Title: Symmetry and Symmetry Breaking in the Periodic Table - Towards a Group-Theoretical Classification of the Chemical Elements
Other Titles: Symmetrie en Symmetriebreking in de Tabel van Mendelejev - Een Groepentheoretische Classificatie van de Chemische Elementen
Authors: Thyssen, Pieter
Issue Date: 24-May-2013
Abstract: During this PhD fellowship, supported by the FWO, we have studied the periodic system (PS) from a group theoretical (GT) point of view in order to deepen our limited understanding of the periodic law. The results obtained have shed new light on the PS and have induced us to write a textbook Shattered Symmetry: From the Eightfold Way to the Periodic Table.1. IntroductionThe PS represents a classification of the manifold of chemical elements. Although the PS has become the undisputed cornerstone of modern chemistry, the overall (quantum mechanical, QM) structure of the PS never been fully understood from an atomic physics point of view. Active research continues to illuminate important aspects of the periodicity phenomena, and makes this subject interesting from both educational, theoretical and philosophical points of view.2. QM structure of the PS in a nutshellThe QM description of the PS is based on three principles: 1. the quantum numbers n and l, 2. the Pauli exclusion principle, and 3. Bohr’s Aufbau principle. The PS however cannot be build on the basis of 1—3, unless an energy ordering rule is provided. Many rules exist:1. The hydrogenic (n, l) rule for H and positively ionized atoms with charge &#8805; 2, according to which the orbitals are filled in order of increasing n, and according to increasing l for fixed values of n;2. The Madelung (n + l, n) rule for neutral atoms, where the orbitals are filled according to increasing N = n + l. For fixed n + l, the orbitals are filled in order of increasing n;3. Intermediate rules exist for ionized atoms with a charge < 2.The hydrogenic rule gives rise to the following orbital sequence:{1s} « {2s ‹ 2p} « {3s ‹ 3p ‹ 3d} « {4s ‹ 4p ‹ 4d ‹ 4f} « ...where orbitals have been grouped according to the same value of n. Taking the possible values of the magnetic and spin quantum numbers into account leads to the following dimensionalities for the above sequence: 2 – 8 – 18 – 32, as summarized by the well-known formula 2n². This yields the following hydrogen spectrum:n dim4 32 {4s ‹ 4p ‹ 4d ‹ 4f}3 18 {3s ‹ 3p ‹ 3d}2 8 {2s ‹ 2p}1 2 {1s}The Madelung rule, in contrast, gives rise to the following sequence:{1s} « {2s} « {2p ‹ 3s} « {3p ‹ 4s} « {3d ‹ 4p ‹ 5s} « {4d ‹ 5p ‹ 6s} « {4f ‹ 5d ‹ 6p ‹ 7s} « {5f ‹ 6d ‹ 7p ‹ 8s} « ...with grouping according to constant n + l. This corresponds to the following dimensionalities: 2 – 2 – 8 – 8 – 18 – 18 – 32 – 32. Interestingly, the hydrogenic dimensions appear twice in the Madelung sequence. By organizing the elements in periods of constant n + l and groups of constant l, ml and ms, one obtains the left-step PS of Charles Janet:n + l dim 1 2 {1s} 2 2 {2s} 3 8 {2p ‹ 3s} 4 8 {3p ‹ 4s} 5 18 {3d ‹ 4p ‹ 5s} 6 18 {4d ‹ 5p ‹ 6s} 7 32 {4f ‹ 5d ‹ 6p ‹ 7s} 8 32 {5f ‹ 6d ‹ 7p ‹ 8s}Both the Madelung sequence and the ‘period doubling’ are characteristic properties of the PS. Yet, neither of these has ever been ab initio explained by QM! Many claims have appeared in the scientific literature, but most have been dismissed. This has been called the Löwdin challenge.3. A novel GT description of the PSThe mathematical theory of abstract groups has been used as a classificatory tool by Gell-mann and Ne’eman to classify the zoo of elementary particles. This GT approach led to the eightfold way and the discovery of the quark structure of hadrons, both of which were described by the unitary SU(3) group.Rationale. Since the foundations of the PS are to be found in the QM theory of multi-electron systems, GT should be able to shed some light on the classification of the chemical elements as well. Following the particle physics tradition, we believe there is a more sophisticated, symmetry-based way of understanding how the chemical elements should be accommodated in the PS, and how the periodic law emerges from its quantum mechanical foundations.The phenomenological study of the global group structure of the PS originated in the 1970s with the pioneering work of a small group of theoretical physicists (e.g. Barut, Fet, Rumer, Ostrovsky, Demkov, and Novaro). Within this fascinating approach, the chemical elements are considered as various states of some ‘atomic matter’ — described by a non-compact spectrum generating dynamical Lie group and its chain of subgroups and Cartan-Weyl subalgebras. From this point of view, the PS represents some sort of metasystem where the different elements constitute the multiplets of a multidimensional group; they form the basis for an infinite-dimensional irreducible representation (unirrep) of the dynamical symmetry group.The identification of the correct symmetry group and its decomposition into subgroups has, however, remained a problem to this very date. It has therefore been our aim to extend this fundamental research during a doctoral fellowship by studying the symmetry breaking (SB) mechanisms that can account for the structure of the PS.The hydrogen atom. In the course of this PhD project, we have elucidated a profound connection between the H atom, the harmonic oscillator, and the PS. It appears that these three systems are manifestations of the same supergroup SO(4,2) although they differ from each other by a different chain of subgroups (i.e. SB mechanisms). The chain of subgroups for the H atom follows the traditional SB mechanism in terms of groups and subgroups, and has been verified to be:SO(4,2) > SO(4) &#8855; SO(2,1) > SO(4) > SO(3) > SO(2)The spherical symmetry group SO(3) describes the spatial symmetry of the angular equation, and relates orbitals of the same n and l (e.g. 2px, 2py, and 2pz). The hyperspherical SO(4) group explains the accidental degeneracy of the hydrogen levels, yielding SO(4) multiplets with dimensionalities that rise as 2, 8, 18, 32 (i.e. 2n²) for the K, L, M and N shell respectively (vide supra). The covering group SO(2,1) finally describes the dynamic symmetry of the radial equation of hydrogen and relates orbitals of different n. The combination of all these symmetries provides shift operators, which allow us to run through the entire set of bound states of hydrogen.The periodic system. Starting from the four-dimensional hidden symmetry and accidental degeneracy of the H atom, as first revealed by Fock in 1935, our research has mainly focussed on the way this SO(4) symmetry of the Coulomb potential gets broken in the PS as a consequence of the transformation of the hydrogenic (n, l) filling order to the Madelung (n + l, n) order due to electronic repulsions, relativistic effects and spin-orbit coupling.The single infinite-dimensional degeneracy space of SO(4,2) has been shown to be applicable to the PS and has been denoted as the baruton. Filling all the states with electrons leads to all the elements of the PS, and in this way the baruton can be looked upon as a massive metaparticle, which covers the full PS. The baruton represents the primeval atom, at the point of the Big Bang when all energies were degenerate. As its symmetry broke, the universe unfolded and phenomena appeared. “C’est la dissymétrie qui crée le phénomène,” dixit Pierre Curie. The observable manifestations of the baruton are the different chemical elements, which are arranged together in the PS. This implies that the structure of the PS is to be found in a particular symmetry breaking of the SO(4,2) group.Removing all operators with an index 4 from the SO(4,2) matrix has yielded an SO(3,2) subgroup which has the interesting property that the spectrum splits into two separate manifolds, depending on whether n + l is even or odd: SO(4,2) > SO(3,2). This leads to a doubling of the Aufbau series. Not unlike spin, SO(3,2) thus adds an additional quantum characteristic which can only take two values: even or odd. We have found that the ‘period doubling’ observed in the PS is nothing else than a manifestation of this binary quantum level.In the final stages of this PhD fellowship, we have looked for a symmetry group which governs the different n + l multiplets of the left-step table. Since the dimensionalities of these multiplets correspond to those of the hydrogen spectrum, it seems plausible that the n + l multiplets result from a new SO(4) group. In contrast with the H atom, however, the Casimir operator of this group has to label the different unirreps by a new quantum number N = n + l. The PS would then be described by the following subgroups:SO(4,2) > SO(3,2) > SO(4)' > SO(3) > SO(2)However, the embedding of the infinite diagonal sequences of hydrogenic states, corresponding e.g. to 1s, 2p, 3d, 4f, 5g, ... as well as the finite counter-diagonal sequences such as 5s, 4p, 3d in the SO(4,2) covering group of Barut hasproved problematical. Although the sequences bear characteristics to supermultiplets of the SO(3,1) Lorentz group, and are reminiscent of Regge trajectories in hadron physics, the standard embedding does not yield the right result, since it leads to a different reduction of the standard hydrogenic basis.The discovery of a new S operator, first introduced by Englefield, has opened the way to a further supersymmetry beyond SO(4,2). We have studied the extended symmetry that results and have demonstrated that this S operator provides the required flexibility to obtain the diagonal Regge sequences. Similar work has be done for the counter-diagonal sequences (Madelung sequences), and have lead to a parallel outcome.The S operator yields under commutation with the generators of the initial SO(4,2) algebra a new set of operators. We believe that these, together with the original L and Q operators will form a modified so(4,2) algebra. We have recently constructed a set of new operators by combining operators from the two algebras. These new operators act as diagonal ladders in the hydrogen spectrum — raising and lowering n and l simultaneously by one unit as desired.The new operators commute among themselves to form a deformed Lorentz so(3,1) algebra. The algebraic structure which is obtained is a kind of nonlinear Lie algebra, where the structure constants are replaced by a function. This function is based on the invariants of subalgebras, and furthermore contains so(3) as a subalgebra. It thus bears a resemblance to the nonlinear extensions of simple Lie algebras, and opens many new perspectives.It is our hope to extend this fundamental research by studying this nonlinear SB mechanism in more detail.
Description: The results obtained in this PhD have shed new light on the periodic system and have induced us to write a textbook Shattered Symmetry: From the Eightfold Way to the Periodic Table which is currently under review at Oxford University Press.
Table of Contents: Abstract
Nederlandstalige Abstract
List of Tables
List of Figures
List of Symbols
List of Abbreviations
Chapter outline

1 Literature study

1.1 Introduction
1.2 Chemical periodicity
1.3 Quantum mechanics of atomic systems
1.4 Quantum mechanics of the periodic system
1.5 The Madelung (n+l,n) rule
1.6 Misapplying the Madelung rule
1.7 Group theory and the periodic system

2 Group theory in a nutshell

2.1 Introduction
2.2 Algebraic group theory
2.3 Symmetry in quantum mechanics
2.4 Lie groups and Lie algebras
2.5 The Cartan-Weyl basis
2.6 The three pillars of group theory

3 SO(4) group

3.1 Accidental degeneracy & dynamical symmetries
3.2 Classical Kepler problem
3.3 Quantum mechanics of the hydrogen atom
3.4 The special orthogonal group SO(4)
3.5 The origin of accidental degeneracies

4 SO(2,1) group

4.1 The road to noninvariance groups
4.2 The pseudo-orthogonal group SO(2,1)
4.3 Hydrogenic realization of the SO(2,1) group
4.4 Dynamical treatment of the radial wave equation

5 SO(4,2) group

5.1 The pseudo-orthogonal group SO(4,2)
5.2 The Cartan-Weyl basis
5.3 Quantum alchemy

6 SO(4,2) and the rules of atomic chess

6.1 Introduction
6.2 Group theoretical classification of the elements
6.3 The rules of atomic chess
6.4 The quest for the chiral bishop
6.5 Nonlinear algebras
6.6 Literature study (cont’d)

ISBN: 978-90-8649-613-6
Publication status: published
KU Leuven publication type: TH
Appears in Collections:Molecular Design and Synthesis
Quantum Chemistry and Physical Chemistry Section

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