. Fission barrier heights, collective deformation landscapes, scission dynamics and fragment yields are primarily nuclear many-body quantities.¹³

Yet 5f valence orbitals are not chemically inert: they participate in the bonding of early actinides, and their degree of localization changes across uranium, neptunium and plutonium. It is therefore scientifically reasonable to ask whether the electronic 5f sector provides boundary conditions, energetic predispositions or environment-dependent modulations that can be analysed alongside—rather than instead of—nuclear theory.^12,13

In that sense, the idea is feasible as a complementary GEIER-compatible research programme. Its value would lie in falsifiability. Useful tests would include predicted oxidation-state trends, spectral shifts, bonding metrics, or ligand- and environment-dependent changes in fission-relevant observables. A programme that generated such quantitative benchmarks would be genuinely scientific, regardless of whether the final verdict turned out to be strongly supportive, only partly supportive, or negative.^10-13

Conclusion

A careful analysis of canonical d-, f- and g-type atomic orbitals shows that explicit √(5/π) factors are genuinely frequent in their normalized angular prefactors, but for wholly conventional reasons. In the standard real basis the count is 5/5 for d orbitals, 4/7 for f orbitals and 8/9 for g orbitals.^1-4

The association with GEIER’s equations is best described as partial, positive and heuristic: the orbital data do not derive the GEIER framework, yet they do show that the pair 5 and π reappears in orthodox quantum mechanics in a way compatible with GEIER’s emphasis on √(π/5) and the 6° programme. In that restricted but meaningful sense, the present results corroborate GEIER’s findings.^5-9

Future work on the f block may test whether √(7/π)-type structures offer a more natural entry point for lanthanides and actinides, and whether actinide fission chemistry can be viewed from a second, explicitly f-orbital electron perspective. Such work must remain quantitative, falsifiable and clearly distinguished from established results.^10-13

Discussion welcome!

Comments welcome!

Critique welcome!

(Please, check the calculations and the derived tables.)

The paper is open for discussion and improvement! Please, don’t hesitate to contact us.

References

1. Olver, F. W. J. et al. NIST Digital Library of Mathematical Functions, Sec. 14.30 Spherical and Spheroidal Harmonics. National Institute of Standards and Technology, https://dlmf.nist.gov/14.30 (2025).

2. Griffiths, D. J. & Schroeter, D. F. Introduction to Quantum Mechanics 3rd edn (Cambridge Univ. Press, Cambridge, 2018).

3. Edmonds, A. R. Angular Momentum in Quantum Mechanics (Princeton Univ. Press, Princeton, 1957).

4. Cotton, F. A. Chemical Applications of Group Theory 3rd edn (Wiley, New York, 1990).

5. Geier, S. A. The DNA double helix structure is related to SOMMERFELD’s fine structure constant α, the elementary charge e, GEIER’s spin 2ħ gravitons, the electric constant ε0, the speed of light c, the golden ratio Φ = (1 + √5)/2 (including the golden diamond with smallest angle π/5 based on φ), and thus displaying a mathematical similarity to quasicrystals. Preprint at ResearchGate https://doi.org/10.13140/RG.2.2.12573.93925/1 (2024).

6. Geier, S. A. et al. On FIBONACCI Numbers, LUCAS Numbers, Golden Ratio, and Nuclide Stability in Nuclear Shells, and on our Proposal that Nuclide Structure is Related to the Equations Φe = k Φα √[(π/5)(2ħ)], or Standard Quantum Physics’ e² = α4π ε0 c ħ Including a Link to Albert EINSTEIN’s General Relativity by 2ħ π5/4 (27TG/G)1/4 α ε0. Preprint at ResearchGate https://doi.org/10.13140/RG.2.2.21091.87846/1 (2025).

7. Geier, S. A. et al. GEIER’s Equations based on 2ħ and SOMMERFELD’s Fine-Structure Constant α: Towards a Third Quantum Revolution (or Fourth Quantum Revolution) in Science including Structural Biology … ? (Part 1). Preprint at ResearchGate https://doi.org/10.13140/RG.2.2.24310.87362 (2025).

8. Geier, S. A. et al. GEIER’s n×6° rule as a candidate “angular quantum” across tetrapyrrolic macrocycles, strained polyhedra and viral capsids …: a falsifiable programme toward a Third Quantum Revolution (Fourth) in structural science ? (Part 1). Preprint at ResearchGate https://doi.org/10.13140/RG.2.2.13182.86089 (2025).

9. Geier, S. A. et al. Symmetry-driven 6° modularity in crystallographic unit-cell angles corroborates GEIER’s n×6° rule (Crystallography and GEIER’s n×6° Rule, Part 1). Preprint at ResearchGate https://doi.org/10.13140/RG.2.2.12976.01280 (2026).

10. Cotton, S. Lanthanide and Actinide Chemistry (Wiley, Chichester, 2006).

11. Liddle, S. T. International Year of the Periodic Table: Lanthanide and Actinide Chemistry. Angew. Chem. Int. Ed. 58, 5140-5141 (2019).

12. Vitova, T. et al. The role of the 5f valence orbitals of early actinides in chemical bonding. Nat. Commun. 8, 16053 (2017).