Content uploaded by Jan Boeyens
Author content
All content in this area was uploaded by Jan Boeyens on Oct 10, 2014
Content may be subject to copyright.
In: Electrostatics: Theory and Applications
Editor: Camille L. Bertrand, pp. 191-215 ISBN 978-1-61668-549-2
c
2010 Nova Science Publishers, Inc.
Chapter 9
EMERGENT PROPERTIES IN BOHMIAN CHEMISTRY
Jan C.A. Boeyens
Unit for Advanced Study, University of Pretoria, South Africa
Abstract
Bohmian mechanics developed from the hydrodynamic interpretation of quantum
events. By this interpretation all dynamic variables retain their classical meaning in
quantum systems. It is of special significance in chemistry as a discipline which is
traditionally based on the point electrons of quantum field theory. It could be more
informative to assume a non-dispersive electronic spinor, or wave packet, with diver-
gent and convergent spherical wave components, and with many properties resembling
those of a point particle.
Complex chemical matter is endowed with three attributes: cohesion, conforma-
tion and affinity, which can be reduced to the three fundamental electronic properties
of charge, angular momentum and quantum potential, known from the wave struc-
ture. The chemical effects of these respective scalar, vector and temporal principles,
all manifest as extremum phenomena. The optimal distribution of electronic charge in
space appears as Pauli’s exclusion principle. Minimization of orbital angular momen-
tum becomes the generator of molecular conformation. Equilization of electronegativ-
ity, the quantum potential of the valence state, dictates chemical affinity. The chemical
environment is said to generate three emergent properties: the exclusion principle,
molecular structure and the second law of thermodynamics. These concepts cannot be
predicted from first fundamental principles.
Only by recognition of the emergent properties of chemistry is it possible to simu-
late chemical behaviour. The exclusion principle controls all forms of chemical cohe-
sion, atomic structure and periodicity; molecular structure underpins vector properties
such as conformational rigidity, optical activity, photochemistry and other stereochem-
ical phenomena; while transport properties and chemical reactivity depend on the sec-
ond law.
Simulation of these chemical concepts by constructionist procedures, starting from
basic physics, is impossible. The ultimate reason is that complex chemical properties
are not represented by quantum-mechanical operators in the same sense as energy and
momentum. The Bohmian interpretation, which enables the introduction of simplify-
ing emergent parameters, in analogy with classical procedures, allows the calculation
of molecular properties by generalized Heitler-London methods, point-charge simula-
tion and molecular mechanics.
192 Jan C.A. Boeyens
1. Introduction
The editorial challenge to address the quantum frontiers of atoms and molecules in
chemistry cuts far deeper than a survey of current activity in quantum chemistry, which for
all practical purposes means ab-initio computational chemistry.
There is no quantum theory of chemistry: Quantum mechanics originated as a theory
to understand radiation and its interaction with sub-atomic matter. It gave birth to the mod-
ern science of spectroscopy, in which form it stimulated the development of sophisticated
observational techniques that revolutionized physics, chemistry and biology. However, the
early promise of a quantum theory of matter in general has not come to fruition. A theory
that fails to elucidate the nature of electrons, atoms and molecules can never lead to an un-
derstanding of chemistry. It may enable computations of unrivalled complexity, but without
a conceptual framework the results have no meaning.
As a theory of spectroscopy, quantum mechanics serves to relate measured quantities,
such as frequency and wavelength to the dynamic concepts of energy and angular momen-
tum, by means of differential operators e.g.:
−i~∂/∂t →E, i~∂/∂ϕ →L.
Although chemically more useful information on molecular structure and electronegativity
is also embedded in the state functions, there are no known operators whereby to extract
this information directly. The current computational alternative fails for the same reason.
Minimization of the scalar quantity, energy, can never generate three-dimensional structure,
which is a vector concept.
The failure to rationalize chemical behaviour is not a failure of quantum theory, but
rather, a failure of the traditional interpretation of the theory. The ruling interpretation of
quantum theory, known as the Copenhagen interpretation, incorporates a number of fea-
tures totally at variance with chemistry. It defines quantum objects, including photons,
electrons, atoms and molecules, as structureless point particles without extension. It re-
duces a continuously varying density, such as the electronic charge distribution in atoms,
molecules and crystals to a probabilistic function. It offers no rationale for the occurrence
of stationary states, apart from a postulate. Non-local effects are forbidden and the doctrine
teaches more about measurement problems and quantum uncertainty than about chemical
interaction.
The unfortunate reality is that Schr¨
odinger’s description of elementary matter as wave
structures, which could not be reconciled with the Copenhagen orthodoxy, was ruled un-
acceptable and re-interpreted in terms of the quantum jumps and probability density of the
particle model. This compromise resulted in the awkward concept of wave-particle duality,
familiar to all modern chemists, but understood by none.
The challenge that we are facing here is to retrace our steps to the point where the
time-honoured concepts of classical chemistry merge in a natural way with the ideas of
wave mechanics and start rebuilding a theory that ′′stimulate(s) the mutual understanding
of the various branches of chemistry and its neighbouring sciences′′, realizing that ′′ the main
stumbling block for the development of a theory of large and complex molecular systems
is not computational but conceptual′′ [1].
Emergent Properties in Bohmian Chemistry 193
I propose to start with an outline of the essential concepts of cohesion, conformation
and affinity; show how these relate to the more fundamental concepts of space, matter and
number; and the derived concepts of interaction, waves and periodicity. Next we examine
the Schr¨
odinger formalism, the Bohmian interpretation and the application of wave me-
chanics, supported by higher-level emergent properties. The photoelectric effect, probably
the most effective argument to have established the particle nature of photons is shown to
be explained more convincingly by the transactional-wave model of electromagnetic inter-
action.
2. The Fundamental Concepts
Any object or event observed in Nature can always be considered as the product of
more primitive events or the interaction between more primitive entities. It is quite natural,
in this reductionist spirit, to ascribe the actions and features of a living organism to some
lower-level activity of biological cells, which in turn is driven by intracellular chemical
interactions. The molecular building blocks of biological cells are assumed to consist of
atoms, held together by electromagnetic forces, while the sub-atomic nucleons interact with
strong interaction, and so on, ad infinitum. Not really. The reduction has to stop somewhere.
As progressively smaller entities are implicated at the more primitive levels, it is reasonable
to conclude that the cascade ends in a void, traditionally assumed to be the same as space,
or the vacuum.
At this point it is easy to get distracted by the interminable philosophical dispute about
the possible existence or non-existence of a void. As a practical alternative I prefer to define
space in geometrical terms and the vacuum as a physical entity.
2.1. Space
The simplest way of looking at space is as a coordinate system that serves to describe
the relative positions of observable objects. The intuitively most obvious is a cartesian sys-
tem on three orthogonal axes. However, this may not necessarily be the most convenient
coordinate system. Although it works well in a laboratory environment, it is well known to
be inappropriate in geographical context, which is simplified by using spherical trigonom-
etry. By the same argument cartesian coordinates may not be the most convenient to map
the relative positions of astronomical objects. To first approximation the planets and most
of their moons have been assumed to move in a single plane, called the ecliptic, around the
sun. Individual elliptic orbits of planets and moons have a simple description in terms of
Kepler’s laws in a plane, but most of these planes are known to make non-zero angles with
the ecliptic.
Moving into interstellar, or even intergalactic, space the situation becomes more com-
plicated when dealing with cosmological distances, times and velocities, which demands
relativistic rather than Galilean kinematics. Special relativity is conveniently formulated in
four-dimensional Minkowski space and general relativity requires non-Euclidean geometry,
i.e. curved space.
In general relativity the geometry of space is described by a curvature tensor, which
is linearly related to a stress tensor that describes the distribution of matter in the cosmos.
194 Jan C.A. Boeyens
This reciprocal relationship shows that an empty universe has zero curvature and that curved
space generates matter. The mechanism whereby curvature generates matter is visualized in
the process of covering a curved surface with an inflexible sheet. The higher the curvature,
the poorer the fit and the more the wrinkles in the cover that cannot be smoothed away. Such
wrinkles in space are interpreted as matter and energy – the content of the stress tensor.
2.1.1. Number
Figure 1. Spiral structure of a fossilized nautilus shell.
Not only the topology of space-time, but also the physical content of the universe, re-
sembles the natural number system in remarkable detail [2]. This explains the unreasonable
effectiveness of mathematics as a scientific tool and the success of number theory to predict
natural phenomena as a manifestation of cosmic symmetry [2, 3]. The physical world, as
an image of the natural numbers, can never be known in more detail than the number sys-
tem. Concepts such as infinity and singularity, poorly understood mathematically, therefore
make no physical sense. On the other hand, the concept of high-dimensional space, readily
manipulated mathematically, and, although physically difficult to visualize, has a legitimate
place in scientific discourse.
To deal with the ubiquitous, but bothersome, infinities of physics, the infinity concept of
projective geometry can be used to define both the number system and the physical cosmos
as closed. Realizing that any closed system is periodic, by definition, a wave structure of
the vacuum and periodicity of matter are inferred.
The implied cosmic symmetry is referred to as self-similarity. The chambered structure
of a nautilus shell, shown in Figure 1, is one of the best-known examples of this symme-
Emergent Properties in Bohmian Chemistry 195
try type. All the chambers have the same shape and only differ in size, which increases
regularly along a golden logarithmic spiral. The same pattern occurs in the arrangement of
growing seed buds in a sunflower head and in the image of a spiral galaxy. Recognition of
the same growth features [2, 4] in atomic nuclei, atoms, covalent molecules and the solar
system, reveals self-similarity on a much wider scale. The number theory of self-similarity
shows that all of these structures are based on the Fibonacci number sequence, which con-
verges to the golden ratio.
2.2. Vacuum
The fabric of space is a matter of conjecture. However, if tangible matter occurs on
curving flat space, flat space is not void. A useful analogy is to picture the vacuum as a
regular undulating expanse filled by waves of constant wavelength. When curved, interfer-
ence of the primary waves produces persistent wave packets, earlier identified as wrinkles.
In local space such a wave packet is conveniently described as the superposition of three-
dimensional spherical waves, converging to and diverging from a centre of mass. These
waves are the retarded and advanced solutions of the general wave equation, which implies
motion, either forward or backward in time.
2.2.1. Wave Packets
A typical wave packet generated by such a superposition of waves is shown in Figure
2. The tangent curve follows the amplitude of the 1/r Coulomb potential, which reflects
the actual charge density, except when r→0. The secondary waves propagate with the
group velocity vgof the system and the primary waves have phase velocity vφ, such that
vgvφ=c2, where c= 1/√ǫ0µ0, is the velocity of light in the vacuum. Such a wave
packet [4]:
Φ = Aeiωt cos kr
kr or sin kr
kr (1)
has been shown [5] to describe elementary waves, equivalent to the postulated elementary
distortions of space, i.e. the elementary particles of atomic physics. Interpreted as an
electron, the distance between nodal points represents λdB =h/mevg, the de Broglie
wavelength of a free electron and λC= 2π/k =h/mec, the Compton wavelength. The
amplitude of the standing wave is proportional to the electronic charge.
Φ0=A, in eqn.(1), represents a wave packet with charge proportional to 0 or ±A.
Electrons and protons, despite their difference in mass have charges of ±e. The neutron
is neutral. The field intensity ΦΦ∗=A2(sin kr/kr)2=C/r2defines the force between
charges, in line with Coulomb’s law, except when r→0. The breakdown of Coulomb’s
law, which occurs naturally for charged wave structures is equivalent to the special renor-
malization postulate in quantum field theory.
2.2.2. Electron and Atom
A particle image, as shown in Figure 3, is obtained by rotating the diagram of Figure
2 about two axes perpendicular to x. The charge density can either contract or expand,
196 Jan C.A. Boeyens
Figure 2. One-dimensional section through a spherical wave packet with components con-
verging on and diverging from x0.
depending on environmental pressure. The minimum radius that can be reached on com-
pression depends on the rest mass of the object. For an electron r0=e2/m0c2.
Figure 3. Wave structure of a free electron with de Broglie wavelength λdB =h/mevg.
Given the inferred flexible structure of an electron as a continuous indivisible charge,
the self-similarity of atoms and the solar system is not obvious. The pioneering work on
the planetary model of atomic structure was firmly based on Kepler’s model of elliptic
orbits. Two crucial parameters that define a Kepler ellipse are the semimajor axis and
the eccentricity, which characterize the size and shape of an orbit. By Newton’s laws these
respective parameters are related to the energy and angular momentum of the orbital motion.
In retrospect Kepler’s laws are seen to embody the general conservation principles for
energy and angular momentum in celestial mechanics. The efforts of Bohr and Sommer-
feld to explain electronic motion in atoms by the same model were spectacularly successful,
despite a few subtle, but fatal, defects. Whereas Kepler’s model is valid in a gravitational
field, it needs modification in an electromagnetic field as an accelerated charge radiates en-
ergy and an accelerated point charge therefore cannot maintain a stable orbit. Conservation
Emergent Properties in Bohmian Chemistry 197
of angular momentum in a central electrostatic field should rather be interpreted as conser-
vation of the spherical shape of a continuous charge. Polar deformation under an external
influence is described by the three-dimensional surface harmonics, or eigenfunctions, of the
circulation Laplacian, with discrete eigenvalues:
L2Yml
l=l(l+ 1)k2Yml
l,
LzYml
l=mlkY ml
l.
This classical result acquires quantum-mechanical meaning by equating the arbitrary con-
stant kwith ~, the elementary unit of angular momentum [6].
Figure 4. Phase-locked cavity with perfectly reflecting walls, filled with radiation in the
form of standing waves.
The difference between atomic and planetary systems goes a long way towards un-
derstanding of cosmic self-similarity. Although electrons in an atom are spread in three
dimensions and planets orbit the sun in an approximately two-dimensional plane, both ar-
rangements depend parametrically on the golden ratio. In the same way sunflower seeds
of varying size are closely packed in a plane, compared to the three-dimensional stacking
of nucleons; both styles conditioned by the golden ratio. The only common factor in all
cases is the general curvature of space. Evidently, the curvature of cosmic space must be a
function of the golden ratio, from the sub-atomic to supergalactic scales.
2.2.3. Mass
Not only the charge, but also the characteristic mass and spin of sub-atomic species are
accounted for by their wave structure. Jennison and Drinkwater [7] demonstrated that mi-
crowave radiation trapped in a phase-locked cavity, as in Figure 4, generates an interaction
pattern which is mathematically equivalent to a system with inertial mass.
Disturbing the equilibrium by a pulse that moves a cavity wall at velocity δv for a pe-
riod δt, which is matched to the wave propagation across the cavity, modifies the internal
pressure by Doppler shifting of the waves and sets the entire system into motion with veloc-
ity 2δv. The radiation pressure on the walls is balanced by an electromagnetic field, which
keeps the system in static equilibrium. In the real vacuum the analogue of the phase-locked
cavity is a standing wave, filled with radiation of Compton wavelength, internal energy E,
in equilibrium with the external radiation (wave) field. Simple calculation [7] shows that
198 Jan C.A. Boeyens
the inertial mass of the wave packet obeys Newton’s law, F=ma, on identifying E/c2
with the rest mass.
2.2.4. Spin
The standing-wave description of an electron defines it as an integral part of the vac-
uum, not obviously free to move without impediment. Linear motion of an electron must
then clearly lead to continuous drag and deformation of both electron and its immediate
environment, culminating in rupture of the vacuum and creation of a turbulent state. An
electron that rotates in the vacuum, although more symmetrical, winds up the connecting
medium until it shears and develops a discontinuity along a cylindrical surface. The only
motion that occurs without distortion of the spherical wave packet or mechanical entangle-
ment of the environment is rotation around a point. Unlike axial rotation this mode is more
like a continuous wobble that returns to the original situation after two complete revolu-
tions. The three dimensions of space participate equally in the motion without the transfer
of rotational energy from the spinning object to the connecting medium. The strain that
builds up during the first part of the rotational cycle relaxes during the second part. Apart
from half-frequency cyclic disturbance in the connecting medium, the electron is free to
move through the vacuum without permanent entanglement.
An object, which performs this type of spherical rotation, is described mathematically
by a spinor, or a quantity that reverses sign on rotation through an odd multiple of 2π
radians.
3. Quantum Theory
Quantum theory started with the discovery of line spectra and Balmer’s observation that
the spectral lines of atomic hydrogen obey a digital formula, later generalized to:
ν=Rc 1
n2
1−1
n2
2, n2>n1= 1,2,3... (2)
The first sensible explanation of the formula was proposed in 1904 by Nagaoka who used
the planet Saturn with its system of rings as the basis of an atomic model, with electrons at
energy levels (rings) in simple numerical order, orbiting a heavy positively charged nucleus.
Experimental confirmation of such an arrangement was found by Rutherford in 1910 and
a dynamic model, based on Planck’s quantum condition, E=hν, was proposed in 1914
by Bohr. Where Nagaoka argued that electrodynamically stable orbits required a standing
electron wave of length λ= 2πr/n at an average distance rfrom the nucleus, Bohr postu-
lated quantum stability for an orbiting electron with angular momentum p=nh/2π≡n~.
With electrostatic and mechanical forces in balance, (using electrostatic units, 4πǫ0= 1):
e2
r2=p2
mr , E =T+V=e2
2r−e2
r=−e2
2r,
Emergent Properties in Bohmian Chemistry 199
the postulate leads to the Balmer formula
hν = ∆E=2π2me4
h21
n2
1−1
n2
2,(3)
En=2π2me4
n2h2, rn=n2h2
4π2me2.(4)
The rest is history. When de Broglie rediscovered the Nagaoka condition in 1924 by pos-
tulating that all matter has an associated wavelength of λ=h/mv, only the mathematical
framework for defining a general wave formalism of electronic behaviour, was lacking.
3.1. Wave Mechanics
The mechanical behaviour of a Newtonian particle is described correctly by three quan-
tities – energy, momentum and angular momentum, which describe the motion as a func-
tion of either time, displacement or rotation. In wave formalism each of these parameters is
specified as a periodic function [8] with respect to time (τ), translation (λ) or rotation (ϕ):
ω= 2π/τ, k = 2π/λ, ml= 2π/ϕ,
E=~ω, p =~k, Lz=~ml.
The carrier of the electromagnetic field is described by the differential wave equation:
∇2Ψ = µǫ∂2Ψ
∂t2=1
c2
∂2Ψ
∂t2.(5)
To remain consistent with the previous relationships the dynamic variables need to be spec-
ified as differential operators:
E→ −~i∂/∂t, p → −~i∇, Lz→~i∂/∂ϕ,
which can be checked by direct substitution. To allow for the first-order temporal depen-
dence of the energy, the equation for matter waves is restricted to processes which only
depend on time through a factor exp(2πiνt), leading to the final form:
~2
2m∇2+VΨ = ±~i∂Ψ
∂t (6)
which formally resembles the classical Hamiltonian definition of total energy, as
H=T+V=p2
2m+V=E. (7)
By defining a density function ρ= ΨΨ∗and a current density
j=~
2mi (Ψ∗∇Ψ−ψ∇Ψ∗)(8)
there follows a continuity equation as in classical hydrodynamics
∂ρ
∂t +divj= 0.(9)
200 Jan C.A. Boeyens
A general expression for a one-electron wave function over all available states
Ψ = X
k
ckψke2πiνt (10)
may be used to calculate the current density over two states kand l:
j=~e
mi X
k,l
ckcl(ψl∇ψk−ψk∇ψl)e2πi(νk−νl)t.(11)
If only a single eigenvibration is excited, the current disappears and the distribution of
electron density remains constant. Otherwise an electron flows from one state to another in
an exchange that involves a photon to keep the energy in balance.
This flow of electricity can hardly be described as a quantum jump. More realistically
the vibrations of the two affected states (emitter and acceptor) are seen to interact and
generate a beat (wave packet) that moves to the state of lower energy. The virtual photon
that links two equilibrium states turn into a real photon that carries the excess energy, either
into or away from the system.
In chemical applications Schr¨
odinger’s equation is best known in its amplitude form,
which is obtained by substituting Ψ = ψexp(2πiνt), followed by elimination of the time
parameter to give:
∇2ψ+2m
~2(E−V)ψ= 0.(12)
In spherical polar coordinates this equation, for the hydrogen problem, separates into inde-
pendent radial and angular equations:
d2R
dr2+2
r
dR
dr+2m
~2E−V(r)−l(l+ 1)
r2R= 0,(13)
1
sin θ
∂
∂θ sin θ∂Y
∂θ +1
sin2θ
∂Y
∂ϕ2+l(l+ 1)Y= 0,(14)
with separation constant λ=l(l+ 1), integer l. The angular part is further separable into:
d2Φ
dϕ2+m2
lΦ = 0 (ml=−l . . . l),(15)
1
sin θ
d
dθsin θdΘ
dθ+l(l+ 1) −m2
l
sin2θΘ = 0.(16)
An electron associated with a stationary proton (V=e2/r)defines the only problem of
some chemical significance for which the radial equation has been solved. Since the proton
is here regarded as a point particle, the system does not represent a wave-mechanical model
of a hydrogen atom, despite contrary claims in all chemistry texts. Like the Bohr model, it
defines a set of quantized energy levels to match most spectroscopic measurements, apart
from the Lamb shift, fairly well.
The angular equations are valid for central-field problems and produce quantized values
of the orbital angular momentum. These eigenvalues should not be confused with the angu-
lar momenta of an orbiting particle. They are, more appropriately, considered as symmetry
Emergent Properties in Bohmian Chemistry 201
parameters, such that ml= 0 defines a spherically symmetrical charge distribution. For
given lthere is always an odd number of 2l+ 1 sub-levels with different quantum numbers
ml, which, for many-electron systems, can be chosen in such a way that Plml= 0, in
all cases. The choice reflects the electrostatic property of the charge distribution to assume
spherical symmetry. A hydrogen atom, by this model, has ml6= 0 only for excited states,
which spontaneously relax to the ml= 0 spherically symmetrical ground state.
3.1.1. Electron Spin
Schr¨
odinger’s equation appears incomplete in the sense of lacking an operator for spin,
only because its eigenfunction solutions are traditionally considered complex variables. The
wave function, interpreted as a column vector, operated on by square matrices, such that
ei(ωt−kx)0
0e−i(ωt−kx) φ1
φ2=φ1ei(ωt−kx)
φ2e−i(ωt−kx).
abbreviated to φ1e+
φ2e−, represents a spinor that moves in the x-direction. By forming the
derivatives:
∂φ
∂t =iω φ1e+
φ2e−,
∂2φ
∂x2=k2φ1e+
φ2e−,
it follows that (in three dimensions):
−i∂φ
∂t =ω
k2∇2φ.
This is Schr¨
odinger’s equation, providing (~k)2= 2m~ω,i.e.
−i∂φ
∂t =~
2m∇2φ, as in (6):V= 0
which shows ~ω=E=p2/2m,k= 2π/λ,p=h/λ. This result is interpreted [4]
to show that a region of the continuum, which rotates in spherical mode, interacts with
its environment by generating a wave-like disturbance at half the angular frequency of the
core. The angular momentum on the surface of a unit sphere is L=mω. At λ= 2π,
k= 1, the spin angular momentum follows as L=~/2, with intrinsic magnetic moment
µ=~e/2mc.
3.2. Bohmian Mechanics
The connection between wave mechanics and hydrodynamics, expressed by equations
(7) and (8), was developed in more detail by Madelung, writing the time dependence of Ψ
202 Jan C.A. Boeyens
as an action function, Ψ = ψe2πiνt →ReiS/~, which seperates (5) into a coupled pair that
resembles the field equations of hydrodynamics:
∂S
∂t +(∇S)2
2m−~2∇2R
2mR +V= 0,(17)
∂R2
∂t +∇ · R2∇S
m= 0,(18)
which describe the irrotational flow of a compressible fluid, assuming R2to represent the
density ρ(x)of a continuous fluid with stream velocity v=∇S/m. It was shown that both
density and flux vary periodically with the same periodicity as νik = (Ei−Ek)/h, that
results from superposition of states iand k. This means that radiation is not due to quantum
jumps, but rather happens by slow transition in a non-stationary state.
An attractive feature of the hydrodynamic model is that it obviates the statistical inter-
pretation of quantum theory, by eliminating the need of a point particle. It is worth noting
that the assumption of a point electron derives from the observation that it responds as a
unit to an electromagnetic signal, which must therefore propagate instantaneously through
the interior of the electron, at variance with the theory of relativity. However, by now it
is known from experiment that non-local (instantaneous) response is possible in quantum
systems and the initial reservation against Madelung’s proposal and Lorentz’s definition of
an electron as a flexible sphere should fall away.
On reinterpretation it was pointed out by David Bohm that equation (17) differed from
the classical Hamilton-Jacobi equation only in the term
Vq=−~2∇2R
2mR .(19)
The quantity Vq,called quantum potential vanishes for classical systems as h/m →0. A
gradual transition from classical to quantum behaviour is inferred to occur for systems of
low mass, such as sub-atomic species. All dynamic properties of classical systems should
therefore be defined equally well for quantum systems, although the relevant parameters
are hidden [10].
3.2.1. Quantum Potential
As for the classical potential, the gradient of quantum potential energy defines a quan-
tum force. A quantum object therefore has an equation of motion, m..
x=−∇V− ∇Vq.
For an object in uniform motion (constant potential) the quantum force must vanish, which
requires Vq= 0 or a constant, −ksay. Vq= 0 defines a classical particle; alternatively1
−(V+Vq) = T, the kinetic energy of the system. Hence ~2∇2R/2mR =−E, which
rearranges into
∇2R+2mE
~2R= 0
Schr¨
odinger’s equation for a free particle.
1It is a common misconception that Vq=Tfor a free electron – compare [11]. Stationary states do not
occur for Vq= 0, but when Vq=−V.
Emergent Properties in Bohmian Chemistry 203
The quantum potential concept is vitally important for understanding the structure of
an electron and of quantum systems in general. The fact that the amplitude function (R)
appears in both the numerator and denominator of Vqimplies that the effect of the wave
field does not necessarily decay with distance and that remote features of the environment
can affect the behaviour of a quantum object.
The quantum potential for a many-body system:
Vq=
n
X
i=1 −~2
2mR ∇2
iR
mi
depends on the quantum state of the entire system. The potential energy between a pair
of entities, Vq(xi, xj)is not uniquely defined by the coordinates, but depends on the wave
function of the entire system, Ψ. This condition defines a holistic system in that the whole
is more than a sum of the parts. The instantaneous motion of one part depends on the coor-
dinates of all other parts at the same time. That defines a non-local interaction of the type
assumed to exist within an indivisible electron, and now inferred to occur in all quantum
systems, including molecules. If the system is distorted locally, the entire system responds
instantaneously. As the quantum potential is not a function of distance, the behaviour of
a composite system depends non-locally on the configuration of all constituents, no mat-
ter how far apart. In a chemical context the properties, structure and rearrangement of
molecules must depend intimately on the quantum potential. It is necessary to give up the
notion that molecular rearrangement involves the breaking and making of bonds and rather
consider it as a modification of the intramolecular electronic wave interference pattern.
However, all systems are not correlated equally well. Whenever a wave function can be
written as a product
Ψ(r1, r2, t) = ΦA(r1, t)ΦB(r2, t)
the quantum potential becomes the sum of two terms:
Vq(r1, r2, t) = VA
q(r1, t) + VB
q(r2, t).
The two sub-systems evidently behave largely independently. That is a good description
of a molecular crystal, or liquid, with relatively weak interaction between molecular units.
Systems like these are better described as partially holistic.
The contentious issue of quantum-particle trajectories is put into perspective by the
Bohmian model. One interpretation is that the quantum electron has an unspecified diffuse
structure, which contracts into a classical point-like object when confined under external
influences. The observed trajectory, as in a cloud chamber, may be considered to follow the
centre of gravity.
In a two-slit experiment an electron wave passes through both slits to recombine, with
interference, but without rupture. The interference pattern disappears on closure of one slit
or when the slits are too far apart, compared to the de Broglie wavelength. It now behaves
exactly like a classical particle, when forced through a single slit2[12].
2The de-Broglie – Bohm formulation of particle plus pilot wave is considered an unnecessary complication
by this author. Instead, Ψmay be thought of as a state of vibration of empty space.
204 Jan C.A. Boeyens
3.2.2. Stationary States
Writing the wave equation in two equivalent forms:
Ψ(x, t) = Ψ0e−iEt/~,
Ψ(x, t) = R(x, t)eiS(x,t)/~,
and noting that R(x, 0) = R0(x);S(x, 0) = S0(x);Ψ0=R0eiS0/~, it follows that:
S(x, t) = S0(x)−Et, (20)
R(x, t) = R0.
The unexpected conclusion is that a real wave function, Ψ0=ψ, implies S0(x) = 0 and
hence the momentum ∇S=p= 0 and E=V+Vq. Those states with ml= 0 all have
real wave functions, which therefore means that such electrons have zero kinetic energy and
are therefore at rest. The classical (electrostatic) and quantum forces on electrons in such
stationary states are therefore balanced and so stabilize the position of the electron with
respect to the nucleus.
For the hydrogen atom in the ground state, R(r) = N e−r/a0and hence,
d2R
dr2=N
a2
0
e−r/a0,
such that, from (19), Vq=~2/2ma2
0. In general
Vq=~2
2mr2,(21)
and the quantum force on the electron:
Fq=∂Vq
∂r =−~2
mr3
whereas the electrostatic force F=e2/r2. These forces are in balance when
~2
mr3=e2
r;r=~2
me2=a0,
the Bohr radius. This means that V=Vqat r=a0/2, halfway between proton and
electron.
3.2.3. Orbital Angular Momentum
Orbital angular momentum is perhaps the most awkward concept to visualize as the
property of a quantum-mechanical point electron, but is readily understood in hydrody-
namic analogy. Like tidal motion, atomic orbital motion in a continuous spherical charge
cloud consists of the propagation of a wave disturbance, without matter circulation, as
first proposed by Nagaoka and described by the quantized spherical surface harmonics,
Yml
l=N P ml
leimlϕ, in terms of Legendre polynomials, P.
Emergent Properties in Bohmian Chemistry 205
In Bohmian formalism angular momentum is described by rotation of the phase func-
tion:
S(x, t) = S0(x)−Et
=ml~ϕ−Et.
The wavefronts S=constant are planes parallel to and rotating about the z-axis, with angular
velocity ∂ϕ/∂t =E/ml~.
Single-valuedness of Ψ = Rexp(iS/~)requires that Ψ(S) = Ψ(S+ 2πn~) = ψ(S+
nh). This is interpreted to mean that n=|ml|wave crests occur during each cycle. Positive
and negative values of mlrepresent anticlockwise and clockwise rotations respectively.
This interpretation of orbital angular momentum has a formal resemblance to the semi-
classical model of Bohr and Sommerfeld, but there is no physical rotation of charge. Two
electrons with magnetic quantum numbers of ±mlhave wave structures that rotate, in
phase, in opposite directions, with resultant distortion of zero. Quenching of orbital an-
gular momentum during chemical interaction between neighbouring atoms happens by the
same principle. The wave pattern in the case where l6= 0 and ml= 0 is to be interpreted
as the three-dimensional analogue of the circular modes of a vibrating drumhead. There is
no axial component to the disturbance. The wave motion is more like spherical vibration,
compared to spherical rotation that causes electron spin and which can be oriented in the
polar direction of a magnetic field.
4. Chemical Change
In the same sense that biological activity is more than chemical change, chemical effects
depend on a number of emergent properties unknown to physics. The concepts of chemical
affinity, cohesion and structure were discovered experimentally and not anticipated from
first principles. Although chemical events can therefore not be inferred from the laws of
physics, the Bohmian interpretation of quantum mechanics provides an attractive frame-
work for their understanding. The fundamental reason for this emergence is the chemical
environment. The interaction between chemical species, partially characterized in isolation,
is as hard to predict as the behaviour of an individual in a crowd. Not being acquainted with
the concepts molecule, phase transition and free energy, there is no possibility of deriving
the laws of chemical affinity, reactivity and composition from the quantum numbers that
quantify the energy and angular momentum of electrons in isolated atoms. The problem is
approached here by examining the possible modes of interaction between charges and the
response of atoms to close confinement.
4.1. Interaction Theory
Interaction at a distance is interpreted in modern theories as a field phenomenon. The
electromagnetic field, described by Maxwell’s equations as waves, propagate through the
vacuum, with a constant velocity that depends on the permittivity and permeability of free
space, c= 1/√µ0ǫ0. The wave equation (4) has solutions Ψ(t)and Ψ(−t), known as
retarded and advanced waves, respectively. The transmission of electromagnetic energy
206 Jan C.A. Boeyens
between an emitter and a distant receptor is assumed to be negotiated by a pair of retarded
and advanced waves. As a spherical wave signal from the emitter reaches an acceptor, it
responds with an advanced return signal that reaches the emitter at the exact moment of first
emission, to establish a standing wave, known as a photon.
Further interaction depends on the potential energy difference between emitter and re-
ceptor. Transfer of excess energy occurs by relaxation of the standing wave, which is exper-
imentally observed as photon emission. Alternatively the standing wave, known as a virtual
photon, that exists between interacting sites, becomes balanced against external factors, at
a distance that defines the electrostatic force of interaction between the charges as:
F=q1q2
4πǫ0r2.
All chemical interactions are of this type [4].
In Bohmian formalism the theory predicts the stability of atomic matter as a function of
the fine-structure constant.
Sommerfeld [13] – (p.107) introduced the fine-structure constant as α=v1/c =
e2/4πǫ0c~(= 2πe2/ch, in esu), where v1is the velocity of an electron in the first Bohr
orbit. More generally, the parameter α′=v/c for a freely moving electron with de Broglie
wavength λdB =h/mv and Compton wavelength λC=h/mc is defined, more appropri-
ately as α′=λC/λdB. An electron in a hydrogenic stationary state has nλdB = 2πn2a0,
hence:
αn=e2
n~c.
In the Bohmian interpretation an atomic stationary state occurs when the potential energy
of the electron, at rest, is balanced by the quantum potential.
The relativistic mass of an electron at the position of the nucleus, with respect to the
rest mass moin the ns state, would be
m=mo
p1−v2/c2=mo
√1−α2
i.e.,
α2=m2−m2
o
m2=4π2e4
n2h2c2=En
mc2,
Hence En= ∆m′c2,where ∆m′=m−m2
o/m ≃m−mo.
This is interpreted here to show that an electron in a stationary state has its mass reduced,
with respect to the nucleus, by an amount ∆m′, which reappears as the binding energy
−En. The same argument explains nuclear binding energy as a mass defect.
Transition of an electron with n > 1to a lower unoccupied energy level by emission
of a photon with energy hν and spin ~, is anticipated. However, in the 1sstate with quan-
tum number l= 0, there is no orbital angular momentum to transfer in promoting photon
emission and the ground state remains stable. The calculation does not imply different ve-
locities for the electron at different energy levels – only a quantized change in de Broglie
wavelength. The mass-energy difference amounts to exchange of a (virtual) photon in the
form of a standing wave between the charge centres.
Emergent Properties in Bohmian Chemistry 207
With the classical radius of the electron defined as r0=e2/mc2it is noted that
r0
a0
=me4
m~2c2=e2
~c2
=α2,
where a0is the Bohr radius. This result follows from the two relationships:
αλC=2πe2
mc2= 2πr0,
λC
α=2π~2
me2= 2πa0=λdB .
Now define λZ= 2πr0. Whereas the wavelength λdB =λC/α represents a wavepacket
with group velocity vg< c, the phase velocity vφ> c is associated with the Zitterbewegung
of wavelength λZ=α·λC;vgvφ=c2[15].
This argument relates to two problematic parameters: αand the classical electron ra-
dius r0which still awaits quantum-mechanical definition. The fine-structure constant ap-
pears firmly associated with the wave nature of an electron, seen as a standing wave that
results from the superposition of diverging and converging spherical components. The inter-
nal wave structure of the electron is observed as high-frequency Zitterbewegung while the
macroscopic effects in an electromagnetic field are fixed by the spread of the wavepacket,
conveniently defined as a de Broglie wavelength. Trapped in the field of a proton the de
Broglie wavelength is quantized to avoid self-destruction, such that
λC
λdB
=αn=e2
n~c.
For an effective charge separation of rn, the ratio αnmay be considered the ratio of two
energies:
e2
n~c=e2
rn·1
hν
an electrostatic and a quantum-mechanical factor. The constant c=λ/τ =λν describes the
virtual photon that occurs as a standing wave (nλ = 2πr) between the charge centres. The
balance between the classical coulombic attraction and the quantum-mechanical repulsion
(the quantum potential) defines the fine-structure constant with a value, fixed by the de
Broglie wavelength of the virtual photon.
In a strong field the size of an electronic wavepacket may be compressed below the
Compton radius to an absolute minimum of λZ, which describes the minimum size to which
an electron may be compressed, measuring r0=λZ/2π, for an electron defined as an
electric charge −edistributed over a sphere of radius r0. The potential energy E=e2/r0
corresponds to r0=e2/moc2, as measured classically.
4.2. Environmental Effects
Schr¨
odinger’s solution for the hydrogen electron serves as the starting point for the qual-
itative discussion of all chemical effects in quantum formalism. It is routinely forgotten that
the simple hydrogen solution ignores all interactions that the electron would experience in
208 Jan C.A. Boeyens
a chemical environment. Even the use of hydrogen energy levels to rationalize the structure
of the periodic table is of limited value.
A useful approach to simulate environmental effects was pioneered by Sommerfeld on
solving Schr¨
odinger’s equation under modified boundary conditions. Non-zero environ-
mental pressure was introduced by assuming that ψ(r)→0as rapproaches some finite
value rc, rather than infinity. All energy levels move to higher values with decreasing rc,
until the ground level reaches the ionization limit at rc=r0, the ionization radius.
It is noted that on reaching the ionization limit by uniform compression the electron
that becomes decoupled from the nucleus finds itself confined to a spherical cavity at zero
potential and kinetic energy. However, the non-zero energy of a free electron in a hollow
sphere, must therefore be interpreted as quantum potential energy. The Helmholtz equation
for such an electron:
∇2+k2ψ= 0 , k =p2mE/~2
has the radial solutions, R=p2kr/π ·kl(kr). At the first zero of the spherical Bessel
function 0= sin(kr)/(kr),kr0=π, and hence (compare 21)
E0=h2
8mr2
0
=Vq.(22)
The Fourier transform of 0is the box function
f(r) = √2π/2r0if |r|< r0,
0if |r|>r0.(23)
It follows that the decoupled (valence) electron of the hydrogen atom, compressed to r0is
uniformly spread across the ionization sphere.
4.3. Emergent Properties
Chemical theory requires insight into more than atomic stability. One-electron quantum
theory provides no guidance beyond the hydrogen atomic ground state and the structure of
many-electron atoms must be inferred from the empirically known periodic table of the
elements. However, the superficial correspondence between the calculated quantum states
of hydrogen and the observed elemental periods strongly suggests a functional relationship
between the two sets. To better appreciate the relationship it is noted that both sets can be
generated by convergent sequences of Fibonacci or Lucas fractions as shown below.
4.3.1. Periodicity
A modular pair of rational fractions h1
k1,h2
k2has the property:
h1h2
k1k2
=±1.
Emergent Properties in Bohmian Chemistry 209
Such a pair is geometrically represented by two Ford circles with radii and y-coordinates
of 1/2k2at x-coordinates of h/k. A series of rational fractions with all neighbouring terms
in unimodular relationship is represented by a set of tangent Ford circles [4]. Examples of
such modular series are the Farey sequences, Fnand the converging Fibonacci and Lucas
fractions on the segment (1
2
3
5
2
3) of F5:
Despite a number of uncertain half-lives, a reasonable estimate of 264 divides the sta-
ble (non-radioactive) nuclides into 11 periods of 24. Plotting the ratio of protons:neutrons
(Z/N)for all isotopes as a function of atomic number, the hem lines that separate the pe-
riods of 24, intersect a reference line, at Z=τ, in Z-coordinates which correspond to
well-known ordinal numbers that define the periodic table of the elements [2,3]. Remark-
ably, the same hem lines intersect a reference line at Z/N = 0.58 in atomic numbers that
correspond to the closure of the calculated wave-mechanical energy levels for hydrogen.
Noting that the radii of unimodular Ford circles are inversely proportional to the number
(2k2)of atoms in elemental periods, we look for converging circles that match the two
forms of periodicity. The primary circle
at x= 0 or 1, rF=1
2, is flanked by two tangent circles at x= 0.5and 1.5≡ −0.5,
rF=1
8; further converging pairs are at x=±2
3,rF=1
18 and x=±3
4,rF=1
32 . This
arrangement mimics the periodic table:
210 Jan C.A. Boeyens
What is probably the aesthetically most pleasing form of the periodic table is obtained
by rearrangement in circular array, as shown in Figure 5 for the hundred naturally occurring
cosmic elements.
Interpreted in terms of electronic distribution it implies twelve 8-fold and three 2-fold
energy levels, with all closed-shell elements grouped together. These are not hydrogen-like
energy levels, but they agree with the valence levels, calculated for compressed atoms in
Hartree-Fock-Slater approximation [14].
The hypothetical arrangement based on the hydrogen solution is recognized in the
nested set of Ford circles at x= 1, predicting consecutive periods of 2n2,n= 1,2,...,
arranged as follows:
If nis interpreted as Schr¨
odinger’s principal quantum number, periods of the correct
length (2n2)are predicted. Each of the periods consists of nsubshells for subsidiary quan-
tum numbers 0≤l < n. The number of elements per subset equals 2(2l+ 1),l≤ml≤l.
This result provides the basis of Pauli’s exclusion postulate, which defines an emergent
property, not of quantum-mechanical origin.
The hypothetical and observed versions of the periodic table are in agreement for el-
ements 1 to 18. The superficial agreement (e.g. for elements 28, 46 and 78; and 29–36)
beyond that point is purely accidental. We conclude that the wave-mechanical hydrogen
model fails to account for elemental periodicity mainly because it ignores all interactions
apart from the central-field unitary electrostatic attraction. The common thesis of chem-
istry textbooks that Schr¨
odinger’s equation, with due allowance for interelectronic effects,
accounts for the periodic table, fails on two important counts. It predicts transition series of
ten elements, compared to the observed eight. The guiding principle, known as the Aufbau
procedure, is valid only for the alkaline-sand pblocks. Less than two-thirds of the nominal
transition elements obey an Aufbau rule. The correct periodic system occurs in an environ-
Emergent Properties in Bohmian Chemistry 211
Figure 5. The Periodic Table of the elements in circular form.
ment that requires the convergence of stable nuclear composition, Z/N to the golden ratio,
τ, and subject to an emergent exclusion principle.
Further new properties are expected to emerge in the analysis of chemical affinity, co-
hesion and conformation, at a higher hierarchical level.
4.3.2. Electronegativity
Chemical affinity is the intuitive qualitative concept that guided experimental chemistry
for centuries. The first quantitative measure of affinity was discovered by Lothar Meyer
as the atomic volume of an element – his basis of periodicity. It served to differentiate
between electropositive and electronegative elements, with a natural affinity between them.
The concept was generalized by Pauling, Mulliken and others, by placing all elements on
212 Jan C.A. Boeyens
Figure 6. Electronegativity as quantum potential of the valence state
a single empirical electronegativity scale. By demonstrating the equivalence of electroneg-
ativity and the quantum potential of the valence state [16] it was finally recognized as an
emergent atomic property, readily reduced to fundamental quantum theory.
Like hydrogen, an atom is said to be in its valence state when ionized by environmen-
tal pressure. The energy of the electron, decoupled from the nucleus but confined to the
ionization sphere, is given by (22). Characteristic ionization radii, r0, are obtained by nu-
merical Hartree-Fock-Slater calculation [14] with boundary conditions modified as for H.
Redefined on this basis, electronegativity, χ, is calculated as
χ2=h2
8mr2
0
,
expressed in eV, such that χrelates to Pauling electronegativities on a linear scale and χ2
to the Mulliken scale. A plot of χ=√E0reveals the same periodicity as Figure 5 and
as Lothar Meyer atomic volumes. The uniform electron density of the valence state, from
(23):
ρ=ψ2(r), ψ(r) = (φ/V0)1
2exp {−(r/r0)p}, p >> 1.(24)
The scale factor, φ, which compensates for an inaccessible core, is proportional to r0and
varies inversely with the number of nodes as defined by an effective principal quantum
number n. Hence, φ=cr0/n. The wave function
ψ(r) = p3c/4πn 1
r0exp {−(r/r0)p}(25)
describes the interaction of an atom with its chemical environment.
Emergent Properties in Bohmian Chemistry 213
4.3.3. Covalence
Chemical cohesion has for many years been the main topic of theoretical chemistry,
conducted as an exercise in computational quantum physics, described by one practitioner
[17] as if repeatedly ′′...validating Schr¨
odinger’s equation!′′. There is a curious conviction
that the Born-Oppenheimer scheme enables molecular structure to be computed ab initio.
An initially assumed structure is treated only as a device to kickstart the calculation. Once
the electronic density has been obtained, the nuclear framework is computed theoretically,
without assumption. It always comes out miraculously close to the assumed structure.
To the uninitiated the procedure appears to be circular and unlikely to produce anything
of physical significance beyond the assumed molecular structure. An obvious alternative
is to model the electron exchange that constitutes atomic pairwise interactions, known as
covalent bonds, before assembly into a three-dimensional structure is attempted.
The computational details for this procedure, which requires atomic wave functions,
have been documented as the well-known Heitler-London method. Appropriate wave func-
tions (25) are obtained from empirically adjusted ionization radii that compensate for steric
factors [4]. H–L calculations predict both dissociation energy and equilibrium interatomic
distance for any first-order covalent interaction. High-order interaction results from the
valence-level screening of the internuclear repulsion.
Figure 7. Covalent binding energy curve for homonuclear diatomics in dimensionless units.
The same set of characteristic atomic radii (r)can be used to model covalent electron
exchange by point-charge simulation, as a function of interatomic separation (d)only. It
214 Jan C.A. Boeyens
is found that with the ratio d/r and binding energy E, expressed in dimensionless units,
all homonuclear diatomic interactions are described by a single interaction curve, shown in
Figure 7.
The curve turns where d/r =τand E=−2τ, at the point where exactly two electron
waves are concentrated in the interatomic region. The condition is seen to reflect the ex-
clusion principle for fermions. Its relationship to the golden ratio defines the origin of the
exclusion principle as the curvature of space-time. Without this emergent property there is
no understanding of covalent interaction.
4.3.4. Molecular Shape
The inability to derive molecular structures from fundamental quantum theory identifies
molecular shape as another emergent property. Although it cannot be inferred from basic
theory it is readily reduced to the conservation of orbital angular momentum.
Conventional computational schemes, designed to minimize energy, with total neglect
of orbital angular momentum, must, by definition converge to a spherically symmetrical ar-
rangement. To prevent this from happening a potential field of lower symmetry is imposed
by assuming a fixed nuclear framework. Instead of imposing an experimentally determined
structure, conservation of orbital angular momentum provides a theoretically more satisfy-
ing algorithm to generate such a structure from first principles.
Polarization of mutually approaching reactants resolve local angular-momentum vec-
tors, just like an applied magnetic field. During the formation of a molecule, the alignment
of reactants that minimizes angular momentum in the local polar direction is favoured.
In many reaction systems there is sufficient symmetry for the orbital angular momentum in
the polar direction to become quenched completely. Where the quenching in low-symmetry
(chiral) systems is incomplete, the residual angular momentum will couple to the magnetic
field of polarized light, causing optical activity. Should quenching be possible only for a
specific angular alignment of neighbouring fragments, a rigid system, which resists tor-
sional deformation, is obtained. So-called double bonds and aromatic systems are common
examples.
The empirical stereochemical rules, pioneered by Kekul´
e, van’t Hoff and others, are
consistent with the principles outlined here and these have been generalized into empirical
computational schemes, collectively known as molecular mechanics. There is no more
fundamental procedure to predict molecular structure.
References
[1] H. Primas, Chemistry, Quantum Mechanics and Reductionism, 2nd ed., Springer-
Verlag, Berlin, 1983.
[2] J. C. A. Boeyens and D.C. Levendis, Number Theory and the Periodicity of Matter,
Springer.com, 2008.
[3] J. C. A. Boeyens, Periodicity of the stable isotopes, J. Radioanal. Nucl. Chem., 2003
(257) 33–41.
Emergent Properties in Bohmian Chemistry 215
[4] J. C. A. Boeyens, Chemistry from First Principles, Springer.com, 2008.
[5] M. Wolff, Beyond the Point Particle – A Wave Structure for the Electron, Galilean
Electrodynamics, 1995 (6) 83–91.
[6] J. C. A. Boeyens, Angular Momentum in Chemistry, Z. Naturforsch. 2007 (62b) 373–
385.
[7] R. C. Jennison and A. J. Drinkwater, An approach to the understanding of inertia from
the physics of the experimental method, J. Phys. A, 1977 (10) 167–179.
[8] J. C. A. Boeyens, Quantum theory of molecular conformation, C.R. Chimie, 8 (2005)
1527 – 1534.
[9] E. Schr¨
odinger, Collected Papers on Wave Mechanics, (Translated from the second
German edition), 2nd ed., Chelsea, NY, 1978.
[10] D. Bohm, A Suggested Interpretation of the Quantum Theory in Terms of ′′Hidden′′
Variables, Phys. Rev., 1952 (85) 166–179, 180–193.
[11] D. W. Belousek, Einstein’s 1927 Unpublished Hidden-Variable Theory: Its Back-
ground, Context and Significance, Stud. Hist. Phil. Mod. Phys., 1996 (27) 437–461.
[12] P. R. Holland, The Quantum Theory of Motion, University Press, Cambridge, 1993.
[13] A. Sommerfeld, Atombau und Spektrallinien, 4th ed., Vieweg, Braunschweig, 1924.
[14] J. C. A. Boeyens, Ionization radii of compressed atoms, J. Chem. Soc. Faraday Trans.,
1994 (90) 3377–3381.
[15] J. C. A. Boeyens, New Theories for Chemistry, Elsevier, Amsterdam, 2005.
[16] J. C. A. Boeyens, The Periodic Electronegativity Table, Z. Naturforsch., 2008 (63b)
199–209.
[17] N. C. Handy, in: R. Broer, P.T.C. Aerts and P.S. Bagus, New Challenges in Computa-
tional Chemistry, University of Groningen, 1994.
