Atoms in molecules

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3
Atoms in Molecules
BY P. L. A. POPELIER, F. M. AICKEN AND S. E. O'BRIEN
1 Introduction
1.1 What Is AIM? ± The theory of ``Atoms in Molecules'' (AIM) is an
interpretative theory which aims to recover chemical insight from modern high-
resolution electron densities.
1
These densities may be of experimental origin or
derived from ab initio wave functions. AIM de®nes two important cornerstones
of chemistry: the atom and the bond. There is a need for such a theory in view of
the widening gap between chemical insight and currently accepted and taught,
on one hand, and the vast ever-growing body of (high-resolution) crystal-
lographic and ab initio data, on the other. Indeed, most chemists still think in
terms of the Lewis model of the 1900s (e.g. octet rule), the Heitler-London-
Pauling Valence Bond model of the 1930s (e.g. resonance), or the Hund-
Mulliken Molecular Orbital of the 1960s (e.g. Mulliken charges). Of course
many of these early concepts have been scrutinised and their limitations are well
documented but a complete, coherent and consistent theory to bridge the gap
between modern solutions of the Schro
Èdinger equation and chemical insight is
still elusive. However, an excellent candidate to ful®l that purpose is AIM. This
theory is often mistaken to be another atomic population analysis, rather than an
extensive and profound theory rooted in quantum mechanics.
2
Being a novel
paradigm
3
it has gained slow acceptance although it has been incorporated as a
vital part of a recent textbook on the chemical bond
4
aimed at undergraduates.
The theoretical community has focused most of its attention on the energy
and its derivatives with respect to nuclear motion, i.e. forces, force constants,
etc. If one accepts that eigenvalues and eigenfunctions are on a par as solutions
of an eigenvalue problem such as the Schro
Èdinger equation, then why is it that
the electron density does not enjoy the same status as the energy? After all the
electron density ris immediately derived from the wave function, which is an
eigenfunction, and the energy is in fact an eigenvalue. This imbalanced view is
corrected by the development and application of AIM, a theory that recognises
and reveals the wealth of information hidden in the electron density and its
derived functions.
Chemical Modelling: Applications and Theory, Volume 1
#The Royal Society of Chemistry, 2000

From a super®cial point of view AIM seems to consist of two sub-theories:
one based on the topology of rand the other on the topology of the Laplacian
of r, denoted as H
2
r. The latter theory was developed in an attempt to analyze
electron pair localisation and takes up about one ®fth of Bader's 1990
monograph summarising AIM.
5
The impression that AIM encompasses two
separate parts is a consequence of historical developments and its present
incomplete state. Indeed, if one views AIM as an interpretative theory which
aims to retrieve chemical insight from modern ab initio wave functions then
both parts of AIM could be further integrated. For example, a systematic study
of the gradient vector ®eld of H
2
r, at the same depth as that of ris technically
possible but has not been carried out yet. Physical quantities could then be
integrated over basins in H
2
r. In the longer term AIM might be expanded into a
complete topological theory of many if not all chemically relevant scalar ®elds.
For example, any of the following ®elds, r(r), H
2
r(r), the kinetic energy densities
K(r) and G(r), might be integrated over basins of another ®eld, or any scalar
®eld evaluated in a critical point of another ®eld. It is often tempting to invoke
molecular orbitals (MO) in the extraction of chemical information from ab initio
calculations, such as in an AIM inspired de®nition of bond order,
6
but AIM
traditionally strives at orbital-free interpretations. Ideally, any de®nition of a
chemical concept should exist for any representation of the electron density,
whether given on a grid, in terms of a basis set expansion or even from
experimental measurement.
Currently AIM is used in many dierent areas of research, many of which we
now sum up with examples of references relating to that area. AIM studies have
been made in relation to high-resolution crystallography,
7±10
surface science,
11
solid state physics,
12
biological chemistry,
13
organometallic chemistry,
14
noble
gas chemistry,
15
physical organic chemistry,
16
transition metal complex
chemistry,
17
boron chemistry,
18
lithium chemistry
19
and other areas. Clearly,
AIM has been applied in a wide range of areas of chemical research but to our
knowledge applications in supramolecular chemistry are currently lacking.
The theory of AIM has been helpful in the re-interpretation of many concepts
such as bond energy,
20
hydrogen bonding,
21
the VSEPR model,
22
the Ligand-
Close-Packing (LCP) model,
23
nucleophilic addition,
24
atomic volume,
25
strain
energy,
26
chemical reactivity,
27
acid and base promoted hydrolysis,
28
polaris-
abilities,
29
molecular similarity,
30,31
atomic charges,
32
semi-empirical valence
electron distributions,
33
non-bonded charge concentrations (lone-pairs),
34
(hyper)conjugation,
35
the characterisation of atomic interactions,
36
F-
centres,
37
electron delocalisation,
38
aromaticity,
39
origin of molecular dipole
moments,
40
structural stability,
41
magnetic properties
42
and others. Many of the
aforementioned concepts have been rigorously rede®ned within the context of
AIM, or the theory has at least shed new light on the current mainstream
de®nitions and posed new challenges.
1.2 Scope. ± The current contribution reports on the literature on ``Atoms in
Molecules'' in one year from June 1998 to June 1999. Since this is the ®rst time
that AIM is discussed in a Specialist Periodical Report we cannot refer to
144 Chemical Modelling: Applications and Theory, Volume 1

previous reports covering the period from the inception of the theory (1972)
until June 1998. Instead we oer a brief historical perspective, highlighting
AIM's roots, and treat the development of AIM over the last three decades via
references to review articles and selected key research papers.
A measure of the impact of AIM is the number of references to Bader's 1990
book, which expounds the theory and applications of AIM.
5
At the time of
writing we found 906 references in the primary literature to this seminal book
since it was published. The distribution of these references on a yearly basis is
given in Figure 1. Except for the curious reduction in 1995 the number of
references has grown steadily, perhaps asymptotically. It should be mentioned
that a substantial number of entries are referring to AIM only in a tangential
way, i.e. in connection with the topology of a scalar ®eld. For example, one
paper
43
referred to Bader's book only because it presented CPs in the
electrostatic potential V(r)(i.e. where HV(r)=0) without making use of AIM
or discussing any of its applications. Of course in this review we have selected
only those papers that make restricted or extensive use of AIM, either at the
application or at the development side (see disclaimer, Section 9). This
selection encompasses 90 papers produced by more than 60 dierent groups
world-wide.
The dilemma one faces in writing a report of this kind is whether to categorise
the papers in terms of chemical classes and activity (e.g. organic, inorganic,
biological, surface science, etc.), or methods, concepts and techniques (e.g.
algorithms, NMR, X-ray, H-bonds, reaction paths, etc.). We have opted to give
priority to method and concepts and have grouped papers accordingly. For
Figure 1 A histogram of the number of literature references per year to Bader's book
``Atoms in Molecules. A Quantum Theory'' (1990)
3: Atoms in Molecules 145

example, a paper which focuses on biochemical compounds, but with emphasis
on hydrogen bonding, will be grouped under the section hydrogen bonding
(Section 6) ®rst and then under the subsection biochemical compounds (Sub-
section 6.9). Also, within each section, priority has been given to concepts and
methods. For example, in the section ``Chemical Bonding'' (Section 5),
compound-oriented subsections only appear after ``5.1 Theory'', ``5.2 LCP'',
``5.3 Hypervalency''. Although there is a degree of arbitrariness in this division,
it is necessary to structure papers in a clear way in view of their mixed and
overlapping content.
1.3 The Roots of AIM. ± The theory of AIM was pioneered by the Bader group
at McMaster University (Canada) in the early seventies. This theory emerged
from studies that interpreted the electron densities of early computer-generated
ab initio wave functions as they became available in the sixties, e.g. from the
theoretical chemistry group in Chicago. Early work of the Bader group in the
sixties focused on the Hellmann-Feynman theorem, in particular its use in
calculating molecular energies
44
and understand chemical binding.
45
Detailed
analyses of the electron density distributions in simple molecules such as
water,
46
ammonia
47
and hydrogen ¯uoride
48
were the subject of following
work. Subsequent studies addressed the forces operative in homonuclear
diatomic molecules
49
and the nature of the chemical bond,
50
in particular the
ionic bond.
51
Further contributions concentrated on the relaxation of the
molecular electron distribution and the vibrational force constant.
52,53
In
summary, in the sixties, prior to the actual development of AIM, the Bader
group was predominately preoccupied with the intricate relationship between
the electron density and the energy and its derivatives with respect to nuclear
coordinates. This relationship was certainly an important theme in that decade
as made evident by research by Hohenberg, Kohn and others culminating in the
Hohenberg-Kohn theorems published in 1964.
54
Prior to the actual paper marking the birth of AIM
55
Bader, Beddall and
Cade published a proposal for the spatial partitioning of a molecular electron
density.
56,57
This method was restricted to linear molecules and partitioned
them via planes perpendicular to the molecular axis. The partitioning planes
were positioned at a point where the electron density reached a minimum along
the internuclear axis. An immediate concern arising in the construction of the
early atomic fragments was that they were not virial fragments.
57
It was known to Bader in 1969 that there are (at least) two types of kinetic
energy densities,K(r) and G(r), which yield dierent kinetic energies
58
when
integrated over arbitrary portions of space. These kinetic energies become
identical when the respective energy densities are integrated over whole space
and over special portions of space called virial fragments. It was only in 1972
that a spatial partitioning of molecular electron distributions was published
55
that did produce virial fragments. This spearheaded the development of the
theory of AIM. Although AIM is often described as the topological analysis of
the electron distribution it is important to realise that its roots lie in a primary
concern to obtain atomic fragments with a well-de®ned energy. In other words,
146 Chemical Modelling: Applications and Theory, Volume 1

the emergence of AIM was driven by the quest for quantum mechanically
meaningful fragments in molecules rather than by the discovery of topological
objects in the electron density, such as the bond critical point (BCP) and the
interatomic surface (IAS).
In 1974 Daudel, Bader and co-workers published
59
a paper on the electron
pair in chemistry. They presented calculations that show that the most probable
partitioning of a system is the one that localises pairs of electrons in well-de®ned
spatial regions or loges. Although loge-partitioning never became part of the
AIM theory its in¯uence led to an extensive study of the pair density and the
Fermi hole density.
60
About a decade later the electron pair localisation
problem was revisited using the Laplacian of r, a simple but information-rich
function, which would provide a physical basis
22
for the VSEPR theory.
61
1.4 The Development of AIM. ± Readers uninitiated to AIM may wish to
consult a didactic and introductory work to the theory
62
because it is beyond the
scope of this report to review AIM's development comprehensively. A recent
introductory text by Bader can be found in the Encyclopedia of Computational
Chemistry.
63
For a formal ab ovo treatment of the theory two more advanced
and classic reviews
64,65
are recommended, of which the more recent one
64
puts
emphasis on applications. An early and informal review
66
was written in 1985
predated by a decade-older thought-provoking review
67
airing the young but
important AIM results on the de®nition of an atom and a bond. Rather than
exhaustively reviewing all contributions to AIM up to the current literature
survey we merely summarise the chronological development of AIM as it
reached a mature stage in the early eighties.
We pick up the thread of the previous section at the point where the Bader
group realised how to de®ne a chemically appealing and conceptually economic
virial fragment in a molecule. In 1973 the virial partitioning method was applied
to second-row diatomic hydrides and a year later to third-row hydrides.
68
In the
following ®ve years attention was mainly directed towards the quantum
mechanical justi®cation of the proposed virial fragments or AIM atoms. The
®rst publication relating to this question discussed sucient conditions for
fragment virial theorems
69
and for an alternative partitioning method intro-
duced by Parr and co-workers.
70
A subsequent paper in this series of contribu-
tions focused on the development of the quantum mechanics of a subspace, with
particular emphasis on a variational treatment.
71
The following doublet of
papers deepened the formulation of subspace mechanics. In the ®rst paper
Srebrenik and co-workers
72
obtained a variational solution of the Schro
Èdinger
equation with the zero-¯ux surface requirement serving as a variational
constraint. Using a variational statement of a time-dependent hypervirial
theorem they described the time dependence of subspace averaged properties.
In the second paper the authors
73
introduced Schwinger's quantum action
principle to obtain a quantum mechanical description of a subspace and its
properties. In this paper the subspace quantum mechanics governing AIM is
rederived using a further level of abstraction introduced by Schwinger for
ordinary full space quantum mechanics.
74
3: Atoms in Molecules 147

Meanwhile the Bader group published a paper
75
which de®ned bond paths in
terms of molecular electron densities. These considerations, the formulation of
subspace quantum mechanics and the ensuing construction of a theory of
molecular structure in terms of the topology of rculminated in an impressive
quadruplet of papers entitled ``Quantum topology of molecular charge distri-
butions''. In the ®rst paper
76
the primary concepts of descriptive chemistry, of
an atom in a molecule with its own set of properties, and of a chemical bond
were introduced. In the second paper,
77
Thom's catastrophe theory
78
was
applied to molecular structure change and stability, while the third paper
79
summarised and streamlined the mechanics of an atom in a molecule, work that
would later be extended to include a variety of atomic theorems.
80
The fourth
and ®nal paper in this series discussed the relation between the topological and
energetic stabilities of molecular structure, de®ning the structural diagram and
the two mechanisms of molecular structure change (con¯ict and bifurcation).
81
The then established topological theory of molecular structure was sub-
sequently the subject of two reviews,
65,82
one
82
of which shared Feynman's
enthusiasm for the importance of the atomic hypothesis.
The Laplacian of the electron density, which forms an integral part of AIM,
has been the subject of a rather independent line of development. It became
clear in the mid seventies that localised pairs of electrons are not evident in the
topological properties of rand the pair density does not, in general, de®ne
regions of space beyond the atomic core.
60
However, it turned out that there is
remarkable but not perfectly faithful mapping between the CPs in H
2
rand the
electron pairs of the Lewis model. The Laplacian is a simple function with
known shortcomings,
83,84
but has been successful as a physical basis
22,85
for the
VSEPR model of Nyholm and Gillespie.
61,86
Functions more sophisticated than
H
2
rhave been constructed to study electron pair localisation, for example
Becke's electron localisation function (ELF)
83
and the Lennard-Jones func-
tion.
87
The Laplacian turns out to be homeomorphic in many cases to ELF
88
and performs well in the characterisation of bonding and non-bonding (or
``lone'') electron pairs for reasons that are not properly understood.
The Laplacian continues to be used in the characterisation of molecules (e.g.
metallocenes,
89
disiloxanes
90
), in particular in connection with exceptions to the
VSEPR model,
91,92
but its scope is wider. The Laplacian is proportional to an
energy density (or pressure) expressing the balance between the kinetic energy
density G(r) and the potential energy density V(r).
36
Furthermore a Laplacian
complementarity principle has been formulated,
93
which, in a way, parallels the
Lewis complementarity de®ned in the context of Lewis's generalisation of acidity
and basicity and their neutralisation. A few dramatic examples of this principle
appear in the prediction of the angle of nucleophilic attack to carbonyl carbons
from crystallographic data,
94
the angle of attack in Michael addition compared
with potential energy surface calculations
95
and the reasonable prediction of a
large number of hydrogen-bonded complexes involving HF.
96
Finally, under
some conditions the Laplacian is able to point out the preferred sites of
protonation, demonstrating its usefulness in both electrophilic and nucleophilic
attack. It was shown that the preferred orientation (cis or trans) of the proton
148 Chemical Modelling: Applications and Theory, Volume 1

with respect to the substituent in a dozen carbonyl compounds (XCHO) is
generally indicated by the relative magnitude of maxima in the Valence Shell
Charge Concentration (VSCC) at the positions of the lone pairs of the oxygen
atom. In summary, the Laplacian is a simple function but rich in information.
1.5 Software. ± Over many years the Bader group has produced a suite of
programs called AIMPAC, which is freely available via http://www/chemistry.
mcmaster.ca/aimpac. The most complex algorithms appear in the atomic
integration, which where designed by the applied mathematician Biegler-Ko
Ènig
et al. in the early eighties. The ®rst algorithm
97
used the elegant idea of
formulating the atomic integration in natural coordinates introducing a com-
plicated Jacobian. Unfortunately this algorithm proved to be inappropriate
because of the highly curved gradient paths near BCPs. As a result the
concomitant program OMEGA became obsolete and was replaced by the more
successful program PROAIM,
98
which approximates the interatomic surfaces
by triangulation. AIMPAC has been modi®ed and streamlined on several
occasions during the nineties, by Laidig, Keith, Popelier, Heard and others.
An independent program to perform a topological analysis of rand its
Laplacian has been developed by Popelier since 1991, called MORPHY. In
contradistinction to AIMPAC this program is a single entity, not a suite,
entirely written by one person except for a few subroutines. The ®rst version
called MORPHY1.0
99
was released in 1996 and deposited in the CPC Program
Library, Queen's University of Belfast, N. Ireland, and the QCPE Library in
Bloomington, Indiana, USA. The more recent version, called MORPHY98,
is also able to perform atomic integration according to two new complemen-
tary but integrated algorithms. One algorithm
100
employs an analytical
expression
101
for the interatomic surfaces and the other
102
determines the
intersection of an integration ray without explicit knowledge of the interatomic
surface. This program also contains a robust and highly automated but ¯exible
CP localiser based on the eigenvector following method.
103
MORPHY98 is
available for a fee from http://www.ch.umist.ac.uk/morphy. Its features and
bene®ts, as well as manual, can be inspected from this web site. A free demo
version with full functionality but restricted dimension is freely available from
this web site without time limitation.
Finally, since 1994 the commercial program GAUSSIAN provides an AIM
option based on work by the Ciosøowski group.
104
This program is available via
http://www.gaussian.com. In our experience the GAUSSIAN94 atomic integra-
tion subroutines can provide erroneous results (or even crash), an uncritical
interpretation of which may unjustly damage the trustworthiness of AIM.
However, important bugs seem to have been corrected in GAUSSIAN98.
2 Theoretical
2.1 Open Systems. ± In his Polanyi Award Lecture Bader
105
asks why there are
atoms in chemistry. He states that Dalton's bold assumption that atoms
3: Atoms in Molecules 149

retained their mass and identity in chemical combination had to await its
justi®cation by Rutherford's nuclear model of the atom. The evolution of
chemistry led to the realisation that atoms also exhibit characteristic additive
properties, enabling the recognition of their presence in a molecule. According
to Bader the theoretical vindication of the model of a functional group as the
carrier of chemical information had to await the work of Feynman and
Schwinger. Their generalisation of physics leads to a unique de®nition of an
atom as an open quantum system and accounts for the short-range nature of the
forces that identify a given group in any environment. His lecture demonstrated
that the proper open systems predicted by the quantum action principle de®ne
the atom and that this de®nition accounts for the retention of an atom's
chemical identity.
In a second contribution Bader
106
re®nes the previous question and asks
whether there can be more than a single de®nition of an atom in a molecule.
Arguments are presented that the theoretical de®nition of an atom in a molecule
or of a functional grouping of atoms that derive from experimental chemistry
must be unique. Bader argues that de®nitions based on the orbital model or, as
recently proposed, in terms of domains de®ned by isovalued density envelopes
fail. They fail for a number of reasons, among them being their incapacity to
enable a quantum mechanical description of the atomic or group properties.
According to Bader chemistry is concerned with the observation and measure-
ment of properties. De®nitions that do not predict the measurable, additive
properties found for atoms in molecules fail to recover the essence of the atomic
concept and can play no operational or predictive role in chemistry. He
concludes that atoms exist in real space and that their form determines their
properties. Bader states that there is but a single de®nition for an atom, free or
bound, that meets this essential requirement.
2.2 Molecular Similarity and QSAR. ± In a ®rst contribution on the design of a
practical, fast and reliable molecular similarity index Popelier
107
proposed a
measure operating in an abstract space spanned by properties evaluated at
BCPs, called BCP space. Molecules are believed to be represented compactly
and reliably in BCP space, as this space extracts the relevant information from
the molecular ab initio wave functions. Typical problems of continuous
quantum similarity measures are hereby avoided. The practical use of this
novel method is adequately illustrated via the Hammett equation for para- and
meta-substituted benzoic acids. On the basis of the author's de®nition of
distances between molecules in BCP space, the experimental sequence of
acidities determined by the well-known sconstant of a set of substituted
congeners is reproduced. Moreover, the approach points out where the
common reactive centre of the molecules is. The generality and feasibility of
this method will enable predictions in medically related Quantitative Structure
Activity Relationships (QSAR). This contribution combines the historically
disparate ®elds of molecular similarity and QSAR.
In a second contribution on quantum molecular similarity O'Brien and
Popelier
108
investigate the relation between properties in BCP space and bond
150 Chemical Modelling: Applications and Theory, Volume 1

length. In their contribution the authors critically examine the dependence of
properties in BCP space on equilibrium bond length, a topic which has been
extensively investigated in the literature. For that purpose they have designed a
data set of 57 molecules yielding 731 BCPs. They con®rm the existence of local
linear relationships provided the bonds vary little in their chemical surround-
ings. Such relationships break down completely for larger subsets of BCPs
encompassing a wider variety of bonds. The patterns observed in the global
picture show so little correlation that one may safely conclude that BCP
properties cannot be trivially recovered or even predicted by knowledge of
bond length alone.
Meurice et al.
109
compare benzodiazepine-like compounds using AIM's
topological analysis and genetic algorithms. Four compounds within a set of
ligands for the benzodiazepine receptors are characterised by their electron
density maps at dierent resolution levels and reconstructed from calculated
structure factors. The resulting complex three-dimensional density maps are
®rst simpli®ed into connected graphs using AIM. Then, an original genetic
algorithm method, GAGS (Genetic Algorithm for Graph Similarity search), is
developed and implemented in order to compare the connected graphs. Finally,
the best solutions of the algorithm are expressed in terms of functional group
superimpositions. The GAGS analysis is applied to dierent resolution levels of
the electron density maps and the resulting models are compared in order to
assess the in¯uence of the resolution on the resulting pharmacophore models.
2.3 Electron Correlation. ±Ciosøowski and Liu
110
discuss the topology of
electron±electron interactions in atoms and molecules, introducing what they
call the ``correlation cage''. The concept of the correlation cage provides new
insights into electron±electron interactions in atoms and molecules. The cage
constitutes the domain in the space of interelectron distance vectors Rwithin
which correlation eects are substantial. Its shape and size are entirely
determined by the topological properties of the electron intracule density I(R),
thus avoiding any references to ill-de®ned ``uncorrelated'' quantities. Integra-
tion of observables related to I(R) over the correlation cage aords quantitative
measures of electron correlation. The number of strongly correlated electron
pairs M
corr
{I}, their electron±electron repulsion energy W
corr
{I}, and the cage
volume V
corr
{I} that characterises the spatial extent of electron correlation are
functionals of I(R). The ratio k{I}ofI(0)V
corr
{I} and M
corr
{I}, which measures
the strength of short-range correlation eects, is small for systems such as H
7
and closer to one for those with weaker correlation eects.
2.4 Transferability. ± Gran
Äa and Mosquera
111
focused on the eect of proto-
nation on the atomic and bond properties of the carbonyl group in aldehydes
and ketones. They studied the protonation in a series of aldehydes and ketones,
R
1
(CÐ
ÐO)R
2
(with R
1
,R
2
= H, Me, Et, Pr and Bu) using AIM to examine the
atomic and bond properties of the CÐ
ÐO group and its relationship to the energy
involved in the protonation process. Based on the results, aldehydes, methyl
ketones and the remaining dialkyl ketones exhibit three dierent types of
3: Atoms in Molecules 151

behaviour. From the many graphs and tables in this paper a few conclusions can
be drawn: protonation results in small dierences in the properties of the CÐ
ÐO
bond; while the atomic charge of CÐ
ÐO hardly changes, the proton bonded to O
exhibits a high positive charge after protonation; and atomic contributions to
the total energy depend on molecular size.
In work on a closely related data set Gran
Äa and Mosquera
112
revisit carbonyl
groups in aldehydes and ketones in connection with transferability using AIM.
The authors calculated the atomic and bond properties of the carbonyl group of
a series of 42 aldehydes and ketones from Hartree-Fock (HF) 6-31++G**//6-
31G* wave functions. They found that the quantities r
b
, the intra-atomic dipole
moments m(O) and m(C), the volume v(C), and l
3
dier between aldehydes and
ketones. These quantities can be said to be transferable within each of these
series, with the exception of formaldehyde. Gran
Äa and Mosquera considered the
populations N(O) and N(C), the ®rst-order electronic charge moments r1(O),
r1(C), v(O), the internuclear distance R, the distance from O to the BCP denoted
by r,e
b
and H
b
as transferable dividing them into three groups: aldehydes,
methyl ketones, and ketones of greater length. Both the total and potential
energies varied in accordance with molecule size and, therefore, cannot be
considered transferable properties of CÐ
ÐO. However, the molecular energies
can be reproduced extremely accurately by means of a group contribution
model which distinguishes the classic fragments: H aldehyde, CÐ
ÐO, CH
2
, and
CH
3
. This reproduction stems from the complementary variation that the
fragments' energies undergo throughout the series of compounds. At their
level of integration accuracy for the oxygen atom, none of the integrated
properties are aected by the value of L(O). However, for C the population
and the ®rst moment depend linearly on L(O), preventing the above properties
from being used directly in the analysis of transferability.
2.5 Multipoles. ± Chipot et al.
113
performed a statistical analysis of distributed
multipoles derived from molecular electrostatic potentials. They described a
simple way to obtain distributed electric multipoles on a selection of sites in a
molecule. The method is based on a statistical analysis of the multipole
components, aimed at reproducing the electrostatic potential created by the
molecular charge distribution on a grid of points around the molecule.
Applications to HF, H
2
O, CF
4
, CCl
4
, CCl
2
F
2
,CH
3
OH, H
2
CO are presented
to illustrate this novel approach.
2.6 Molecular Dynamics. ± Soetens et al.
114
performed a molecular dynamics
simulation of liquid CCl
4
with a new polarisable model based on the Topological
Partitioning of Electrostatic Properties (TPEP) method. This novel method is
rooted in AIM and on Stone's work
115,116
and was developed very
recently.
29,117,118
It consists of dividing the molecular volume into disjoint
regions usually centred around the nuclear sites. A multipole expansion on
each site for the electron density response function leads to multi-centre
multipolar polarisabilities, which include charge-¯ow terms as well local and
non-local atom±atom dipole±dipole (and higher) polarisabilities. It was found
152 Chemical Modelling: Applications and Theory, Volume 1

before
29
that distributed polarisabilities de®ned within AIM remain remarkably
stable with respect to basis set extension. This is a consequence of the fact that
these polarisabilities have not been obtained by partitioning in the Hilbert space
of basis functions, but in real space.
The van der Waals parameters of the potential of Soetens et al. have been
derived from ab initio interaction energies of selected dimer structures. The
computed thermodynamical and dynamical properties are in good agreement
with the experimental values.
2.7 Partitioning. ± An alternative partitioning scheme of molecular density into
atoms is proposed by Rico et al.
119
Their minimal deformation criterion,
previously proposed for the partition of the molecular density into atomic
contributions, is updated and extended. The authors claim that the decomposi-
tion of the molecular density r(r) into atomic densities, r(r)=PAr
A
(r)is
intrinsically arbitrary. As a consequence the above equation has an in®nite
number of solutions. The relation between their method and AIM is not
discussed.
3 The Laplacian
3.1 Alternative Wave Functions. ± Further advances in the understanding and
application of the Laplacian have been made, in particular its use in conjunc-
tion with the CNDO method. Results for CH
4
,CH
3
Cl, CCl
4
,H
2
S and PH
3
show that the CNDO electron density can be constructed such that the
topology of 7H
2
robtained from full-electron ab initio calculations is qualita-
tively reproduced. Following their method Sierraalta et al.
120
also evaluated
the topology of the Laplacian of the spin density (r
a
(r)±r
b
(r)) using CNDO
calculations in a study of modelled catalysts. The location of the CPs was
associated with the most reactive sites on a Ni
5
surface in order to predict the
adsorption of C and the most convenient orientation of H
2
for dissociation on
Mo
3
S
14
H
4
.
Bader et al.
121
resolve an apparent contradiction arising from the use of
eective core potentials (ECP) in transition metal atoms. The recent experi-
mental determination of the geometry of Ti(CH
3
)
2
Cl
2
shows it to be incon-
sistent with the VSEPR model, a result not uncommon for molecules containing
transition metal atoms. The VSCCs that appear as maxima in L(r) =7H
2
r(r),
provide a physical basis for the VSEPR model of molecular geometry for main
group molecules. The same model accounts for the geometry of transition metal
molecules provided the VSCCs are formed within the outer shell of the core of
the metal atom, as de®ned by the shell structure of L(r). This observation
appears to be in con¯ict with calculations for Ti(CH
3
)
2
Cl
2
, showing that its
geometry can be predicted using an ECP for the metal atom, a procedure that
would appear to preclude the presence of core distortions. The apparent
contradiction is resolved by distinguishing between the de®nition of the core
using L(r) and one based on the orbital model. As a result the suggestion that
3: Atoms in Molecules 153

the core distortions in metal atoms arise from the distortion of the density of the
3s
2
3p
6
outer core orbitals cannot be sustained.
86
3.2 Relation to Bohm Quantum Potential. ± In 1997 Levit and Sarfatti
122
noted
the great similarity between the plots of the Laplacian of rand the Bohm
quantum potential Q in the case of H
2
O. They realised that Qis identical to
Hunter's one-electron potential (OEP),
123
which is de®ned as 7H
2

r
p/(2 
r
p).
Hamilton, however, visually compared the Bader Laplacian and the Bohm
quantum potential for molecules containing atoms from beyond the third row
and concluded that for these molecules the two distributions can be qualita-
tively dierent
124
(Figure 2). In a reply to Hamilton's study Levit and Sarfatti
argued that for all molecules studied all topological features (maxima, minima
and saddles) appearing in the Laplacian also appear in the Bohm quantum
potential,
125
but the converse is not always true.
3.3 Protonation. ± The Bohm quantum potential Qhas been applied (intro-
duced as the one-electron potential (OEP), which is identical to Q) in the context
Figure 2 Contour plots of (a) H
2
rfor MgOH; (b) Q for MgOH; (c) H
2
rfor CaOH and
(d) Q for CaOH. Solid contours are negative
(Reproduced by permission of Elsevier)
154 Chemical Modelling: Applications and Theory, Volume 1

of proton anities. It has been known for some time
126
that proton anity
correlates linearly with 7H
2
rat the non-bonding (``lone-pair'') maximum in
7H
2
rfor a series of alkylated amines. Now Chan and Hamilton
127
have shown
that proton anities of a series of alkyl amines and posphines also exhibit a
linear correlation with valence minima in the distribution of the OEP.
Polyoxometalates are relevant to catalysis, analytical chemistry, biology,
medicine and material science. Their properties are related to their high
charges and the strong basicity of the oxygen surfaces. The ®rst ab initio
calculations on hexametalate anions were reported by Maestre et al.,
128
who
studied the cis and trans forms of Nb
2
W
4
O
19
47
(Figure 3) and several isomers
of the protonated anion Nb
2
W
4
O
19
H
37
. Although these calculations are
expensive they are superior to X-ray determinations, which often suer from
disorder problems. The central oxygen (O
9
) has been described as an O
27
species inside a cage, an observation that is compatible with the high AIM
atomic charge of 71.59e. Net charges of +3.12e and +3.46e are assigned to
the Nb and W respectively, con®rming the high ionic character of the metal±
oxygen interaction. The authors ®nd that the relative protonation energy of
an oxygen site appears to be strongly correlated to the corresponding
minimum in the electrostatic potential and to the Bader charge of the
protonated oxygen atom. Furthermore it was observed that the net charges
con®rm that protonation induces a low electronic perturbation to the cluster.
Finally the authors use the Laplacian of rto prove that the polarisation of
the oxygen atoms is small and hence consistent with their high negative
charge.
Figure 3 Schematic representation of the polyoxometalate M
2
W
4
O
19
47
(M = Nb or V),
a niobotungstate. The speckled spheres are tungsten nuclei, and the small open
spheres oxygens
(Reproduced by permission of the American Chemical Society)
3: Atoms in Molecules 155

4 Electron Densities from High-resolution X-ray Diraction
4.1 State of the Art. ± In a paper entitled ``Charge-density analysis at the turn of
the century'',
129
Coppens states that X-ray diraction is now used in an
increasingly routine fashion for the measurement of electron densities in solids.
In addition to representing the detailed electron distribution, the results can be
used to derive electrostatic moments of molecules and d-orbital populations of
transition metal atoms. A topological analysis of the experimental density yields
quite reproducible results when related molecules are compared, but dierences
with Hartree-Fock theory remain, in particular for more polar bonds.
Accurate experimental electronic properties can now be obtained in only one
day with synchrotron radiation and a charge-coupled device area detection
technique. Recently spectacular electron densities were acquired on DLDL-proline
monohydrate at 100 K.
130
The accuracy of the data is comparable or even
superior to the accuracy obtained from a 6-week experiment on DLDL-aspartic acid
with conventional X-ray diraction methods. A data acquisition time of one
day is comparable to the time needed for an ab initio calculation on isolated
molecules. This technique renders larger molecular systems of biological
importance accessible to electron density experiments. The impact of the rapid
collection of accurate data should not be underestimated. Indeed, if properly
allocated a dedicated synchrotron source could now routinely produce accurate
experimental densities at a dramatically increased rate.
4.2 Comparison between Experimental and Theoretical Densities. ± Systematic
comparisons between theoretical and experimental densities provide an ample
testing ground for the validity of the Schro
Èdinger equation, although agreement
is still only fair rather than excellent with current computational schemes and
technology. Nevertheless the new generation of rapid high-resolution X-ray
studies furthers crystallography beyond its current status of a generator of
molecular geometries. From routine X-ray studies one cannot obtain chemical
insight beyond that of the nuclear geometry. With the recent advent of, in
principle, routine high-resolution crystallography chemical insight may be
retrieved from the electron density. Although this insight (e.g. the covalent
character of a hydrogen bond) has traditionally been obtained using deforma-
tion densities
131
more workers are turning to AIM because it eliminates the
necessity for an (arbitrary) reference density.
One of the advantages of AIM is that it can be used on experimental densities.
Although a fully streamlined and well-maintained software interface between
an AIM analyzer and a crystallographic package still does not exist, AIM serves
an increasing number of crystallographers in their interpretation of experi-
mental electron densities. In the period reported on here about two dozen
papers appeared using AIM in the context of X-ray and neutron diraction.
The topological analysis is rarely invoked to obtain atomic properties, such as
the charge, probably due to technical diculties and computational cost.
However it is frequently used to characterise bonds in terms of their covalent
and ionic character, albeit not always in a critical manner.
156 Chemical Modelling: Applications and Theory, Volume 1

The theoretical and experimental electron densities of bis(diiminosuccino-
nitrilo)nickel, Ni(disn)
2
(Figure 4) has been compared by Hwang and Wang.
132
In their work they show a rare comparison between the calculated and
experimental gradient vector ®eld (Figure 5). A quantitative description of the
chemical bond in Ni(disn)
2
is made in terms of the topological properties of
electron densities. The asphericity in raround the Ni atom is observable from
the Laplacian of rwith density accumulation in the dp-direction but density
depletion in the ds-(Ni±N)direction. On the basis of the topological properties
at BCPs, the bonding between Ni and the imino nitrogen atom is classi®ed as
mainly a closed-shell interaction but with some covalent character. The bonds
within the ligand (disn), are all shared interactions. The authors claim that the
bond order is re¯ected in the density at the BCPs. For many topological
properties the agreement between theory and experiment is reasonable but
fairly large discrepancies are found in H
2
r
b
, the Laplacian evaluated at the BCP.
Another study focusing on the comparison between theoretical and experi-
mental densities is that of Tsirelson et al. on MgO.
133
Here precise X-ray and
high-energy transmission electron diraction methods were used in the explora-
tion of rand the electrostatic potential. The structure amplitudes were
determined and their accuracy estimated using ab initio Hartree-Fock structure
amplitudes. The kmodel of electron density was adjusted to X-ray experimental
structure amplitudes and those calculated by the Hartree-Fock model. The
electrostatic potential, deformation density and H
2
rwere calculated with this
model. The CPs in both experimental and theoretical model electron densities
were found and compared with those of procrystals from spherical atoms and
ions. A disagreement concerning the type of CP at (1
4,1
4, 0) in the area of low,
near-uniform electron density is observed. The authors noted that the topolo-
gical analysis of rin crystals can be related with a close-packing concept.
In the introduction of their extensive paper on the topological analysis of the
experimental electron density of DLDL-aspartic acid at 20 K Flaig et al.
134
state
that the extent to which topological properties are reproducible depends not
only on the experimental conditions but also on the interpretation of the data.
They also claim that proper treatment of the diraction data needs increasing
Figure 4 Molecular geometry and atomic labeling of bis(diiminosuccinonitrilo)nickel,
Ni(disn)
2
(Reproduced by permission of the American Chemical Society)
3: Atoms in Molecules 157

support from theory, next to the opposite requirement, the urgent need for
experimental veri®cation of the theoretical results. This view is compatible with
the increasing incorporation of theoretical calculations into the interpretation
of experimental work in areas other than crystallography. The epoch in which
calculations lagged behind experiment has been replaced by an era of symbiosis
between theory and experiment.
The paper of Flaig et al.
134
discussed here is part of a systematic study on the
20 naturally occurring a-amino acids, which have been mainly studied by
Figure 5 Gradient vector ®eld of rfor Ni(disn)
2
from (a) experiment and (b) calculation
(Reproduced by permission of the American Chemical Society)
158 Chemical Modelling: Applications and Theory, Volume 1

conventional techniques at room temperature or under liquid nitrogen cooling.
Despite the importance of this class of compounds very few studies have been
performed on the determination of their experimental charge distribution. In
the study on DLDL-aspartic acid special attention was paid to proper thermal
convolution, the treatment of the hydrogen atoms and chemical constraints in
the least-square re®nement procedures. The density and its Laplacian extracted
from the data are analyzed in terms of the topological properties of covalent
bonds and non-bonded interactions. The results are compared to those
calculated at Hartree-Fock level and to those obtained experimentally for
analogous molecules.
There are three main conclusions: the theoretical density of an isolated
molecule is topologically equivalent to that extracted from the crystal through
modeling the diraction pattern. In other words, the number and types of CPs
in rfound for the energy-optimised stationary system in the experimental
geometry are the same as obtained for the molecule being in thermal equi-
librium in the crystalline state. Secondly, the locations of the BCPs can dier
markedly between the theoretical and experimental densities, especially in polar
bonds. The value of the density at the BCP typically agrees well between theory
and experiment, but this is because rusually possesses a ¯at minimum between
the atoms sharing the electrons, according to Flaig et al. Since theory and
experiment can lead to considerably dierent CP locations in polar bonds, the
concomitant AIM population analysis may give dierent results as well. Finally
the eect of the crystal ®eld on the Laplacian was detectable. In summary, Flaig
et al. join other authors
135,136
in stressing that theoretical and experimental
densities are not comparable because of their dierent nature. They conclude
that the extent to which ®ne details in the gradient vector ®eld of raect
integrated atomic properties should be subject to further studies. It is curious
that this careful study (and others) do not include electron correlation in their
theoretical calculations, although Zadovnik et al.
137
do.
Keeping up with their research program to perform high-resolution dirac-
tion experiments on amino acids Flaig et al.
138
also studied LL-Asn, DLDL-Glu, DLDL-
Ser and LL-Thr with the fast synchrotron method.
130
Their comparative electron
density determinations in the group of 20 naturally occurring amino acids
focused on the reproducibility and transferability in the vicinity of the C
a
atom.
At the same time the authors support computational work
139,140
and prove
experimentally the transferability of electronic properties and submolecular
fragments from this class of compounds onto larger systems like oligopeptides.
The observed variations for a given bond range from 1±5% for rand 7±19% for
H
2
r, indicating a high degree of reproducibility and transferability. Although
integrated AIM properties were not determined, the similarity of the topology
of chemically equivalent bonds found in this study yields an indirect proof for
the transferability of atomic and group properties.
Another comparison between theoretical and experimental densities can be
found in the topological analysis of 2-amino-5-nitropyridinium dihydrogen
posphate carried out by Puig-Molina et al.
141
The BCP properties of the total
experimental density agree fairly well with ab initio Hartree-Fock calculations
3: Atoms in Molecules 159

for the isolated ions. The analysis of the hydrogen-bond BCPs shows that the
crystal hydrogen bond framework involves four anions and one cation. All the
hydrogen BCPs display small positive H
2
r
b
values, consistent with ionic closed-
shell interactions between the participant atoms.
Zadovnik et al.
137
included electron correlation in the calculated electron
density of urea and compared it with the experimental density obtained by high-
precision single-crystal X-ray diraction at 148 K. The displacement para-
meters agree quite well with results from neutron diraction. Orbital calcu-
lations were carried out at the Hartree-Fock level, DFT/LDA and DFT/GGA
level. The agreement between experimental and theoretical results is excellent,
judged by the deformation density and the structure factors. The agreement
with respect to the results of the topological analysis was only fair. Density-
functional calculations seem to yield slightly better results than Hartree-Fock
calculations. The authors conclude that there is a semi-quantitative agreement
between the experimental and theoretical (i.e. topological) characteristics of the
intramolecular and intermolecular interactions.
A further theory-experiment comparison invoking AIM is made in the area of
minerals by Rosso et al.
142
BCP properties calculated for representative Si
5
O
16
moieties of the structure coesite are compared with those observed and
calculated for the bulk crystal. The values calculated for the moieties agree
with those observed to within 5% on average, whereas those calculated for the
crystal agree to within 10%. As the SiOSi angles increase and the SiO bonds
shorten, there is a progressive build-up in the calculated electron density along
the bonds. This eect is accompanied by an increase in the curvatures of r
b
(both perpendicular and parallel to each bond) and in H
2
r
b
. Whereas Flaig et al.
scrutinise the treatment of crystallographic data in an attempt to understand the
discrepancy between theory and experiment Rosso et al. hold responsible
possible problems in X-ray data recording, basis set limitations, the exclusion
of relativistic eects and improper representation of correlation.
In their search for homoaromatic semibullvalenes Williams et al.
143
investi-
gated the X-ray electron density of 1,5-dimethyl-2,4,6,8-semibullvalenetetra-
carboxylic dianhydride in the temperature range of 123 K to 15 K. This paper
encompasses another extensive application of AIM to experimental densities
showing excellent agreement with the Hartree-Fock electron density. The region
between the C
2
and C
8
carbons revealed a ``normal cyclopropyl s-bond'' for this
bisannelated semibullvalene, while a very sparse density was found between the
C
4
and C
6
carbons. These results con®rm that in the solid state the semibullva-
lene is not homoaromatic. Topological equivalence of experimental and
theoretical densities is shown to be evident although agreement for the H
2
r
b
values is not excellent (in contradistinction to the r
b
values). This discrepancy is
claimed to be caused by the fact that the used basis set (6±31G**) cannot furnish
a satisfactory representation of the Laplacian in polar bonds.
4.3 Hydrogen Bonding. ± Three studies focused on the features of hydrogen
bonding drawn from experimental electron densities using AIM tools. The ®rst
study investigated the intramolecular hydrogen bonding in benzoylacetone (or
160 Chemical Modelling: Applications and Theory, Volume 1

1-phenylbutane-1,3-dione) by means of a 8 K X-ray and 20 K neutron dirac-
tion experiment. Madsen et al.
144
showed that from the neutron data we
ascertain that the hydrogen, between the two oxygens in the keto-enol part of
the molecule, is asymmetrically placed in a large ¯at potential well. In Figure 6
we see that the hydrogen bond hydrogen nucleus is slightly closer to O2 (1.25 A
Ê)
than to O1 (1.33 A
Ê). Evidence for extensive p-delocalisation in the keto-enol
group was obtained from the use of multipolar functions and topological
methods. The multipole populations show that there are large formal charges
on the oxygens and the enol hydrogen, which impart polar character to the
hydrogen bond. This eect is also evident in the Laplacian and in the
electrostatic potential calculated from the X-ray data. It is found that the
hydrogen position is stabilised by both the electrostatic and covalent bonding
contributions at each side of the hydrogen atom. The authors note that the
electronic nature of the hydrogen bond cannot be inferred solely from the
distance between oxygen atoms of the hydrogen bond. They base their
argument on a comparison with a similar keto-enol system found in citrinin.
Recently much attention has been devoted to the detailed mechanism by
which the class of enzymes called serine proteases work. These enzymes catalyze
the ubiquitous and paramount cleavage of peptide bonds and all have the so-
called catalytic triad (His-Asp-Ser) in common. A number of studies have
suggested that a low barrier hydrogen bond (LBHB) is involved in the reaction
mechanism as a partial proton transfer between His and Asp (N±H ...O). In
Figure 6 Contour plot of L(r) vs. 7H
2
r(r) in the plane of the keto-enol group in 1-
phenylbutane-1,3-dione. The contours are drawn at logarithmic intervals of
1.0 62
n
e/A
Ê
5
. The dotted line is the zero contour, solid lines are positive and
broken lines negative contours. The ®rst two positive and negative contours are
omitted for clarity. BCPs are marked with solid in circles
(Reproduced by permission of the American Chemical Society)
3: Atoms in Molecules 161

order to investigate the electronic structure of short N±H...O bonds Overgaard
et al.
145
carried out a combined low-temperature (28 K) X-ray and neutron
diraction study of the cocrystal between betaine, imidazole and picric acid.
This complex serves as a model for the catalytic triad (biomimetic) (Figure 7).
The data show that all the hydrogen atoms are localised at the nitrogen atoms,
and that none are involved in LHBHs. The absence of LHBHs is supported by
the topological analysis, which shows that the O...H interactions are predomi-
nantly electrostatic interactions (H
2
r
b
40) between closed-shell atoms. This
study demonstrates that short, strong hydrogen bonds do not have to be low-
barrier. The Espinosa hydrogen bond energy estimate
146
is strictly valid for
electrostatic interactions (based on a large number of normal hydrogen bonds)
and breaks down for the present hydrogen bonds. The authors are performing
high-level ab initio calculations on their model to gain further insight about the
catalytic triad.
A new type of hydrogen bond was discovered in the mid-nineties by
Richardson et al. called the dihydrogen bond,
147
in which the acceptor atom is
a (hydridic) hydrogen atom. A few years later it was shown by Popelier
148
that
the dihydrogen bond complied with all the hydrogen bond criteria previously
proposed by Koch and Popelier
21
based on observations of the (H
3
BNH
3
)
2
moiety. Popelier's study also showed that the boron in this complex is very
positive (q(O) = 2.15) which is diametrically opposed to the Mulliken charge
(q=70.26) quoted by Richardson et al.,
147
thereby rendering their explanation
of the geometry of the complex incompatible with AIM.
Figure 7 Cocrystal of betaine, imidazole, and picric acid (ORTEP drawing at 95%
probability level based on neutron diraction data at 28 K). Hydrogen atoms
have been omitted for clarity. A short C±H...O interaction exists between
C8B±H8B...O7A, which is important for the crystal packing
(Reproduced by permission of Angewandte Chemie, VCH)
162 Chemical Modelling: Applications and Theory, Volume 1

Abramov et al.
149
carried out a low temperature neutron and high-resolution
X-ray diraction study on another system, cis-HMn(CO)
4
PPh
3
, that contains a
dihydrogen bond. The hydride ligand H(1) (Figure 8) is nucleophilic in nature
and makes a short contact (2.1 A
Ê) with an electrophilic ortho phenyl hydrogen.
The electrostatic component of the H
d+
...H
d7
interaction energy is calculated
to be 23.9 kJ mol
71
from the experimental data. This electrostatic evidence
coupled with the geometry and the identi®cation of an H...H bond path in r
strongly supports the characterisation of this interaction as an intramolecular
C±H...H±Mn hydrogen bond. Both the deformation density and the topologi-
cal study clearly illustrate the s-donor nature of both the H±Mn and PPh
3
±Mn
interactions, and the s-donor/p-acceptor nature of the Mn±CO bonds. The
topological study further con®rms the decrease in the CO bond order upon
coordination to the metal. This method demonstrates for the ®rst time that the
metal±ligand bonds have a signi®cant dative covalent component, although
they show characteristics of a closed-shell interaction.
4.4 Organic Compounds. ± The electron de®cient and multicentre nature of B±B
and C±C bonds in 8,9,10,12-tetra¯uoro-o-carborane prompted a topological
study on its electron distribution obtained by high-resolution X-ray diraction
at 120 K. The detailed analysis by Lyssenko et al.
150
of rand its Laplacian at
the BCPs revealed unexpected features in the bonding pattern, i.e. the positive
value of the Laplacian in the homopolar C±C bond and the negative values for
the B±C bonds (Figure 9). These data were compared with corresponding ab
initio calculations on small deltahedral boranes and carboranes. The authors
Figure 8 Molecular geometry of HMn(CO)
4
PPh
3
shown with 60% probability ellipsoids
based on neutron diraction re®nement. Dashed line indicates the C±H...H±Mn
hydrogen bond
(Reproduced by permission of the American Chemical Society)
3: Atoms in Molecules 163

concluded that the electron-withdrawing eect of the ¯uorine atoms causes a
considerable redistribution of the electron density in the molecule. In particular
this is re¯ected in the shift of r
b
from the more electron-rich C±C bonds to the
B±C bonds. Deformation density maps showed as well that ris essentially
delocalised over the surface of the cage and locally depleted in its centre. All B±
B and B±C bonds in the polyhedron are characterised by signi®cant bending,
which is evident in shifts of their BCPs from the straight lines between the
respective nuclei.
Puig-Molina et al.
141
compared the theoretical and experimental electron
density in the nonlinear optical material 2-amino-5-nitropyridinium dihydrogen
phosphate, 2A5NPDP. The experimental rwas determined from X-ray dirac-
tion data interpreted in terms of the Hansen & Coppens pseudoatom formalism.
The BCP properties of the total experimental electron density agree fairly well
with Hartree-Fock calculations for the isolated ions. The analysis of the
Figure 9 Laplacian of the electron density through the centre of the icosahedron of
8,9,10,12-tetra¯uoro-o-carborane in the plane, containing two carbon and
boron nuclei. Logarithmic scale, positive contours are dashed
(Reproduced by permission of the American Chemical Society)
164 Chemical Modelling: Applications and Theory, Volume 1

hydrogen-bond CPs shows the crystal H-bond framework to involve four
anions and one cation. All the H-bond CPs show small positive H
2
r
b
values,
consistent with ionic closed-shell interactions between the participant atoms.
Bianchi
151
studied the experimental electron density study of 4-cyanoimida-
zolium-5-olate (Figure 10) at 120 K obtained from accurate X-ray data. The
interpretation chemical of the data was aided by a multipole model (``POP'',
Stewart
152
), where ris represented by an expansion in terms of rigid pseudo-
atoms, an ab initio Hartree-Fock calculation on the isolated molecule and AIM.
The topological analysis reveals the presence of three HBs with ionic character
that contribute to stabilizing the crystal structure. The title compound can be
described as the juxtaposition of two separated p-conjugated systems, linked to
each other by two nearly single C±N bonds (N1±C5 and N3±C4). The Mulliken
atomic charges and molecular dipole moment derived from the Hartree-Fock
calculations are close to those obtained from the multipole model. However, the
authors admit that if a partitioning were carried out by means of IASs a more
signi®cant set of atomic populations would be obtained.
Finally we mention the study of Aguilar-Martõ
Ânez et al. who synthesised and
analyzed the substituent eects on the redox properties of 19 compounds of 3'-
(meta) and 4'-(para) substituted 2-{(R-phenyl)amine}-1,4-naphtalenediones in
acetonitrile.
153
Beside an UV-vis analysis and a voltammetric study the authors
performed semi-empirical (PM3) and DFT calculations (B3LYP with double-z
Figure 10 (a) An ORTEP drawing of the molecular structure of 4-cyanoimidazolium-5-
olate at 120 K (ellipsoids are drawn at the 50% probability level). (b) The same
molecule in the crystal, connected by hydrogen bonds
(Reproduced by permission of International Union of Crystallography)
3: Atoms in Molecules 165

split valence basis set). They claim that H
2
ris very sensitive to electronic
delocalisation, which allows them to dierentiate eectively between the two
quinone carbonyls. In six compounds two unexpected bond paths were
detected. One corresponds to the hydrogen bond between the amine proton
and a quinone carbonyl. The second unexpected CP corresponds, according to
Aguilar-Martõ
Ânez et al., to an interaction line expressing steric crowding.
154
This interpretation is controversial in the light of a paper by Bader
155
which we
discuss below. Each interaction generates one ring critical point (RCP) but with
dierent local Hessian eigenvectors. In the case of the RCP associated with the
hydrogen bond, the major ellipticity axis is practically parallel to the ring plane.
In the RCP generated by the alleged H±H steric repulsion, however, the axis is
perpendicular to the ring plane.
4.5 Transition Metal Compounds. ± Macchi et al. performed the ®rst experi-
mental electron density study of a p-ligand Z
2
-coordinated to a metal atom.
156
They considered this work as an ``experimental test'' of the Dewar-Chatt-
Duncanson (DCD) bonding formalism and expected that their work would
provide information concerning the p-complex versus the metallacycle dicho-
tomy. The authors claim that the successful application of AIM to the
experimentally determined rhas been the most important step in the coupling
of X-ray studies and theoretical chemistry. Their paper reports on the determi-
nation of an accurate electron density of crystalline bis(1,5-cyclooctadiene)-
nickel, Ni(COD)
2
(Figure 11) by X-ray diraction at 125 K.
Figure 11 ORTEP view of the Ni(COD)
2
molecule; thermal ellipsoid for non-H atoms
are drawn at the 50% probability level while H atoms are idealised
(Reproduced by permission of the American Chemical Society)
166 Chemical Modelling: Applications and Theory, Volume 1

The overall bonding picture (Figure 12) that emerges from the AIM analysis
is in agreement with the DCD model. The Ni±C bond paths are inwardly
curved, which is interpreted as s-donation, but the bond paths are well
separated, which is a sign of p-back-donation. The topology supports a p-
complex with a concave ring structure intermediate between a T-shape and a
convex ring. The former would imply electrostatic interaction and the latter a
fully covalent model. Previous theoretical work on Ni(C
2
H
4
)
n
(n = 1±4)
157
demonstrated that the DCD approach is substantially correct. The authors note
that an AIM analysis provides information that goes beyond a simple geome-
trical approach, which cannot address separately the two main bonding eects,
or an orbital study, which cannot de®ne a clear picture of the interactions (here
represented by bond paths).
The same authors (Macchi et al.) investigated metal±metal (MM) and metal±
ligand (ML) bonds in Co
2
(CO)
6
(AsPh
3
)
2
(Figure 13) via deformation densities
and an AIM analysis from an accurate X-ray electron density at 123 K.
158
The
existence of a true MM bond remained controversial partially due to the
interpretation of rather noisy deformation density maps. It was recognised
Figure 12 The complete bond path for the p-system. Note the eight lines that link Ni to
each C(sp
2
) atom
(Reproduced by permission of the American Chemical Society)
3: Atoms in Molecules 167

that the major weakness of the deformation density approach is caused by the
choice of a proper promolecule. This choice is particularly delicate when the
total density between the two atoms is small or when atoms have more than
half-®lled shells (a problem which was ®rst observed with the F
2
molecule). The
authors claim that AIM oers a better and less ambiguous theoretical under-
standing of MM interactions. They conclude that the ``expected'' lack of charge
accumulation in the deformation density map is ``contradicted'' by the presence
of a BCP and a bond path linking the two Co atoms.
The major conclusion of their paper is the experimental proof of the presence
of a genuine, covalent Co±Co bond. However, their conclusion is not reached in
a straightforward manner because based on the Laplacian (Figure 14) one
might conclude that the Co±Co interaction is not shared. Also, the electron
density at the BCP between the two cobalt atoms is small. Based on the
observations made in F
2
and invoking the functions H(r)
159
[H(r)=V(r)+
G(r)] and G(r)/r(r) Macchi et al. conclude that ``the Co±Co is far from the
closed-shell limit and we do not ®nd any reason for not considering it a shared
interaction as suggested by common chemical sense.''
However, the ®rst topological analysis of the experimental electron density in
a binuclear metal complex concludes that the metal±metal interaction is
unshared. In their high-resolution X-ray diraction study at 120 K of
Mn
2
(CO)
10
(Figure 15) Bianchi et al.
160
quote a positive value of H
2
rat the
Figure 13 View of the molecular geometry of Co
2
(CO)
6
(AsPh
3
)
2
; ellipsoids are drawn
at the 50% probability level
(Reproduced by permission of the American Chemical Society)
168 Chemical Modelling: Applications and Theory, Volume 1

Figure 14 The distribution of L(r) in the plane containing Co1, Co2, As1 and C1 (contour
levels at 2.0 610
n
, 4.0 610
n
, 8.0 610
n
e/A
Ê
5
,n=72, 71, 0, +1; the solid
lines represent positive values)
(Reproduced by permission of the American Chemical Society)
Figure 15 An ORTEP plot (30%) of the entire molecule of Mn
2
(CO)
10
. The molecule
has crystallographic C
2
symmetry with the two-fold axis passing through the
middle of the Mn±Mn bond
(Reproduced by permission of the Royal Society of Chemistry)
3: Atoms in Molecules 169

Mn±Mn BCP and observe that the density is contracted towards each Mn
nucleus. It should be mentioned that the deformation maps showed no evidence
of signi®cant density accumulation in the MM bonding region.
Scherer et al.
161
embarked on a topological analysis of the experimental and
theoretical electron densities in EtTiCl
3
(dmpe) (dmpe = 1,2-bis(dimethylphos-
phino)ethane) in order to pin down agostic interactions in a precise way.
Agostic interactions are of particular interest in organotransition-metal chem-
istry in view of their role in C±H activation, but reliable ways of detecting and
characterising these interactions are still at a premium. Work by Popelier and
Logothetis
14
suggested to employ AIM as a means to identify agostic inter-
actions. Scherer et al. agree that some agostic interactions may be identi®ed
solely on the basis of electron densities. Nevertheless they are concerned about
the fact that a Ti...H BCP representing an agostic bond may not be detectable if
the agostic interaction is weaker than in the compound they studied. Therefore
the authors propose the non-linearity of the Ti±C
a
bond as a more robust
criterion of b-agostic interaction.
An interdisciplinary study on transition metal coordinated Al(X)L
27
and
Ga(X)L
27
fragments has been carried out by Fischer et al.
162
Their work
describes the new intermetallic systems by means of elemental analysis, IR,
Raman, NMR, mass spectroscopy, single-crystal X-ray diraction and ab initio
calculations at MP2 level. The Laplacian is used to characterise the W±Al bond
in the calculated model compound (CO)
5
WAlH (Figure 16), as well as other
bonds occurring in the systems under study. The area of electron concentration
in the Al atom represents the lone pair of electrons. The electron concentration
at Al in (CO)
5
WAlH becomes deformed and shifted toward the p-bonding
region as a result of Al±W formation. A comparison of the electron concentra-
tion in the W±Al bonding region of (CO)
5
WAlH(NH
3
)
2
and (CO)
5
WAlH shows
that in the former the concentration is more in the p-bonding region along the
W±Al bond path. Along similar lines one concludes that there is stronger
W?Al back donation in (CO)
5
WAlH.
4.6 Minerals. ± Ivanov et al.
163
performed a hybrid study (static deformation
density and AIM) study on the high-resolution X-ray diraction electron
density of topaz. The electron deformation density, positive values of H
2
r
b
and
the net atomic charges indicate a closed-shell type interaction in the SiO
4
polyhedra. Anion valence-shell charge depletions are revealed and it is found
that maxima in H
2
rare displayed towards the close-packed plane owing to the
mutual repulsion of the anion valence shells. The relationship between r`s
topology and the close-packing concept is discussed. Shifts of the CPs from the
internuclear vectors re¯ect the strain in the structure.
Another application of AIM can be found in the work of Kuntzinger et al.,
164
who determined the high-resolution X-ray electron distribution of scolcite
(CaAl
2
Si
3
O
10
.3H
2
O). The densities on the Si±O±Si and Si±O±Al bridges have
been characterised using H
2
r
b
values. The Si±O and Al±O bond features are
related to the atomic environment and to the Si±O(Al,Si) geometries.
170 Chemical Modelling: Applications and Theory, Volume 1

5 Chemical Bonding
5.1 Theory. ± Recently bond paths in certain systems have been interpreted as
indicative of a ``repulsive interaction'' rather than a bonded interaction.
165,166
In
response Bader rebutted this proposal stating that a bond path is the universal
indicator of bonded interactions.
155
He argues that the ``attractor interaction
line between nonbonded atoms''
165
is an oxymoron making clear that they
restrict their de®nition of bonding to the Lewis model of the electron pair. Bader
reasons that Ciosøowski et al. use subjective judgements of relative bond
strengths to determine when a bond path represents a bond or a repulsive
interaction. For example, Bader deduces that the bond path responsible for the
formation of the Cl
2
dimer and for the cohesive energy of solid chlorine is
described by them as a repulsive interaction when present in perchlorocyclohex-
ane.
Bader's main argument is that there is a homeomorphism between rand the
virial ®eld,
167
which determines the system's potential energy. As a result every
bond path is mirrored by a virial path, along which the potential energy density
is maximally negative, i.e. maximally stabilising with respect to any neighbour-
ing line.
Figure 16 Contour diagram of H
2
rin a plane through (CO)
5
WAlH calculated from an
MP2 ab initio wave function. Dashed lines indicate charge depletion, solid lines
charge concentration. The solid lines connecting the atomic nuclei are the bond
paths, those separating the nuclei are intersection of the zero-¯ux surfaces with
the plotting plane
(Reproduced by permission of the American Chemical Society)
3: Atoms in Molecules 171

5.2 Ligand Close Packing (LCP) Model. ± In an educational contribution
Gillespie
168
proposes an alternative method of interpreting simple ¯uorides
using AIM. For example, he criticises the textbook explanation of the electronic
structure of BF
3
by means of resonance canonicals containing a double bond
(invoking so-called ``back-bonding''). Because atomic charges, which were
previously held to be small, can now be obtained through use of AIM, Gillespie
proposes that some ``covalent'' molecules are in fact predominantly ionic. Other
evidence comes from the F±F interligand distances, which are remarkably
constant and independent of coordination number. According to Gillespie
such molecules are best regarded as a central cation-like nucleus with a close-
packed arrangement of anion-like ligands surrounding. Size determines whether
crystal structures are possible or not. Finally, the description of these molecules
as ionic does not violate the octet ``rule''.
In a second contribution Gillespie et al.
169
discuss the bonding and geometry
of OCF
3
7
, ONF
3
, and related molecules in terms of the Ligand Close Packing
(LCP) model. Inspired by AIM this model is an extension and re®nement of a
model proposed by Bartell in the sixties,
170
which recognised the importance of
nonbonded interactions. The authors reinvestigated the nature of the bonding in
these and some related molecules by analyzing their calculated electron density
distributions. The results show that the bonding in the series OBF
3
27
, OCF
3
7
,
ONF
3
ranges from predominately ionic in OBF
3
27
to predominately covalent in
ONF
3
. Also the interligand distances are consistent with the close packing of the
ligands around the central atom. The diculties of trying to describe the
bonding in these molecules in terms of Lewis structures are discussed.
In a ®nal contribution Gillespie et al.
171
further criticise Pauling's suggestion
to interpret the lengths of X±F and X±O bonds in terms of multiple character.
This multiple character would result from back-bonding, after ``correction'' of
the observed bond lengths for polarity on the basis of the so-called Schomaker-
Stevenson equation. In a recent paper Robinson et al.
172
show that there is no
justi®cation for the purely empirical Schomaker-Stevenson equation and that
there is little convincing evidence for the supposed double-bond character in
molecules such as BF
3
or SiF
4
. In their study Gillespie et al. have surveyed the
experimental data for oxo, hydroxo, and alkoxo molecules of Be, B, and C and
have shown that the intramolecular interligand distances for a given central
atom are remarkably constant and independent of coordination number and of
the presence of other ligands. AIM charges for a large selection of molecules of
this type have shown that these molecules are predominately ionic. The authors
suggest that the bond lengths and geometries of these molecules can be best
understood in terms of a model in which anion-like ligands are close-packed
around a cation-like central atom.
5.3 Hypervalency. ± Dobado et al.
173
used AIM to focus on chemical bonding
in hypervalent molecules such as Y
3
X and Y
3
XZ (Y = H or CH
3
;X=N,Por
As; Z = O or S). The nature of the P±O bond in particular has been extensively
reviewed and explained in terms of a combination of two dierent descriptions,
R
3
P
+
±O
7
and R
3
PÐ
ÐO. The former structure obeys the octet rule but requires
172 Chemical Modelling: Applications and Theory, Volume 1

back-donation from oxygen to phosphorus. Three plausible models were
contrasted with the AIM analysis: (i) one s-bond and two p-back-bonds
(negative hyperconjugation), (ii) one s-bond and three p-back-bonds, and
(iii) three O(banana) bonds. The topological analysis [based upon r
b
,H
2
r
b
,e
b
and populations N(O)] is consistent with a highly polarised s-bond with bond
strength dependent on the electrostatic interactions. The bonding nature is close
to model (ii) but no p-back donation can clearly be found. Note that Dobado et
al. do not reach the same far-reaching conclusion as Gillespie et al. against the
validity of the concept of negative hyperconjugation.
169
The quantitative and
qualitative AIM results obtained from the dierent wave functions (MP2 and
B3LYP) were in strong agreement, suggesting that the Bader analysis was
independent of the theoretical method by which the densities were generated.
In a second related study Dobado et al.
174
revisited bonding in hypervalent
molecules involving only chalcogen (O, S, Se) substituents. They applied AIM
to Y
2
XZ and Y
2
XZ
2
(Y = H, F or CH
3
; X = O, S or Se; Z = O or S)
compounds. The topological analyses (based upon the r
b
,H
2
r
b
,e
b
and E
b
, the
local energy density, and the atomic charges) clearly displayed the dependence
of the bonding properties on the central atom. In particular, when the central
atom is O, the main electron charge concentration remains in the surroundings
of the central atom, yielding a very weak coordinate bond. On the other hand,
bonding to the central S and Se is consistent with a model of a highly polarised
s-bond, its strength depending mainly on electrostatic interactions. In other
words, no evidence was found for double bonding, which has so far been the
conventional way to describe the interaction in these systems. The equilibrium
geometries were optimised at B3LYP and MP2(full) level using the 6±311+G*
basis set.
Chesnut
175
produced an ab initio NMR and AIM study of the PO bond in
phosphine oxides and reached the conclusion that posphines are best pictured as
R
3
P
+
±O
7
, a result to be contrasted with that of Dobado et al. Chestnut found
that ab initio NMR calculations on the eect of correlation on phosphorus
shielding in the phosphine oxides clearly suggest the absence of conventional
multiple bonding in the PO bond. AIM studies that yield AIM-based localised
MOs indicate one highly polarised s-bond plus strong back-bonding of the
oxygen p-orbitals, a picture consistent with a number of prior investigations.
While it has been argued that the strong character of the PO bond in the
phosphine oxides is highlighted best by the R
3
PÐ
ÐO formula, the present study
indicates that the situation is better pictured as R
3
P
+
±O
7
.
5.4 Organic Compounds. ± In their outspoken and upbeat study on the 2-
norbornyl cation Werstiuk and Muchall
176
apply AIM to solve the controversy
on the exact nature of this species. Since 1949 this cation has been described by
many terms, such as equilibrating classical, s-bridged, edge-protonated or face-
centred species, corner-protonated nortricyclene and p-complex. The authors
conclude that the 2-norbornyl cation is not a nonclassical, s-bridged species but
ap-complex, based on their AIM analysis. Similarly, there is nothing non-
classical about the 2-bicyclo[2.2.2]octyl cation. The study concludes that no
3: Atoms in Molecules 173

``new bonding principle'' is required to explain observed bonds. The studied
cations have an unstable topology and a small movement of any atom results in
a change in structure.
Cho and Park
177
performed an ab initio study (including BLYP, BP86, MP2
and various basis sets) on the bonding nature of the N±N bonds in 1,2,5-
trinitroimidazole and 1,2,4,5-tetranitroimidazole. Bond properties including r
b
are strongly dependent on the equilibrium bond length. Thus, accurate predic-
tion of geometric parameters is of particular importance in deriving reliable
bond properties. A substantial dierence in bonding properties is observed
when electron correlation is included. According to CHELPG charges the N±N
bonds of both compounds appear to have a signi®cant ionic nature, and the
l-nitro group bears a considerable positive charge. This group also exhibits
attractive electrostatic interactions with O atoms of adjacent nitro groups.
Signi®cantly long N±N bond lengths calculated at MP2 and DFT level imply a
strong hyperconjugation eect, which may explain the ease with which these
compounds form a salt.
Glaser et al.
178
have been interested in deamination reactions and their
relation to modi®cations of DNA. Small aliphatic and aromatic diazonium
ions can play a role in DNA alkylation. In their work the authors focus on the
anomalous behaviour of dediazonation reactions for which the dual substituent
parameter relations yield reaction constants of opposing sign. Glaser et al.
computed AIM quantities based on the total electron density, which is an
observable, as well as the sand pcomponents of the atomic populations. While
these components are not ``observable'' in the strict quantum mechanical sense,
the concept of s/pseparation is a successful one and it is central to all models
invoking dative and backdative bonding. The main conclusion is that the
classical tool of p-electron pushing is not sucient to provide a correct
account of the electronic structures. This conclusion aects the way we ought
to think of the majority of donor-acceptor substituted conjugated dyes and
nonlinear optical materials. In particular, the analysis resolves the apparent
paradox that the amino group can function as an electron donor even though it
is negatively charged.
5.5 Transition Metal Compounds. ± Dobado et al.
179
used AIM to study the
electronic properties of seven isomers of three-coordinated copper(I) thiocya-
nates, calculated at MP2 and B3LYP level using the 6±311+G* basis set. The
results indicate that in the gas phase N-bonding is preferred to S-bonding. The
coordination bond between the Cu(I) cation and the donor atoms is strongly
polarised, almost ionic. The charge depletion around the Cu(I) cation is in
accordance with sp
2
hybridisation. Moreover, the canonical form for the non-
coordinated as well as S-coordinated thiocyanates is mainly S±CÐ
ÐN, whereas
the N-bonded thiocyanates have also NÐ
ÐCÐ
ÐS contribution.
Jansen et al.
180
examined the metal±metal bond polarity in heterobimetallic
complexes containing Ti±Co and Zr±Co bonds. The concept of bond polarity,
which is used in main group chemistry in the context of electronegativity, is less
rapidly applied in a quantitative way to bonds between transition metals. The
174 Chemical Modelling: Applications and Theory, Volume 1

increased level of sophistication required in a meaningful discussion of bond
polarity in M±M'bonds necessitates elaborate quantum chemical methods. The
authors reported the synthesis and X-ray structure of {CH
3
Si{SiMe
2
N(4-
CH
3
C
6
H
4
)}
3
M±Co(CO)
3
(L)} (M = Ti or Zr; L = CO, PPh
3
or Ptol
3
) and
performed DFT calculations (BLYP, B3LYP) on the curtailed model systems
(Figure 17) (H
2
N)
3
Ti±Co(CO)
4
and (H
2
N)
3
Ti±Co(CO)
3
(PH
3
). The authors
provide a thorough theoretical analysis of the electronic structure of the
bimetallic bond, using AIM charges and bond orders, natural population
analysis (NPA), charge decomposition analysis (CDA), and the electron
localisation function (ELF). Both the orbital-based NPA and CDA schemes
and the essentially orbital-independent AIM and ELF analysis suggest a
description of the Ti±Co bond as being a highly polar covalent single bond.
The combination of AIM and ELF is employed for the ®rst time to analyze
metal±metal bond polarity and appears to be a powerful theoretical tool for the
description of bond polarity in potentially ambiguous situations.
Boehme and Frenking
181
applied AIM to characterise bonds in complexes
containing copper, silver and gold, warranting the use of relativistic ECPs and
large valence basis sets (at MP2 level). Their work on N-heterocyclic carbene,
Figure 17 Views of the two model structures [(H
2
N)
3
Ti±Co(CO)
4
] and [(H
2
N)
3
Ti±
Co(CO)
3
(PH
3
)] optimised by DFT calculations
(Reproduced by permission of the American Chemical Society)
3: Atoms in Molecules 175

silylene, and germylene complexes of MCl (M = Cu, Ag or Au) evaluated the
quantities r
b
,H
2
r
b
and H
b
. The metal±ligand bonds have a strong ionic
character, which comes from the Coulomb attraction between the positively
charged metal atom and the s-electron pair of the donor atom. The covalent
part of the bonding shows little p-back-bonding from the metal to the ligand.
The aromaticity of the N-heterocyclic ligands is slightly enhanced in the metal
complexes.
Pavankumar et al.
182
carried out extensive ab initio calculations on the
important anti-cancer drug cis-diaminedichloroplatinum(II) (cisplatin), com-
prehensively reporting on its structure, bonding, electron density, and vibra-
tional frequencies. Although the discussion of the bonding in the title
compound was centred around MO concepts, the authors used AIM to
complete their analysis. The charge density and the Laplacian of charge
density of cisplatin were calculated to determine its bonding relationships.
Pavankumar et al. were unable to locate a BCP between Cl and H (Figure 18)
that they expected on the basis of the contour lines.
The last 15 years have seen phosphaalkynes (P :CR) transformed from
chemical curiosities to versatile synthetic building blocks. The compounds have
been shown to undergo head-to-tail or head-to-head dimerisations at transition
metal centres to form 1,3- and 1,2-diphosphacyclobutadiene complexes.
Howard and Jones
183
accomplished an ab initio study on the analogous
Figure 18 Contour plot of the electron density of cisplatin using the MP2/6±
311++G(2d,2pd) basis set with an ECP on Pt. The numbers indicate the
positions of the BCPs in cisplatin: 1is the Pt±Cl BCP, 2is the Pt±N BCP, and
3is the N±H BCP
(Reproduced by permission of Wiley)
176 Chemical Modelling: Applications and Theory, Volume 1

diarsacyclobutadienes, on which there had been no reports until recently. They
report MP2/6±311G* and MP4SDQ/6±311G* calculations for the various
possible conformers of the simple 1,2- and 1,3-diarsacyclobutadienes in order
to determine their geometries, the nature of the bonding, and the relative
stabilities of dierent isomers. These calculations revealed some qualitative
dierences between the analogous 1,2- and 1,3-diphosphacyclobutadienes. The
authors use AIM to characterise As±C and As±As bonds. Howard and Jones
conclude that the As±As BCP appearing in one system's transition state cannot
be interpreted as a bond because it does not occur in a stable equilibrium
geometry.
36
Atomic charges were determined with both the NPA and AIM
schemes, which give remarkably similar values for As.
5.6 Minerals. ± SoscuÂn et al.
184
present the ®rst investigation of topological
properties of rof a series of hydroxyl acid sites in zeolites at ab initio level.
The zeolite acid sites were modelled by using the following molecular clusters:
silanol H
3
SiOH and the clusters H
3
SiO(H)AlH
3
, (OH)
3
SiO(H)Al(OH)
3
and
H
3
SiO(H)Al(OH)
2
SiH
3
. The calculations showed that the frequency of the OH
vibrational modes of acid sites is linearly related to r
b
at the OH BCPs. These
results indicate that r
b
(OH) can be used as a tool for interpreting the structural
and electronic features of the zeolite hydroxyl groups. This is not possible using
the Mulliken charge of H in the OH bond, which gives a poor correlation with
the frequency. From the H
2
r
b
values of the bonds of the acid sites the authors
conclude that the zeolite structure is dominated by a network of Si±O and Al±O
ionic interactions. The O±H bonds are characterised as covalent bonds, with
dierent extents of charge concentration.
The work of Gibbs et al.
185
invokes AIM in the area of the physics and
chemistry of minerals. The topological properties of rfor more than 20
hydroxyacid, geometry optimised molecules with SiO and GeO bonds with
3-, 4-, 6- and 8-coordinate Si and Ge cations were calculated. Electronegativ-
ities calculated from the BCP properties indicate that the electronegativity of
Ge (+1.85) is slightly larger than that of Si (+1.80) for a given coordination
number. The electronegativities of both atoms increase with decreasing bond
length. With an increase in r
b
the quantities H
2
r
b
, and l
i
(i= 1, 2 or 3) of each
bond increase with decreasing bond length. The covalent character of the
bonds is assessed, using BCP properties and electronegativity values calculated
from the electron density distributions. A mapping of the (3, 73) CPs of the
valence shell concentrations of the oxide anions for bridging SiOSi and
GeOGe dimers reveals a location and disposition of localised nonbonding
electron pairs that is consistent with the bridging angles observed for silicates
and germanates. The BCP properties of the SiO bonds calculated for
representative molecular models of the coesite structure agree with average
values obtained in X-ray diraction studies of coesite and danburite to
within 5%.
In a second mineralogical contribution Gibbs et al.
186
advocate to their
community the use of the Laplacian in delineating those regions of a mineral
surface that are potentially susceptible to electrophilic and nucleophilic attack.
3: Atoms in Molecules 177

They compute Laplacian and bond critical point properties of sul®de bonds
containing ®rst- and second-row main group M-cations and compare them with
oxide bonds. The Laplacian maps of the distributions show that the VSCC of
the sul®de anion is highly polarised and extends into the internuclear region of
the M±S bonds, coalescing with the VSCCs of the more electronegative ®rst-row
cations. On the other hand, maps for a corresponding set of oxide molecules
show that the oxide anion tends to be less polarised and more locally
concentrated in the vicinity of its valence shell, particularly when bonded to
second-row M-cations.
A search for extrema in 7H
2
rreveals maxima in the VSCCs that can be
ascribed to bonded and nonbonded electron pairs. The dierent and distinctive
properties of sul®des and oxides are examined in terms of the number and the
positions of the electron pairs and the topographic features of the Laplacian
maps. The evidence provided by rand its topological properties indicates that
the bonded interactions in sul®des are more directional, for a given M-cation,
than in oxides. The r
b
value and the length of a given M±S bond are reliable
measures of a bonded interaction: the greater the accumulation of rand the
shorter the bond, the greater its shared (covalent) interaction.
A ®nal mineralogical paper by Feth et al.
187
focuses on bonded interactions
in nitride molecules in terms of BCP properties and relative electronegativities.
This study discusses an AIM analysis, which is very similar to the one in the
previous study, but now the MN bonds are compared with MO bonds in
oxides.
5.7 Solid State. ± In their work on ionic materials Pendas et al.
188
enthusiasti-
cally embrace AIM as the de®nitive frame to assign an accurate meaning to
geometric solids. The authors have turned to AIM because they believe that
many of the unsatisfactory characteristics with the ionic radius concept stem
from its rather slippery de®nition in quantum mechanics. They paid particular
attention to the concept of ionic radius, in relation to the shapes of ions in
crystals, and to the various correlations among atomic properties such as
electronegativity and deformability. Using a simple model to ®t their results to
a theoretical frame, Pendas et al. show that ionic bonds display properties in
complete parallelism to those known in covalent bonds. This enables the
unambiguous de®nition of the strength of an ionic bond, which is found to
correspond to Pauling's bond valence. Topological families of ions are dis-
cussed and r-bond length relationships mentioned. According to the authors
the development of physics and chemistry of the solid state has been intimately
linked to the vague concept of atomic size, but using AIM the traditional
concept of ionic radius has been illuminated. AIM uncovers the complexity and
richness of atomic or ionic shapes in the crystalline state. Figure 19 shows an
example of a prototypical ionic basin for one of the three topological families in
the B2 phase of alkali halides.
5.8 Compounds of Atmospheric Interest. ± Alcami and Cooper
189
performed ab
initio calculations on neutral bromine oxide and dioxides and their correspond-
178 Chemical Modelling: Applications and Theory, Volume 1

ing anions in view of their importance in atmospheric chemistry. Both single-
con®guration-based methods {MP2, QCISD, and QCISD(T)} and multi-
con®guration-based methods (CASSCF and CASMP2) have been used. A
topological analysis of r(r
b
,H
2
r
b
,H
b
,e
b
) shows that the nature of the BrO
bond is very dierent within OBrO and BrOO. The authors use H
2
r
b
and H
b
to
gauge the covalent character of bonds.
5.9 Van der Waals Complexes. ± Rayon and Sordo
190
report on the nature of
the interaction in donor±acceptor van der Waals complexes. Ab initio calcula-
tions at the MP2/6±31G** level have been carried out on BH
3
...CO, BF
3
...CO,
BH
3
...NH
3
and BF
3
...NH
3
. They found a good correlation between charge
transfer, as measured by a method based on the expansion of the MOs of a
complex in terms of the MOs of its fragments, and the corresponding bond
lengths. Additional consideration of the energy gaps between the frontier
orbitals involved in the charge transfer allows for rationalisation of the strength
of the donor±acceptor bonds. However, analyses based on natural bond
orbitals and energy density at the BCP suggest that no correlation exists
between charge transfer and bond strength.
6 Hydrogen Bonding
6.1 Reviews. ± In a recent review Alkorta et al.
191
discuss non-conventional
hydrogen bonds (HB), which are much more exotic than the once unconven-
tional C±H...O hydrogen bond. Their survey includes HBs such as C±H...C,
where isocyanides, CO or carbanions and zwitterions act as HB acceptors.
Figure 19 Prototypical ionic basin for one of the three topological families in the B2 phase
of the alkali halides. The small basin corresponds to the cation and the larger
one to the anion
(Reproduced by permission of the American Chemical Society)
3: Atoms in Molecules 179

Furthermore the authors describe: C...H±C bonds, involving highly reactive
carbenes, and similar systems encompassing silylenes and carbynes. They
discuss HB systems that contain pacceptors and dihydrogen bonds, and even
suggest so-called inverse HB complexes, where the H atom provides electrons
and another non-hydrogen atom accepts them as in Li±H...Li±H for example.
Alkorta et al. expand on the work presented in the authoritative review by
Hibbert and Elmsley
192
and embrace the HB criteria proposed by Koch and
Popelier
21
as a more accurate and precise perspective in the understanding of
currently exotic HBs.
6.2 Relationships. ± Espinosa et al.
146
investigated the relationship between the
energetic properties of the HB interaction and the topological overlapping of the
electronic clouds at the H...O BCP. Their study involves a total of 83 X±H...O
(X = C, N, O) HBs, which have been described in terms of the following
topological properties: H
2
r
b
,e
b
,G
b
,V
b
,l
i
(i= 1, 2 or 3), where Gis a kinetic
energy density, Vthe potential energy density, and lan eigenvalue of the
Hessian of r. Espinosa et al. show that the kinetic energy density of the electrons
around the BCP is proportional to the curvature of rat the BCP (Figure 20a),
and that the potential energy density of the electrons around the BCP is linearly
related to the negative curvatures at the BCP (Figure 20b). The topological
variation of the curvatures at the BCP, and therefore changes in the H...O
overlapping, are related to the onset of the repulsion between the electronic
clouds of the basic and acidic atoms.
In their study Alkorta et al.
193
look at the additive properties of r
b
in HB
complexes. A large variety of hydrogen-bonded complexes (protic, hydric and
protic-hydric) leads to the conclusion that r
b
is an additive property when it is
expressed relatively, or more precisely as a dimensionless quantity. The authors
propose an equation, involving r
b
at covalent bonds in the isolated molecules,
which probes the additivity.
6.3 Cooperative Eect. ± The ab initio study of Parra and Zeng
194
focused on
the cooperative eect in mixed dimers and trimers of methanol and tri¯uoro-
methanol. Cooperativity is the enhancement of of the dipole moment in a HB
complex compared to the sum of the dipole moments of the constituents. The
authors interpreted the value of r
b
for the C±O and O±H bonds in four dimers
and four trimers in terms of bond strength. A weakening of the O±H bond is
seen to be favourable for hydrogen bonding. Another indication of cooperativ-
ity is noticed in the strengthening of the HB in the trimer as opposed to the
dimer. Their AIM observations provide a consistent picture in support of the
cooperative eect.
Another study which used AIM in the context of the cooperative eect was
performed by Masella and Flament.
195
In their ab initio computations at the
MP2 level on ®ve dimers and ®ve cyclic trimers, drawn from water, ammonia,
and formaldehyde, they evaluated the density only at the HB BCPs. The authors
mainly use AIM to detect HBs but do not characterise them via topological
properties. Instead the cooperative eect is described in terms of geometry
180 Chemical Modelling: Applications and Theory, Volume 1

Figure 20 (a) Phenomenological behaviour of G
b
versus the positive curvature of rat the
HB BCP. The solid line corresponds to the linear ®tting G
b
= 15.3 l
3
, the
correlation factor being 0.98. (b) Phenomenological behaviour of V
b
versus the
sum of the negative curvatures of rat the HB BCP. The solid line corresponds
to the linear ®tting V
b
= 35.1(l
1
+l
2
) the correlation factor being 0.95
(Reproduced by permission of Elsevier)
3: Atoms in Molecules 181

changes and shifts in vibrational frequencies. Masella and Flament claim that
their results exhibit the great incidence of cooperative eects on the properties
of X±H...Y interactions, which are of importance to understand the properties
of biochemical systems.
6.4 Bifurcated Hydrogen Bonds. ± Rozas, Alkorta and Elguero studied
196
the
nature of bifurcated HBs or three-centred HBs. They point out that the term
bifurcated HB can designate two entirely dierent situations (Figure 21). The
three-centred HBs the authors investigate explain a large number of biological
structures. They are commonly used by biochemists and biologists to account
for certain interactions in biomolecules, such as zwitterionic amino acids, where
they were ®rst seen in 1939. The authors chose dierent families of compounds:
monomers with intramolecular three-centred HBs, dimers with a HB donor and
a molecule with two HB acceptor groups, and trimers with one HB donor and
two HB acceptors. All the systems were optimised at the B3LYP/6±31G* level,
and, in the case of the complexes, the interaction energies were evaluated and
corrected with the BSSE.
The study of Rozas et al. relies on the HB criteria proposed by Koch and
Popelier
21
and proves the existence of bifurcated bond paths and distinguishes
four dierent types of interactions based on the degree of symmetry and
magnitude of the two r
b
values. Therefore, looking at the geometry, electron
density, and energy results, the nature of these HBs as three-centred interactions
has been con®rmed.
6.5 Low-barrier Hydrogen Bonds. ± Schiott et al. investigated the electronic
nature of low-barrier hydrogen bonds (LBHBs) in enzymatic reactions and
published their work in two very similar articles.
197,198
The intramolecular
hydrogen bond in their model system, benzoylacetone (Figure 22), has been
studied with high-level ab initio Hartree-Fock and density functional theory
methods. The transition state (double-well potential) for intramolecular hydro-
gen transfer was located with the barrier estimated to be about 8 kJ mol
71
,
consistent with a LBHB. Upon addition of the zero-point vibration energies to
the total potential energy, the internal barrier vanished, overall suggesting that
the intramolecular hydrogen bond in benzoylacetone is a very strong HB,
Figure 21 Dierent con®gurations to which the term bifurcated has been applied. The
term ``three-centred interaction'' only corresponds to con®guration 1
(Reproduced by permission of the American Chemical Society)
182 Chemical Modelling: Applications and Theory, Volume 1

estimated at 67 kJ mol
71
. To determine the type of bonding in the O...H...O
region the authors looked at the classical triplet (r
b
,H
2
r
b
,e
b
). In general, very
good agreement between the theoretical and X-ray electron density is found,
except for e
b
in the polar C±O bond.
The authors take up the suggestion of Cheeseman et al.
199
that, in the case of
heteroatomic bonds having a large charge transfer, e
b
is not a sensitive indicator
of p-contributions. Rather Cheeseman et al. suggest looking at the ellipticity
over the entire bond. Schiott et al. conclude that there is only weak p-
delocalisation over the O±H bonds and view the O...O...H system in benzoyl-
acetone as a 3-centre, 4-electron s-bond with considerable polar character.
Therefore, the HB gains stabilisation from both covalency and from the normal
electrostatic interactions found for long, weak HBs. Based on comparisons with
other systems having short-strong hydrogen bonds or LBHBs, it is proposed
that all short-strong and LBHB systems possess similar electronic features in the
Figure 22 Optimised structures for the two hydrogen-bonded cis-b-keto-enol isomers at
the B3LYP/6±311G(d,p) level of theory
(Reproduced by permission of the American Chemical Society)
3: Atoms in Molecules 183

hydrogen-bonded region, namely polar covalent bonds between the hydrogen
atom and both heteroatoms in question.
6.6 Dihydrogen Bonds. ± Recent theoretical and experimental studies on
transition metal complexes involving a new type of interaction EH...HX
(where E is a transition or alkali metal or boron and X is any electronegative
atom or group) have stimulated much interest. Kulkarni et al.
200
systematically
studied the dihydrogen bonds in the complexes and dimers of complexes
involving the main group elements (LiH, BH
3
, AlH
3
with HF, H
2
O, NH
3
and
their dimers) using ab initio calculations at the MP2 level. The H...H bond
energy in (BH
3
HF)
2
, (BH
3
H
2
O)
2
, and (AlH
3
H
2
O)
2
is analogous to the conven-
tional moderate or weak hydrogen bond. The bonding features of these
complexes and their dimers are analyzed via the quantities r
b
,H
2
r
b
and the
ellipticity e
b
. The decomposition analysis of interaction energies of dimers
reveals the predominance of electrostatic contributions followed by charge
transfer and polarisation.
6.7 Very Strong Hydrogen Bonds. ± In their contribution on very strong HBs
Gonzalez et al.
201
conclude that the dimerisation energies of posphinic acid (PA)
and its dimethyl derivative (DMPA) are the highest reported so far for neutral
homodimers in the gas phase (order 100 kJ mol
71
). To the authors' surprise
there is a complete lack of theoretical studies on these dimers, which is why they
performed high-level ab initio calculations on these systems. Gonzalez et al. ®nd
that for O...H the r
b
values can be as high as 0.054 a.u., although they reported
0.046 a.u. for the formic acid dimer, which is signi®cantly larger according to
the authors.
6.8 Organic Compounds. ± Kovacs and Hargittai
202
studied the potential energy
hypersurface of 2-tri¯uoromethylresorcinol and 2,6-bis(tri¯uoromethyl)phenol
at HF/6±31G** and MP2/6±31G** levels. The global minimum is stabilised by
two HBs in 2-tri¯uoromethylresorcinol and by one in 2,6-bis(tri¯uoromethyl)-
phenol. The authors provide a list of detailed geometrical changes in these
molecules due to the formation of the OH...F HBs compared to their ``parent''
compounds, phenol, tri¯uoromethylbenzene, resorcinol and meta-bis(tri¯uoro-
methyl)benzene. It is concluded that all the HBs in the title compounds can be
characterised by a BCP between the interacting F and H nuclei and by a bent
shape of the bond path. No atomic (integrated) properties are calculated in this
work.
According to Alkorta et al.
203
radicals are poor HB acceptors and the
strength of the HBs qualitatively correlates with the molecular electrostatic
potential (MEP) minimum of the isolated radicals. They studied the ability of
carbon radicals to act as HB acceptors using three well established ab initio
methods, i.e. B3LYP, MP2 and QCISD. The complexes formed by four
radicals {H
3
(d), C
2
H
4
(t), (CH
3
)
2
C(t) and (CH
3
)
3
C(d)} with four standard
hydrogen bond donors {H, H
2
O, HCN and H
3
N} were studied and their
geometry, interaction energy, and electronic properties were analyzed within
184 Chemical Modelling: Applications and Theory, Volume 1

the AIM framework. Atomic properties indicate that HBs complexes invol-
ving radicals behave dierently from other HBs formed between neutral
molecules.
In another contribution
204
Alkorta et al. focused on nine strained hydro-
carbon compounds, ®ve simple hydrocarbons and HCN, each forming a
complex with ammonia. The strained compounds have been selected on the
basis of an increasing number of three- and four-membered rings. The ability of
the strained hydrocarbons to act as HB donors has been explored at B3LYP
level using the 6±31G* and 6±311++G** basis sets. The characteristics of the
HB formed were correlated with geometrical parameters (number of three- and
four-membered rings, bond angles, and HB bond distances), electronic charac-
teristics of the complexes and isolated monomers (r
b
, atomic charges, and
dipolar magnetisation), and other properties (gas-phase acidities and atomic
volume and energy). For example, an excellent linear relationship is found
between rat the HB BCP and the value of the Laplacian there. The results have
been rationalised on the basis of a simple strain model and compared with
nonsaturated hydrocarbons with donor C±H groups.
In their work on three-membered heterocycles Alcami et al.
205
aimed to ®nd
out if stabilisation takes place by cation association to the zwitterionic
tautomers of the studied compounds. The authors chose Li
+
as a suitable
reference acid and include species derived from aziridine, diaziridine, triaziri-
dine, oxirane, dioxirane, oxadiaziridine and dioxaaziridine in their survey using
high level ab initio methods. The eects of the tautomerisation on the bonding
characteristics of the dierent parent compounds were investigated mainly by
means of the Laplacian of r.
6.9 Biochemical Compounds. ± AIM has been invoked in a study in the ®eld of
molecular biology
206
where Luisi et al. attempt to understand why opposing
exocyclic amino groups in duplex DNA (Figure 23a,b) may form close NH...N
contacts. In order to comprehend the nature of such interactions, the authors
examined the CSD database of high-resolution crystal structures of small
molecules. They also performed ab initio calculations on model complexes,
which indicate that the hydrogen±amino contact is improved energetically when
the amino group moves from the conventional geometry, where all atoms are
co-planar with the base, to one in which the hydrogen atoms lie out of the plane
and the nitrogen is at the apex of a pyramid, resulting in polarisation of the
amino group. Only a few AIM criteria are invoked to characterise an N±H...N
HB (Figure 23c,d) in the ortho-amino-pyridine dimer. The authors speculate
that the amino group can accept HBs under special circumstances in macro-
molecules, and that this ability might play a mechanistic role in catalytic
processes such as deamination or amino transfer.
Epibatidine, an alkaloid discovered in 1974, has attracted considerable
interest because it appears to be the ®rst compound exhibiting analgesic activity
as a selective and potent nicotinic receptor agonist. Campillo et al.
207
performed
a theoretical study on the conformational pro®le of epibatidine and its
protonated form (Figure 24) using molecular mechanics, semi-empirical and ab
3: Atoms in Molecules 185

initio methods. The stability of the minima has been explained using the AIM
methodology. In particular, the intramolecular HBs, although weak as shown
by the low r
b
values, are able to explain the stability of the A conformers versus
the B conformers. The authors discovered a second interaction, denoted as
H...H and traditionally associated with repulsive interactions. However, a
model calculation of the ammonium/benzene complex, with one of the
ammonia hydrogens pointing towards a benzene hydrogen, yielded an opti-
mised geometry with H...H distances close to those observed in the protonated
epibatidine. This model calculation indicates that the epibatidine H ...H inter-
actions can be attractive, according to Campillo et al.
Hernandez-Laguna et al.
208
examined the theoretical proton anities of
histamine, amthamine and some substituted derivatives. The internal HBs
Figure 23 (a) and (b) Schematic representation of a base-pair showing the propeller twist
operation. (c) A possible case for pyramidalisation of the amino group in the
deanine-5-methylcytosine contact, based on calculations. (d) An exaggerated
schematic representation of pyramidalisation in aminobenzene, to illustrate the
HB to the nitrogen lone pair that is proposed to occur in (c)
(Reproduced by permission of Academic Press)
Figure 24 Neutral (forms A and B) and protonated epibatidine with numbering
(Reproduced by permission of Elsevier)
186 Chemical Modelling: Applications and Theory, Volume 1

occurring in some conformers of these important biological molecules have
been characterised with the help of an AIM analysis.
6.10 Compounds of Atmospheric Importance. ± Berski et al.
209
applied AIM to
molecules of atmospheric interest, such the dimers of hypo¯uorous acid,
(HOF)
2
, which was studied at CCSD(T)/6±311++G(2d,2p)//MP2/6±
311++G(2d,2p) level. The authors noticed that Mulliken charges for the
atoms in HOF are not realistic, even at MP2/6±311++G(3df,3pd) level,
because the oxygen's atomic charge turns out to be more negative than that of
¯uorine. The CHELP method
210
also fails to be compatible with expected
electronegativity scales. However, the net charge on F according to AIM is
70.23e and 70.15e for O. Berski et al. found two structures, a cyclic dimer
(planar, C
2h
), which is only 1.5 kJ mol
71
higher than the linear dimer (C
1
). The
authors introduce the ELF in a comprehensive way and show that it reveals a
shift in electron density from the F±O bonds to the regions of the free, valence
electron pairs located at the ¯uorine atoms in the cyclic dimer (Figure 25). The
cyclic structure which assumed planar (C
2h
) geometry was found to be unstable
(transition state) within a dielectric medium.
Figure 25 Two-dimensional representation of the ELF map for the (HOF)
2
in the cyclic
conformation. Contour lines are spaced at 0.05 au; distance is measured in Bohr
on both axes
(Reproduced by permission of the American Chemical Society)
3: Atoms in Molecules 187

7 Reactions
7.1 Organic Compounds. ± Calvo-Losada et al.
211
report a detailed investiga-
tion of the reaction path for the thermal rearrangement of 3,4-dihydro-1aH-
azirine[2,3-c]pyrrol-2-one to yield the cyanoketene-formaldimine complex,
carried out at MP2/6±31G* and B3LYP/6±31G* level (Figure 26). The
authors used AIM to characterise bonds in 2H-azirine, in particular the
ellipticity of the C
3
±N bond. The authors designate the C
3
±N single bond as a
moderate bonding interaction intermediate between a typical shared and closed-
shell interaction. Calvo-Losada et al. continue their AIM analysis with the BCP
properties and ®tted bond orders of the azirine intermediate. From the
eigenvalues of the Hessian of rthey conclude that the electronic rearrangement
through the transition state skeleton should be interpreted as a preferential p
accumulation of the electron density.
Episulfonium ions (ESI) are three-membered sulfur-containing cationic
species that have been postulated as intermediates in many reactions. Mechan-
isms of the reaction of 1-alkoxy-2-(arylsulfanyl)alkyl halides with dierent
nucleophiles in the presence of a Lewis acid appear to be more complex
because of the possibility of forming either ESIs or oxonium ions (IOS) as
intermediates (Figure 27). Dudley
212
reported equilibrium structures and
harmonic vibrational frequencies of two model compounds related to the
intermediates of nucleophilic attack on 1-alkoxy-2-(arylsulfanyl)alkyl halides
Figure 26 The thermal rearrangement of 3,4-dihydro-1aH-azirine[2,3-c]pyrrol-2-one to
yield the cyanoketene±formaldimine complex
(Reproduced by permission of Wiley)
Figure 27 Reaction scheme involving the 1-alkoxy-2-sulfanylethan-1-yl cation
(Reproduced by permission of the American Chemical Society)
188 Chemical Modelling: Applications and Theory, Volume 1

and 2-(arylsulfanyl)pyranosyl halides at HF and MP2 level. Mulliken and AIM
atomic charges in the model ions were compared with corresponding sites in
oxonium and episulfonium. The studies of the model ions suggest that the
intermediate is more like an oxonium ion than an episulfonium ion, although
the sulfur is critical to the stereoselectivity. The authors observed that a typical
positive formal charge on the oxygen atom ought to be assigned a fairly large
negative charge according to the calculations. Not only are the partial charges
contrary to the standard view but also the C±O bonds in the studied structures
may not be ``classical'' sand pbonds.
Formamide is the simplest amide containing a prototype HNCÐ
ÐO linkage,
and is therefore frequently used as a model to understand proton exchange
processes in peptides and proteins. The aim of the work of Luna et al.
213
was to
investigate whether the isomerisation of formamide to its tautomers, formami-
dic acid and (aminohydroxy)carbene can be catalyzed by association with
closed-shell transition-metal monocations such as Cu
+
. The structures, relative
stabilities, and bonding characteristics of complexes of formamide, formamidic
acid, and (aminohydroxy)carbene with Cu
+
have been investigated through the
use of high-level density functional theory (DFT) calculations. The values of r
b
,
H
2
r
b
and H
b
were invoked to investigate the bonding features of the complexes.
The AIM analysis show that the electron densities at the N±Cu, O±Cu and
C±Cu BCPs are almost an order of magnitude larger than the typical values
(0.012±0.038) found in ionic linkages.
Fang and Fu
214
invoked AIM as a suitable tool for studying whether or not
cyclisation has occurred in the cycloaddition between a ketene and an imine.
The reaction of a ketene with pyridine for example would give rise to a ketene-
pyridine ylide.
Scheme 1 Reproduced with permission of Elsevier
The formal positive charge on nitrogen in the ylide is not compatible with
Mulliken or AIM charges, which agree in sign for the stationary points in the
examined reaction. However, overall there is charge transfer (about 0.3e) from
the pyridine moiety to the ketene moiety in the ylide, leaving a net positive
charge on the pyridine, and con®rming that the ylide has some zwitterionic
character. The quadruplet (r
b
,H
2
r
b
,e
b
and H
b
) was used to extract information
about the nature of the bonds, for example in the ylide (Figure 28)
The last paper mentioned in this subsection is not a study of a reaction pro®le,
but a contribution which cannot be categorised in any other section of this
report, other than the current one, in view of its relevance to the general
understanding of organic addition reactions. Halonium ions have played vital
roles as intermediates in organic chemistry. Bridged species such as the
bromonium ion were proposed in 1939 to explain the stereochemistry of Br
2
3: Atoms in Molecules 189

addition to alkenes. Such ions have only been observed directly after the
development of experimental conditions for stabilising cations in superacids.
Damrauer et al.
215
present a computational study of halonium ions of
cyclohexene and cyclopentene. Figure 29 shows the atomic AIM charges for
some geometries of C
6
H
10
X
+
(X = F, Cl or Br). According to the authors it
seems clear that the trade-o between halogen electronegativity and so-called
back-bonding eects is very ®nely tuned in these systems. It is striking that in
both cation series, whether for open or bridged cations or for 1,2- or 1,4-bridges,
that the F atoms have quite negative charges, the chlorine atoms are nearly
neutral and the bromine atoms are more positive still. A detailed analysis
suggests that electronegativity is quite dramatic and that even in the 1-halo
cation the C±F bond is still polarised C
+
±F
7
. Inspection of the electron
distributions suggests that there are electrostatic and size eects that dominate
the stability of the cations.
7.2 Inorganic Compounds. ± Hamilton
216
characterised the proton transfer in
the isoelectronic species HO
2
7
and HF
2
+
. Electron densities were calculated at
the QCISD/6±311++G(2d, 2p) level for the nonlinear equilibrium geometry
and the C
2v
saddle point and linear saddle point geometries. AIM is applied to
partition rinto its atomic components and atomic and molecular properties are
calculated. These quantities are used to characterise the proton dynamics as
similar to internal rotation.
Kulkarni and Koga
217
performed a similar study to the previous one from the
point of view of AIM, i.e. the characterisation of bonds in reaction schemes.
This time the (r
b
,H
2
r
b
and e
b
) triplet was used in the mechanistic investigation
of samarium(III)-catalyzed ole®n hydroboration reaction via ab initio calcula-
Figure 28 Molecular graphs and Laplacian distribution in the stationary point corre-
sponding to the ylide
(Reproduced by permission of Wiley)
190 Chemical Modelling: Applications and Theory, Volume 1

tions. In their work the bonding features of all stationary structures on the
reaction path are obtained from the topological analysis of the corresponding
electron density distributions. The authors note that the small core ECP
they employed leads to correct topological behaviour, for it is known that
the omission of considerable core density may lead to severely corrupted
topologies.
A reaction mechanism involving a transition metal was studied by Decker
and Klobukowski
218
who investigated the role of the acetylene ligand from a
density functional perspective in M(CO)
4
(C
2
H
2
) (M = Fe, Ru or Os). Recent
kinetics experiments have shown that the rate of CO substitution in complexes
of the type M(CO)
4
(C
2
R
2
) is accelerated by factors of 10
2
±10
13
over their
respective pentacarbonyl complexes. These substitution reactions have been
shown to be dissociative in nature and show a marked metal dependence on the
rate. The origin of the increased reactivity of these alkyne complexes was
Figure 29 Atomic AIM charges for the C
6
H
10
X
+
(X = F, Cl or Br) some geometries
obtained at MP2 level
(Reproduced by permission of the American Chemical Society)
3: Atoms in Molecules 191

studied with BLYP, using both ECP and all-electron basis sets, in conjunction
with Frenking's charge decomposition analysis (CDA) scheme and AIM. By
using the CDA scheme the nature of the acetylene ligand was characterised in
both the reactant and the unsaturated dissociation product. Acetylene was
found to act as a two-electron donor in the reactant complex and as a four-
electron donor increasing the stability of the otherwise 16-electron unsaturated
dissociation product. The predicted structural changes along with the results of
the AIM analysis fully support the CDA ®ndings.
8 Conclusion
Since the early seventies the theory of AIM has grown and gained acceptance in
a wide community of users, ranging from mineralogists to enzyme crystal-
lographers. Although controversies occasionally arise on some aspects of AIM
the theory has reached the stage where it is used as a practical tool by
crystallographers and experimental (synthetic) groups with access to an ab
initio package. AIM's use to characterise bonds is a prominent application but
discussions based on atomic populations and structural stability feature as well.
Already now, over a hundred research papers appear yearly that devote
extensive or even exclusive space to AIM.
Provided continuous investments into properly interfaced, well-documented,
well-maintained, aordable and user-friendly software are made and provided
computer hardware keeps improving at accelerated pace AIM has the potential
to become a prime method of extracting chemical information from the electron
density, a fundamental quantity that has hitherto been almost neglected in
chemistry.
9 Disclaimer
The primary set of references for this manuscript was retrieved from the Science
Citation Index, accessed via BIDS (Bath Information and Data Services,
University of Bath, UK). The completeness of the ®rst automated selection of
papers depends on the accuracy of the information present in the Science
Citation Index, the determination of the non-trivial publication time window
(June 1998 to June 1999) set by our publisher, and the accuracy of the reference
to Bader's 1990 monograph in the original paper. All papers appearing in
readily obtainable journals were screened in full and included if they were of
sucient interest from the point of view of AIM. When in doubt it was decided
to exclude a possible paper in favour of other papers with higher AIM content.
This choice was determined by space restrictions and the readability of the
report. We apologise to authors for the possibly erroneous omission of their
work, but believe that we have provided a rather complete and unbiased
re¯ection of the research activity in the AIM area.
192 Chemical Modelling: Applications and Theory, Volume 1

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198 Chemical Modelling: Applications and Theory, Volume 1