√(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
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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 or, 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 e-2ħ-Φ-π/5 programe and the relations Φe = k_G Φα √(π/5) √(2ħ) and variants. Furthermore, 3 of the 4 electrons ot the 5f-orbitals of Uranium fit the three lower energy not-√(π/5)-associated 5f-orbitals, and the 6 electrons ot the 5f-orbitals of Plutonium might do so, too (I invite collegues to provide precise calculations for Plutonium*.).

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).

*Supplementary Notes:

Supplementary summary: In the 5f5/2 manifolds of U and Pu the prefactors include √(5/π) due to the spin-orbit coupling (Clebsch-Gordan-coefficients due to Paul DIRAC's chi part of the Dirac Spinor equivalent to d-orbitals); this corroborates my considerations at least to some extent. The frequencies of occupation are analyzed below:

Supplement 1: Elemental plutonium: spectroscopy-based occupation of the 5f5/2 and 5f7/2 manifolds by Stefan A. Geier

Short supplement 1 focused on the mixed-valence, 5f5-leaning interpretation of real elemental Pu

Key quantitative takeaway. Published elemental-Pu analyses place most occupied 5f weight in the lower-spin-orbit 5f5/2 manifold. Using the tabulated Pu-metal EELS/sum-rule numbers (n5/2 = 4.32, n7/2 = 0.67) gives 86.6% of the occupied 5f weight in 5f5/2; using the δ-Pu XAS/ED and XAS/sum-rule values (n5/2 = 4.79–4.80, n7/2 = 0.38–0.37) gives 92.6–92.8% in 5f5/2. These are j-manifold occupations, not occupations of a unique set of seven named real spatial 5f orbitals. [1,2]

This supplement 1 isolates the most robust spectroscopy-based statement about real elemental plutonium: in both Pu-metal EELS/sum-rule work and later model-assisted XAS analysis for δ-Pu, the occupied 5f weight is strongly concentrated in the 5f5/2 manifold. The exact percentage depends on the dataset and fit model, but it lies in the high-80s to low-90s percent range. That conclusion is more stable than any single decomposition into 5f4, 5f5 and 5f6 configuration weights, which vary more from method to method. [1-3]

Figure 1 | Share of the occupied 5f weight carried by the 5f5/2 and 5f7/2 manifolds in representative elemental-Pu analyses. Percentages were computed from n_j/(n5/2 + n7/2). [1,2]

 

Table 1 | Spectroscopy-based j-manifold occupations for elemental Pu

The values below are the clearest quantitative support for the statement that real elemental Pu is mixed-valent but still strongly 5f5/2-dominated.

System	Method	n5/2	n7/2	n_f	5f5/2 share	5f7/2 share	Per-spinor occupation
Pu metal	N4,5 EELS + spin-orbit sum rule (tabulated by Moore et al.) [1]	4.32	0.67	4.99	86.6%	13.4%	5/2: 0.720 7/2: 0.084
δ-Pu	XAS + exact diagonalization (ED ground state) [2]	4.79	0.38	5.17	92.6%	7.4%	5/2: 0.798 7/2: 0.048
δ-Pu	XAS + spin-orbit sum rule [2]	4.80	0.37	5.17	92.8%	7.2%	5/2: 0.800 7/2: 0.046

Reading guide. The per-spinor occupation is n5/2/6 for the six m_j states of j = 5/2 and n7/2/8 for the eight m_j states of j = 7/2. For δ-Pu, the sum-rule and ED values agree closely, which is why the resulting percentages differ by only about two tenths of a percentage point. [2]

Table 2 | Mixed-valence configuration weights: useful context, but less stable than the j split

Configuration-space decompositions reinforce that δ-Pu is not a pure 5f5 system, but they vary more strongly across methods than the j-manifold split does. That is why the 5f5/2 fraction is the cleaner cross-method summary.

System	Method	5f4 (%)	5f5 (%)	5f6 (%)	5f7 (%)
δ-Pu	DMFT [3]	12	66	21	—
δ-Pu	RXES [3]	8(2)	46(3)	46(3)	—
δ-Pu	CHPES [3]	6(1)	66(7)	28(3)	—
δ-Pu	XAS/ED fit [2]	13	56	29	2

Interpretation. DMFT and CHPES favor a clearly 5f5-leaning multiconfigurational state for δ-Pu, while RXES places more weight in 5f6 than those two methods do. Even with that spread in configuration weights, the j-resolved occupancy remains heavily 5f5/2-dominated in the elemental-Pu analyses summarized above. [2,3]

Scope note

These spectroscopies constrain j-resolved occupations of the spin-orbit-split 5f manifolds. They do not uniquely determine occupations of seven named real spatial 5f orbitals in a solid. For plutonium, strong spin-orbit coupling, hybridization and crystal-field effects mean that the 5f5/2 / 5f7/2 language is the better experimentally constrained description. [2,4]

References

[1] Moore, K. T., van der Laan, G., Haire, R. G., Wall, M. A., Schwartz, A. J. & Söderlind, P. Emergence of strong exchange interaction in the actinide series: the driving force for magnetic stabilization of curium. Phys. Rev. Lett. 98, 236402 (2007). DOI: 10.1103/PhysRevLett.98.236402.

[2] Chiu, W.-t., Tutchton, R. M., Resta, G., Lee, T.-H., Bauer, E. D., Ronning, F., Scalettar, R. T. & Zhu, J.-X. Hybridization effect on the x-ray absorption spectra for actinide materials: application to PuB4. Phys. Rev. B 102, 085150 (2020). DOI: 10.1103/PhysRevB.102.085150.

[3] Janoschek, M. et al. The valence-fluctuating ground state of plutonium. Sci. Adv. 1, e1500188 (2015). DOI: 10.1126/sciadv.1500188.

[4] Moore, K. T., Wall, M. A., Schwartz, A. J., Chung, B. W., Shuh, D. K., Schulze, R. K. & Tobin, J. G. Failure of Russell-Saunders coupling in the 5f states of plutonium. Phys. Rev. Lett. 90, 196404 (2003). DOI: 10.1103/PhysRevLett.90.196404.

[5] van der Laan, G., Moore, K. T., Tobin, J. G., Chung, B. W., Wall, M. A. & Schwartz, A. J. Applicability of the spin-orbit sum rule for the actinide 5f states. Phys. Rev. Lett. 93, 097401 (2004). DOI: 10.1103/PhysRevLett.93.097401.

Supplement 2: Elemental uranium in comparison with plutonium:

Uranium versus plutonium 5f5/2 occupation
SO-MCSCF model frames and spectroscopy-based metal values

Scope. This handout compares uranium and plutonium in two scientifically distinct frames: (i) localized trivalent ions, where a multireference treatment is naturally formulated as state-averaged CASSCF/CAS(n,7) followed by spin-orbit coupling; and (ii) elemental metals, where XAS and EELS constrain the occupied 5f5/2 and 5f7/2 weights.

Key comparison • Localized-ion / SO-MCSCF frame: U(III) is more 5f5/2-pure than Pu(III) (about 93.0% versus 84.6% of occupied 5f weight in 5f5/2). • Elemental-metal spectroscopy: Pu metal is more 5f5/2-dominated than U metal (about 84.0-86.6% versus 72.7-76.3%). • Do not mix these two frames numerically: free-ion multireference targets and elemental-metal spectroscopy answer different questions.

 

Figure 1 | Comparison of p5/2 across two frames. The first two bars are localized free-ion target values expressed in an SO-MCSCF/CAS(n,7)+SOC frame. The last two bars are elemental-metal XAS/EELS midpoints; the error bars show the XAS-EELS spread.

How the probabilities were computed

Definitions. For every row, the reported probabilities were computed from published occupancies using p5/2 = n5/2 / (n5/2 + n7/2) and p7/2 = n7/2 / (n5/2 + n7/2). Individual spinor occupancies are n5/2 / 6 and n7/2 / 8.

Important note: the localized-ion rows below combine two literature ingredients. The CAS active spaces and ground terms come from a modern multireference benchmark, while the n5/2 and n7/2 values are atomic intermediate-coupling target occupancies used in spectroscopy analyses. They are presented together because this is the correct localized free-ion comparison frame for SO-MCSCF work on U(III) and Pu(III).

Localized-ion SO-MCSCF frame

System	Active space	Ground term	n5/2	n7/2	p5/2	p7/2
U(III)	CAS(3,7) + SOC	4I9/2	2.796	0.210	0.930	0.070
Pu(III)	CAS(5,7) + SOC	6H5/2	4.230	0.770	0.846	0.154

Per-spinor occupancies. U(III): 5f5/2 spinor = 0.466, 5f7/2 spinor = 0.026; Pu(III): 5f5/2 spinor = 0.705, 5f7/2 spinor = 0.096.

Interpretation. Within the localized free-ion comparison frame, uranium is more 5f5/2-pure than plutonium. The physical reason is simple: the lower j = 5/2 manifold can hold six electrons, so a 5f3 ion still sits deep in that lower manifold, whereas a 5f5 ion is already much closer to saturating it and therefore carries more 5f7/2 admixture.

Elemental-metal spectroscopy

These rows are calculated directly from XAS and EELS occupancies reported for uranium and plutonium metals. Because some published rows are rounded, p5/2 and p7/2 were normalized to n5/2 + n7/2 in each row.

Material / probe	nf	n5/2	n7/2	p5/2	p7/2
U metal (XAS)	3.00	2.18	0.82	0.727	0.273
U metal (EELS)	3.00	2.28	0.71	0.763	0.237
Pu metal (XAS)	5.00	4.20	0.80	0.840	0.160
Pu metal (EELS)	4.99	4.32	0.67	0.866	0.134

Per-spinor occupancies. U metal XAS: 0.363 / 0.102; U metal EELS: 0.380 / 0.089; Pu metal XAS: 0.700 / 0.100; Pu metal EELS: 0.720 / 0.084.

Interpretation. In the metals the trend reverses: plutonium is more 5f5/2-dominated than uranium. This is consistent with the long-standing spectroscopy picture that U metal sits closer to an LS-like / more delocalized regime, whereas Pu metal is much more strongly driven toward intermediate coupling.

Context note: why some newer delta-Pu numbers look even more 5f5/2-rich

Recent fully relativistic many-body band-structure work on delta-Pu reported MT-sphere occupancies n5/2 = 4.769 and n7/2 = 0.250 in scGW, and n5/2 = 4.569 and n7/2 = 0.340 in sc(GW+G3W2). These correspond to about 95.0% and 93.1% 5f5/2 weight inside the chosen muffin-tin spheres. The same paper explicitly warns that muffin-tin geometry changes the absolute occupancies, so these values are best read as context rather than direct replacements for the older XAS/EELS extractions.

A separate 2024 plutonium reassessment lowered the total elemental-Pu 5f count to about 5.0 +/- 0.1 for alpha-Pu and 4.9 +/- 0.2 for delta-Pu, but that reassessment does not by itself provide one new unique spectroscopy pair n5/2 / n7/2 that can simply replace the older table values.

Bottom line

• If you want a localized-ion SO-MCSCF reference, compare U(III) CAS(3,7)+SOC with Pu(III) CAS(5,7)+SOC: uranium is more purely 5f5/2 in that frame.

• If you want real elemental-metal occupations, use XAS/EELS-derived n5/2 and n7/2 values: plutonium is more 5f5/2-dominated than uranium in the metal.

• The apparent reversal is physical, not contradictory: free-ion and elemental-metal comparisons probe different regimes of localization, hybridization, and spin-orbit mixing.

References

Sarkar, A. & Gagliardi, L. Multiconfiguration Pair-Density Functional Theory for Vertical Excitation Energies in Actinide Molecules. J. Phys. Chem. A 127, 9389-9397 (2023).
Moore, K. T., van der Laan, G., Tobin, J. G., Chung, B. W., Wall, M. A. & Schwartz, A. J. Probing the population of the spin-orbit split levels in the actinide 5f states. Ultramicroscopy 106, 261-268 (2006).
Moore, K. T. & van der Laan, G. Nature of the 5f states in actinide metals. Rev. Mod. Phys. 81, 235-298 (2009).
Kutepov, A. L. et al. New experimental and theoretical insights into electronic structure of alpha-U and delta-Pu. J. Phys.: Condens. Matter (2023).
Tobin, J. G. A Reassessment of 5f Occupation in Plutonium. arXiv 2408.14517 (2024).

Supplement 3: 5f5/2 manifolds and the prefactor √(5/π) due to the spin-orbit coupling (Clebsch-Gordan-coefficients)

Part 1:
a) Paul DIRAC spinor: 5f5/2 manifolds: The small component chi of the DIRAC spinor (bispinor) is based on l'=2 and thus brings in 5^1/2.

b) Racah-normalized spherical tensors when computing 5f5/2 quadrupole moments or electric field gradients (key for Pu structural anomalies): this factor appears explicitly.