[Paper Review] On a group-theoretical approach to the periodic table of chemical elements

Maurice Kibler|ArXiv.org|Aug 16, 2004

History and advancements in chemistry3 citations

TL;DR

This paper proposes a group-theoretical framework using SO(4,2) ⊗ SU(2) to rationalize the periodic table based on the Madelung rule. It establishes a complete set of 11 commuting operators—9 from SO(4,2) and 2 from SU(2)—to label electron states and derive quantitative phenomenological laws for chemical properties via fitting to experimental data.

ABSTRACT

This paper is concerned with the application of the group SO(4,2)xSU(2) to the periodic table of chemical elements. It is shown how the Madelung rule of the atomic shell model can be used for setting up a periodic table that can be further rationalized via the group SO(4,2)xSU(2) and some of its subgroups. Qualitative results are obtained from the table and the general lines of a programme for a quantitative approach to the properties of chemical elements are developed on the basis of the group SO(4,2)xSU(2).

Motivation & Objective

• To establish a rigorous group-theoretical foundation for the periodic table using SO(4,2) ⊗ SU(2) as the dynamical noninvariance group.
• To connect the Madelung rule (n + ℓ ordering) with the representation theory of SO(4,2) ⊗ SU(2).
• To develop a systematic programme for deriving quantitative phenomenological laws of chemical elements from group-theoretical operators.
• To extend techniques from nuclear and particle physics (e.g., interacting boson model) to atomic and chemical systems.
• To enable prediction of unmeasured properties of chemical elements through operator-based fitting procedures.

Proposed method

• Utilize the Madelung rule (n + ℓ ordering) to construct a periodic table with ordered electron configurations.
• Rationalize the table using the Lie group SO(4,2) ⊗ SU(2), with SO(4,2) as the dynamical noninvariance group of the hydrogen-like atom.
• Solve the state labelling problem for SO(4,2) by identifying 9 commuting operators: 3 Cartan generators, 3 Casimir operators, and 3 additional operators from Racah's completeness condition.
• Include SU(2) for electron spin, contributing 2 additional commuting operators (1 Cartan, 1 Casimir), yielding a total of 11 independent commuting operators.
• Use the 11 operators as an integrity basis to express physical observables (e.g., ionization energy, electronegativity) as linear combinations.
• Apply fitting procedures to experimental data—either per period or per group—to derive phenomenological formulas for predicting unmeasured properties.

Experimental results

Research questions

• RQ1 How can the Madelung rule (n + ℓ ordering) be systematically rationalized using the group SO(4,2) ⊗ SU(2)?
• RQ2 What is the complete set of commuting operators needed to label irreducible representations of SO(4,2) ⊗ SU(2) for atomic states?
• RQ3 Can the 11 commuting operators derived from SO(4,2) ⊗ SU(2) serve as a basis for constructing quantitative models of chemical properties?
• RQ4 To what extent can phenomenological laws for atomic properties be derived by fitting linear combinations of these operators to experimental data?
• RQ5 How might this group-theoretical framework be extended to isotopes and molecular systems?

Key findings

• The group SO(4,2) ⊗ SU(2) provides a unified framework for rationalizing the periodic table based on the Madelung rule.
• A complete set of 11 commuting operators is established: 9 from SO(4,2) (3 Cartan generators, 3 Casimir operators, and 3 additional operators from Racah’s completeness condition) and 2 from SU(2).
• The 11 operators form an integrity basis for labeling electron states and constructing observables in the representation space of SO(4,2) ⊗ SU(2).
• Chemical properties such as ionization energy, electron affinity, and electronegativity can be expressed as linear combinations of operators built from this integrity basis.
• Fitting procedures applied to experimental data for elements in a period or group yield phenomenological formulas capable of predicting unmeasured properties.
• The programme, termed the KGR programme, extends methods from nuclear and particle physics to atomic and chemical systems, offering a new quantitative approach to periodic trends.