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Challenges to the Structural Conception
of Chemical Bonding∗

Michael Weisberg
University of Pennsylvania

April 11, 2007

Abstract

While the covalent bond plays a central role in chemical predictions, in-
terventions, and explanations, it is a di!cult concept to de"ne precisely. #is
paper investigates the structural conception of the covalent bond, which says
that bonding is a directional, sub-molecular region of electron density lo-
cated between individual atomic centers that is responsible for holding the
atoms together. Several approaches to constructing molecular models are
considered in order to determine which features of the structural concep-
tion of bonding, if any, are robust across these models. #e paper concludes
that key components of the structural conception are absent in all but the
simplest quantum mechanical models of molecular structure, seriously chal-
lenging the conception’s viability.

Crucial to chemical practice and discourse is the notion of the chemical bond,
speci"cally the covalent bond. For most synthetic and analytical purposes, molecules
are conceived of not just as collections of atoms, but as structural entities consist-
ing of directionally oriented atomic centers connected, in some way or other, by

∗Earlier versions of this paper were presented at the Australia National University, the Philoso-
phy of Science Association, and the Center for Philosophy of Science, University of Pittsburgh. I am
grateful to these audiences for their enthusiasm and very helpful comments. Special thanks to Jerry
Berson, Barry Carpenter, Dave Chalmers, Peter Godfrey-Smith, Clark Glymour, Robin Hendry,
Roald Ho$mann, Paul Needham, John Norton, Janet Stemwedel, and Deena Skolnick Weisberg for
extremely helpful feedback.

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covalent bonds. Hydrogen gas is said to be composed of two hydrogen atoms held
together by a single, covalent bond; oxygen gas, two oxygen atoms and a double
bond; methane, four equivalent C–H single bonds.

While the chemical bond plays a central role in chemical predictions, interven-
tions, and explanations, it is a di!cult concept to de"ne precisely. Fundamental
disagreements exist between classical and quantum mechanical conceptions of the
chemical bond, and even between di$erent quantum mechanical models. Once
one moves beyond introductory textbooks to advanced treatments, one "nds many
theoretical approaches to bonding, but few if any de"nitions or direct characteri-
zations of the bond itself. While some might attribute this lack of de"nitional clar-
ity to common background knowledge shared among all chemists, I believe this
re%ects uncertainty or maybe even ambivalence about the status of the chemical
bond itself.

#is kind of ambiguity has led chemists to di$erent conclusions. Most com-
monly, chemists adopt a pragmatic stance and simply demand that bonding theo-
ries be useful for making predictions and aiding in the synthesis of new molecules.
#e underlying ontological status of the bond holds little interest. Although useful
to everyday chemical practice, this approach is deeply unsatisfying to the philoso-
pher of chemistry because it leaves unanswered fundamental questions about the
nature of the chemical bond.

Other chemists, including ones of deep realist commitment, draw a di$erent
and more skeptical conclusion from these facts. Robert Mulliken, a founder of
quantum chemistry said, “I believe the chemical bond is not so simple as some
people seem to think.” (Coulson, 1960) Charles Coulson, another founder wrote
that the chemical bond “is a "gment of our imagination.” (1952) #ese chemists
believe that progress in quantum chemistry shows that there is something deeply
%awed about the chemist’s notion of the bond. While this degree of skepticism
may be warranted in the end, it is a hard thing for most chemists to accept given
the ubiquity and usefulness of the bond concept.

Whatever the ultimate verdict, entertaining Coulson and Mulliken’s skepticism

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raises many deep philosophical questions about the nature of chemical bonds and
molecular structure. #ese questions include: Are chemical bonds real? If so,
should we think of them as entities? If bonds are not real, is the phenomenon
of bonding real? If so, how should this phenomenon be characterized? What is the
relationship between molecular structure and bonding?

#ese issues are complex and interrelated, and philosophers of chemistry are
beginning to consider them. (Woody, 1998; Needham, forthcoming; Berson, forth-
coming; Stemwedel, forthcoming; Hendry, forthcoming) Approaches to answer-
ing these questions necessarily involve the combination of philosophical analysis
and argumentation with a close examination of chemical theory itself. #e latter
has proved especially di!cult because the last 80 years have seen myriad theoret-
ical models of varying complexity developed to account for molecular structure.
Much of the diversity comes from di$erent approximations and idealizations that
are made in the course of developing such models. Di$erent models say di$erent
things about the phenomenon of covalent bonding and no single model is obvi-
ously the correct one, making the task of philosophical analysis more di!cult. #e
problem of unclear foundations for the chemical bond is thus compounded by the
diversity of models of structure and bonding.

#is paper addresses one of these issues: It asks what the myriad models of
molecular structure collectively say about the nature of bonding. Speci"cally, it
asks whether models of molecular structure endorse a structural conception of co-
valent bonds. To cope with the multiplicity of models, I will follow a modi"ed form
of robustness analysis. (Levins, 1966; Wimsatt, 1981; Weisberg, 2006; Weisberg &
Reisman, ms) As in standard robustness analysis, I will examine multiple models
of molecular structure, looking for features and predictions common to models.
Where this paper will di$er from standard robustness analysis is that we are already
in possession of an external calibration point for the predictions of each model: ex-
perimentally measurable quantities. Using this external standard, we can examine
successively more predictively accurate model types, investigating the general fea-
tures that these models have and which ones they lack. Approaching the problem

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in this way will let us determine which bonding features of molecular models are on
the surest theoretical footing, the features which ought to guide our inquiry about
the foundations of covalency.

1 !e Structural Conception of Bonding

#e chemical literature o&en describes covalent bonds in structural terms, employ-
ing what I will call the structural conception of bonding. In subsequent sections,
I will investigate whether this conception is robust and if not, which features of
bonding are robust. In order to carry out this analysis, I will rely on a working
de"nition for the structural conception of bonding. Characterized in purely func-
tional terms, the structural conception of bonding says:

A covalent bond is a directional, submolecular relationship between
individual atomic centers that is responsible for holding the atoms to-
gether.

As this initial de"nition is functional, it does not say what realizes the role of
the bond. De"ning the bond this way will be useful in historical studies because
bonds were an important part of chemical theory before the discovery that elec-
trons played a role in bonding.

In modern chemistry, it is completely uncontroversial to see electrons or elec-
tron density as what realizes the submolecular relationship. #is leads to the fol-
lowing structural conception of bonding:

A covalent bond is a directional, sub-molecular region of electron den-
sity located between individual atomic centers that is responsible for
holding the atoms together.

#is conception tells us three important things about the nature of covalent
bonds. First, it distinguishes covalent bonds from ionic bonds with the directional-
ity restriction. Ionic bonds are omni-directional electrostatic interactions between

4



positively and negatively charged ions. Covalent bonds are regions of electron den-
sity that bind atoms together along particular trajectories. Second, this conception
says that bonding is a sub-molecular phenomenon, con"ned to regions between
the atoms. #is eliminates the possibility that bonds are a molecule-wide phe-
nomenon. #ird, the region of electron density between the atomic centers has
to hold these centers together, which I will interpret to mean that they are closer
together then they would have been in the absence of the bond. In other words,
bonds are a stabilizing force for the molecule, and this stabilization will manifest
itself in the amount of energy required to separate the atomic centers.1 #e next
sections ask how robustly this conception of the chemical bond is represented in
models of molecular structure.

2 Valence Bond Models

Valence bond models are the "rst type of quantum mechanical treatment of molec-
ular structure I will discuss. #ese models can be thought of as “bringing together
complete atoms and then allowing them to interact to form bonds.” (Carroll, 1998,
24) In other words, covalent bonds are formed when two isolated atoms, each pos-
sessing a full complement of protons, neutrons, and electrons, are brought together
and allowed to interact. #e change in electron distribution and the resulting en-
ergetic stabilization from this change is taken to be the bond. All of this has to
be unpacked in some detail, but we need to begin by considering what a quantum
mechanical description of an atom looks like.

#e quantum mechanical description of atoms and molecules are given as wave-
function of the form √(x, y, z, t). Di$erent molecular modeling techniques will
generate somewhat di$erent wavefunctions for the molecule in question because
of the di$erent approximations they employ. In all of the bonding models I will
discuss, several simplifying assumptions will be made: First, I will only focus on the

1For further discussion about the structural conception as well as other conception of covalent
bonding, see Hendry, forthcoming.

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electrons and hence on the electronic wavefunctions.2 Second, I will only examine
time-independent wavefunctions, taking into account the spatial properties of the
electronic wavefunctions, but not how they evolve in time. #ird, I will only discuss
models of molecular hydrogen (H2), a two-proton, two-electron system. Because
this system only has two electrons, it is safe to simply assume that the electrons
have opposite spin. #is will prevent us from having to explicitly keep track of the
spin component of electronic wavefunctions, which would make the analysis more
complicated.

I will begin the robustness analysis with the simplest possible valence bond
model of hydrogen, one that most closely embodies the structural conception of
bonding.

2.1 Simple Valence Bond Model

#e "rst model of H2 is constructed by multiplying together two wavefunctions
corresponding to the electrons originally associated with two individual hydrogen
atoms. #e electrons are still “tied” to the nuclei that they came to the bond with.

Let a and b denote electronic wavefunctions centered around two distinct hy-
drogen nuclei. #ey will each bring along an electron which we will designate as
1 and 2. #is simple model keeps the "rst electron (1) in the vicinity of the "rst
hydrogen nucleus (a) and the second electron (2) being in the vicinity of the sec-
ond nucleus (b), but multiplies them together to represent the mixing of the wave-
functions upon bring the atoms together. Mathematically, this can be expressed as
follows, where c is a normalization parameter:

√1 = c a(1)b(2) (1)

Using this wavefunction, the Hamiltonian for the molecule, and the Schrödinger
equation, we can calculate the dissociation energy of the molecule described by this

2#e assumption that the nuclear and electronic wavefunctions can be computed separately is
called the Born-Oppenheimer approximation. (Born & Oppenheimer, 1927)

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wavefunction. #e dissociation energy is the energy required to pull the atoms
apart, removing the interaction between them and hence destroying the covalent
bond. Using this model, the dissociation energy is calculated to be 24kJ/mol. We
can also calculate the equilibrium distance between the atomic nuclei. #is gives us
what is usually called the bond length. In this model, the equilibrium bond length
is calculated to be 90 pm.

We can check these predictions against experimental measurements of these
properties. #e simple valence bond model predicts 24 kJ/mol and 90 pm, but
experimental investigation has measured the bond length to be 74.1 pm and the
dissociation energy to be 458.0 kJ/mol. We are in the right ball park, but clearly
there is much room to improve our model. #e "rst improvement will be to loosen
the spatial restriction of each electron.

2.2 !e Heitler-London Model

A substantial improvement in accuracy can be made to our simple model of the
H–H bond. #e model was "rst proposed by Heiter and London (1927). It starts
from the model discussed in §2.1, but modi"es it by making the following observa-
tion: #e electrons are indistinguishable and hence it is arbitrary to have put elec-
tron 1 around atomic center A and electron 2 around atomic center B. #e model
thus considers a compound wavefunction, generated from the one discussed in
§2.1 and the following:

√2 = c a(2)b(1) (2)

#is corresponds to the electrons “swapping places.”
In quantum mechanics, complex wavefunctions can be built up by taking linear

combinations of simpler wavefunctions, which are sometimes called bases or basis
sets. #us our full wavefunction will be the linear combinations of (1) and (2).
#ese linear combinations can be written as follows:

√BOND = c a(1)b(2) + c a(2)b(1) (3)

7



√ANTI = c a(1)b(2) − c a(2)b(1) (4)

Since we will be con"ning our discussion to the ground state energies of the
electrons, we will not consider (4) further, as this corresponds to an excited state,
where the bond would actually be broken. Con"ning our attention then to (3),
let’s consider what the wavefunction tells us qualitatively about the distribution
of electrons. If we think about them as particles, it tells us that they are free to
move between the atoms, but in a coordinated fashion. #ey must always be near
opposite atomic centers, but they still have mobility. It is impossible for a verbal
description to be fully accurate, but a somewhat better description is that when
we calculate the probability distribution of the electrons through space (™™∗), we
"nd that there is a buildup of electron density between the two atomic centers, with
maximum electron density centered on each of the two atomic centers. #e biggest
di$erence between this model and the simpler one considered in the previous sec-
tion is that the electrons have greater mobility and there is a greater distribution of
density throughout the molecule.

How well does the Heitler-London model predict the measurable properties of
H–H bonds? Considerably better than the simple valence bond model, it turns
out. #e bond distance is calculated to be 86.9 pm, which is a little better than the
simple model, but the dissociation energy is calculated to be 303 kJ/mol, which
now brings it to the same order of magnitude as the measured value. By allowing
for electron mobility, the model is brought in to closer alignment with the exper-
imentally measured values. #is is a theme that I shall return to throughout the
paper.

2.3 Improving Valence-Bond Models

#e Heitler-London model can be improved further if we take the Heitler-London
wavefunction to be a just one part of the basis for a more complex wavefunction.
In other words, we add additional terms corresponding to other characteristics the

8



wavefunction might have. I will only discuss one example of this, but it illustrates
a very general principle at the heart of modern model building in quantum chem-
istry.

One thing that the Heitler-London model does not do is take in to account the
possibility that the molecule could have ionic character. In other words, it does not
allow for the possibility that there is a greater distribution of electron density on
one side of the molecule or the other, however brie%y or however slightly.

We can take this possibility in to account by adding in additional terms corre-
sponding to an ionic distribution of electron density, which at the extreme would
correspond to the electrons both being on one side of the molecule or on the other
side (e.g. a(1)b(1)). At the same time, we do not want to over-compensate, adding
too much ionic character. One way around this is to add an adjustable parameter
∏ to each term. #e wavefunction is then optimized by setting ∏ to some value,
calculating the energy of molecule, then trying another value for ∏, and keep it-
erating until we "nd the lowest value, making the well-motivated assumption that
the ground state electronic wavefunction will be the one with the lowest energy.3

#e new model has the following wavefunction:

√ionic = c a(1)b(2) + c a(2)b(1) + ∏c a(1)b(1) + ∏c a(2)b(2) (5)

Adding these ionic terms considerably improves the model, allowing us to calculate
388 kJ/mol and 74.9pm, bringing the dissociation energy of the model even closer
to the experimental value.

Having looked at three successively improved valence bond models, let’s con-
sider what they have told us collectively about the structural conception of bonding.
First, by letting the electrons have greater mobility, as in the Heitler-London ap-
proach, the model came closer to agreement with experimentally measured quanti-

3Technically, we also rely on the variation theorem which tells us that this procedure will only
every overestimate the energy, but not under estimate. #us we can keep varying parameters until
we get the energy minimum, without fear that we are underestimating. (Levine, 1991)

9



ties. When the model was made even more %exible, allowing the electrons to travel
further a"eld and giving the model ionic character, the model came in even better
agreement with experimental measurements. #is trend continues with increasing
electron mobility: In 1968, Kolos and Wolniewicz used a 100-term, highly %exible
wavefunction, and a&er optimizing the parameters, they came within 0.01kJ/mol
of the measured values. (Kolols & Wolniewicz, 1968) #us the result most strongly
robust among these many models of the covalent bond is that greater electron mo-
bility leads to greater stabilization and closer agreement to experiment.

What do these valence bond models say about our three criteria for the struc-
tural conception of bonds? First, they are directional and allow us to distinguish
between ionic and covalent bonds. However, as we have just seen, adding ionic
character actually stabilizes many molecular models such those for H2 model. Sec-
ond, while the simple valence bond models start out with sub-molecular regions
of electron density that we can identify as the bond, they are improved by allowing
the electrons to have increased mobility, spreading some of the density throughout
the molecule. Finally, the valence bond models do nothing to challenge the idea
that bonds hold the molecule together.

3 Molecular Orbital Models

Our robustness analysis will continue with a second, distinct approach to con-
structing molecular models: the molecular orbital method. Rather than looking at
bonding as what happens when we bring together two atoms and let them interact,
we start o$ with atomic centers at equilibrium distances, then calculate electron
wavefunctions for the entire molecule. #e electrons are a$ected by a potential
"eld generated by the nuclei collectively.

As with valence bond models, one can start with an arbitrary form for the
molecular orbital wavefunction, and then improve the model by adding more terms
and adding adjustable parameters on these terms. We will start with a simple trial
wavefunction for H2 to illustrate some of the key features of molecular orbitals. A

10



prudent way to begin is to use wavefunctions corresponding to linear combina-
tions of the atomic orbitals of our two hydrogen atoms, which we will call A and
B. So we have:

√bond = √A + √B (6)

√anti = √A − √B (7)

Equation 6 gives us the lower energy bonding wavefunction, while equation 7
gives us the higher energy anti-bonding wavefunction. If we want to calculate the
total electronic wavefunction for the molecule, corresponding to adding the two
electrons into the hydrogen molecular orbital, we take the product of the wave-
function for each orbital.

√H2 = √bond(1)√bond(2) (8)

Combining equation 6 with equation 8, we get the following expansions:

√H2 = [√A(1) + √B(1)][√A(2) + √B(2)] (9)

√H2 = √A(1)√A(2) + √B(1)√B(2) + √A(1)√B(2) + √A(2)√B(1) (10)

Interesting structure emerges from the terms in this equation. #e third and
fourth terms (√A(1)√B(2) and √A(2)√B(1)) are exactly the same terms as those found
in the Heitler-London model of the H2 molecule. #is corresponds to a single
electron being located in the vicinity of each nucleus. However, there are two ad-
ditional terms √A(1)√A(2) and √B(1)√B(2) which correspond to the electrons si-
multaneously positioned near the A nucleus or the B nucleus. #ese ionic terms
corresponding to asymmetric electronic distributions had to be added manually to
the Heitler-London model. So right from the start, this model includes the possi-
bility of ionic character in what would classically be treated as a non-polar, covalent

11



bond. We will come back to this point below.
How does this simple molecular orbital model conform to experiment? If we

calculate the dissociation energy and bond distance predicted by this model, we
come up with -260kJ/mol and 85pm respectively, values which accord reasonably
well with the measured values of -458.0kJ/mol and 74.1pm. #ese values are similar
to those calculated with the Heitler-London model, but considerably better than
those calculated with the simplest valence bond model.

One of the interesting things about molecular orbital models of bonding is that
very little classical structure is built in from the beginning. Essentially, the nu-
clear centers are oriented in space, then electronic wavefunctions are constructed,
relying only on the potential "eld generated by these atomic centers. In accurate
calculations, we also search for the equilibrium distances of the atomic centers by
moving them around to "nd the minimum total energy for the molecule. Even in
this simple model, we see some interesting and non-classical structural features.

First of all, there is the importance of the aforementioned ionic terms. #is
suggests that there is far greater electron mobility than in a case of two isolated
atoms. Further, the electronic density is at its peak in the areas immediately sur-
rounding the nuclear centers. Figure 1 is a cross-section of the amplitude of the
wavefunction of H2 calculated from our model. We can see that the highest den-
sity is just around the nuclear centers, with considerable density in between the
atomic centers, but also a reasonable amount of electronic density on the far sides
of the atomic centers.

#ere are many ways to improve this simple molecular orbital model, some of
which will be discussed in the next section. But even from this very simple model
and the Heitler-London model discussed in the last section, we can see the contin-
uation of the pattern that has been emerging. Electron mobility or delocalization
of electron density is a prominent feature of this model of molecular structure. #e
electrons are not sitting between the two atoms holding everything together; rather,
they are distributed throughout the molecule. #is puts pressure on the second
part of the structural conception of bonds, the idea that bonds are sub-molecular

12



Figure 1: A cross-section of the electronic density distribution calculated from a
molecular orbital model of hydrogen. #is diagram is not to scale.

13



phenomena. Similarly, delocalization makes the bonds less directional. #e nuclei
are still located in particular places that minimize the total energy of the molecule;
however, the bonds themselves are considerably more di$use. As we will see in the
next section, the trend continues with more complex molecular orbital models.

4 Modern Molecular Orbital Models

#e methods considered so far were worked out for simple molecules in the "rst
half of the twentieth century. Since that time, there has been considerable progress
in developing models which can yield more accurate calculations of molecular
properties. To review all of these methods here would be impossible and unneces-
sary; however, many modern methods can be seen as systematic improvements to
molecular orbital models along two dimensions relevant to the structural concep-
tion of bonds.

#e "rst type of improvement involves using more complex basis functions as
trial wavefunctions. #ese basis functions are expanded by taking in to account
many possibilities (with adjustable parameters), including greater delocalization
of the electrons as well as occupation of higher energy electronic states. Just as
ionic terms are “mixed” in to the valence bond wavefunctions, so too these spatially
di$use, delocalized, and higher energy states are mixed in to the starting molecular
wavefunctions.

#e second dimension on which modern methods improve upon simple molec-
ular orbital models is in the nature of the optimization of the wavefunction, given
the basis set. I have already alluded to the way calculations are typically performed
using simple molecular models, a method called the Hatree-Fock method. #e
method works as follows:

1. Make an initial guess of the position of the nuclei.

2. Start with initial values of the parameters on all the electronic terms in the
trial wavefunction.

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3. Choose the terms corresponding to a single electron, then vary the parmaters
on those terms in order to minimize the energy.

4. Repeat the step 3 for each electron in sequence.

5. Repeat step 4, cycling through all of the electrons until the energy values no
longer change, having reached self-consistency of the orbitals with the "eld
generated by the electrons.

6. Perturb the nuclei a small distance and repeat steps 2–5, searching for an
energy minimum. (Levine, 1991; Ansyln & Dougherty, 2005)

Although wavefunctions calculated with the Hatree-Fock method can yield ex-
cellent results, the method does not take in to account electron correlation. Recall
that electronic wavefunctions are optimized from the point of view of a single elec-
tron. #e method imagines that each electron is in%uenced by a "eld generated
by the stationary distribution of all of the other electrons. It moves systemati-
cally through the electrons, treating them in this fashion, until self consistency
is achieved. But any two electrons in the vicinity of one another will repel one
another and this is not taken in to account in the Hartree-Fock method. Hence
a more accurate calculation would take this electronic correlation in to account.
One method for doing this is called con!guration interaction.

Con"guration interaction approximates the a$ect of electron correlation by
constructing a new molecular wavefunction from linear combinations of the ground
state and electronically excited molecular wavefunctions. Each of these terms has
a coe!cient that can be varied in order to minimize the total energy. #e resultant
wavefunction from a con"guration interaction calculation will take the following
form:

™CI = c1™1 + c2™2 + c3™3 + . . . (11)

A full con"guration interaction calculation would take in to account all of the ex-
cited states of the molecule. In practice, only a few excited states are usually taken

15



in to account because of the great computational cost of these calculations.
#ese approximation methods and others are interesting, but take us away from

the important question that is needed for robustness: What do successively better
calculations that take in to account CI tell us about bonds?

We can answer this question by re%ecting on why con"guration interaction
allows us to build very good molecular models that take into account electron cor-
relation. Remember that correlation is the property of the electrons “avoiding” one
another because of electron-electron repulsions. Conceptually, adding in these re-
pulsions will change the shape of the molecular orbital, because the electrons may
spread apart further in space. A systematic way to alter the shape of molecular or-
bitals is to mix in orbitals corresponding to excited states, which will have di$erent
shapes and symmetry properties from the ground state orbital. When enough of
these are mixed together, we will have the %exibility to accurately describe the re-
sultant molecular orbitals for the molecules, orbitals which correctly account for
correlation as well as the nucleus-electron attractions that account for the electronic
distribution. (Ansyln & Dougherty, 2005, 824)

Con"guration interaction is another example of how moving electrons out of
the classical bonding region stabilizes the molecular model and brings it in to closer
agreement with experimentally determined values. When we allow for greater
delocalization, taking in to account electrons temporarily entering higher energy
states so as to avoid one another, molecular models increasingly agree with exper-
imentally measurable quantities.

5 Robust Properties of Chemical Bonds

We have now come to the conclusion of our whirlwind tour of molecular models,
searching for the robust properties of chemical bonds. In this concluding section, I
will review what we have discovered, examining both what is and what is not robust
across models of molecular structure.

Our examination of molecular models began with the simple valence bond

16



model, the one that most faithfully illustrates the structural conception of bonding.
In this model, electrons were associated with particular nuclei and as a result, the
molecule was represented by bringing together two atoms, letting their electrons
interact, and taking advantage of the stabilization created by this interaction. #e
electrons remained associated with a particular nucleus and could be seen as being
shared between two atoms to form a bond.

#is idea of electrons coming along with nuclei and being shared between two
nuclei is resolutely not robust across bonding models. As we move in the direction
of models which correspond more accurately to experimentally measurable values,
this property diminishes and seems to vanish altogether in modern models.

#e structural conception does not demand that electrons are shared or even
fully located in the region between the atoms. Indeed it only demands that bonds
can regularly be identi"ed as directional, sub-molecular regions of electron density.
So what we really need to know is whether such regions of density can be identi"ed
in molecular models. Answering this question is considerably more di!cult and
this paper certainly gives no de"nitive answer to it; however, the properties which
we have found to be robust go a long way towards answering it.

Among the bonding models, several things have been shown to be robust. In
the examples we have considered, the models became energetically stabilized by
forming the bond. Energy is liberated when the bond is formed — conversely, it
takes energy to break a bond. #is means that for atoms that readily form covalent
bonds, it is energetically more stable for them to be in the bonding state than sep-
arated from each other such that the electrons cannot interact. Bonding seems to
fundamentally involve energetic stabilization.

#e second robust feature of the models is that greater electron mobility or de-
localization leads to stabilzation. When we moved from the simple valence bond
model to the Heitler-London model, we saw considerable stabilization. #is ac-
companied relaxing the restriction that pinned electron 1 to nucleus A and elec-
tron 2 to nucleus B. We further improved that model by adding terms to the basis
function which allowed greater %exibility to the resultant wavefunction, which has

17



the physical signi"cance of allowing greater delocalization.
#e same is true in the molecular orbital models. Even the simple molecular

orbital models represent the electrons as delocalized across the molecule. Improve-
ments to the molecular orbital model involved using basis functions with greater
numbers of terms, which also allows more %exibility in describing the electronic
distribution, a distribution which is not tied to the classical positioning of the elec-
trons in between the nuclei.

#ese robust features of molecular models put considerable pressure on the
structural conception of bonds, and are no doubt the sorts of results that led Coul-
son and Mulliken to their skepticism about the reality of bonds. To be explicit, the
robust conception says that covalent bonds are directional, sub-molecular regions
of electron density that hold molecules together. #roughout this paper, we have
seen that delocalization—density spread beyond the sub-molecular region between
the atoms—is the norm in molecular models. Further, the addition of ionic char-
acter, correlation, and the mixing of higher energy states all seem to improve the
model, but make the bonds less directional and sub-molecular.

I believe that this clearly rules out any straightforward version of the struc-
tural conception of covalent bonding. However, while the features I have pointed
to that improve bonding models including correlation and ionic character involve
deviations from local, sub-molecular regions of electron density, they themselves
describe electronic characteristics of molecules in structural terms. #ey may in
fact be compatible with a revised structural conception of bonding. While future
analysis is required to either vindicate or fully reject the structural conception of
bonding, the considerations in this paper go a long way towards motivating Coul-
son and Mulliken’s skepticism about the reality of the covalent bond.

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