Content uploaded by Paul L A Popelier
Author content
All content in this area was uploaded by Paul L A Popelier on May 04, 2018
Content may be subject to copyright.
1
8
The QTAIM Perspective of Chemical Bonding
Paul Lode Albert Popelier
8.1
Introduction
In this chapter the ‘‘Quantum Theory of Atoms in Molecules (QTAIM)’’ is
discussed in the spirit of this book, which combines an educational style with
an awareness of current scientific boundaries, while avoiding too many equations
in the main text. Because QTAIM has been reviewed many times before it
has become harder to add value to what has already been written. Still, the
current text seeks to achieve added value, by an alternative angle of exposition
of QTAIM, by including both a historic narrative as well as a pointer to future
research.
Before starting to explain QTAIM we mention background texts that readers
may want to consult. An authoritative source on QTAIM is a book [1] written
by Richard Bader, who inspired and oversaw the development of QTAIM since
its inception. Apart from a didactic monograph [2] on QTAIM there are also a
number of accounts, reviews, and edited books on the subject. An early review [3]
on so-called molecular virial fragments was written in 1975. This account covered
only one aspect of QTAIM, namely that of the virial fragment. We will discuss
this concept in detail later in the text but for now it is best to think of a virial
fragment as ‘‘the atom’’ in QTAIM. Obviously this is an important cornerstone of
QTAIM, and was the early driving force for QTAIM’s development. It is important
to realize, already now, that the definition of an atom (within the framework of
QTAIM) is energetic in nature. It demands that not only should atomic energy
exist but it should also be well defined. This 1975 account did not introduce
bonds nor did it define them; in fact, it shied away from the general idea of
a bond, as implied by its title Molecular Fragments or Chemical Bonds?. Here
Bader suggested that, in the search to understand the properties of a total system
in terms of its parts, one can choose atoms rather than bonds as fundamental
parts. Later, in a more mature version of QTAIM, this ‘‘either or’’ view was not
sustained.
Ten years later, in 1985, a second account [4] was published in the same journal
by the same author. This account presented QTAIM in its entirety, including
The Chemical Bond: Fundamental Aspects of Chemical Bonding, First Edition.
Edited by Gernot Frenking, Sason Shaik.
c
2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.
28 The QTAIM Perspective of Chemical Bonding
a treatment of bonding through the topology of the electron density. What we
mean by topology will become clear later in this chapter. In his 1985 account,
Bader was careful enough to write that It is the atom and its properties that are
defined by quantum mechanics. The bond paths (BPs) and the structure they define
just mirror and summarize in a convenient way what the atoms are doing. The 1985
account would effectively form the basis for Bader’s book on QTAIM, which
was published 5 years later. This book appeared more or less simultaneously
with a long review [5], presenting QTAIM as deduced from first principles.
Ten years later a more introductory and didactic book [2] appeared, followed
by two reports soon after [6, 7] listing the by then hundreds of applications
of QTAIM in fields ranging from mineralogy to biochemistry, together with
developments, both methodological and algorithmic. In a philosophical review [8]
of QTAIM as a quantum mechanical basis of conceptual chemistry, Bader listed
all atomic theorems [9] derived to date. Two years later, in 2007, an extensive
compilation of QTAIM contributions appeared in a book [10] edited by Matta and
Boyd. This book’s subtitle From Solid State to DNA and Drug Design reaffirmed
the enormous breadth of application that QTAIM had acquired by the mid
2000s. In another personal account entitled Everyman’s Derivation of the Theory
of Atoms in Molecules Bader presented QTAIM as an extension of quantum
mechanics to subspaces. Finally, in his swansong publication [11], posthumously
published in Foundations of Chemistry, Bader emphasized the bounded, space-
filling, and nonoverlapping feature of QTAIM atoms, a topic discussed [12] at
great length seven years earlier, and in the context of short-range intermolecular
interactions.
Three final remarks are in order. First, the references cited in this Introduc-
tion are not exhaustive but merely provide background material to make up
for the brevity of the QTAIM exposition in this chapter. These references were
ordered chronologically, in line with the historical character of the exposition
below. Second, another chapter in this book, written by Scherer et al. (Chapter 9),
assumes background knowledge of QTAIM in their account on chemical bond-
ing from the point of view of high-resolution crystallography. Third and finally,
the way QTAIM partitions the electron density, through the concept of the gra-
dient path (GP), has been adopted by other approaches such as the topological
analysis [13] of the electron Localization function (ELF) [14]. It makes sense to
bundle the topological analysis of the electron density, its Laplacian (which are
both part of QTAIM), ELF, and many other quantum mechanical functions into
one approach called Quantum Chemical Topology (QCT). This name has been
justified in detail over the last decade in various sources, such as footnote 19
in Ref. [15], Section 2 of Ref. [12], the Appendix of Ref. [16] and the Introduc-
tion of Ref. [17]. More details on QCT can be found in Box 8.1 of this chapter.
Box 8.1 lists the various quantum mechanical functions that have been investigated
via the language of topology and dynamical systems, and thereby outlines what
QCT is.
8.1 Introduction 3
Box 8.1 Quantum Chemical Topology (QCT)
There is a growing body of work that uses the language of topology to extract
chemical information from modern ab initio wave functions. The important point
is that such topological analyses use three-dimensional quantum mechanical
functions other than the electron density. A well-documented example of a
function that features as a chapter in this book is the ELF. In 1994, bonds were
for the first time classified [13] by introducing the topology of ELF. Central to that
work was again the concept of the GP. A GP is a trajectory of steepest ascent
through a given three-dimensional (3D) function. The full collection of GPs,
which draws itself in a molecule or cluster of molecules, is called the gradient
vector field. This field was first studied as it operated on the electron density.
This is how QTAIM started: the gradient vector field of the electron density
partitioned a molecule into atoms (that have a well-defined kinetic energy).
However, many 3D quantum mechanical functions, other than the electron
density, also have maxima that attract GPs. Therefore one can find basins
(i.e., subspaces containing GPs terminating at an attractor) in these other 3D
functions. The identity of the partitioned function will determine which type of
chemical information one recovers. It is important that all activity and results
that use the central idea of a gradient vector field partitioning are collectively
referred to under one single name. A sensible name has already been proposed
[15] in 2003, which is QCT.
A non-exhaustive list of 3D topologically partitioned quantum mechanical
functions includes:
•Electron density 𝜌(r) (started with Ref. [20]).
•The Laplacian of 𝜌,∇2𝜌(r) (started with Refs [68, 69] and full topology first
explored in Refs [70–72]).
•(Bare) nuclear potential Vnuc(r) (early start with Ref. [73] but elaborate and
self-contained study [17]).
•ELF [14] (started with Ref. [13] and reviewed in Ref. [74]).
•Electrostatic potential [75] (started with thorough studies [76, 77] and continued
with Ref. [78], applied in the area of chemical reactions [79], Lewis acidity [80]
or electron diffraction study [81]).
•Virial field (or trace of the Schr¨
odinger stress tensor) (topology explored in Ref.
[57].).
•Magnetically induced molecular current distributions (started with [82]).
•Intracule density (started with Ref. [83], which reveals correlation cages).
•Ehrenfest force field (topology first investigated [84] in 2012).
•Energy partitioning (beyond the kinetic energy and atom virial theorem)
(Coulomb potential energy partitioning started with [37] and culminated into
the theory of IQA [24], leading to energetic underpinning for the topological
expression of chemical bonding [85]).
48 The QTAIM Perspective of Chemical Bonding
Alternative methods of interpretative quantum chemistry [86–92] do not share
the central concept of the gradient vector field. This crucial difference draws
together the topological analysis of the various functions under the heading of
QCT, which is thus distinct from non-QCT methods.
8.2
Birth of QTAIM: the Quantum Atom
The birth of QTAIM was marked by work that, with hindsight, could be called a
false start actually. Two publications appeared in 1971, one [18] in Chemical Physics
Letters and one [19] in Journal of the American Chemical Society,whereBaderand
coworkers explained their ‘‘natural partitioning.’’ It is interesting to pause here
and highlight what these publications convey because this reveals the driving force
behind QTAIM and what it is essentially. The opening sentence of the JACS paper
states that ‘‘There is a history of attempts (…) to partition the total electronic charge
between the nuclei in the system. The prime reason for proposing any scheme which
assigns some number of electrons to each nucleus in a molecule is to provide a measure
ofthechargetransfer…’’ Let us make clear here, and keep in mind for later in this
chapter, that a population analysis indeed provides a measure of charge transfer,
nothing more and nothing less.
In the abstract of the 1971 JACS paper the authors term their method a ‘‘natural
partitioning’’, ‘‘…as it is suggested by the nature of the charge distribution itself; the
point along the internuclear axis at which the charge density attains its minimum value
between a pair of bonded nuclei defines the position of the partitioning surface.’’ They
immediately illustrated their partitioning recipe on the linear molecule FCN. In
this system a first partitioning plane was positioned at the point on the molecular
axis with the lowest electron density value between the F nucleus and the C nucleus.
The second partitioning plane was positioned in the same manner but now in
between the C nucleus and the N nucleus. Figure 8.1 of the 1971 JACS paper
shows two panels, which we do not reproduce here. In the top panel there is a one-
dimensional profile of the electron density along the molecular axis, partitioned in
the way described just earlier. The bottom panel shows these partitioning planes
again but now cutting through a two-dimensional representation consisting of
contour lines of constant electron density. Instead of reproducing their original
figure we show a partitioned mountain landscape in Figure 8.1. The reason for this
example from macroscopic reality is that it reminds us how the brain would separate
two (mountain) peaks that are mathematically similar to the peaks appearing in a
1D profile of the electron density.
Mountain landscapes also appear in the context of chemical reactivity where
saddle points or paths of steepest ascent need to be illustrated, often in connection
with a new configuration sampling algorithm. Here the metaphor of the mountain
landscape simply fulfills the role of emphasizing how ‘‘the human eye’’ naturally
partitions an object into two nonoverlapping parts. No one would readily think
8.2 Birth of QTAIM: the Quantum Atom 5
Figure 8.1 A system of two mountains partitioned into a left mountain (L) and a right
mountain (R) by a red line going through the point of lowest altitude between the two
peaks.
of mountain L as an object that overlaps with mountain R. There is a sharp
boundary between the two objects, and few would argue with this assertion. In
macroscopic reality there are many examples of sharp boundaries separating two
entities, ranging from borders between countries to cell walls. Yet, when it comes
to atoms in a molecule, a sharply delineated atom does not (yet) constitute the
mainstream view.
We now return to the question why the natural partitioning proposed in 1971 was
a false start. The concomitant 1971 Letter [18] applied the ‘‘natural partitioning’’
method to another linear molecule: carbon dioxide or O=C=O. An important
new concept was invoked, which is transferability. In the broadest sense, this
transferability gauges the extent to which a molecule can be constructed from
atoms or molecular fragments. The Letter first looked at transferability in terms
of ‘‘bond properties,’’ which were restricted to bonded and nonbonded electron
populations. It is easy to imagine that a different type of partitioning plane, this
time through a nucleus, can separate a bonded population from a nonbonded one.
The blue line in Figure 8.2 marks an example of such a plane, going through
O here. Left of this plane is situated the nonbonded electron density, which is
segregated from the bonded density appearing to the right of this blue plane.
Envisage that one wants to first obtain the total population of the left oxygen in
O=C=O. For that purpose the electron density is integrated from infinity (at the
left) up to the partitioning plane (red line in Figure 8.2) at the minimum point
between the left O nucleus and the C nucleus next to it. Both partitionings (red
and blue line) may appear cavalier in the light of the mature theory that QTAIM
later became. However, the 1971 Letter already gives the impression that, in the
long run, ‘‘not all is well’’ with such a red partitioning. Understanding the reason
for this is important to comprehend the essence of QTAIM and what it strives to
achieve.
Noting the relative insensitivity of populations to changes in bonding (e.g., going
from CO to CO2) the authors directed their attention to the transferability of a
fragment’s energy. They focused on the kinetic energy only, working with two
68 The QTAIM Perspective of Chemical Bonding
OO
C
Figure 8.2 Partitioning of the linear
molecule carbon dioxide (CO2)superim-
posed on a set of contours of constant elec-
tron density. The red line marks the ‘‘false
start’’ or ‘‘natural partitioning’’ between
O and C, while the green curve marks the
correct QTAIM boundary between C and O.
The blue line marks a way to separate non-
bonded (left) and bonded (right) electron
density, as suggested in the 1971 Letter (see
main text).
alternative kinetic energy densities, denoted K(r)andG(r) (see Box 8.2). They
showed that when K(r)andG(r) are integrated over a subspace bounded by the red
line then one obtains very different numerical values. In order for these two values
to be identical, the 1971 Letter correctly concluded that the subspace, denoted by
Ω, had to be such that the Laplacian of the electron density, ∇2𝜌, vanishes when
integrated over the subspace’s volume (see Box 8.2),
Ω
dV∇2𝜌(𝐫)=0 (8.1)
However, the 1971 Letter failed to see this fact through to its final conclusion and
thereby failed to draw the green boundary in Figure 8.2. This had to wait until
the next paper [20] appeared in 1972, containing a consistent treatment. This 1972
paper can be considered as actually heralding the birth of QTAIM. Here, for the
first time, the characteristic shape of a molecular fragment (i.e., atom) with a well-
defined kinetic energy was born. This paper illustrated this molecular fragment
by means of simple lithium-containing diatomics, both neutral and positively
charged. Figure 8.2 shows this characteristic shape by means of the green curve.
When reflected to the left side of the molecule, this green curve provides a complete
partitioning of CO2into three atomic subspaces with a nonplanar shape. We could
call such a subspace a kinetic atom, because it has a well-defined kinetic energy.
However, one could give this subspace the more general name quantum atom,
which has been used elsewhere in the literature (e.g., Chapter 7 in Ref. [2]. and in
Refs [21– 24].). This name is appropriate because much work of the Bader group
during the rest of the 1970s aimed at strengthening the link between this quantum
atom and quantum mechanics itself. The need for strengthening this link follows
8.2 Birth of QTAIM: the Quantum Atom 7
Box 8.2 A Molecular Fragment with a Well-defined Kinetic Energy
The purpose of this box is to show how to define a molecular fragment that
has a well-defined kinetic energy. The starting point is having a definition of
local kinetic energy, which is the kinetic energy at a particular point per unit
volume. This quantity is thus a kinetic energy density, which when integrated over
a volume, gives the kinetic energy of the electrons in that volume. If this volume
corresponds to that of a molecular fragment then one recovers the kinetic energy
of that fragment.
A potential problem immediately arises because there is a quantum mechanical
ambiguity [93] in defining this kinetic energy density. Although there are an infinite
number of expressions for the kinetic energy density it is sufficient to choose only
two possible expressions to develop the argument of this box. Two alternative
definitions, previously used [94] by Bader and coworkers in this argument, are
given by Eqs. (B2.1) and (B2.2),
K(𝐫)=−
1
4Nd𝜏′[𝜓∗∇2𝜓+𝜓∇2𝜓∗](B2.1)
G(𝐫)= 1
2Nd𝜏′∇𝜓∗⋅∇𝜓(B2.2)
where Nis the total number of electrons in the system, 𝜓the system’s
N-electron wave function, and d𝜏′the integration over all electrons except
one. This integration reduces the 3Ndimensional character of the integrand to
a three-dimensional function. To simplify matters spin is not considered here.
In a way, G(r) resembles classical kinetic energy in that it is always positive. It is
easy to show that these two kinetic energy densities are linked via the Laplacian
of the electron density, ∇2𝜌,
K(𝐫)=G(𝐫)− 1
4∇2𝜌(𝐫)(B2.3)
When integrated over whole space the Laplacian vanishes, or
whole space
dV∇2𝜌(𝐫)=0(B2.4)
As a result, integrating both sides of Eq. (B2.3) over whole space, yields
K(molecule)=G(molecule)=T(molecule)(B2.5)
where Texpresses the kinetic energy regardless of whether it was calculated from
K(r)orG(r). Because a single molecule in the gas phase occupies the whole
space, one indeed recovers the kinetic energy of the molecule by integration over
the whole space. This energy is well-defined in that it is unique: both K(r)and
G(r) give the same answer.
The main question is now if this unique result can also be obtained for a
molecular fragment. Let us consider the subspace of an arbitrary molecular
88 The QTAIM Perspective of Chemical Bonding
fragment, denoted by ⊗. For such an arbitrary subspace one finds that
⊗
dV∇2𝜌(𝐫)0(B2.6)
Hence, one will not recover a unique kinetic energy for such a fragment, or
K(⊗)G(⊗)(B2.7)
However, there are special subspaces Ωfor which
Ω
dV∇2𝜌(𝐫)=0(B2.8)
such that it makes sense to speak of a unique and hence well-defined kinetic
energy T(Ω) associated with those subspaces,
K(Ω) = G(Ω) = T(Ω) (B2.9)
from a concern raised by Bader and Beddall themselves at the end of their 1972
‘‘QTAIM birth’’ paper [20]. They wrote that While the concept of the energy of a
fragment appears to be at variance with the quantum mechanical definition of the energy
as the expectation value of the Hamiltonian operator averaged over all space, we are
led to this concept by our observations. As is perhaps always the case with mature
and fruitful theories (such as QTAIM) they have their roots in observation and
practicality. As another example, one could quote Feynman path integrals, which
were used successfully long before it was shown they were mathematically allowed.
The Dirac delta function is another example.
Returning to QTAIM, Srebrenik joined the Bader group in the mid 1970s and
started the development of subspace quantum mechanics, a topic also pursued by
others at that time. When the expectation value of an operator is evaluated in full
space quantum mechanics, one obtains a real value when the operator is Hermitian.
However, the evaluation of an expectation value over a subspace generally yields a
complex number. This problem can be solved by taking the mean of the (non-real)
expectation value and its complex conjugate, thereby eliminating the purely imag-
inary part. More details can be found in Ref. [25], Chapter 7 of Ref. [2] and Ref. [1].
In the next section we introduce the language of dynamical systems to reinterpret
the quantum atom from a topological view.
8.3
The Topological Atom: is it also a Quantum Atom?
In order to make some new concepts concrete right away we work with a simple
molecule, HFC=O, which is still general enough. The fact that this molecule is
planar eliminates some unnecessary complexity at the outset. Figure 8.3 shows
how the electron density varies within HFC=O. The electron density determining
8.3 The Topological Atom: is it also a Quantum Atom? 9
O
C
FH
Figure 8.3 Electron density contour plot
of HFC=O in the plane of symmetry. The
molecule has been geometry optimized at
the MP2/cc-pVDZ level of theory. The carbon
is placed at the origin and the bold square
box marks the −5au and +5 au horizontal
and vertical boundaries of the plot. The
electron density values of the contour lines
are 1 ×10n,2×10n,4×10n,and8×10nau
where nstarts at −3 and increases with
unity increments.
the contour lines (roughly) doubles its value in going from one contour line to the
next. From the (almost) linear spacing of the contour lines one deduces that the
electron density increases exponentially, reaching a local peak at each of the four
nuclei.
These peaks dominate the electron density landscape to such an extent that
one may have the impression that the positions of the nuclei are all that the
electron density indicates about the molecule. This information is what a routine
crystallographic study gathers and this may be all one needs to know if one is only
interested in the geometry of a newly synthesized molecule, for example. However,
the field of high-resolution crystallography [26] has pushed the interpretation
of the electron density further. A natural way of eliminating the dominance of
the molecular electron density near the nuclei is subtracting from it another
electron density with large peaks. The latter electron density corresponds to that
of a superposition of isolated atoms that have not been chemically hybridized
or polarized. The resulting difference density is called the deformation (electron)
density. In principle, it reveals the chemical features that the molecular electron
density contains but which are drowned by the peaks at the nuclei. Deformation
densities constitute a useful idea at first sight, and have indeed been used much
in the 1980s, but open issues surrounding the exact nature of the isolated atoms
(i.e., which quantum mechanical state or which part of their electron density,
valence or core) can spoil the interpretation of deformation densities. The current
10 8 The QTAIM Perspective of Chemical Bonding
section will show that there is no need for a reference electron density (i.e., a
superposition of isolated atoms). Indeed, one can use the internal difference within
a single molecular electron density to extract chemical information from it. The
key concept to achieve this is the gradient vector, as explained later, but first we
focus on obtaining topological atoms in HFC=O.
Let us look again at the constant electron density contour lines of Figure 8.3, and
focus on the outer line, which corresponds to 𝜌=0.001 au. Because this contour
can be considered as the practical edge of the molecule [27] it encompasses the
whole molecule. If the value of the constant electron density is now increased the
rather ‘‘blobby’’ and shapeless outer contour line becomes increasingly shaped.
Higher electron density contour lines start protruding toward the region between
the bonded nuclei. Like a corset, the contour lines start increasingly following the
shape of the individual atoms that form the molecule. Figure 8.4 zooms in on
the essential characteristics of the patterns emerging from the contour lines that
enclose individual atoms or groups of atoms. Starting with the lowest electron
density value, 𝜌=0.25 au, the first contour line we encounter encloses the whole
molecule. In fact, this electron density value is the highest possible value yielding
a contour line that still encloses the whole molecule; any higher value of 𝜌will
enclose parts of the molecule. More precisely, a contour line corresponding to
𝜌>0.25 au will enclose the fluorine atom on its own. This atom now becomes a
ρ = 0.43 au
ρ = 0.29 au
ρ = 0.25 au
H
C
O
F
Figure 8.4 Diagram illustrating how HFC=O
falls apart into four atoms, by following three
disconnection events, at three different elec-
tron density contour values. First, when
𝜌=0.25 au, F becomes disconnected from
HC=O. The blue zone marks contour lines
with 𝜌>0.25 au but still <0.29 au, at which
point the H atom splits off from C=O, when
𝜌=0.29 au. The yellow zone marks contour
lines with 𝜌>0.29 au but still <0.43 au, at
which point the C and O atoms also become
disconnected. Finally, contour lines with
𝜌>0.43 au enclose all atoms as completely
separated entities.
8.3 The Topological Atom: is it also a Quantum Atom? 11
separate object that is disconnected from the rest of the molecule. The full meaning
of the special point in Figure 8.4, marked by the little square where 𝜌=0.25 au, will
be explained later.
A little black square in Figure 8.4 is reminiscent of the point of lowest altitude
between two mountains, where the red line in Figure 8.1 partitions the overall
landscape in two separate mountains. Of course, in the light of the discussion on
defining atomic kinetic energy it would be uninformed to still draw such a line
through each of the black squares in order to partition the molecule into atoms.
Instead, the equivalent of the green boundary in Figure 8.2 needs to be found,
which is shown in Figure 8.5. In this figure three bold black curves mark the
edges of the four topological atoms in HFC=O. The little black squares appear
in the center of each of these three bold curves marking the sharp boundaries of
the atoms. These curves are actually intersections between the plotting plane and
surfaces in three dimensions called interatomic surfaces.
There are five comments to make on Figure 8.5. First, one can prove again
that the Laplacian of the electron density, ∇2𝜌, vanishes when integrated over the
subspace’s volume of each of the four atoms. Therefore, the atoms displayed in
Figure 8.5 are quantum atoms. In the current section we establish why we can also
refer to the atoms in Figure 8.5 as topological atoms. It should be clear already that
all topological atoms are quantum atoms but not all possible quantum atoms are
topological atoms [21]. Second, the topological atoms in Figure 8.5 do not overlap.
This nonoverlapping picture may appear alien to many (quantum) chemists. A
prevailing picture appears to be one of fuzzy atoms that penetrate each other.
Although this image is possibly inspired by the cloudlike nature of an electron
O
C
H
F
Figure 8.5 The molecule HFC=O is partitioned into four topological atoms: F (green), C
(gray), O (red), and H (white).
12 8 The QTAIM Perspective of Chemical Bonding
density, one should keep in mind that topological atoms also emerge from that
same electron density. Third, topological atoms do not leave any gaps in between.
This space-filling feature of topological atoms is also at variance with a more
mainstream picture where some regions of space are not allocated to any atom.
This is especially common in molecular complexes held together by hydrogen
bonds and/or van der Waals interactions. For example, a ligand that docks into
an enzyme’s active site is typically thought of as a molecule of finite size, even
if it finds itself in a condensed matter situation, which is the case when inside
the active site. In this picture, therefore, there are regions of space, between the
ligand and the enzyme, that are unassigned to any molecule and thus any atom.
Similarly, water molecules in the liquid state do not leave gaps between them: all
of space belongs to one water molecule or another. This is a result of each water
molecule being completely bounded by interatomic surfaces. Fourth, the fluorine
atom distorts the carbon atom more than the hydrogen atom. QTAIM provides
a picture of atoms reminding us of the macroscopic world where objects can be
distorted to whichever degree of malleability they possess. Fifth and finally, there is
‘‘no direct contact’’ between the oxygen and the fluorine, that is, topologically they
do not touch. In fact, they are kept apart by a very thin wedge of space belonging
to the carbon atom, although it is not visible in this plot. Similarly, there is no
direct contact between the oxygen and the hydrogen, which are again separated by
a wedge of carbon that is now more clearly visible than in the previous case. The
same is true for fluorine and hydrogen. In any of these three ligand···ligand pairs
(F···H, O···H, and O···F), direct contact would mean that there is a little black
square at the center of the interatomic surface separating the atoms of interest. If
the nuclear configuration is sufficiently changed, such little black squares can be
created.
Figure 8.6 reveals the three-dimensional shape of the three interatomic surfaces,
in two different types of graphical representation and in a slightly different
orientation (Figure 8.6a,b). In order to obtain a deeper insight into the nature of a
topological atom we need to introduce the fundamental concept of the GP.
(a) (b)
Figure 8.6 Three-dimensional representation of the topological atoms in FHC=O, using
the same colors as in Figure 8.5. (a) Semitransparent solid surfaces and (b) wireframe sur-
faces. The small purple spheres correspond to the black squares in Figure 8.5.
8.3 The Topological Atom: is it also a Quantum Atom? 13
AGP is a path of steepest ascent through a multidimensional function, here
one depending on three spatial coordinates. In the part of QTAIM that we discuss
first, this function is the electron density 𝜌(r). In the second part of QTAIM, this
function will be the Laplacian of the electron density. In general, this function
can still be another quantum mechanical function of interest (see Box 8.1). The
GP is the cornerstone of QCT, which was briefly mentioned in the Introduction.
Returning to the GP, it ‘‘wastes no time’’ in reaching a local maximum. Indeed, the
GP pierces through surfaces of constant electron density such that it reaches a local
maximum in the quickest possible way. The direction a GP takes is orthogonal to
a very small (i.e., infinitesimal) plane, locally representing a surface of constant 𝜌
value. Because the gradient vector is also orthogonal to such a plane one deduces
that a GP is in fact a succession of gradient vectors. Imagine the gradient vector
of the electron density, denoted ∇𝜌, being evaluated at a given point. This vector
points in the direction of steepest ascent. Follow the gradient vector ∇𝜌over a very
short (i.e., infinitesimal) stretch and reevaluate ∇𝜌at its endpoint. Continue this
process iteratively. The resulting succession of very short gradient vectors can be
regarded as a smooth curve. This curve is the GP.
Earlier we stated that a deformation density introduces a reference electron
density. QTAIM eliminates the need for this reference density. Indeed, the gradient,
as a differential, operates on a single electron density. This differential does not
represent a difference between two densities but effectively constitutes an internal
difference, within the same density, which is the molecular electron density 𝜌.
This is an example of how QTAIM obeys Occam’s razor or the principle of
minimalism.
Figure 8.7 illustrates a multitude of GPs in HFC=O. The majority of the GPs
originate at infinity and travel through space until they reach a local maximum,
which practically coincides with a nuclear position. A collection of such types of
GPs forms a spider web-like pattern, marking the subspace of each topological
atom. Because the GPs ‘‘draw themselves,’’ the topological atom also draws itself.
The space it occupies is naturally carved out by the minimal prescription of tracing
GPs, wherever they lead us. The resulting atomic shapes can be complicated but the
prescription that generated them is minimal. By loose analogy, Newton’s second
law is also minimal but it also generates many complicated patterns of motion
(e.g., of celestial bodies in the solar system).
It is clear from Figure 8.7 that the web-like patterns of GPs forming the topological
atom are bounded by the interatomic surfaces. In other words, some GPs graze
along an interatomic surface, sometimes indiscernibly closely. These grazing GPs
move toward the center of an interatomic surface, marked by the little black square,
but then, as they approach this center, they suddenly bend and terminate at a
nucleus. This behavior suggests that the interatomic surface itself consists of GPs
that do not terminate at a nucleus but at a mysterious little black square. In that
sense, the trajectories of GPs constituting an interatomic surface can be considered
as a limiting case.
It is time to zoom in on what these little black squares are, and thereby reveal a
more complete picture of topological relationships.
14 8 The QTAIM Perspective of Chemical Bonding
O
C
H
F
Figure 8.7 Gradient paths traversing the subspace of each topological atom in HFC=O.
The interatomic surfaces shown in Figure 8.5 are repeated here in bold.
8.4
The Bond Critical Point and the Bond Path
Let us start with the GP, which was the central concept of the last section. We know
that a GP is oriented: it has a beginning and an end. We also know that a great
many GPs can come together at one point, practically coinciding with a nuclear
position. We also know from Box 8.3 that, if at a given point ∇𝜌0, then there is
only one GP containing this point. Thus, by the logical construction modus tollens,
we can state that if a given point belongs to more than one GP then, at that point,
it is not true that ∇𝜌0. In other words, a point where more than one GP meets
has the property that ∇𝜌=0. Such a special point is called a critical point (CP).
Box 8.3 The Gradient Path and Natural Coordinates
The gradient vector is all that is needed to map out the potentially complicated
internal structure of a molecule in terms of the topological atoms that make up
the molecule. A GP can be seen as succession of very short gradient vectors,
in the simplest possible description. Thus, the tangent to such a GP, in a given
point, is a gradient vector. This statement can be rewritten as a system of three
ordinary differential equations,
d𝐫
𝑑𝓁=∇𝜌
|∇𝜌|(B8.1)
where ris a position vector describing the path and 𝓁is the path length. The
simplest way to solve this system is to use the Euler method, which is essentially
8.4 The Bond Critical Point and the Bond Path 15
tracing the paths by following the gradient vector over a very short stretch
and reevaluating it at every new end point. Although this method may suffice
for visual purposes it is not that accurate. The Runge–Kutta method (and its
more modern variants such as the Cash–Karp method) is more appropriate and
so is a predictor-correct method. Unpublished research showed that the more
modern Bulirsch–Stoer method is not performing better than the Runge–Kutta
or Cash–Karp method. Analytical expressions for the solutions of Eq. (B2.1) have
been investigated before [95]. Note that one can trace only one GP through
a point where the gradient of the electron density does not vanish. Slightly
rephrased, if at a given point ∇𝜌0, then there is only one GP containing this
point, or in other words, going through this point.
Let us introduce a path coordinate s, which is related to the actual path length
𝓁via the equation ds=d𝓁/|∇𝜌|. One can then rewrite Eq. (B8.1) in a slightly
simpler way,
d𝐫(s)
ds=∇𝜌(B8.2)
Foragiveninitialpointr0(s=0) (where ∇𝜌0), a unique GP follows from
solving Eq. (B8.2). At the origin of the GP s=−∞ while at its terminus s=+∞.
The parameter scan be regarded as a coordinate describing the path over its
full length. We know from the main text in Section 8.4 (Figure 8.8) that a GP
always originates at a CP and terminates at another CP. If the distance between
these two CPs is finite then the GP’s length will also be finite. This means that
a finite curve in Cartesian space is described by a path coordinate that spans
the full range of [−∞,+∞]. To complete the set of coordinates that describe the
topological atom to three coordinates we need two more coordinates. Imagine
a sphere of radius 𝛽, centered at the nucleus within a given topological atom.
The radius 𝛽is such that the sphere lies completely within the volume of the
atom. Any point on the surface of the sphere can be uniquely described by 𝜃
and 𝜑, the familiar angular spherical coordinates. Equation (B8.3) makes clear
GP
CP
CP
Figure 8.8 Each gradient path (GP) has a direction, originates at a critical point (CP)
and terminates in a CP.
16 8 The QTAIM Perspective of Chemical Bonding
how an initial point r0(x0,y0,z0) is described as the GP is traced in the forward
(ascending) and backward (descending) direction,
x0=𝛽sin 𝜃cos 𝜑
y0=𝛽sin 𝜃sin 𝜑
z0=𝛽cos 𝜃(B8.3)
Now we have three coordinates that each cover their full range, that is, s∈[−∞,
+∞], 𝜃∈[0, π]and𝜑∈[0, 2π]. These are called natural coordinates. The interesting
point is that they describe the topological atom ‘‘from within,’’ as if it was a full
(and unbounded) space. This ‘‘complete world’’ is of course not complete in
Cartesian space, where its description in terms of the x,y,zcoordinates clearly
shows areas where the atom is absent. The coordinate mapping (s,𝜃,𝜑)→
(x,y,z) could be interesting from the point of view of a mathematical branch
called differential geometry. Such an analysis has already been carried out [96] on
interatomic surfaces, which are described by their own two natural coordinates.
Because a gradient vector ceases to exist at a CP, this point is not associated
with a direction pointing at a higher electron density value. Thus, the CP can be
considered as the end point of a GP: as the path arrives at a CP there is no gradient
vector to guide it beyond this point. Now, one can ask if a CP can also serve as
the starting point of a GP. The answer is yes and again, this situation arises in
a straightforward manner. If a GP starts in a given point then there cannot be a
GP trajectory preceding it. This is exactly the case if ∇𝜌=0atthatpoint,andthat
confirms that the GP started at a CP. Figure 8.8 summarizes the explanation above,
introducing GP as a convenient acronym for gradient path and CP for critical point.
The most common type of GP is the one that originates at infinity and terminates
at a maximum in 𝜌, which for all practical purposes coincides with a nucleus.
There are many examples of this type of GP in Figure 8.7, which collectively
form the atomic basin, that is, the subspace carved out inside a larger quantum
system, forming the topological atom. From this picture one deduces that there is
an ocean of CPs at infinity. The maximum in 𝜌at the nucleus is a maximum in
three dimensions. In other words, in whichever of the three mutually orthogonal
directions one approaches the maximum, 𝜌will increase.
Each interatomic surface in Figure 8.7 also consists of a bundle of GPs, sharing
with the GPs forming the atomic basin, the fact that they originate at infinity.
However, the GPs of the interatomic surface terminate at a so-called bond critical
point (BCP), which is the proper name for the mysterious black squares in Figure 8.7
(and some earlier figures). Locally, the interatomic surface may be regarded as a
plane containing the BCP. GPs travel a very short stretch in this plane, on their
way to the BCP, which acts a local maximum in this plane. The BCP is thus a
maximum in two dimensions. In this case these two dimensions describe the
molecular plane. Is the BCP a maximum or a minimum in the third dimension,
which is characterized by the direction orthogonal to the plane?
8.4 The Bond Critical Point and the Bond Path 17
BCP
Figure 8.9 Examples of bond critical points
(BCP), which are the three black squares in
HFC=O, one of which is explicitly marked by
the acronym BCP, and lies in between C and
H. One gradient path originates at this BCP
and terminates at C, while another gradient
path originates at the same BCP but termi-
nates at H. Together they form the bond
path linking C and H.
Figure 8.9 answers this question by focusing on one BCP, lying in between
C and H. The behavior of the third dimension is marked by two arrows, which
show the direction of steepest ascent. It is clear that the BCP is a minimum in
this third dimension. The BCP is thus a saddle point, or a CP with mixed local
behavior in terms of the directions in which it is a minimum or a maximum. In
three-dimensional space there are four types of CPs in this sense: a maximum, a
BCP, a minimum and a second type of saddle point that we will discuss later. In
general, there are (n+1) types of CPs in an n-dimensional space. Box 8.4 provides
more details on CP classification and local curvature. This Box also explains the
various types of GPs, as determined by the type of CP they link.
Box 8.4 Classification of Critical Points
In two dimensions there can be only three types of CP: a maximum, a minimum
or a saddle point. A saddle point is a CP that acts both as a minimum and a
maximum depending on the direction one scans in. In three dimensions there
are four types of CP: a maximum, a minimum and two types of saddle point. Any
CP is typified by the local curvature of the function in which the CP appears. For
our purpose, this function is the electron density, denoted by 𝜌,andtheCPisa
point where the gradient vanishes, or ∇𝜌=0. The statements made here, about
the number of CPs in two or three dimensions, are only true if this curvature is
18 8 The QTAIM Perspective of Chemical Bonding
not zero, that is, the function is not flat at the CP. This curvature is quantified by
the eigenvalues of the Hessian of the 3D function of interest. The Hessian is a
matrix of partial second derivatives, each matrix entry Hij being defined by
H𝑖𝑗 =∂2f
∂qi∂qj
where iand j=1,2,or3 (B4.1)
where q1,q2,andq3stands for x, y, and z, respectively, if the Hessian is con-
structed in three dimensions. A second derivative naturally measures curvature
but the individual Hessian matrix elements cannot be analyzed directly as gauges
of local curvature. The reason is that the numerical values of the entries in the
Hessian matrix themselves depend on the choice of the coordinate system with
respect to which the coordinates x, y,andzare expressed. However, the Hessian’s
eigenvalues are independent of this choice and thus solve this problem.
To each eigenvalue corresponds an eigenvector that points out the direction
in which this eigenvalue determines the curvature. In order to illustrate this
statement, let us take one type of CP, the BCP, because it is the most relevant
for this chapter. The figure given here shows the three eigenvectors u1,u2,and
u3, associated with a BCP. The latter eigenvector points in the direction in which
the CP is a minimum. Accordingly, the corresponding eigenvalue 𝜆3is positive.
The figure shows how the electron density, starting at nucleus A, decreases along
the BP and reaches its lowest value at the BCP, to then increase again to reach a
local maximum value at the nucleus B.
B
A
BP
u1
u3
u2
ρ
BCP
The figure also shows how eigenvectors u1and u2mark the directions in which
the BCP is a maximum, as amplified by the density of points in the plane spanned
by these eigenvectors. As expected, the corresponding eigenvalues are negative,
or 𝜆1<λ2<0.
8.4 The Bond Critical Point and the Bond Path 19
The other type of saddle point is called a ring critical point (RCP). It occurs in the
center of a benzene ring, for example. Clearly, the electron density is a minimum
at the RCP with respect to any displacement away from it in the molecular plane.
Hence, there are two positive Hessian eigenvalues. In the direction orthogonal
to the molecular plane the RCP acts as a maximum. Hence, the remaining
eigenvalue is negative.
A compact notation, denoted (r,s), is often used to refer to any of the four
types of CPs. The rank (r) of a CP refers to the number of nonzero eigenvalues
and the signature (s) to the sum of the signs of the eigenvalues. More precisely,
the latter means that a negative eigenvalue is associated with ‘‘minus one’’ (−1),
and a positive eigenvalue with ‘‘plus one’’ (+1). The minus/plus ones can then
be added. CPs of rank 3 are the most common, that is, those points with strictly
nonzero Hessian eigenvalues. In this notation, the BCP is written as (3, −1)
because it has three nonzero eigenvalues and because the ‘‘sum of eigenvalue
signs’’ is (−1) +(−1) +1=−1. Another useful number is the so-called index,
which is the number of positive eigenvalues of the Hessian. In summary, the
four types of rank-3 CPs in three dimensions are denoted (3, −3), (3, −1), (3,
+1), and (3, +3) corresponding to the index values 0, 1, 2, and 3.
Figure 8.8 made clear that GPs always link two CPs. This fact can be used
to characterize all possible GPs. This was achieved [72], for the first time, in
2003. All details can be found in this paper but the final result is reproduced
here in Table B4.1, which lists all possible types of GP in the electron density.
This classification is universal, beyond the electron density, and is valid for
other 3D scalar fields, such as the Laplacian of the electron density. The key to
understanding how many GPs can originate or terminate at a CP is determined by
the eigenvalues and associated eigenvectors of the Hessian evaluated at the CP.
A negative eigenvalue corresponds to a direction in which the CP is a maximum
and hence acts as a sink (or terminus) for GPs. Conversely, a positive eigenvalue
corresponds to a direction in which the CP is a minimum. In this direction the
CP acts as a source (or origin) of GPs.
As an example, let us look at the second row in Table B4.1, which shows
the case of GPs in an interatomic surface (the name for this manifold if one
works with the electron density). The notation for such a manifold is a [3, 1],
or [Isource,I
sink] in general, where the source and sink CPs are characterized by
their index. Here, the first CP at the origin acts a source in all three directions
and therefore does not restrict the dimensionality of the manifold that connects
this CP and the CP at the terminus (or sink). It is clear that the dimensionality
of the connecting manifold is determined by the CP at the terminus. Because
the terminal CP can receive GPs in two directions the dimensionality of the
connecting manifold is two. Obviously, the number of directions the terminal CP
can receive GPs from, determines the dimensionality of the first three GP types
in the Table, that is, [3, 2], [3, 1], and [3, 0]. In general, the dimensionality of a
manifold connecting two CPs is the dimensionality of the source CP or that of the
sink CP, whichever one is lowest. Formally, dimmanifold =min(dimsource, dimsink ).
20 8 The QTAIM Perspective of Chemical Bonding
Table B4.1 Survey of all nine possible typesaof gradient path (GP) in the electron density
(or any 3D scalar field).
Dimensionalityb
GP typec(r,s)origin →(r,s)terminus Origin Terminus ManifolddExample (name is
for 𝝆only)
[3, 2] (3, +3) →(3, +1) 3 1 1 Ring line
[3, 1] (3, +3) →(3, −1) 3 2 2 Interatomic
surface
[3, 0] (3, +3) →(3, −3) 3 3 3 Atomic basin
[2, 2] (3, +1) →(3, +1) 2 1 1
[2, 1] (3, +1) →(3, −1) 2 2 2
[2, 0] (3, +1) →(3, −3) 2 3 2 Ring surface
[1,2] (3, −1) →(3, +1) 1 1 1
[1, 1] (3, −1) →(3, −1) 1 2 1 GP in 𝜌in a
conflict structure
[1, 0] (3, −1) →(3, −3) 1 3 1 Bond path
aOnly critical points (CPs) of rank 3 are considered.
bFor a CP at the origin this is the number of directions in which the CP can ‘‘send’’ GPs, while for a
CP at the terminus this is the number of directions in which the CP can ‘‘receive’’ GPs. For the
manifold this is the dimension of the connecting topological object, that is, curve, surface or basin,
corresponding to one, two or three dimensions, respectively.
cEach gradient path is characterized by two critical points, an origin and a terminus. The critical
points are denoted by their indices, Iorigin and Iterminus,respectively,andthegradientpathas[I
origin,
Iterminus].
dThis is the set of gradient paths that connect two given critical points.
Figure 8.9 shows a GP that originates at the one explicitly marked BCP and
that terminates at the carbon nucleus. The other GP, originating at the same BCP,
terminates at the hydrogen nucleus. Together they form the so-called BP linking C
and H. This name was first proposed [28] by the Bader group in 1977, many years
after the work on the topological atom was published. This 1977 article immediately
spotted the chemically intuitive character of the BP, when it is curved, for example,
as in the case of ring strain. Figure 8.10 shows the simple example of cyclopropane,
in which the CC BPs bulge outward. This deviation from a straight line connecting
the nuclei can be quantified by straightforward geometrical measures.
At this juncture, a moment of reflection serves to obtain a better understanding
of the overall philosophy and development of QTAIM. In his 1975 Accounts, Bader
wrote that While one cannot deny the important role played by the bond concept, we
propose, as an alternative, a return to what is essentially the ‘‘atoms in molecules’’
approach to chemistry. Specifically, one seeks to understand or predict the properties of a
total system in terms of the properties of its parts. Rather than bonds, we choose as our
8.4 The Bond Critical Point and the Bond Path 21
Figure 8.10 Contour plot of the electron
density in the plane of the carbon nuclei
in cyclopropane. The interatomic surfaces
are marked by dashed line and the bond
paths by solid lines. The little arrows mark
the direction of the gradient paths in the
vicinity of the bottom CC bond critical point.
(Reproduced from Ref. [28].)
fundamental parts mononuclear fragments of the system with boundaries defined in real
space.
About 2 years after this statement was published, an alternative view was
proposed by the Bader group, this time very much in line with the traditional
picture of a chemical bonding. The Bader group had observed that the topology of
the electron density offered an interpretation of a bond that associated it with the
BP, a topological object coexisting with topological atoms. So, whereas QTAIM
started off as an energy partitioning scheme and viewed bonding as a result of
atomic energies, the theory was now tempted to reinterpret bonding guided by the
presence of BCPs and BPs. This dual approach, both resulting from the full topology
of the electron density, should not be seen as a dilemma. Instead, both pictures
should ideally be complementary aspects of the same thing: a chemical bond. In
fact, in the last part of this chapter we spend time on exactly this complementarity.
A question of essence is the following: can we link the QTAIM energy partitioning
with BPs? We will show in Section 8.5 that this is indeed possible as demonstrated
by a seminal paper published [29] in 2007, entitled BPs as Privileged Exchange
Channels.
First, however, we look at how evaluating quantities at the BCP can offer an insight
into the type of bond that it is a signature of. Box 8.5 provides details on an important
function called the Laplacian of the electron density, ∇2𝜌, which has received much
attentioninQTAIM.ThefunctionL(r)=−∇
2𝜌(r) often appears in the literature
when bonds are classified along the lines discussed in Box 8.5. This is because it is
more intuitive to think of a positive L(r) value as one associated with a concentration
of electron density, while depletion is naturally reminiscent of a negative value.
The sign of L(r) at the BCP can be used as an indicator to characterize a bond,
22 8 The QTAIM Perspective of Chemical Bonding
Box 8.5 The Laplacian of the Electron Density
The Laplacian of the electron density, ∇2𝜌, is the second quantum mechanical
function ever to be analyzed topologically, after 𝜌itself. However, the first paper
on the Laplacian (in a QTAIM context), which was finally published [97] in 1984,
did not (yet) explore its topology. Instead, this work plotted ∇2𝜌its value for
a variety of systems including covalent diatomics, hydrogen-bonded complexes,
ionic systems, and van der Waals (noble gas) complexes. This work interpreted
the sign of ∇2𝜌in order to make a connection with the energetics of atomic
interactions. A variant of Eq. (B2.3), listed below as Eq. (B5.1), made this
connection possible,
2G(𝐫)+V(𝐫)= 1
4∇2𝜌(𝐫)(B5.1)
where G(r) is the kinetic energy density (defined in Eq. (B2.2)) and V(r)isthe
electronic potential energy density. Note that this is a local relationship; when
integrated over the volume of a topological atom Eq. (B5.1) leads to the atomic
virial theorem, due to the fact that the Laplacian then vanishes (see Eq. (B2.8)).
In regions of a (molecular) system where ∇2𝜌<0, the electron density is
locally concentrated. This interpretation of concentration is a general property
of the Laplacian in a (3D) scalar field, disconnected from quantum mechanics.
This interpretation can be seen as an extension of the second derivative in
one dimension. Indeed, the Laplacian is basically a second derivative, and
it essentially quantifies a local curvature. It is well known that, in any one
dimension, a negative curvature corresponds to a local maximum. Therefore,
a negative Laplacian corresponds a local concentration, even if this does not
happen at a stationary point in the Laplacian (where ∇(∇2𝜌)=0). From Eq.
(B5.1) one can deduce that the potential energy density then dominates if
∇2𝜌<0, because G(r) is always positive and can therefore never contribute to
an overall negative sign. When ∇2𝜌< 0 is found in the internuclear region then
one classifies the interatomic interaction as shared. Covalent and polar bonds
fall in that class, and the interaction between atoms in such bonds is caused
by a contraction of the electron density toward the interaction line linking the
two nuclei. We imagine this situation at the BCP. This local state of affairs can
be understood semiquantitatively by invoking the fact that ∇2𝜌=𝜆1+𝜆2+𝜆3,
where the three terms are eigenvalues of the Hessian, as discussed in Box 8.4.
The Laplacian can only be negative if the two manifestly negative eigenvalues, 𝜆1
and 𝜆2, dominate (in magnitude) the one positive eigenvalue, 𝜆3. This, in turn,
means that the electron density is concentrated toward the BCP.
The opposite situation arises when ∇2𝜌>0, in which case the electron density
is locally depleted. If we imagine this situation to arise again at a BCP then
this time the positive eigenvalue 𝜆3dominates. This means that the electron
density is rapidly increasing away from the BCP. This situation arises in so-called
closed-shell interactions such as ionic bonds, hydrogen bonds and van der Waals
8.4 The Bond Critical Point and the Bond Path 23
complexes. By virtue of Eq. (B5.1), the kinetic energy density is now in charge of
the mechanics of the interaction.
At the time the 1984 paper [97] appeared the local energy density H(r)was
proposed [98] to complete the description of bonding along the lines of the
earlier discussion. This function is defined in Eq. (B5.2),
H(𝐫)=G(𝐫)+V(𝐫)(B5.2)
The driver to introduce this function was the observation that for about 100
covalent bonds (including F–F in F2)H(rb) is always negative. The opposite is
true for ionic bonds, hydrogen bonds, and van der Waals bonds, thereby turning
H(rb) into a convenient discriminator between covalent and non-covalent bonds.
The second paper on the Laplacian [68] this type looked the topology of this
function, starting with its CPs. The numbers and locations of the bonded and
nonbonded concentrations of charge in the valence shell of a bonded atom, as
determined by the maxima (CPs) of L(r)=−∇
2𝜌(r), were found to be in general
agreement with Gillespie’s valence shell electron pair repulsion (VSEPR) model
[99, 100]. It should be noted that this interpretation is based on observations
only, and actually at variance with the conclusions based on an analysis of the
Fermi hole [101] by Bader and Stephens in 1975. They found that regions that
maximize Fermi correlation to yield localized groupings of electrons correspond
to atomic cores, and not, in general, to localized pairs of bonded and nonbonded
electrons as anticipated on the basis of the Lewis model.
In 1985, the concept of an atomic graph was proposed [102] in the context of
reactivity. The idea of an atomic graph is loosely analogous to that of a molecular
graph. Both represent a connectivity scheme of CPs, completely determined by
their topological links, as embodied by GPs. Discussing this in detail is beyond
the scope of this chapter but it is worth mentioning though that atomic graphs
were introduced [103] in 1992, in the context of complementarity in chemistry and
molecular recognition. Later, more complete insight into the Laplacian’s topology
(in terms of CPs and their connectivity only) was published [70, 104] and how this
topology changes in response to a change in molecular geometry. A thorough
study on a multitude of atomic graphs covering most common functional groups
of organic chemistry revealed 16 highly transferable atomic graphs [105]. This
work offers a physicalization of chemical graph theory, introducing building
blocks rooted in quantum mechanics, through properties of the orbital-invariant
electron density. In 2003, Laplacian basins were shown for the first time [106]
thanks to the construction of a novel algorithm (based on the ‘‘octal tree’’) that
also enabled integration the electron density over their volume. This led to the
first ever study [71] using the full topology of the Laplacian, which scrutinized
it as a physical basis for the VSEPR model. This study confirmed, by computed
volume of Laplacian basins, that a nonbonding domain is indeed larger than
a bonding one, in full accord with the VSEPR model. Second, these volume
quantities for the first time being available also showed that a multiple-bond
domain is indeed larger than a single-bond one, again in agreement with VSEPR.
24 8 The QTAIM Perspective of Chemical Bonding
However, the calculations did not corroborate the effect of the electronegativity of
the central atom or ligand on the volume of bonding domains. More worryingly,
this study yielded unexpectedly small electronic populations (by integration of 𝜌
over Laplacian basins), nearer to one than to two, for non-hydrogen cores and
bonding domains, while nonbonding domains could have populations much
larger than two.
although it is safer to use it in combination with other indicators, the simplest
being the value of 𝜌at the BCP, denoted by 𝜌(rb). In a typical ionic bond, which
is a closed-shell interaction, 𝜌(rb) is typically of the order of 0.01 au whereas for a
shared interaction it is typically about an order of magnitude larger, that is, 0.1 au.
The need to characterize a bond by more than just L(rb) is clear from the existence
of bonds such as COorF
2, which are characterized as intermediate interactions.
This is because it has mixed features: a high 𝜌(rb) value (normally found when
L(rb)>0) but L(rb)<0. Box 8.5 introduces the measure H(rb), which circumvents
the appearance of intermediate interactions by always assigning a negative value to
H(rb) for any recognized covalent bond, including that in F2. When complemented
with the extra indicator G(rb), 𝜌(rb) atomic interaction can be classified in a way
more sophisticated than in the 1980s. One such proposal came in 1998 from the
world of experiment, in particular, high-resolution X-ray crystallography, where
bonds in a transition metal dimer were classified [30] by topological indicators.
Table 8.1 describes the main features of several types of interaction including
between light (L) elements (e.g., H, C, B) and heavy (H) elements (e.g., As, Co).
Note that this table takes into account the position of the BCP with respect to the
behavior of L(r) in terms of sign, sign change, or extrema. One of the conclusions
of this extensive crystallographic study on Co2(CO)6(AsPh3)2is that there is a
covalent Co– Co bond, in spite of the lack of accumulation of the deformation
Table 8.1 Summary of topological indicators and features that characterize the atomic
interaction.
𝝆(rb) Position of rbwrt L(r)
along the BP
L(rb)G(r
b)/𝝆(rb)H(r
b)
Open-shell (covalent bonds) Large Close to a
minimum
>0<1<0
Intermediate interactions (polar
shared bonds, e.g., CO)
Large Close to a nodal
surface
Arbitrary 1<0
Closed-shell Small Inside a flat
region
<01>0
Shared (e.g., Co– Co) Small Close to a
maximum
∼0<1<0
Donor– acceptor (e.g., Co–As) Small Close to a nodal
surface
<0∼1<0
8.5 Energy Partitioning Revisited 25
density map. Note that all values in Table 8.1 were obtained experimentally but the
kinetic energy density was indirectly obtained via a semiempirical method [31] that
estimates G(rb) from a function of 𝜌and its Laplacian.
An otherwise popular indicator that does not feature in Table 8.1 is the so-called
ellipticity 𝜀, which is defined as 𝜆1/𝜆2−1. Note that the ellipticity is not bounded
from above but, because 𝜆1<λ2<0, its minimum value is zero. This corresponds
to a cylindrically symmetric electron density, which is perfectly reached at the
carbon– carbon BCP in ethyne. The BCP of the central CC bond in butane has
an ellipticity of 0.01, which is almost zero. The ellipticity reaches 0.3 at the C=C
BCP in ethene, and is interpreted as a measure of the πcharacter. Any CC
BCP in benzene has the lower value of 𝜀=0.18, which is in agreement with the
expected reduction in the πcharacter compared to the ‘‘pure’’ double bond in
ethene. Remarkably, the substantial ellipticity of 0.4 at a CC BCP in cyclopropane
(Figure 8.10) is reminiscent of the widely recognized double bond character of
(strained) three-membered carbon rings, in terms of their reactive behavior (e.g.,
rates of solvolysis).
Finally, we point out that the topological descriptors were introduced [32] in the
area of molecular similarity through the concept of BCP space. In its original form,
BCP space represented BCPs as points in a 3D space, spanned by 𝜌(rb), −L(rb)
and 𝜀(rb). This compact description sufficed, compared to descriptions from whole
electron densities, in providing descriptors for linear free energy relationships such
as the one proposed by Hammett on acidity. Further research showed that the
BCP space proved a reliable source of compact quantum mechanical descriptors
for QSARs (quantitative structure–activity relationship) in a medicinal, ecological,
physical organic (pKaprediction), or toxicological context, culminating in a method
called Quantum Topological Molecular Similarity (QTMS) [33].
8.5
Energy Partitioning Revisited
An early important result in the development of QTAIM was the establishment
that an atom in a molecule has its own atomic virial theorem. This means that, for
a single atom, there is a relation between the kinetic energy of this (topological)
atom and its potential energy. This relationship can be obtained by integrating Eq.
(B5.1) over the volume of a topological atom Ω,or
Ω
d𝐫[2G(𝐫)+V(𝐫)] = 1
4Ω
d𝐫∇2𝜌(𝐫)=0=2T(Ω) + V(Ω) (8.2)
where we emphasize again that topological atoms are remarkable subspaces in that
they have a well-defined kinetic energy, denoted T(Ω). Thanks to this relationship
there was no need to calculate V(Ω) in the early days of QTAIM. Calculating T(Ω)
was sufficient, which required a 3D volume integration [34] over the volume of Ω.
The literature on the integration algorithms over topological subspaces is fairly large
and has been briefly reviewed in a recent paper [35] on fully analytical integration
(over a spherical core inside a topological atom), a body of work to which one can
26 8 The QTAIM Perspective of Chemical Bonding
add a most recent contribution [36]. In spite of the technical challenges of such 3D
integration over topological atoms, they are computationally less demanding than
the 6D integrations required to calculate V(Ω) (independently from the atomic
virial theorem). The existence of an atomic virial theorem enables one to calculate
only T(Ω) and then deduce V(Ω) from it as −2T(Ω). As a result, the total atomic
energy E(Ω)=T(Ω)+V(Ω)=−T(Ω). One can the sum the total atomic energies
in a molecule and obtain the total molecular energy. Although this is convenient,
the latter statement is true only if the forces on the nuclei all vanish, that is, at
a stationary energy point, typically an energy minimum. A calculation of V(Ω),
independent of the atomic virial theorem, would free QTAIM energy partitioning
from being confined to equilibrium geometries only.
The latter was achieved [37] for the first time in 2001, in a paper that calculated
the Coulomb interaction energy defined in Eq. (8.3),
EAB
Coul =ΩA
d𝐫1ΩB
d𝐫2
𝜌tot(𝐫1)𝜌tot (𝐫2)
r12
(8.3)
where the total charge density, 𝜌tot(𝐫), is the sum of the electron density (i.e.,
electronic) 𝜌(r) and the nuclear charge density, and r12 is the distance between two
infinitesimal pieces of charge density. This work was further developed completing
the calculation of non-Coulomb interaction energies [38, 39]. However, in 2004, an
efficient algorithm [40] made possible a full and systematic analysis of molecular
energy in terms of intra-atomic and interatomic (pairwise) energy contributions,
dubbed Interacting Quantum Atoms (IQA) [24]. All electron–electron interactions
are determined by the diagonal of the second order reduced density matrix, 𝜌2(r1,r2).
The electron– electron interaction energy between two topological atoms ΩAand
ΩBcan then be written as in Eq. (8.4),
VAB
ee =ΩA
d𝐫1ΩB
d𝐫2
𝜌2(𝐫1,𝐫2)
r12
(8.4)
Interestingly, the electron– electron interaction can always be split into a classical
(Coulombic), and a nonclassical (quantum mechanical) exchange-correlation (xc)
part, as follows,
𝜌2(𝐫1,𝐫2)=𝜌1(𝐫1)𝜌1(𝐫2)+𝜌xc(𝐫1,𝐫2)(8.5)
where the subscript ‘‘1’’ can be omitted if no emphasis is needed on the fact 𝜌(r)is
in fact the diagonal of the first-order reduced density matrix. Substituting Eq. (8.5)
into Eq. (8.4) leads to
VAB
ee =VAB
cl +VAB
xc (8.6)
where VAB
cl is the purely electronic part of EAB
Coul, that is, without any contributions
of nuclear charge. Stimulatingly, this part measures the ionic-like contributions,
while the second term VAB
xc measures the covalent-like contribution to a given
atom– atom interaction (no matter how far apart the atoms are). This term is then
of course defined as,
VAB
xc =ΩA
d𝐫1ΩB
d𝐫2
𝜌xc(𝐫1,𝐫2)
r12
(8.7)
8.5 Energy Partitioning Revisited 27
This genuine quantum mechanical index is key to enduring progress in the ‘‘holy
grail’’ question of how (if at all) to extract chemical bonds from wave functions. It
should be emphasized that this index is an energetic term, and therefore enables
one to analyse bonding directly in terms of energy, which is the ultimate arbiter
of chemical stability, even of chemical fragments. Second, the index VAB
xc does not
invoke a reference state, and hence any (risky) decision as to what that state might
be. An analysis based on VAB
xc benefits from the minimalism discussed earlier.
We now discuss an observation of paramount importance [29], made in 2007,
which revealed a remarkable link between the presence of a BCP and VAB
xc .The
importance of this observation lies in directly relating the existence of the topological
object that decides on a bond to atomically partitioned covalent energy. How this
works will be explained using the example of water. We consider the minimum-
energy dissociative adiabatic pathway (maintaining C2v symmetry) that leads to the
formation of an oxygen atom and a hydrogen molecule, H2O(
1A1)→O(
1D) +
H2(1Σg+). A sequence of gradient vector fields for relevant snapshots along this
pathway has already been published [41] in 1981, in the context of the QTAIM
theory of structural change, which invokes catastrophe theory. We are interested
in following this pathway in the reverse direction, that is, to form a water molecule
by H2approaching an oxygen atom. This formation is controlled by a single
(reaction) parameter d, which is the distance between the oxygen nucleus and the
midpoint of H2. Figure 8.11 plots this parameter as the x-axis against a y-axis
of VAB
xc energies. Starting at the right-hand side of this plot, H2is an essentially
unperturbed molecule. Here, H2is a BCP expressing the fact that the two hydrogen
atoms are bonded. In this regime VHH
xc is markedly lower than VOH
xc and this is why
the BCP appears between H and H. As the control parameter ddecreases, suddenly
the VOH
xc energy becomes lower than that of its competitor VHH
xc . This happens near
kJ mol−1
0
−200
−400
−600
0.0 0.5 1.0 1.5 2.0 2.5
VAB
XC
VOH
XC
VHH
XC
d/Å
O
O
HH
H
H
d
d = 1.38 Å
Figure 8.11 Competition between VHH
xc and
VOH
xc in the formation of ground state water
(left) from H2and an oxygen atom (right).
Moving from the right to the left, the con-
trol parameter ddecreases, and an intersec-
tion is hit at d=1.38 ˚
A, beyond which VOH
xc
becomes lower than VHH
xc . Around this inter-
section, the O–H BCP is formed and the
H–H BCP point is destroyed (ignoring the
complication of a very short-lived ring critical
point near this transition point).
28 8 The QTAIM Perspective of Chemical Bonding
d=1.38 ˚
A, at which point, the O–H BCP is formed and the H–H BCP is destroyed.
Matters are actually slightly more complicated because the evolution of the gradient
vector field passes through a very short-lived ring structure between d=1.391 ˚
A,
and d=1.389 ˚
A. Note that in the ring structure both O–H BCPs and H–H BCPs
coexist. The same energy-topology of the 𝜌relationship has been established both
in the HCN to CNH isomerization and in a process in which a fluoride anion
moves around LiF in a circle, passing from FLiF to LiFF.
The general conclusion of the 2007 paper [29] was confirmed very recently, in
2013, by Tognetti and Joubert [42]. They thoroughly analyzed admittedly approxi-
mate values for VAB
xc (ignoring electron correlation and for the first time employing
density functional theory (DFT), all calculations being performed in the mono-
determinantal Kohn– Sham ansatz using B3LYP). They computed what they call
Ex
AB values for a large number of intramolecular O···X interactions (where X is an
O, S, or halogen atom) occurring in a set of 36 molecules, which also contained
C=C bonds. The authors address many natural questions about the nature of the
BCPs, for example, if they would always appear between any two nuclei. This
misleading impression is often had when a one-dimensional interpretation of a
minimum lying between two maxima (see Figure 8.1 and the mountain landscape)
is wrongly extrapolated to three dimensions. One should not forget that for a
BCP to exist, all three components of the gradient of the electron density must
vanish, not just one component along a line connecting the two nuclei. A second
type of misconception arises when one presumes that a BCP is merely a reflection
of internuclear distance. Again, this is na¨
ıve: a BCP is not a docile signature
of internuclear distance. There are surprizing cases where two O···O distances,
identical up to three significant figures, appear in two different molecules where
one displays a BCP in between these two oxygens and the other molecule does not.
In fact, when surveying the many cases studied in this paper it becomes clear that
a BCP, by its presence or absence, is a highly informative yet compact signature of
the exchange energy between two given atoms. For the first time, numerical criteria
are proposed. The authors conclude that the existence of BCPs indeed depends on
the competition of what the 2007 paper calls ‘‘exchange channels.’’ The authors go
further and state that the controversies about the existence or not of BCPs ( …)are
not sufficient ( …) to undermine QTAIM’s foundations. Given these most interesting
and intriguing observations there is a need for more studies of this ilk.
A natural and important question is then: can QTAIM, through patterns in
VAB
xc values, extract a Lewis diagram from a given molecular wave function? A
very recent study, published [43] in 2013, set out to exactly answer this question,
and the answer is yes, so far. In that paper the interatomic exchange-correlation
energy VAB
xc was investigated exhaustively for all atom– atom interactions in 31
small covalent molecules (including ions) and three van der Waals complexes. For
the first time, clear clusters were revealed in the values of VAB
xc , clusters separated by
almost an order of magnitude (depending on the system at hand). This quantitative
information, justified by a precise and minimal physical picture of topological
energy partitioning, underpins the idea of a molecular graph. Such a full analysis
8.5 Energy Partitioning Revisited 29
54
6
34
117
222 385
54
Figure 8.12 A ball-and-stick diagram of diborane, B2H6, expanded with numerical values of
−VAB
xc in kJ mol−1.
of VAB
xc , for all atom– atom interactions in a system, reveals where to draw the lines
in a Lewis diagram.
A useful example illustrating this success is that of diborane, B2H6,whichat
one time was controversial in terms of its Lewis structure. Figure 8.12 shows
a ball-and-stick diagram of B2H6, endowed with −VAB
xc values (in kJ mol−1and
for HF/6-311G(d,p) wave functions). The two largest values (in terms of absolute
values) are 385 and 222 kJ mol−1, corresponding to the covalent bonds BHterm
and BHbridge, respectively. They provide the ‘‘sticks’’ of the full molecular graph
because it can be completely built from just these two types of B–H bonds. The
next strongest interaction is between the bridging hydrogens (117 kJ mol−1). This
relatively large value is compatible with diborane forming a H–H BCP upon
small geometrical distortions, rather than forming a B–B BCP (note that the B–B
interaction is more than three times weaker than the H–H interaction). Some one
versed in molecular theory may find this fact surprising.
In summary, VAB
xc displays a hierarchy of values, expressing both coarse-grained
and fine-grained information about the covalent electronic structure of a molecule,
without using molecular orbitals. The quantity VAB
xc is (still) expensive to compute
and numerical errors of a kilojoule per mole are common (due to the quadrature
of large energy values). However, the fundamental patterns that VAB
xc reveals are
beyond this numerical error. The quantity VAB
xc urgently needs to be computed
in larger molecules and in molecular complexes. The most important feature
of a VAB
xc analysis is that it does not impose a priori chemical views (e.g., steric
repulsion) onto the chemical bonding puzzle or controversy at hand, and that it
does not invoke a reference state or arbitrary parameters either. Unfortunately,
these two desirable features are absent in some recent studies [44– 48] that propose
interpretations of chemical bonding without using VAB
xc . Many of these studies
were vigorously refuted by Bader himself [49] and we will not repeat the arguments
and his counterarguments here. However, we briefly discuss a case study that
fully represents this fundamentally important debate on whether a particular
atom– atom interaction is a bond or not, namely that of H···H interactions in
30 8 The QTAIM Perspective of Chemical Bonding
aromatic hydrocarbons. We chose this example for three reasons: this case study is
prototypical, it is the oldest controversy of this kind, and a preliminary VAB
xc analysis
of this type of interaction exists.
The full narrative, up to 2004, on the debate on the meaning of H···H BCPs
(excluding the now widely accepted dihydrogen bond [50, 51]) is given in a book
chapter (Section 4.3 of Ref. [12])). Salient element of this story will be repeated
here. Between 1990 and 1992, a flurry of seven papers appeared on BPs between
unexpected atoms (H and H, O and O, N and O), ending with the last one
[52] on ortho-fluoro-substituted biphenyls. During the course of publishing this
flurry, the authors changed their mind about whether the BPs they observed
were indicative of bonds rather than ‘‘steric interactions.’’ The fourth paper in the
sequence [53] investigated kekulene, a highly symmetric polyaromatic consisting
of a hexagon of 12 fused benzene rings. Remarkable highly curved BPslinking
the six inner hydrogens were dismissed under the banner that they were unable to
attach any physical significance to these bonds. In the sixth paper [54], focusing on
‘‘hydrogen–hydrogen’’ nonbonding interactions, the authors, now more confident,
stated that the term BP should be reserved for the interaction lines describing ordinary
strong bonds. Here they varied the central torsion angle 𝜑(C2C1C1′C2′)inbiphenyl
to study its effect on the local topology of the interaction between the hydrogen
bonded to C2and the hydrogen bonded to C2′. The corresponding H–H distance
acts as a control variable determining if a BCP has appeared between these two
hydrogen atoms. A BCP appears in the planar transition state (𝜑=0◦), but not in the
local energy minimum, when 𝜑=45◦. Eventually, in the seventh and last paper, the
authors settle their conundrum in favor of an interpretation that projects traditional
chemical intuition onto their set of topological observations. They propose that when
the distance between two atoms is smaller than their contact interatomic separation,
a(…) BCP appears, indicating a nonbonding repulsive interaction. The authors
claimthisisarigorousdefinitionof sterically crowded molecules, superior to that
obtained from van der Waals radii. Then, in 1995, an eighth and final paper [55]
appeared on the matter, further elaborating steric crowding, now in perhalogenated
hydrocarbons.
In a single author paper [56] Bader completely rejected the notion of steric
repulsion as allegedly expressed by a BCP. He argued that the question is not how
the final geometry is attained in some mental process involving passage over a repulsive
barrier, a situation that is in fact common to most chemical changes, but rather how
the mechanics determines the final distribution of charge. In technical terms, Bader
based his rebuttal on the claim that the virial field and the electron density are
homeomorphic [57]. This homeomorphism is not mathematically proven and
only observed for only 15 molecules at that time. The second issue is that the
homeomorphism is not perfect because it has several exceptions. For example,
amongst others, the two fields behave differently in Li2and also in B2H6, where the
virial graph shows a path linking the two borons, which is absent in the electron
density. Because of the imperfection of the homeomorphism, and because the
virial field is judged [56] to have the ultimate authority to decide on bonds, one may
wonder why the electron density is still consulted on matters of bonding, in the first
8.5 Energy Partitioning Revisited 31
place. Nevertheless, the Bader group used exactly this angle of attack of the virial
graph to present [58] a hydrogen–hydrogen BCP as a mark of stabilizing interaction
in molecules and crystals. This type of bonding accounts for the existence of solid
hydrogen, and as it falls in the class of ‘‘van der Waals’’ interactions, no different
in kind, for example, from the intermolecular Cl–Cl bonding in solid chlorine
[59]. The authors point out that these H–H interactions are ubiquitous and that
they should not be confused with dihydrogen bonds (see above). According to their
approach there is no steric repulsion between the ortho-hydrogens in biphenyl;
rather the resultant H–H bonding contributes a stabilizing effect to the molecule’s
energy.
The debate continued when, in 2006, a paper [47] appeared that literally stated
in its title that Hydrogen-Hydrogen Bonding in Planar Biphenyl, Predicted by Atoms-
in-Molecules Theory, Does Not Exist. Of course, one could not be clearer than that,
which is why Bader immediately wrote a rebuttal paper [60] in the same year, with
an equally clear title stating Pauli Repulsions Exist Only in the Eye of the Beholder.The
essence of Bader’s rebuttal is indeed that the arguments presented in the attacking
paper are based on an arbitrary partitioning of the energy into contributions from
physically unrealizable states of the system. Bader invokes a statement [61] from
Morokuma, one of the first architects of energy decomposition analyses, his own
ostentatiously referred to as Energy Decomposition Analysis (EDA). Morokuma
himself wrote that There is no unique choice for the intermediate wave functions, and
they do not correspond to reality (e.g., ‘‘not all’’ satisfy the Pauli principle!.Inthe
many EDA schemes that followed, several variants of such imagined reference
states appeared. The IQA method, which was explained above, is of course also an
energy decomposition analysis, triggered by the pioneering calculation [37] of the
Coulomb energy between two topological atoms (see Eq. (8.3)). However, following
the principle of minimalism, IQA does not invoke any reference state at all, let
alone one that does not exist in Nature.
The seminal 2007 paper [29] on BPs as Privileged Exchange Channels,alsojoined
the debate on controversial H–H BCPs, providing new data, unlike Bader in his
2006 paper. Five systems were investigated, R–H···H–R (where R =H,Li,orCH
3),
cis-butadiene and biphenyl itself, all at Hartree– Fock level, which has proven to be
sufficient for the debate at hand. The distance between the two hydrogen atoms was
taken as a control parameter d, generating a number of geometries, and against
which the VHH
xc energies could be plotted. The profiles of all five systems turned
out to be remarkably similar. Hence, this observed universality puts the H···H
interaction in biphenyl on a par with the well-accepted attractive interaction that
H···HembodiesintheR–H···H–R systems. In summary, the H···H interactions
in biphenyl are not destabilizing (i.e., repulsive) but stabilizing.
One should note that the origin of the rotation barrier in biphenyl was not
determined in the 2007 paper because this would require a full IQA analysis,
which was lacking at the time. Indeed, many atomic self-energy and atom– atom
interaction energies change substantially between molecular configurations and
all have to be captured. The most important point to remember is that chemical
behavior emerges as a net result (i.e., sum) of many different large and often
32 8 The QTAIM Perspective of Chemical Bonding
opposing energy terms. At no point does IQA ascribe any chemical behavior (i.e.,
rotation barrier) to the interaction between just two atoms. Such ascription (i.e.,
isolation of fragment behavior and projection onto the whole) often happens in
Chemistry (for a case study criticizing the secondary interaction hypothesis [62]).
We note that IQA does not fall into the trap of singling out highly local interactions
and then projecting the overall behavior onto them. In summary, we can say that
the overall repulsion barrier in biphenyl is not due to a H···H BCP representing a
locally repulsive interaction.
Finally, a brief discussion of another attacking paper [44], written by a number of
German authors in 2009, is in order here. This case study on phenanthrene studied
experimentally (using Raman spectroscopy), gives the impression of settling the
matter on how to interpret a H···H BCP (this time appearing in the ‘‘bay region’’
of the molecule). This paper was vigorously disputed [49] by Bader, again in
a single-author paper, and again immediately, in the same year. The core of
his counterattack amounts to a perfectly valid point on the interpretation of the
‘‘interaction constant’’ (better called the coupling force constant)k, but unfortunately
it takes up to only one column of text. This main and powerful argument might
better have displaced much of the philosophy and ‘‘back-to-physics’’ outbursts that
wrapped this argument in the rest of the paper.
Moreover, it should be emphasized that the German authors invoked the
‘‘architecture’’ of the MM3 force field in order to separate molecular potential
energy into ‘‘chemically meaningful parts.’’ Of course, this is the wrong way
around. Indeed, popular force fields such as AMBER (or MM3 for that matter) have
little authority, if any, when it comes to partitioning energy in a physically rigorous
manner [63, 64]. The right way around is to construct a rigorous force field from
an actual quantum mechanical energy partitioning scheme. Pivotally, however, the
German authors used the arbitrary energy partitioning that underpins MM3 to
support their main argument of the repulsive nature of H···H interactions. There
is another and, quite frankly, better route however. If one is happy with the main
idea of topological partitioning, which is that of QCT, then one can trust IQA and
proceed with it. The minimal quantity VAB
xc that IQA offers provides a route to cut
the vicious circle of interpreting bonding with schemes that impose bonding. Is it
not safer to calculate a minimal and physically well-defined quantity and observe
the bonding pattern it reveals rather than perpetuate primitive chemical intuition,
by deciding a priori which interactions are bonds and which are not?
8.6
Conclusion
A relatively underexposed paper published in 1972 marked the birth of a completely
novel way of thinking about how to partition a quantum mechanical system such
as a molecule. This paved the way to what is often called the Quantum Theory of
Atoms in Molecule (QTAIM). After an incubation period, the key vision in this 1972
paper developed into a methodology that fully embraced the language of dynamical
References 33
systems as a vehicle to extract chemical information from wave functions. The idea
of using a (gradient) vector field to partition full space into subspaces was initially
applied to the electron density, leading to the concept of topological atoms. Stating
that these are the atoms of chemistry is perhaps based on a leap of faith, but then
one guided by beauty and minimalism.
The idea of ‘‘vector field partitioning’’ remains curiously confined to the field
of Chemistry. Instead, one would expect this idea to be more generally applied in
completely different knowledge fields, such as sociology or meteorology, given its
universality. Still, within chemistry, this partitioning method has been applied to
quantum functions other than the electron density, leading to the more general
and accurate name ‘‘Quantum Chemical Topology (QCT).’’ Moreover, in the area
of high-resolution X-ray crystallography, QCT is now mainstream.
An area of future QCT research is that of intermolecular interactions. Under-
standing and quantifying these interactions is crucial for the future of Chemistry,
which increasingly develops as a science of molecular assembly. This is where
the concept of the chemical bond needs to be scrutinized, widened, or even
(re)introduced, beyond the limitation of hydrogen bonds or the corset of covalent
interactions. QCT provides an appealing potential to tackle the question of detec-
tion, classification, and quantification of intermolecular interactions. The reason
is that understanding and defining bonding patterns, in the widest sense, must
ultimately be based on the atomic partitioning of energy. Topological atoms offer
a successful route to guide and execute this partitioning; let us not forget that they
deliver well-defined atomic kinetic energies. Moreover, they offer a visual picture
that will appeal to those willing to open their mind, and see chemistry as a ballet of
open and malleable boxes, with fluctuating shapes of amazing variety.
Acknowledgment
Mark Griffiths is thanked for preparing Figure 8.6 using in-house software [65– 67].
References
1. Bader, R.F.W. (1990) Atoms in
Molecules. A Quantum Theory, Oxford
University Press, Oxford.
2. Popelier, P.L.A. (2000) Atoms in
Molecules. An Introduction, Pearson
Education, London.
3. Bader, R.F.W. (1975) Molecular frag-
ments or chemical bonds? Acc. Chem.
Res.,8, 34– 40.
4. Bader, R.F.W. (1985) Atoms in
molecules. Acc. Chem. Res.,18,
9– 15.
5. Bader, R.F.W. (1991) A quantum-
theory of molecular-structure and its
applications. Chem. Rev.,91, 893– 928.
6. Popelier, P.L.A., Aicken, F.M., and
O’Brien, S.E. (2000) Chemical Mod-
elling: Applications and Theory, Royal
Socity of Chemistry.
7. Popelier, P.L.A. and Smith, P.J. (2002)
in Chemical Modelling: Applications
and Theory, Vol. 2: Royal Society of
Chemistry Specialist Periodical Report,
Chapter 8 (ed. A. Hinchliffe), Royal
34 8 The QTAIM Perspective of Chemical Bonding
Society of Chemistry, Cambridge, pp.
391– 448.
8. Bader, R.F.W. (2005) The quantum
mechanical basis of conceptual chem-
istry. Monatsh. Chem.,136, 819– 854.
9. Bader, R.F.W. and Popelier, P.L.A.
(1993) Atomic theorems. Int. J. Quan-
tum Chem.,45, 189– 207.
10. Matta, C.F. and Boyd, R.J. (2007) The
Quantum Theory of Atoms in Molecules.
From Solid State to DNA and Drug
Design, Wiley-VCH Verlag GmbH,
Weinheim.
11. Bader, R.F.W. and Matta, C.F.
(2013) Atoms in molecules as non-
overlapping, bounded, space-filling
open quantum systems. Found. Chem.,
15, 253– 276. doi: 10.1007/s10698-012-
9153-1
12. Popelier, P.L.A. (2005) in Structure and
Bonding. Intermolecular Forces and Clus-
ters, vol. 115 (ed. D.J. Wales), Springer,
Heidelberg, pp. 1– 56.
13. Silvi, B. and Savin, A. (1994) Classi-
fication of chemical bonds based on
topological analysis of electron localiza-
tion functions. Nature (London),371,
683– 686.
14. Becke, A.D. and Edgecombe, K.E.
(1990) A simple measure of electron
localization in atomic and molecular
systems. J. Chem. Phys.,92, 5397– 5403.
15. Popelier, P.L.A. and Aicken, F.M.
(2003) Atomic properties of amino
acids: computed atom types as a
guide for future force field design.
ChemPhysChem,4, 824– 829.
16. Devereux, M., Popelier, P.L.A.,
and McLay, I.M. (2009) Towards
an ab initio fragment database for
bio-isosterism: dependence of qct prop-
erties on level of theory, conformation
and chemical environment. J. Comput.
Chem.,30, 1300– 1318.
17. Popelier, P.L.A. and Br´
emond, ´
E.A.G.
(2009) Geometrically faithful home-
omorphisms between the electron
density and the bare nuclear poten-
tial. Int. J. Quantum Chem.,109,
2542– 2553.
18. Bader, R.F.W. and Beddall, P.M. (1971)
The spatial partitioning and transfer-
ability of molecular energies. Chem.
Phys. Lett.,8, 29– 36.
19. Bader, R.F.W., Beddall, P.M., and
Cade, P.E. (1971) Partitioning and
characterization of molecular charge
disitributions. J. Am. Chem. Soc.,93,
3095– 3107.
20. Bader, R.F.W. and Beddall, P.M. (1972)
Virial field relationship for molecular
charge distributions and the spatial
partitioning of molecular properties. J.
Chem. Phys.,56, 3320– 3329.
21. Nasertayoob, P. and Shahbazian, S.
(2008) Revisiting the foundations
of Quantum Theory of Atoms in
Molecules (QTAIM):the variational
procedure and the zero-flux condi-
tions. Int. J. Quantum Chem.,108,
1477– 1484.
22. Delle Site, L. (2002) Bader’s inter-
atomic surfaces are unique. Theor.
Chem. Acc.,107, 378– 380.
23. Bader, R.F.W. (2010) Definition of
molecular structure: by choice or by
appeal to observation? J. Phys. Chem. A,
114, 7431– 7444.
24. Blanco, M.A., Pendas, A.M., and
Francisco, E. (2005) Interacting
quantum atoms: a correlated energy
decomposition scheme based on the
quantum theory of atoms in molecules.
J. Chem. Theor. Comp.,1, 1096– 1109.
25. Srebrenik, S. and Bader, R.F.W. (1975)
Towards the development of the quan-
tum mechanics of a subspace. J. Chem.
Phys.,63, 3945– 3961.
26. Koritsanszky, T.S. and Coppens, P.
(2001) Chemical applications of x-ray
charge-density analysis. Chem. Rev.,
1583– 1627.
27. Bader, R.F.W., Carroll, M.T.,
Cheeseman, J.R., and Chang,
C. (1987) Properties of atoms in
molecules – atomic volumes. J. Am.
Chem. Soc.,109, 7968– 7979.
28. Runtz, G.R., Bader, R.F.W., and
Messer, R.R. (1977) Definition of bond
paths and bond directions in terms of
the molecular charge distribution. Can.
J. Chem.,55, 3040– 3045.
29. Mart´
ın-Pend´
as, A., Francisco, E.,
Blanco, M.A., and Gatti, C. (2007)
Bond paths as privileged exchange
channels. Chem. Eur. J.,13, 9362– 9371.
30. Macchi, P., Proserpio, D.M., and
Sironi, A. (1998) Experimental electron
References 35
density in a transition metal dimer:
metal- metal and metal-ligand bonds. J.
Am. Chem. Soc.,120, 13429– 13435.
31. Abramov, Y.A. (1997) On the possibility
of kinetic energy density evaluation
from the experimental electron density
distribution. Acta Crystallogr.,A53,
264– 272.
32. Popelier, P.L.A. (1999) Quantum
molecular similarity. 1. BCP space. J.
Phys. Chem. A,103, 2883– 2890.
33. Popelier, P.L.A. (2010) in Quantum
Biochemistry: Electronic Structure and
Biological Activity,vol.PartIV(ed.C.F.
Matta), Wiley-VCH Verlag GmbH,
Weinheim, pp. 669– 692.
34. Biegler-Koenig, F.W., Bader, R.F.W.,
and Tang, T.H. (1982) Calculation of
the average properties of atoms in
molecules. 2. J. Comput. Chem.,3,
317– 328.
35. Popelier, P.L.A. (2011) Fully analyt-
ical integration over the 3D volume
bounded by the 𝛽sphere in topo-
logical atoms. J. Phys. Chem. A,115,
13169– 13179.
36. Polestshuk, P.M. (2013) Accurate inte-
gration over atomic regions bounded
by zero-flux surfaces. J. Comput. Chem.,
34, 206– 219.
37. Popelier, P.L.A. and Kosov, D.S. (2001)
Atom-atom partitioning of intramolec-
ular and intermolecular Coulomb
energy. J. Chem. Phys.,114, 6539– 6547.
38. Rafat, M. and Popelier, P.L.A. (2007)
Atom-atom partitioning of total
(super)molecular energy: the hid-
den terms of classical force fields. J.
Comput. Chem.,28, 292– 301.
39. Rafat, M. and Popelier, P.L.A. (2007) in
Quantum Theory of Atoms in Molecules,
vol. 5 (eds C.F. Matta and R.J. Boyd),
Wiley-VCH Verlag GmbH, Weinheim,
pp. 121– 140.
40. Martin-Pendas, A., Blanco, M.A., and
Fransisco, E. (2003) Two-electron inte-
grations in the quantum theory of
atoms in molecules. J. Chem. Phys.,
120, 4581– 4592.
41. Bader, R.F.W., Nguyen-Dang, T.T., and
Tal, Y. (1981) A topological theory of
molecular-structure. Rep. Prog. Phys.,
44, 893– 948.
42. Tognetti, V. and Joubert, L. (2013) On
the physical role of exchange in the for-
mation of an intramolecular bond path
between two electronegative atoms. J.
Chem. Phys.,138, 024102.
43. Garcia-Revilla, M., Francisco, E.,
Popelier, P.L.A., and Martin-Pendas,
A.M. (2013) Domain-averaged exchange
correlation energies as a physical
underpinning for chemical graphs.
ChemPhysChem,14, 1211– 1218.
44. Grimme, S., M¨
uck-Lichtenfeld, C.,
Erker, G., Kehr, G., Wang, H., Beckers,
H., and Willner, H. (2009) When do
interacting atoms form a chemical
bond? Spectroscopic measurements
and theoretical analyses of dideuterio-
phenanthrene. Angew. Chem. Int. Ed.,
48, 2592– 2595.
45. Cerpa, E., Krapp, A., Vela, A., and
Merino, G. (2008) The implications
of symmetry of the external poten-
tial on bond paths. Chem. Eur. J.,14,
10232– 10234.
46. Cerpa, E., Krapp, A., Flores-Moreno,
R., Donald, K.J., and Merino, G. (2009)
Influence of endohedral confinement
on the electronic interaction between
He atoms: A He2@C20H20 case study.
Chem. Eur. J.,15, 1985– 1990.
47. Poater, J., Sola, M., and Bickelhaupt,
F.M. (2006) Hydrogen–Hydrogen bond-
ing in planar biphenyl, predicted by
atoms-in-molecules theory, does not
exist. Chem. Eur. J.,12, 2889– 2895.
48. Haaland, A., Shorokhov, D.J., and
Tverdova, N.V. (2004) Topological
analysis of electron densities: is the
presence of an atomic interaction line
in an equilibrium geometry a suffi-
cient condition for the existence of
a chemical bond? Chem. Eur. J.,10,
4416– 4421.
49. Bader, R.F.W. (2009) Bond paths are
not chemical bonds. J. Phys. Chem. A,
113, 10391– 10396.
50. Richardson, T.B., de Gala, S.,
Crabtree, R.H., and Siegbahn,
P.E.M. (1995) Unconventional
hydrogen bonds: intermolecular
B-H.cntdot..cntdot..cntdot.H-N
interactions. J. Am. Chem. Soc.,117,
12875– 12876.
36 8 The QTAIM Perspective of Chemical Bonding
51. Popelier, P.L.A. (1998) Characterization
of a dihydrogen bond on the basis of
the electron density. J. Phys. Chem. A,
102, 1873– 1878.
52. Cioslowski, J. and Mixon, S.T. (1992)
Topological properties of electron den-
sity in search of steric interactions in
molecules: electronic structure calcula-
tions in ortho-substituted biphenyls. J.
Am. Chem. Soc.,114, 4382– 4387.
53. Cioslowski, J., O’Connor, P.B., and
Fleischmann, E.D. (1991) Is superben-
zene superaromatic? J. Am. Chem. Soc.,
113, 1086– 1089.
54. Cioslowski, J. and Mixon, S.T. (1992)
Universality among topological prop-
erties of electron density associated
with the hydrogen-hydrogen nonbond-
ing interactions. Can. J. Chem.,70,
443– 449.
55. Cioslowski, J., Edgington, L., and
Stefanov, B.B. (1995) Steric overcrowd-
ing in perhalogenated cyclohexanes,
dodecahedranes, and [60]fulleranes. J.
Am. Chem. Soc.,117, 10381– 10384.
56. Bader, R.F.W. (1998) A bond path:
a universal indicator of bonded
interactions. J. Phys. Chem. A,102,
7314– 7323.
57. Keith, T.A., Bader, R.F.W., and Aray,
Y. (1996) Structural homeomorphism
between the electron density and the
virial field. Int. J. Quantum Chem.,57,
183– 198.
58. Matta, C.F., Hernandez-Trujillo, J.,
Tang, T.-H., and Bader, R.F.W. (2003)
Hydrogen-hydrogen bonding: a sta-
bilizing interaction in molecules and
crystals. Chem. Eur. J.,9, 1940– 1951.
59. Tsirelson, V.G., Zou, P.F., Tang,
T.H., and Bader, R.F.W. (1995)
Topological definition of crystal-
structure – determination of the
bonded interactions in solid molec-
ular chlorine. Acta Crystallogr. A,51,
143– 153.
60. Bader, R.F.W. (2006) Pauli repulsions
exist only in the eye of the beholder.
Chem. Eur. J.,12, 2896– 2901.
61. Morokuma, K. (1971) Molecular
orbital studies of hydrogen bonds.
III C=O…H-O hydrogen bond in
H2CO …H2OandH
2CO …2H2O. J.
Chem. Phys.,55, 1236– 1244.
62. Popelier, P.L.A. and Joubert, L. (2002)
The elusive atomic rationale for DNA
base pair stability. J. Am. Chem. Soc.,
124, 8725– 8729.
63. Popelier, P.L.A. (2012) in Modern
Charge-Density Analysis,vol.14(eds
C. Gatti and P. Macchi), Springer, pp.
505– 526.
64. Popelier, P.L.A. (2012) Quantum chem-
ical topology: knowledgeable atoms
in peptides. AIP Conf. Proc.,1456,
261– 268.
65. Popelier, P.L.A. (1996) MORPHY, a
program for an automated ‘‘atoms in
molecules’’ analysis. Comput. Phys.
Commun.,93, 212– 240.
66. Rafat, M., Devereux, M., and Popelier,
P.L.A. (2005) Rendering of quantum
topological atoms and bonds. J. Mol.
Graphisc Modell.,24, 111– 120.
67. Rafat, M. and Popelier, P.L.A. (2007)
Visualisation and integration of quan-
tum topological atoms by spatial
discretisation into finite elements.
J. Comput. Chem.,28, 2602– 2617.
68. Bader, R.F.W., MacDougall, P.J., and
Lau, C.D.H. (1984) Bonded and non-
bonded charge concentrations and
their relation to molecular-geometry
and reactivity. J. Am. Chem. Soc.,106,
1594– 1605.
69. Bader, R.F.W., Gillespie, R.J., and
MacDougall, P.J. (1988) A physical
basis for the VSEPR model of molec-
ular geometry. J. Am. Chem. Soc.,110,
7329– 7336.
70. Popelier, P.L.A. (2000) On the full
topology of the laplacian of the elec-
tron density. Coord. Chem. Rev.,197,
169– 189.
71. Malcolm, N.O.J. and Popelier, P.L.A.
(2003) The full topology of the
Laplacian of the electron density:
scrutinising a physical basis for the
VSEPR model. Faraday Discuss.,124,
353– 363.
72. Malcolm, N.O.J. and Popelier, P.L.A.
(2003) An improved algorithm to locate
critical points in a 3D scalar field as
implemented in the program MOR-
PHY. J. Comput. Chem.,24, 437– 442.
73. Tal, Y., Bader, R.F.W., and Erkku, J.
(1980) Structural homeomorphism
between the electron density and
References 37
the nuclear potential of a molecular
system. Phys. Rev. A,21, 1– 11.
74. Polo, V., Andres, J., Berski, S.,
Domingo, L.R., and Silvi, B. (2008)
Understanding reaction mechanisms
in organic chemistry from catastrophe
theory applied to the electron localiza-
tion function topology. J. Phys. Chem.
A,112, 7128– 7136.
75. Naray-Szabo, G. and Ferenczy, G.G.
(1995) Molecular electrostatics. Chem.
Rev.,95, 829– 847.
76. Gadre, S.R., Kulkarni, S.A., and
Shrivastava, I.H. (1992) Molecular
electrostatic potentials: a topographical
study. J. Chem. Phys.,96, 5253– 5261.
77. Gadre, S.R. and Shrivastava, I.H.
(1991) Shapes and sizes of molecular
aniosn via topographical analysis of
electrostatic potential. J. Chem. Phys.,
94, 4384– 4391.
78. Balanarayan, P. and Gadre, S.R. (2003)
Topography of molecular scalar fields.
I. Algorithm and Poincare-Hopf rela-
tion. J. Chem. Phys.,119, 5037– 5043.
79. Balanarayan, P., Kavathekar, R.,
and Gadre, S.R. (2007) Electrostatic
potential topography for exploring elec-
tronic reorganizations in 1,3 dipolar
cycloadditions. J. Phys. Chem. A,111,
2733– 2738.
80. Aray, Y., Rodriguez, J., Coll, S.,
Rodr´
ıguez-Arias, E.N., and Vega, D.
(2005) Nature of the lewis acid sites on
molybdenum and ruthenium sulfides:
an electrostatic potential study. J. Phys.
Chem. B,109, 23564– 23570.
81. Tsirelson, V.G., Avilov, A.S., Lepeshov,
G.G., Kulygin, A.K., Stahn, J., Pietsch,
U., and Spence, J.C.H. (2001) Quan-
titative analysis of the electrostatic
potential in rock-salt crystals using
accurate electron diffraction data. J.
Phys. Chem. B,105, 5068– 5074.
82. Keith, T.A. and Bader, R.F.W. (1993)
Topological analysis of magnetically
induced molecular current distribu-
tions. J. Chem. Phys.,99, 3669– 3682.
83. Cioslowski, J. and Liu, G.H. (1999)
Topology of electron-electron interac-
tions in atoms and molecules. II. The
correlation cage. J. Chem. Phys.,110,
1882– 1887.
84. Pendas, A.M. and Hernandez-Trujillo,
J. (2012) The Ehrenfest force field:
topology and consequences for the
definitionofanatominamolecule.J.
Chem. Phys.,137, 134101.
85. Pendas, A.M., Francisco, E., Blanco,
M.A., and Gatti, C. (2007) Bond paths
as privileged exchange channels. Chem.
A Eur. J.,13, 9362– 9371.
86. Webster, B. (1990) Chemical Bonding
Theory, Blackwell, Oxford.
87. Reed, A.E., Curtiss, L.A., and
Weinhold, F. (1988) Intermolecu-
lar interactions from a natural bond
orbital, donor-acceptor viewpoint.
Chem. Rev.,88, 899– 926.
88. Stone, A.J. (1981) Distributed multipole
analysis, or how to describe a molec-
ular charge-distribution. Chem. Phys.
Lett.,83, 233– 239.
89. Mulliken, R.S. (1955) Electronic popu-
lation analysis on LCAO-MO molecular
wave functions. I. J. Chem. Phys.,23,
1833– 1840.
90. Pearson, R.G. (2007) Applying the con-
cepts of density functional theory to
simple systems. Int. J. Quantum Chem.,
108, 821– 826.
91. Kovacs, A., Esterhuysen, C., and
Frenking, G. (2005) The nature
of the chemical bond revisited: an
energy-partitioning analysis of nonpolar
bonds. Chem. Eur. J.,11, 1813– 1825.
92. McWeeny, R. (1992) Methods of Molec-
ular Quantum Mechanics, 2nd edn,
Academic Press, SanDiego, CA.
93. Cohen, L. (1978) Local kinetic energy
in quantum mechanics. J. Chem. Phys.,
70, 788– 799.
94. Bader, R.F.W. and Preston, H.J.T.
(1969) The kinetic energy of molecular
charge distributions and molecular
stability. Int. J. Quantum Chem.,3,
327– 347.
95. Popelier, P.L.A. (1994) An analytical
expression for interatomic surfaces
in the theory of atoms in molecules.
Theor. Chim. Acta,87, 465– 476.
96. Popelier, P.L.A. (1996) On the differen-
tial geometry of interatomic surfaces.
Can. J. Chem.,74, 829– 838.
97. Bader, R.F.W. and Essen, H. (1984)
The characterization of atomic interac-
tions. J. Chem. Phys.,80, 1943– 1960.
38 8 The QTAIM Perspective of Chemical Bonding
98. Cremer, D. and Kraka, E. (1984)
Chemical bonds without bonding
electron density - does the difference
electron-density analysis suffice for
a description of the chemical bond?
Angew. Chem.,23, 627– 628.
99. Gillespie, R.J. (1972) Molecular
Geometry, Van Nostrand Reinhold,
London.
100. Gillespie, R.J. (1998) The physical basis
of the VSEPR model. Struct. Chem.,9,
73– 76.
101. Bader, R.F.W. and Stephens, M.E.
(1975) Spatial localization of the elec-
tronic pair and number distributions
in molecules. J. Am. Chem. Soc.,97,
7391– 7399.
102. Bader, R.F.W. and MacDougall,
P.J. (1985) Toward a theory of
chemical-reactivity based on the
charge-density. J. Am. Chem. Soc.,107,
6788– 6795.
103. Bader, R.F.W., Popelier, P.L.A., and
Chang, C. (1992) Similarity and com-
plementarity in chemistry. J. Mol.
Struct. (THEOCHEM),255, 145– 171.
104. Malcolm, N.O.J. and Popelier, P.L.A.
(2001) On the full topology of the
Laplacian of the electron density II:
umbrella inversion of the ammo-
nia molecule. J. Phys. Chem. A,105,
7638– 7645.
105. Popelier, P.L.A., Burke, J., and
Malcolm, N.O.J. (2003) Functional
groups expressed as graphs extracted
from the laplacian of the electron
density. Int. J. Quantum Chem.,92,
326– 336.
106. Malcolm, N.O.J. and Popelier, P.L.A.
(2003) An algorithm to delineate and
integrate topological basins in a three-
dimensional quantum mechanical
density function. J. Comput. Chem.,24,
1276– 1282.
