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The Chemistry of Matter
Waves
Jan C. A. Boeyens
Centre for Advancement of Scholarship
University of Pretoria, South Africa
e-mail: jan.boeyens@up.ac.za
ii
Preface
The spectacular successes such as the construction of lasers and magnetic
resonance instruments, commonly credited to quantum physics and spec-
troscopy, make the expectation of a quantum theory of chemistry almost
irresistable. Equally spectacular failures to account for high-temperature su-
perconductivity, cold fusion, molecular diffraction, optical activity and molec-
ular shape are conveniently ignored. Even the emergent concept of spin, cor-
rectly considered the most non-classical property of elementary matter, has
never been explained in terms of first-principle quantum theory.
It is therefore not surprising to find that beyond the Bohr-Sommerfeld
model of the atom quantum mechanics has caused more confusion than en-
lightenment in theoretical chemistry. However, to turn away from the fantasy
of quantum chemistry, after a century of expectation, could be as traumatic
as renouncing the prospects of alchemical transmutation.
Chemistry is the prodigy of alchemy as modified by the theories of mod-
ern physics. Even so, it still has not resolved the ancient enigma around the
nature and origin of matter. Alchemy itself is the product of ancient her-
menistic philosophies, traces of which have survived the metamorphosis into
chemistry. Elements of the number-based Pythagorean cosmology are clearly
discernible, even in the most modern theories of chemical affinity. Briefly [1]:
The cosmic unit is polarized into two antagonistic halves (male
and female) which interact through a third irrational diagonal
component that contains the sum of the first male and female
numbers (3+2) and divides the four-element (earth, water, fire,
air) world in the divine proportion of τ=p5/4−1
2.
τ
1
2
iii
iv
In Pythagorean parlance, any chemical interaction is essentially of the type
HCl + NaOH →NaCl + H2O.
It is facilitated by the affinity between opposites to produce a product that
symbolizes the principle of substantiality, in harmonious equilibrium with
the total environment.
All harmonic proportions and relationships are said to derive from the
roots of 2, 3 and 5, the number of life. In modern terminology, the harmony
that results from the interplay of integers and irrationals manifests at all
levels of reality. It is colloquially refered to as self similarity, well known
to be mediated by the golden ratio and golden logarithmic spirals. Modern
theories perform little better in describing ponderable matter as resulting
from the interaction between cold dark matter and a universal Higgs field.
The mathematical model that underpins the theory is as mysterious as the
divine proportion.
Chemistry distinguishes between space and time, and between matter and
energy. The seminal theories of physics, independently developed by Newton
and Huygens made the distinction between particles and waves. Hamilton’s
refinement of classical mechanics demonstrated some common ground be-
tween the two theories, but Maxwell’s formulation of the electromagnetic
field revealed a fundamental difference in their respective laws of motion. It
was the unified transformation of Lorentz that finally established the four-
dimensional nature of Minkowski space-time and the equivalence of mass and
energy. The gravitational and electromagnetic fields remained poles apart.
However, both of these could be shown, by Einstein’s general relativity and
the notion of gauge invariance as developed by Weyl and Schr¨odinger, to be
products of Riemann’s non-Euclidean geometry. Ultimate unification of the
fields was achieved in terms of Veblen’s projective relativity.
Analysis of the interaction between matter and radiation and the the-
ories of chemistry were pursued in Euclidean space and remained at vari-
ance with the theory of relativity, culminating in the awkward compromise
of wave-particle duality. It is only the recognition of spin as a strictly four-
dimensional concept that holds the promise of wave structures, which behave
like particles. Formulated as a quaternion structure it defines the common
ground between relativity and quantum theories. The electron, defined as a
nonlinear construct, known as a soliton, recognizes the importance of space-
time curvature and represents final unification of its initially antagonistic
attributes.
It is the theme of this book to show how refinement of the concepts matter
and wave would lead to a consistent description of chemical systems without
v
the confusion of probability densities and quantum jumps. The final model
is that of Schr¨odinger, extended to four dimensions in nonlinear formulation.
The major effect of this more general proposed formulation is that the
procedure of linear combination of atomic orbitals, at the basis of all “quan-
tum chemistry” completely looses its validity and it needs to be replaced
by entirely new modelling strategies. One alternative, already in place, is
molecular mechanics, an empirical procedure based on classical mechanics
and classical notions of molecular structure. It is encouraging to note that
the same number-theoretic simulations, which are effective as a basis of ele-
mental periodicity, are commensurate with molecular mechanics.
The number-theory simulation of chemical systems originated with the
observation that the periodicity of atomic matter depends on the number
ratio of atomic protons to neutrons that converges to τas a function of
either A,Z,A−Zor A−2Z. The same pattern is revealed by the golden
proton excess x=Z−τN . By demonstrating that this convergence is a
function of general space-time curvature the observed cosmic self-similarity
is inferred to depend in equal measure on space-time curvature, the golden
ratio and the shape of the golden logarithmic spiral.
To put the whole scheme into perspective it is noted that, because of cur-
vature, the geometry of space time is non-Euclidean and different from the
commonly perceived Euclidean geometry. Topologists distinguish between an
underlying, globally curved space-time manifold and the local, approximately
Euclidean, three-dimensional, tangent space and universal time. Any analy-
sis performed in tangent space, using a model such as Newtonian mechanics
or Schr¨odinger’s linear equation, produces a good, but incomplete, approxi-
mation, compared to possibly more refined descriptions in four-dimensional
detail.
To compensate for the neglect of curvature the golden parameter τ, or
optimization in terms of golden logarithmic spirals, provides an immediate
corrective, in the simulation of chemical systems by linear procedures. The
very existence of matter is seen to depend on the nonlinear deformation of
a hypothetical, Euclidean, four-dimensional energy field as described by the
theory of general relativity. The product is a non-dispersive solitary wave
packet, known as a soliton. Different modes of deformation lead to the for-
mation of solitons of different symmetry, colloquially known as elementary
particles. Dependent on mass, charge and spin these units are of different sta-
bility and in combination with those of complementary affinity develop into
the different forms of ponderable matter — atoms, molecules, crystals, fluids
and higher aggregates. The imprint of space-time curvature and the golden
ratio remains with all matter, exhibiting a common self-similar symmetry.
The periodicity of matter arises as the product of a closed numerical
vi
system with a natural involution that relates matter to antimatter. In four
dimensions such a function defines elliptic space in the form of projective
space-time, as used by Veblen in the unification of the electromagnetic and
gravitational fields.
The hard sell of convincing chemists that quantum mechanics in its
present guise is too restrictive as a theory of chemistry necessarily involves
unfamiliar mathematical arguments that may turn out to be counterpro-
ductive. To be convincing it is unavoidable to introduce various aspects of
physics and applied mathematics traditionally considered to be way outside
the chemistry paradigm. The bland alternative of starting from “well es-
tablished” mathematical physics appears equally problematical. This is the
exact strategy that created the present dilemma in the first place.
The most daunting prospect is to argue convincingly for the adoption
of a four-dimensional world view, against the millions of three-dimensional
molecular structures derived by sophisticated experimental techniques. To
complicate matters by the introduction of nonlinear effects would surely be
considered as meaningless, unless it can be supported with concrete examples.
The anticipated response is difficult to predict.
The conservative respect for authority creates another problem. It comes
naturally to reject, without thinking, dissident views that contradict the
time-honoured ideas of respected pioneers. A prime example is in the han-
dling of high-temperature superconductivity. The BCS theory, which ascribes
superconduction to the formation of bosonic electron pairs, mediated by lat-
tice phonons, offers no insight into the mechanism that operates in ceramic
materials. Even the correlation of low-temperature metallic superconduc-
tion with normal-state properties remains an empirical observation without
theoretical support. A reported room-temperature superconducting state is
simply denied as theoretically impossible.
The credibility of the quantum-based BCS theory rests entirely on the
reputation of its authors. Reluctance to abandon the model relates to the
mistaken perception that it is supported by the mathematical simulation of
a superconduction transition as the breakdown of gauge symmetry on cool-
ing. However, the symmetry model applies to all forms of superconductivity
whereas the phonon interaction is an empirical conjecture for one special case
only.
The readily demonstrated dependence of superconductivity on the compo-
sition of atomic nuclei favours an alternative description of the phenomenon
as a nuclear, rather than a strictly electronic, property. Special stability of
the nuclear composition that corresponds to the Z/N ratio of τimplies a
positively charged surface shell that correlates remarkably well with anoma-
lous nuclear spin and superconduction. With this surface excess as a guide
vii
an alternative mechanism that effects all forms of superconductivity is rec-
ognized.
At a more speculative level the phenomenon of electrolytic “cold fusion”,
appears to occur at cathodes, rich in high-spin isotopes of the same type.
In this case the active process appears as neutron capture that converts
symmetry-distorted nuclides to lower-energy forms.
These examples all point at the unpalatable conclusion that quantum
theory, in its present form, falls far short of popular perceptions. It is not
the all-embracing panacea that stretches beyond science and inspires the
non-local metaphysics of fundamental acausality, probability and comple-
mentarity, which blossomed into multiverse cosmology. An “inner voice”
told Einstein that something was amiss, but he lacked the data to support
his intuition.
The central issue that defied comprehension was the apparent dual na-
ture of both elementary matter and radiation. Efforts to account for this
uncertainty resulted in concepts, universally accepted by now, such as an
observer’s role in creating patterns from the conceptually unknown. This
confusion between subject and object resonates with the musings of psychol-
ogists and philosophers, groping for an understanding of reality in terms of
medieval mysticism through quantum theory [2].
The unfortunate conviction that inspires such pursuits, although hard to
gainsay philosophically, has a simple resolution:
There is no such thing as an elementary point particle.
Matter, as the product of intrinsically nonlinear four-dimensionally curved
space-time, or “codensation of the vacuum (æther)”, has a wave structure.
Not in the form of dispersive wave packets, but as non-dispersive persistent
solitary waves, or solitons, only known to occur in shallow water at the time
when quantum theory was formulated.
Solitons are flexible and under certain circumstances may appear to be-
have like point particles. Futile efforts to account for a soliton’s wave-like
behaviour with a particle model result in the weird constructs, generally be-
lieved to reflect quantum effects. This statement is a concise summary of the
argument to be developed in the following.
Acknowledgement
I thank Demi Levendis and Vimal Iccharam for their continued interest
in this venture and Faan Naude for his frienly information retrieval service.
viii
Bibliography
[1] J.C.A. Boeyens and D.C. Levendis, The Structure Lacuna, Int. J. Mol,
Sci. 13 (2012) 9081–9096.
[2] R.M. Pirsig, Subjects, Objects, Data and Values, in [3] p 79–98.
[3] D. Aerts, J. Broekaert and E. Mathijs (eds), Einstein meets Magritte: An
Interdisciplinary Reflection, Springer.com, 1999.
Contents
1 Of Electrons and Molecules 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Electrons in Chemistry . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Wave-particle Duality . . . . . . . . . . . . . . . . . . 3
1.2.2 The Schr¨odinger Approximation . . . . . . . . . . . . . 4
1.2.3 Four-Dimensional Waves . . . . . . . . . . . . . . . . . 5
1.2.4 Nonlinear Schr¨odinger Equation . . . . . . . . . . . . . 5
1.3 Molecular Structure . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.1 Molecular Modelling . . . . . . . . . . . . . . . . . . . 7
1.3.2 Atomic and Molecular Structure . . . . . . . . . . . . . 9
2 The Classical Background 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 The Copernican Revolution . . . . . . . . . . . . . . . 16
2.2 Newtonian Physics . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Daltonian Chemistry . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 The Aftermath . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4.1 Dalton’s Legacy . . . . . . . . . . . . . . . . . . . . . . 22
2.4.2 Classical Mechanics . . . . . . . . . . . . . . . . . . . . 24
3 Great Discoveries 31
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Periodic Table of the Elements . . . . . . . . . . . . . . . . . . 32
3.2.1 Static Model of Chemical Affinity . . . . . . . . . . . . 36
3.2.2 The Planetary Quantum Model . . . . . . . . . . . . . 40
3.2.3 The New Periodic Table . . . . . . . . . . . . . . . . . 44
3.3 The Electromagnetic Field . . . . . . . . . . . . . . . . . . . . 45
3.3.1 Wave Theory of Light . . . . . . . . . . . . . . . . . . 45
3.3.2 Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3.3 Electrostatics . . . . . . . . . . . . . . . . . . . . . . . 49
3.3.4 Electromagnetism . . . . . . . . . . . . . . . . . . . . . 53
ix
xCONTENTS
3.3.5 Maxwell’s Theory . . . . . . . . . . . . . . . . . . . . . 55
3.4 Electromagnetic Radiation . . . . . . . . . . . . . . . . . . . . 56
3.4.1 General Theory of Wave Motion . . . . . . . . . . . . . 58
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4 Theoretical Response 65
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.1.1 The Electromagnetic Field . . . . . . . . . . . . . . . . 66
4.1.2 Periodicity of Atomic Matter . . . . . . . . . . . . . . 66
4.1.3 Theories in Conflict . . . . . . . . . . . . . . . . . . . . 67
4.2 The Theory of Relativity . . . . . . . . . . . . . . . . . . . . . 67
4.2.1 Special Relativity . . . . . . . . . . . . . . . . . . . . . 68
4.2.2 General Relativity . . . . . . . . . . . . . . . . . . . . 74
4.3 Quantum Theory . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.3.1 Global Gauge Invariance . . . . . . . . . . . . . . . . . 78
4.3.2 Wave Mechanics . . . . . . . . . . . . . . . . . . . . . 81
4.3.3 Local Gauge Invariance . . . . . . . . . . . . . . . . . . 86
4.3.4 Space-time Manifold and Tangent Space . . . . . . . . 88
4.3.5 The Periodic Function . . . . . . . . . . . . . . . . . . 89
5 State of the Art 93
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 Chemistry at the Crossroads . . . . . . . . . . . . . . . . . . . 96
5.2.1 The Bonding Model . . . . . . . . . . . . . . . . . . . . 97
5.2.2 Molecular Structure . . . . . . . . . . . . . . . . . . . . 99
5.2.3 Stereochemistry . . . . . . . . . . . . . . . . . . . . . . 101
5.2.4 The Particle Problem . . . . . . . . . . . . . . . . . . . 102
5.2.5 Reaction Mechanisms . . . . . . . . . . . . . . . . . . . 102
5.2.6 Atomic Periodicity . . . . . . . . . . . . . . . . . . . . 104
5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6 The Forgotten Dimension 109
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.2 The Classical World . . . . . . . . . . . . . . . . . . . . . . . 111
6.3 Non-classical World . . . . . . . . . . . . . . . . . . . . . . . . 112
6.3.1 Potential Theory . . . . . . . . . . . . . . . . . . . . . 113
6.4 The Spin Function . . . . . . . . . . . . . . . . . . . . . . . . 114
6.4.1 Four-dimensional Action . . . . . . . . . . . . . . . . . 117
6.4.2 Spin Correlation . . . . . . . . . . . . . . . . . . . . . 117
6.5 The Time Enigma . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.5.1 Quantum Potential . . . . . . . . . . . . . . . . . . . . 119
CONTENTS xi
6.5.2 Time Flow . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.6 Space-Time Curvature . . . . . . . . . . . . . . . . . . . . . . 122
6.6.1 Space-Time Topology . . . . . . . . . . . . . . . . . . . 123
6.7 Quantum Effects . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.7.1 Exclusion Principle . . . . . . . . . . . . . . . . . . . . 126
6.7.2 Wave-Particle Duality . . . . . . . . . . . . . . . . . . 127
6.7.3 Quantum Probability . . . . . . . . . . . . . . . . . . . 128
6.7.4 Measurement Problem . . . . . . . . . . . . . . . . . . 131
6.7.5 Uncertainty Principle . . . . . . . . . . . . . . . . . . . 133
6.7.6 Fine-structure Constant . . . . . . . . . . . . . . . . . 135
7 Nonlinear Chemistry 143
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.2 Wave Model of the Electron . . . . . . . . . . . . . . . . . . . 144
7.2.1 Wave Mechanics . . . . . . . . . . . . . . . . . . . . . 145
7.2.2 Matter Waves . . . . . . . . . . . . . . . . . . . . . . . 148
7.2.3 Two-wave Models . . . . . . . . . . . . . . . . . . . . . 155
7.2.4 Fine-stucture Parameter . . . . . . . . . . . . . . . . . 156
7.3 Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . 159
7.3.1 Hydrodynamic Analogy . . . . . . . . . . . . . . . . . 160
7.3.2 Schr¨odinger Oscillator . . . . . . . . . . . . . . . . . . 160
7.3.3 Korteweg – de Vries Equation . . . . . . . . . . . . . . 162
7.3.4 Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . 163
7.3.5 Soliton Eigenvalues . . . . . . . . . . . . . . . . . . . . 165
7.3.6 Soliton Models . . . . . . . . . . . . . . . . . . . . . . 167
7.3.7 Electronic Solitons . . . . . . . . . . . . . . . . . . . . 169
7.4 Chemical Aspects . . . . . . . . . . . . . . . . . . . . . . . . . 177
7.4.1 Solving the Equation . . . . . . . . . . . . . . . . . . . 178
7.4.2 Chemical Interaction . . . . . . . . . . . . . . . . . . . 179
8 Matter-Wave Mechanics 187
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
8.2 The Aether and Matter . . . . . . . . . . . . . . . . . . . . . . 190
8.2.1 Alarming Phenomena . . . . . . . . . . . . . . . . . . . 191
8.2.2 Generation of Mass . . . . . . . . . . . . . . . . . . . . 192
8.2.3 Space-time Topology . . . . . . . . . . . . . . . . . . . 192
8.2.4 The Vacuum . . . . . . . . . . . . . . . . . . . . . . . . 200
8.3 The Wave Model . . . . . . . . . . . . . . . . . . . . . . . . . 202
8.3.1 Projective Solution . . . . . . . . . . . . . . . . . . . . 203
8.4 Matter in Space-Time . . . . . . . . . . . . . . . . . . . . . . 207
8.4.1 Fibonacci Numbers . . . . . . . . . . . . . . . . . . . . 208
xii CONTENTS
9 Chemical Wave Structures 221
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
9.2 Electronic Structures . . . . . . . . . . . . . . . . . . . . . . . 222
9.2.1 Numbers and Waves . . . . . . . . . . . . . . . . . . . 224
9.3 Atomic Structure . . . . . . . . . . . . . . . . . . . . . . . . . 225
9.4 Chemical Concepts . . . . . . . . . . . . . . . . . . . . . . . . 227
9.4.1 Atomic Size . . . . . . . . . . . . . . . . . . . . . . . . 228
9.4.2 The Bohr–de Broglie Model . . . . . . . . . . . . . . . 229
9.4.3 Ionization Radii . . . . . . . . . . . . . . . . . . . . . . 232
9.4.4 Electronegativity . . . . . . . . . . . . . . . . . . . . . 233
9.4.5 Covalent Interaction . . . . . . . . . . . . . . . . . . . 235
9.4.6 Bond Order . . . . . . . . . . . . . . . . . . . . . . . . 236
9.4.7 General Covalence . . . . . . . . . . . . . . . . . . . . 237
9.4.8 Atomic Polarizability . . . . . . . . . . . . . . . . . . . 239
9.4.9 Atomic radii . . . . . . . . . . . . . . . . . . . . . . . . 241
9.4.10 Final Results . . . . . . . . . . . . . . . . . . . . . . . 244
9.5 Molecular Structure . . . . . . . . . . . . . . . . . . . . . . . . 245
9.5.1 Molecular Modelling . . . . . . . . . . . . . . . . . . . 246
9.6 Reaction Mechanism . . . . . . . . . . . . . . . . . . . . . . . 247
10 A Fresh Start 253
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
10.2 The Copenhagen Interpretation . . . . . . . . . . . . . . . . . 255
10.2.1 Quantum Mechanics . . . . . . . . . . . . . . . . . . . 255
10.2.2 The Quantum Postulate . . . . . . . . . . . . . . . . . 257
10.2.3 Atomic Model . . . . . . . . . . . . . . . . . . . . . . . 261
10.2.4 Quantum Chemistry . . . . . . . . . . . . . . . . . . . 264
10.3 Two New Models . . . . . . . . . . . . . . . . . . . . . . . . . 265
10.3.1 Superconductivity . . . . . . . . . . . . . . . . . . . . . 265
10.3.2 Cold Fusion . . . . . . . . . . . . . . . . . . . . . . . . 267
10.4 The Common Wave Model . . . . . . . . . . . . . . . . . . . . 272
10.4.1 The Periodic Function . . . . . . . . . . . . . . . . . . 273
10.5 New Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
10.5.1 Nanostructures . . . . . . . . . . . . . . . . . . . . . . 274
10.5.2 Quasicrystals . . . . . . . . . . . . . . . . . . . . . . . 280
10.6 Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . 282
10.6.1 The Space-time Vacuum . . . . . . . . . . . . . . . . . 283
10.6.2 Perceptions in Linear Tangent Space . . . . . . . . . . 283
10.6.3 Four-dimensional Reality . . . . . . . . . . . . . . . . . 284
