Toward a Consistent Interpretation of the QTAIM: Tortuous Link between Chemical Bonds, Interactions, and Bond/Line Paths

Abstract

Currently, bonding analysis of molecules based on the Quantum Theory of Atoms in Molecules (QTAIM) is popular; however, “misinterpretations” of the QTAIM analysis are also very frequent. In this contribution the chemical relevance of the bond path as one of the key topological entities emerging from the QTAIM’s topological analysis of the one-electron density is reconsidered. The role of nuclear vibrations on the topological analysis is investigated demonstrating that the bond paths are not indicators of chemical bonds. Also, it is argued that the detection of the bond paths is not necessary for the “interaction” to be present between two atoms in a molecule. The conceptual disentanglement of chemical bonds/interactions from the bonds paths, which are alternatively termed “line paths” in this contribution, dismisses many superficial inconsistencies. Such inconsistencies emerge from the presence/absence of the line paths in places of a molecule in which chemical intuition or alternative bonding analysis does not support the presence/absence of a chemical bond. Moreover, computational QTAIM studies have been performed on some “problematic” molecules, which were considered previously by other authors, and the role of nuclear vibrations on presence/absence of the line paths is studied demonstrating that a bonding pattern consistent with other theoretical schemes appears after a careful QTAIM analysis and a new “interpretation” of data is performed.

Full text

&
Bond Theory
Toward a Consistent Interpretation of the QTAIM: Tortuous Link
between Chemical Bonds, Interactions, and Bond/Line Paths**
Cina Foroutan-Nejad,[a] Shant Shahbazian,*[b] and Radek Marek[a, c]
Abstract: Currently, bonding analysis of molecules based on
the Quantum Theory of Atoms in Molecules (QTAIM) is pop-
ular; however, “misinterpretations” of the QTAIM analysis are
also very frequent. In this contribution the chemical rele-
vance of the bond path as one of the key topological enti-
ties emerging from the QTAIM’s topological analysis of the
one-electron density is reconsidered. The role of nuclear vi-
brations on the topological analysis is investigated demon-
strating that the bond paths are not indicators of chemical
bonds. Also, it is argued that the detection of the bond
paths is not necessary for the “interaction” to be present be-
tween two atoms in a molecule. The conceptual disentan-
glement of chemical bonds/interactions from the bonds
paths, which are alternatively termed “line paths” in this con-
tribution, dismisses many superficial inconsistencies. Such in-
consistencies emerge from the presence/absence of the line
paths in places of a molecule in which chemical intuition or
alternative bonding analysis does not support the presence/
absence of a chemical bond. Moreover, computational
QTAIM studies have been performed on some “problematic”
molecules, which were considered previously by other au-
thors, and the role of nuclear vibrations on presence/ab-
sence of the line paths is studied demonstrating that a bond-
ing pattern consistent with other theoretical schemes ap-
pears after a careful QTAIM analysis and a new “interpreta-
tion” of data is performed.
Introduction
Stating the problem
Advanced theories in physics and chemistry are usually com-
posed of two aspects: A mathematical structure/formalism dis-
closing the basic entities of the theory and their mathematical
relationships, and an “interpretative” recipe of basic entities of
the theory. The latter discloses the qualitative meaning of the
basic entities and their relation to other known entities within
and beyond the theory. The connection between the mathe-
matical formalism of a theory and its interpretation is always
subtle. This is best illustrated in the case of the quantum me-
chanics. Although, almost all physicists agree on its formalism,
after ninety years from its emergence, large numbers of inter-
pretations have been flourished that each introduces a new
“meaning” for the “mathematical symbols/entities” of the
theory.[1] Quantum mechanics is the best example demonstrat-
ing the fact that generally, formalism does not in itself impose
an interpretation. In other words, formalism may be compati-
ble/coexist with a number of interpretations. In contrast to this
relative autonomy of formalism and interpretation, an accepta-
ble interpretation of a theory must be “self-consistent”, free
from “contradictions”. A proposed meaning for an entity
cannot lead to conclusions that challenge the meaning of
other entities of a theory.
The Quantum Theory of Atoms in Molecules (QTAIM) is also
composed of a mathematical formalism and a “chemical inter-
pretation” that introduces the glue linking the formalism and
chemical concepts previously defined in the chemical dis-
course.[2–4] The ignorance of making a distinction between the
formalism and the interpretation has been one of the matters
behind the most misunderstandings and controversies around
the QTAIM and its chemical applications. Indeed, this difference
has not always been clearly stressed and disclosed in the origi-
nal literature of the QTAIM. The methodology of recognizing
chemical bonds within the context of the QTAIM is one of
such “interpretative” problems that are scrutinized in this
report. Particularly, the premise of equivalence between chemi-
cal bonds and the bond paths (BPs) (see below) is one of
these “misinterpretations” that is criticized in detail.
In view of these interpretative problems, the ingredients of
a recently proposed interpretation are discussed and scruti-
nized in this report in more detail.[5] First, it is demonstrated
that the detection of the BPs between two atoms in a mole-
cule, emerging from natural alignment of the gradient vector
[a] Dr. C. Foroutan-Nejad, Prof. Dr. R. Marek
National Center for Biomolecular Research
Faculty of Science, Masaryk University
Kamenice 5A4, 62500 Brno (Czech Republic)
[b] Prof. Dr. S. Shahbazian
Faculty of Chemistry, Shahid Beheshti University
G. C., Evin, Tehran (Iran) 19839
P.O. Box 19395-4716
Fax: (+98) 21-22431661
E-mail: chemist_shant@yahoo.com
[c] Prof. Dr. R. Marek
CEITEC-Central European Institute of Technology
Masaryk University, Kamenice 5A4, 62500 Brno (Czech Republic)
[**] QTAIM=Quantum Theory of Atoms in Molecules.
Supporting information for this article is available on the WWW under
http://dx.doi.org/10.1002/chem.201402177.
Chem. Eur. J. 2014,20, 10140 – 10152  2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim10140
Full PaperDOI: 10.1002/chem.201402177

field of the one-electron density of a molecule,[2–4] is neither
necessary nor sufficient condition for the presence of a chemi-
cal bond between those atoms. Accordingly, the conceptual
“disentanglement” of chemical bonds and bond paths not only
resolves many misinterpretations regarding the applications of
the QTAIM, but also reveals a clear need for a fresh viewpoint
regarding the nature of interactions of atoms in molecules
(AIMs) as is discussed in subsequent sections. It is important to
emphasize that this disentanglement has been somehow men-
tioned previously, though sometimes just implicitly, by some
researchers in various contexts.[6–12] However, in contrast to the
previous discussions, our line of reasoning in this paper is
solely based on the formalism of the orthodox QTAIM itself.
What is particularly emphasized is the fact that the assumed
equivalence between chemical bonds and BPs does not lead
to a legitimate ingredient of a consistent interpretation of the
QTAIM formalism. The relation of BPs to weak interactions be-
tween AIMs, usually termed non-bonded interactions, are also
addressed briefly with an emphasis on the point that like in
the case of chemical bonds, BPs do not directly recognize in-
teracting AIMs. Therefore, it is argued that energetic criterion
is needed for a transparent description of the nature of weakly
interacting AIMs. Rather than claiming for “the” final interpreta-
tion, one may hope that the proposed “reinterpretation” in
this paper, paves the way towards “a” consistent interpreta-
tion.
The paper is organized as follows: in the subsequent section
it is argued, solely based on the fabric of the topological analy-
sis, that a number of molecular graphs (MGs) are associated
with given local minimum structures. Thus, the tacit assump-
tion that there is a one-to-one relation between each molecule
in local minimum and its MG, derived at the equilibrium geom-
etry, is not generally compatible with the formalism of the
QTAIM. The implications of this observation on the chemical in-
terpretation of the QTAIM are emphasized. In the next section,
computational examples are considered demonstrating the im-
portance of the molecular vibrations in materializing molecules
with Multiple-MGs (M-MGs) status. It is demonstrated that in
several molecules certain BPs appear and disappear during
molecular vibrations. Also, it is shown that not all AIMs with
significant interactions are linked with BPs. Finally, in the last
two sections it is argued that the connection between chemi-
cal bonds and the concepts introduced within the context of
QTAIM is not straightforward, calling for new theoretical tools
yet to be invented or developed further. It is important to em-
phasize that there is an extensive literature on the chemical
applications of the QTAIM and just a handful of them are cited
in the reference list that are of primary importance to the pur-
pose of this report.
Discussion
Molecular graphs and bond paths: Dynamic versus static
viewpoint and its chemical implications
The distribution of electron density of a molecule parametrical-
ly depends on the nuclear geometry, thus the result of the rel-
evant topological analysis of the gradient of the one-electron
density also depends on the chosen geometry.[2–4] The theoreti-
cal framework of the topological analysis dictates that “contin-
uous” changes of the geometrical parameters are accom-
plished with sudden, “discontinuous”, appearance/disappear-
ance of the critical points (CPs) accompanying the variations of
MGs. Accordingly, in a systematic topological analysis of water
molecule at various geometries (in its electronic ground state),
as a classic example, one observes various distinct MGs and
concomitant BPs (see subsection 3.2.1 in ref. [2]). Even in
simple diatomics, systematic survey of various inter-nuclear dis-
tances reveals a general pattern that upon contraction of inter-
nuclear distances, non-nuclear attractors always emerge dem-
onstrating that to each diatomic, two (sometimes even more)
MGs are associated.[13, 14] Usually, such small inter-nuclear dis-
tances and concomitant MGs are typically accessible only upon
using extreme hydrostatic pressures confirming unambiguous-
ly the “dynamic” nature of MGs in diatomics, that is, appear-
ance/disappearance of non-nuclear attractors.[15] These obser-
vations tacitly imply that molecular vibrations may also trigger
geometrical variations, upon contraction/extension of inter-nu-
clear distances, yielding new MGs distinct from that of the
equilibrium geometry.
In contrast to these observations and their implications,
most computational QTAIM studies are usually restricted to the
equilibrium geometries implicitly assuming that the amplitudes
of the nuclear vibrations do not usually suffice to vary the MGs
observed at these geometries. Accordingly, attributing a single
MG, just computed at the equilibrium geometry, to each local-
minimum structure is commonplace. Notwithstanding, there is
no reason to believe that this assumption is universally valid
and in the subsequent section specific examples are given
demonstrating the contrary. Based on the theory of molecular
vibrations,[16] every local minimum is “surrounded” by a set of
accessible non-equilibrium geometries and their topological
analysis may reveal more than a single MG. As the nuclear vi-
brational dynamics may alter the MG of the equilibrium geom-
etry, yielding new MGs at some non-equilibrium geometries,
one is faced with a “dynamic” viewpoint of MGs against the
usual “static” single MG viewpoint. However, before proceed-
ing to the case studies describing real examples, it is legitimate
to discuss chemical implications of the presence of such M-
MGs molecules.
Let us assume that there are two or more MGs each associ-
ated with a subset of accessible geometries; for such molecule,
the MG of the equilibrium geometry is just one accessible MG.
Roughly speaking, this situation is similar to the case of reso-
nance structures used to represent the electronic structure.
However, in contrast to the resonance hybrid, no “mean”/
hybrid MG is introduced within the context of the QTAIM and
one is faced with M-MGs case. A question may naturally arise
as to what will happen to BPs of these M-MGs cases? Obvious-
ly, since MGs are different at various geometries, some BPs
must disappear in some MGs while appearing in others.
Coming back to the originally stated problem, assuming the
equivalence of chemical bonds and BPs, an unpleasant situa-
tion is emerging upon the molecular vibrations: Chemical
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bonds are appearing and disappearing without any “external”
influences. This is in sharp contrast with the classic chemical
discourse assuming that chemical bonds are formed and dis-
rupted in chemical reactions, whereas nuclear vibrations and
internal rotations are just part of the “internal” dynamics of
a molecule. Clearly, insisting on the equivalence between
chemical bonds and BPs does not lead to a “consistent” picture
because one must propose quite unnaturally that nuclear vi-
brations are capable of triggering the formation or disruption
of chemical bonds. Alternatively, one may propose that solely
the MG of the equilibrium geometry has to be used for bond-
ing analysis of a molecule, which is completely unjustified as
well. Particularly, ignoring the internal dynamics is not legiti-
mate because recently, the nuclear vibrational dynamics has
been incorporated explicitly within the context of the newly
developed multicomponent QTAIM.[17–22] Thus, it is safe to con-
clude unequivocally that BPs are not chemical bonds.[23] The
fact that BPs are usually observed between the AIMs that are
chemically bonded does not invalidate this conclusion but it
merely demonstrates that in most but not all cases chemically
bonded AIMs are linked by BPs.[24] Accordingly, using the termi-
nology used by Farrugia and co-workers,[25] observing chemical
bonds without “chemical bonding”, namely, finding chemical
bonds between AIMs that are not linked by BPs, is not genu-
inely surprising and certainly not against the basic formalism
of the QTAIM. Indeed, the theory of electron localization/deloc-
alization within/between AIMs clearly demonstrates that the
AIMs, which are not linked by BPs, are also “communicating”
(exchanging electrons) and in certain cases these communica-
tions are not damped regularly with the separation distance
(see, in particular, references [28–38] for the case of the para
carbon atomic basins in benzene).[26–38] The localization/deloc-
alization indices are deduced from the second-order reduced
density matrices, as the carriers of the two-electron correlation
information.[39, 40] So, there is no reason, at least not in the
mathematical fabric of the QTAIM, to believe that the “lower
order” one-electron density function and its topological charac-
teristics, for example, BPs, are the only means to be used to
deduce the nature of the interactions between AIMs.[41] Of
course the reverse is also true and there are cases when two
AIMs are linked by BPs, but are neither appreciably exchanging
electrons nor participating in stabilizing electrostatic interac-
tions; the neutral van der Waals complexes and hydrocarbon
contacts in crystals provide excellent examples.[42, 43] Therefore,
using terms like “missed” or “expected” BPs, inferred from
chemical intuition, are just portraying a misunderstanding re-
garding the role of BPs within the context of the QTAIM. More
generally, one may conclude that the QTAIM has its own
jargon, whereas emerging entities, for example, BPs, generally
do not have a one-to-one relation with chemical concepts, for
example, chemical bonds, introduced within classic chemical
discourse. Accordingly, one may go a step further and stress
that MGs are also not chemical/molecular structures because
BPs are not chemical bonds. Once again this conclusion does
not oppose the fact that in many molecules MGs are topologi-
cally equivalent to the molecular structures but it disapproves
the notion of “universal” equivalence between MGs and the
chemical structures.[24] The variations of MGs upon nuclear vi-
brations demonstrate that no such universal claim is accepta-
ble as it would imply that chemical structures are varied with-
out external effects/forces just by nuclear vibrations.
Since the terminology related to the word “bond” may be
a source of confusion in the interpretation of QTAIM, in the
rest of this paper the word “line” is used instead, in agreement
with the usage of the words “ring” and “cage”, emphasizing
the geometrical nature rather than the chemical relevance. Ac-
cordingly, bond critical points (BCPs) are called “line critical
points” (LCPs), whereas BPs are called “line paths” (LPs)
(beyond equilibrium geometries, the phrase “atomic interac-
tion lines” are used instead of BPs ;[2–4] however, in this report
no distinction is being made between topological entities at
equilibrium and non-equilibrium geometries thus, LPs are used
regardless of the nature of given geometry). Furthermore, in-
stead of the phrase “bonded atoms”, denoting two AIMs shar-
ing an inter-atomic surface, they are just called “neighbors”,
whereas the phrase “passionate neighbors” is used for AIMs
with an inter-atomic surface appearing and disappearing
during the nuclear vibrations.[5]
Let us also briefly mention and discard immediately an alter-
native “chemical interpretation” of LPs, scattered in literature,
which assumes the presence of LPs denotes that the corre-
sponding AIMs are “interacting”. At first glance, this is seem-
ingly a weaker claim compared to the previous discussion be-
cause it replaces the chemical bond between AIMs, an icon of
strong interaction, with interacting AIMs that do not need to
be strongly affecting each other. Apart from all previously dis-
cussed obstacles, a new inconvenience emerges as in the
theory of intermolecular forces, each of two atoms in a mole-
cule/complex, regardless of their inter-atomic distance, are in-
teracting at least through long-range London-type forces.[44]
Therefore, restricting the interaction of AIMs just to the neigh-
bors is quite artificial and one may further declare that LPs are
not denoting “interacting AIMs”, though it is probable, but not
inevitable, in general that each atom in a molecule interacts,
be it stabilizing or destabilizing, more strongly with its neigh-
bors rather than distant atoms. All in all, the mutual interac-
tions of AIMs are inevitably tied to the energetic aspects of the
analysis of chemical bonds that are discussed in the last two
sections in more detail.
Multiple-MGs versus single-MG molecules
The exotic (B6C)2has been demonstrated to be an interesting
M-MGs example with extremely flat electron density between
the peripheral boron atoms and the central carbon basins.[45–47]
In this molecule a large number of MGs are accessible even
through the zero-point vibrations that are operative at the ab-
solute zero temperature.[46] Although this single example of
a topologically floppy molecule suffices to dismiss the univer-
sal validity of the single-MG paradigm, the peculiarity of this il-
lusive species may obscure the main reasoning behind the M-
MGs paradigm. However, examples considered in this section
are typical molecules, which have been analyzed previously by
other researchers, thus casting no doubt on the importance of
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M-MGs paradigm and its implications on the consistent inter-
pretation of the QTAIM. For brevity, the details of all used com-
putational levels (method/basis set), employed computational
packages as well as related references are found in the Sup-
porting Information. However, the strategy used to prepare
non-equilibrium geometries are disclosed in detail.
Every molecule experiences non-equilibrium geometries be-
cause of the nuclear vibrations; in polyatomic molecules these
vibrations are described by the normal modes.[16] To generate
non-equilibrium geometries for our topological analysis, we
surveyed the vibrational spectrum of each molecule seeking
for large-amplitude and low-frequency normal modes with
small force constants (f<0.7 mDyn1). Then, a set of “testing”
non-equilibrium geometries were generated along each
normal mode. Only those members of the testing set were
used for topological analysis whose energy difference to the
total energy of the equilibrium geometry was at most equal to
the energy of the first excited vibrational state of the corre-
sponding normal mode, which is populated significantly at
room temperature. The molecules revealing altered MGs at
some of their non-equilibrium geometries, differing from that
of the equilibrium geometry were classified as M-MGs. On the
contrary, if nuclear vibrations did
not alter MG of the equilibrium
geometry, the molecule was
classified as single-MG. However,
for such cases the search was ex-
panded seeking for higher
energy non-equilibrium geome-
tries containing new MGs
though they are not accessible
at room temperature. In reality
each nucleus has a complicated
motion since it participates in all
normal modes not just a single
mode. As a supplementary step,
to simulate this complicated dy-
namics and the perturbation in-
duced by other modes, nuclear
positions of the non-equilibrium
geometries along the normal
modes containing new MGs
were “randomly drifted”. A handful of these non-equilibrium
geometries, which retain new MGs but are energetically near
to the equilibrium geometry, are used for the quantitative
analysis in following discussion.
Trimethylenemethane complexes
The bonding modes in a number of trimethylenemethane
(TMM) complexes ([(CO)3X-TMM] (X=Fe, Ru, Os, Rh+), [h6-
C6H6X-TMM] (X=Fe, Ru, Os), [h5-C5H5X-TMM] (X =Co, Rh, Ir),
and [h4-C4H4X-TMM] (X=Ni, Pd, Pt)) have been considered in
detail in recent years.[25, 48, 49] Particularly, Mousavi and Frenking
have done a comprehensive analysis on the bonding of TMM
and metal-hydrocarbon unit, based on the energy decomposi-
tion analysis (EDA), considering the Laplacian maps of the elec-
tron density as well as quantitative molecular orbital dia-
grams.[48, 49] Their analysis demonstrates that the central metal
atom in each complex is chemically bonded to both the “cen-
tral” and “terminal” carbon atoms of the TMM unit and particu-
larly the bonding interaction with terminal carbon atoms was
found to be significant. However, the authors did not find any
LCPs and LPs linking the terminal carbon atoms and the cen-
tral metal. Assuming abovementioned direct analogy between
the chemical bond and LPs, Mousavi and Frenking, seemingly
confused by the “missing BPs”, concluded : “This (missing BPs)
clearly shows that AIMs analysis does not faithfully represent
the strongest pairwise interactions between the atoms in
a molecule” (the text in italics has been added by the present
authors). Some of these complexes had been reconsidered in
a previous communication and it was shown that the so-called
missing LPs are indeed observable in some accessible low-
energy non-equilibrium geometries.[5] In this section some
other complexes are scrutinized.
Figure 1 (as well as Figure S1 in the Supporting Information)
depicts MGs of [h6-C6H6Fe-TMM], [h5-C5H5Co-TMM], [h4-C4H4Ni-
TMM] complexes all computed employing BP86/Def2-TZVPP
computational level, as used in the original paper,[49] at the
equilibrium geometries (for details of the equilibrium geome-
tries see Figure 1 in ref. [49] or Supporting information).
Indeed, no LPs are observed between the metals and the ter-
minal carbon atoms of TMM at the equilibrium geometries.
Therefore, these complexes are fine examples containing two
atoms involved in a chemical bond but not linked by LPs.
However, considering the inter-atomic surfaces of the central
carbon in Figure 1 (also check Figure 5 in ref. [49]), it is evident
that the atomic basins of the metal and the terminal carbon
are quite near and just a “tiny volume” of the basin of the cen-
tral carbon penetrates in between them. Pursuing the reason-
ing advocated previously, some non-equilibrium geometries
and their associated MGs were produced. The primary desire
behind searching for non-equilibrium geometries was to
obtain and analyze non-equilibrium geometries with a smaller
Figure 1. A two-dimensional view of the basin of the central carbon of the TMM unit in a) [h6-C6H6Fe-TMM],
b) [h5-C5H5Co-TMM], and c) [h4-C4H4Ni-TMM] complexes. Both the contours as well as the gradient paths of the
electron density are depicted in a plane containing the nuclei of the central carbon of TMM, the metal, and one
of the terminal carbons of the TMM. The LCPs are shown as dots, whereas LPs are depicted as black lines linking
nuclei. The ring and cage CPs are eliminated for clarity. For a color version of this Figure see Figure S1 in the Sup-
porting Information.
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metal–terminal carbon inter-nuclear distance. Figure 2 (as well
as Figure S2 in the Supporting Information) depicts displace-
ment vectors of some selected vibrational normal modes and
Figure 3 (as well as Figure S3 in the Supporting Information)
shows the MGs of some selected
non-equilibrium geometries,
whereas Table 1 lists the relevant
information for the analysis (for
details of the non-equilibrium
geometries, see the Supporting
Information).
In the case of [h6-C6H6Fe-TMM]
complex, from a large set of con-
sidered non-equilibrium geome-
tries, three have been selected
as typical examples (denoted as
A, B, and C) with a MG different
from that of the equilibrium ge-
ometry. From Figure 3 and
Table 1, one may conclude that
there are non-equilibrium geo-
metries, with energies quite similar to that of the equilibrium
geometry, with MGs containing the abovementioned “missing”
LPs between the iron and the terminal carbon atoms. At least
three new MGs appear all indicating LPs between the iron and
Figure 2. Selected low frequency and low force constant normal modes of a) [h6-C6H6Fe-TMM], b) [h5-C5H5Co-
TMM], and c) [h4-C4H4Ni-TMM] complexes; during corresponding vibrations the inter-nuclear metal–terminal
carbon distances decreases. For a color version of this Figure see Figure S2 in the Supporting Information.
Figure 3. The MGs of some selected non-equilibrium geometries (denoted by MA, MB, and MC; M=Fe, Co, Ni) of [h6-C6H6Fe-TMM], [h5-C5H5Co-TMM] , [h4-
C4H4Ni-TMM] complexes (for details of geometries, see the Supporting Information). The LCPs are shown with light-gray dots, dark-gray dots are used for
RCPs, and LPs are depicted as black lines linking nuclei. The cage CPs are eliminated for clarity. For a color version of this Figure see Figure S3 in the Support-
ing Information.
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each of the terminal carbons. Thus, the [h6-C6H6Fe-TMM] com-
plex is an example of M-MGs case, whereas the terminal
carbon atoms and the iron basins are examples of the passion-
ate neighbors. The computed electronic delocalization index
(d) between the iron atom and the central as well as the termi-
nal carbon atoms are relatively insensitive to the geometrical
variations (Table 1). Interestingly, the delocalization of electrons
is larger between the terminal carbons and the iron atom in
comparison with that of the central carbon and the iron atom.
Clearly, “abrupt” variations of MGs are chemically unimportant
and the “continuity” reveals itself in the delocalization index as
a “basin property” (see below).
The properties of [h5-C5H5Co-TMM] and [h4-C4H4Ni-TMM]
complexes are analogous and, as is evident from Figure 3 and
Table 1, the same pattern emerges, namely, the M-MGs mole-
cules, which are not reiterated. All the considered complexes
clearly demonstrate the dynamic nature of MGs.
[HS(CH)(CH2)]
Another M-MGs molecule is [HS(CH)(CH2)], depicted in Figure 4
(as well as Figure S4 in the Supporting Information), which has
been considered previously by Stalke and co-workers as well
as Jacobsen.[50, 51] Both studies, based on the natural bond orbi-
tal method as well as the localized-orbital-locator and the ki-
netic energy density analysis, came to the conclusion that
there are chemical bonds between each of the carbon atoms
and the sulfur atom. However, Stalke and co-workers were un-
successful in locating LPs between the carbon atom of the CH
group (hereafter denoted as C1) and the sulfur atom, from the
one-electron densities derived from various ab initio and densi-
ty functional-based methods. In contrast, Jacobsen was able to
locate the LPs using the BP86 density functional but the LPs
were absent from the MG at other considered computational
levels. In particular, at the CCSD/TZVP level, as the highest em-
ployed computational level, LPs were not detected. Jacobsen
also noticed that the differences in the geometries of the de-
rived equilibrium structures at each computational level are
more important than the resulting one-electron densities. He
stressed: “The fact that the presence of BCP is indeed related
to differences in geometry, and not due to the particular com-
putational approach, has been tested in a set of E/TZVP//
CCSD/TZVP and E/TZVP//PBE/TZVP calculations, E =PBE, PB86,
BLYP, B3LYP, B3PW91, CCSD. For all calculations that utilize the
CCSD-geometry with a large SC1inter-nuclear distance of
2.007 , no BCP between S and C1atoms could be located.
Furthermore, all calculations based on the PBE-geometry with
a short SC1inter-nuclear distance of 1.916  predict the pres-
ence of an SC1bond” (the word “bond” has been used by Ja-
cobsen as an alternative for the presence of a LCP).[51] In their
Table 1. Some geometrical parameters (inter-nuclear distance are given
in ) as well as relative energies [kcal mol1] and the electron delocaliza-
tion index (d) computed at both the equilibrium (denoted by Opt suffix)
and selected non-equilibrium geometries of the complexes [h6-C6H6Fe-
TMM], [h5-C5H5Co-TMM] , and [h4-C4H4Ni-TMM] .[a]
FeC(t) Fe-C(c)-C(t) DE[b] d(FeC(c)) d(FeC(t))[c]
FeOpt 2.10 76.0 0.00 0.46 0.70
FeA 2.00 71.5 1.20 0.45 0.75
FeB 2.02 72.5 0.68 0.46 0.74
FeC 2.04 72.5 0.46 0.45 0.74
CoC(t) Co-C(c)-C(t) DE[b] d(CoC(c)) d(CoC(t))[c]
CoOpt 2.07 75.5 0 0.47 0.69
CoA 1.97 71.0 1.23 0.46 0.75
CoB 1.99 72.0 0.72 0.46 0.73
CoC 2.00 72.0 0.48 0.46 0.73
NiC(t) Ni-C(c)-C(t) DE[b] d(NiC(c)) d(NiC(t))[c]
NiOpt 2.11 77.0 0 0.43 0.57
NiA 1.99 72.0 1.36 0.42 0.63
NiB 2.00 72.5 0.97 0.42 0.62
NiC 2.02 72.5 0.61 0.42 0.63
[a] The letters c and t denote the central and the terminal carbons, re-
spectively. [b] Relative to the energy of the equilibrium geometry. [c] This
is the terminal carbon atom; its inter-nuclear distance is varied according
to the first column. The delocalization index of the remaining terminal
carbons with iron is practically equal to that observed at the equilibrium
geometry.
Figure 4. a) The MG of the equilibrium geometry; b) the basin of the C2
carbon; c) a selected low frequency/force constant normal mode ; d) the
MGs of two selected non-equilibrium geometries (denoted as A and B) of
HS(CH)(CH2) molecule. The light-gray dots in panels (a), (b), and (d) are LCPs
and the dark-gray dot in (d) is a RCP. The black lines in panels (a), (b), and
(d) are LPs. In panel (b) the contours as well as the gradient paths of the
electron density are depicted in a plane containing the nuclei of the carbons
and the sulfur atom. For a color version of this Figure see Figure S4 in the
Supporting Information.
Chem. Eur. J. 2014,20, 10140 – 10152 www.chemeurj.org  2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim10145
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detailed QTAIM analysis, Stalke and co-workers were able to
demonstrate that a tiny volume of the basin of the carbon
atom of the CH2unit (hereafter denoted as C2), which is closer
to the sulfur atom, penetrates in between the sulfur and the
C1 carbon basin prohibiting the appearance of the “expected”
LCP (See panel (b) of Figure 4 or Figure 4 in ref. [50]).
For the equilibrium geometry, derived at the CCSD/TZVP
computational level (for details of geometry, see references [50,
51] or the Supporting Information), in line with the previous
studies, LPs were not detected between the sulfur and the C1
atom whereas LPs were observed between the sulfur and the
C2 basin. Upon considering some non-equilibrium geometries
(denoted by A and B), a new MG emerges that is also present-
ed in the panel (d) in Figure 4. Inspection of Table 2 demon-
strates that in line with Jacobsen’s finding, this MG emerges
only when the SC1 inter-nuclear distance contracts considera-
bly with respect to its reference value at the equilibrium ge-
ometry (>0.07 ). However, the resulting non-equilibrium geo-
metries are still energetically
close to the equilibrium geome-
try and quite accessible even at
room temperature. Accordingly,
the confusions raised in the orig-
inal studies have been the result
of the fact that [HS(CH)(CH2)] is
another example of the M-MGs
systems. On the other hand, the
computed electron delocaliza-
tion index between the sulfur
and each of the carbons are not
much different and relatively in-
sensitive to the geometrical var-
iations demonstrating the fact
that chemically, the appearance/
disappearance of LPs is of no
significance.
[Co2(CO)8]
Various structural, spectroscopic,
and inter-conversion aspects of
cobalt carbonyls have been in-
vestigated extensively by various
computational methods.[52, 53]
However, deciphering the nature of the cobalt–cobalt “interac-
tion” in the case of [Co2(CO)8] as a typical bimetallic example
of this class of metal-carbonyls, with a C2vsymmetry point
group (see Figure 5 as well as Figure S5 in the Supporting In-
formation), is not straightforward. In older literature, based on
the 18-electron rule, a bond between the cobalt atoms was
proposed. Nevertheless, the reorganization of the 18-electron
rule introducing three-center two-electron bond (3c–2e), in-
volving two metal atoms and a carbonyl, relegates the need
for a “direct” CoCo bond.[54] Detailed experimental and theo-
retical studies on the electron density of this complex,[55, 56] in-
cluding the density difference maps and topological analysis,
are generally all in line with the absence of a chemical bond
between the metal atoms. Even so, Hall and co-workers
stressed that a “weak CoCo interaction” seems to be survived
in this complex.[56] More recent QTAIM-based studies are am-
biguous, indicative of both direct,[57] and indirect,[6, 58] bonding
mode depending on the “interpretation” of the QTAIM data.
Probably, one of the most accurate computational studies on
this complex is the analysis of the domain averaged Fermi
hole by Ponec and co-workers.[59] This study not only confirms
the 3c–2e bonding mode, but also points to the absence of
CoCo covalent bonding at least if one invokes an electron
pair as “the” indicator of a covalent bond.[59] One may conclude
that at the present state of knowledge there is a weak direct
interaction between cobalt atoms that does not deserve to be
termed a covalent bond.[6, 54, 59]
According to Figure 5, and in line with all the previous stud-
ies,[6,56, 57] the equilibrium-geometry MG of this complex (for de-
tails of geometry, see the Supporting Information), computed
at BP86/Def2-TZVP level, does not indicate the LPs connecting
Table 2. Some geometrical parameters (inter-nuclear distance are given
in ), as well as relative energies [kcal mol1] and the electron delocaliza-
tion index (d) computed at both the equilibrium (denoted as Opt) and se-
lected non-equilibrium geometries of [HS(CH)(CH2)] .
SC1 S-C2-C1 DE[a] d(SC1) d(SC2)
Opt 2.01 72.8 0 0.60 0.79
A 1.92 69.0 0.73 0.68 0.78
B 1.94 69.0 0.41 0.67 0.76
[a] Relative to the energy of the equilibrium geometry.
Figure 5. a) The MG of the equilibrium geometry; b) the MGs of the three selected non-equilibrium geometries
(denoted as A, B and C); c) the inter-atomic surfaces separating cobalt atoms and the bridged carbon basins in
[Co2(CO)8] molecule. The light-gray dots in panels (a), (b), and (c) are LCPs and RCPs are denoted by dark-gray
dots. The black lines in panels (a), (b), and (c) are LPs. In panel (c) the surfaces are the inter-atomic surfaces. For
a color version of this Figure see Figure S5 in the Supporting Information.
Chem. Eur. J. 2014,20, 10140 – 10152 www.chemeurj.org  2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim10146
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the cobalt basins. Instead, a ring critical point (RCP) emerges in
the middle of the ring composed of the two metal atoms and
the two carbon basins of the bridged carbonyls. To check the
role of nuclear vibrations, various non-equilibrium geometries
were constructed varying in the CoCo and CoC(b) inter-nu-
clear distances as well as the CoC(b)Co angle (“b” denotes
the carbon of the bridged carbonyl). Selected geometries, de-
noted as A, B, and C, are considered (for details of geometries,
see the Supporting Information). In contrast to the previous
examples considered in this work, the non-equilibrium geome-
tries containing a LCP in between the cobalt basins are ener-
getically quite distinct from the equilibrium geometry and are
not accessible at room temperature (Table 3). Indeed, Figure 5
demonstrates that the basins of the bridged carbon atoms
penetrate between the cobalt atoms prohibiting the formation
of an inter-atomic surface between the two metal atoms.
Table 3 reveals that just by either a large contraction of the
CoCo inter-nuclear distance (>0.2 ) or a large expansion of
the CoC(b) inter-nuclear distances (>0.2 ) or both, the LCP
and its concomitant inter-atomic surface appear between the
cobalt basins. Therefore, the cobalt atoms are neither neigh-
bors nor passionate neighbors. Interestingly, the small varia-
tions of the CoCo delocalization index, which is less than 0.1,
demonstrates that upon appearance/disappearance of the LCP
and concomitant geometrical changes the electron delocaliza-
tion does not alter significantly. Clearly, the Co-Co electron de-
localization index is large ( 0.35), even at the equilibrium ge-
ometry, revealing the fact that there is a small but quite dis-
tinct interaction between these two atomic basins. The cobalt
basins in [Co2(CO)8] may serve as examples of cases in which
two basins have non-negligible interactions but the “expected”
LP is “missing” between the interacting basins. This observa-
tion indeed confirms the previously stated theoretical assertion
that the presence of LPs is not necessary for non-negligible in-
teraction between atomic basins.
[(F3C)F2SiONMe2]
The 1,3 geminal silicon–nitrogen interaction, usually called a-
effect, is the subject of numerous theoretical and experimental
investigations.[60–65] [(F3C)F2SiONMe2] is a proper model for this
interaction that has been synthesized and extensively studied
by Mitzel and co-workers using various aspects of its structural,
energetic, and electron density characteristics.[61] The electron
diffraction analysis in a gas phase demonstrates that the mole-
cule is composed of two distinguishable conformers, termed
gauche and anti; here only the anti-conformer is reconsidered
(see Figure 6 as well as Figure S6 in the Supporting Informa-
tion). In older literature, the a-effect had been described as
originating in a dative bond between the silicon and the nitro-
gen atoms. Two comprehensive studies on [F3SiONMe2],[60] and
[(F3C)F2SiONMe2],[61] by Mitzel and co-workers based on details
of electronic structure and one-electron density of these mole-
cules ruled out the presence of dative bond between the nitro-
gen and the silicon atoms. Instead, the atomic charges of the
silicon and nitrogen atoms, variation of atomic charges by
Table 3. Some geometrical parameters (inter-nuclear distance are given
in ), as well as relative energies [kcalmol1], and the electron delocaliza-
tion index (d) computed at both the equilibrium (denoted as Opt) and
some selected non-equilibrium geometries of [Co2(CO)8].
CoCo CoC Co-C-Co DE[b] d(CoCo) d(CoC(b))[a]
Opt 2.55 1.96 81.3 0 0.36 0.69
A 2.55 2.17 72.0 21.0 0.43 0.56
B 2.47 2.11 71.7 14.2 0.44 0.60
C 2.35 1.93 75.0 4.5 0.46 0.71
[a] The letter b is an abbreviation for the carbon atom of the bridged car-
bonyl. [b] Relative to the energy of the equilibrium geometry.
Figure 6. a) The MG of the equilibrium geometry; b) the basins of the silicon,
nitrogen, and oxygen atoms at the equilibrium geometry ; c) the basins of
the silicon, nitrogen, and oxygen atoms at the selected non-equilibrium ge-
ometry of [(F3C)F2SiONMe2] molecule. The light gray dots in all panels are
LCPs and the RCP is denoted by dark-gray dot. The black lines are LPs. In
panels (b) and (c) the contours of the electron density are depicted in
a plane containing the nuclei of the silicon, nitrogen, and oxygen atoms. For
a color version of this Figure see Figure S6 in the Supporting Information.
Chem. Eur. J. 2014,20, 10140 – 10152 www.chemeurj.org  2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim10147
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changing the Si-O-N angle, the magnitude of electric dipole
moment of the molecule, flatness of the potential energy sur-
face upon variations of the Si-O-N angle, and certain qualita-
tive features of the Laplacian maps of the one-electron density
are all consistent with an electrostatic interaction (probably
with some contribution from the dispersion interaction). One
may conclude that although the geminal interaction probably
does not deserve to be called a bond, at least in the Lewis
type paradigm, its effect on various characteristics of the mole-
cule casts no doubt that the interaction between the silicon
and nitrogen atoms is non-negligible.
The anti-conformer of [(F3C)F2SiONMe2] was optimized at the
MP2/6-311++g(d,p) level (for details of the geometry, see the
Supporting Information); panel (a) in Figure 6 depicts the MG
of the equilibrium geometry. In line with a previous study,[61]
no LCP and associated LPs are observed between the Si and N
basins. Panel (b) of Figure 6 demonstrates that the oxygen
basin penetrates in between the Si and N basins preventing
the formation of an inter-atomic surface. Like for the previous
cases, various non-equilibrium geometries were constructed,
however, geometries containing LPs, linking the Si and N
basins are energetically high above the energy of the equilibri-
um geometry and, therefore, not accessible at room tempera-
ture. Table 4 contains the relevant computed data of selected
non-equilibrium geometry (for details of the geometry, see the
Supporting Information). Accordingly, at current state of
knowledge, [(F3C)F2SiONMe2] is better described as a single-MG
molecule. The small values of the electron delocalization index
for Si and N basins, at both the equilibrium and non-equilibri-
um geometries, demonstrate that marginal electron delocaliza-
tion is taking place between the two basins. This is certainly
consistent with the absence of a covalent SiN bond. However,
both the atomic charges of Si and N as well as the basin elec-
tric dipoles are large enough to be consistent with an electro-
static interaction between Si and N atoms. Particularly, at the
equilibrium geometry, the electric dipole of the N basin is the
second largest atomic dipole (almost equal to the dipole of
oxygen). All of these observations are indeed in line with pre-
dominant electrostatic nature of Si···N interaction and con-
forms to the picture of the a-effect as articulated previous-
ly.[60–65] This is another example demonstrating that LPs are not
essential for two AIMs to be interacting. Furthermore, LPs
cannot be used to “gauge” the strength of the interaction.
Whereas LPs are observed between AIMs in many van der
Waals complexes,[42] just involved in very weak dispersion inter-
actions (like the case of the helium dimer),[5] they are not ob-
served in the case of the much stronger geminal interaction.
Chemical bonds and QTAIM: The combinatorial/synthetic
approach
Based on what was discussed in the previous sections, it must
be concluded that no “golden”/unique index of chemical bond
is defined within the context of the QTAIM; indeed, the QTAIM
is not “the” theory of chemical bond. Then, if the QTAIM is not
“the” theory of chemical bond, how must one “discover” and
classify chemical bonds with the toolbox of the QTAIM? In the
rest of this section a general Scheme is scrutinized briefly refer-
ring to previous appropriate references. However, before an-
swering this question, it is instructive to briefly mention how
a new bond is introduced in chemistry and then identified in
molecules.
There is a tendency among theoretical chemists as well as
historians of chemistry to simplify and incarnate the concept
of chemical bond in the Lewis electron pair paradigm and its
subsequent “quantization” by Pauling and others.[66–77] Though
there is no question that the Lewis paradigm is an important
ingredient in current chemical discourse, “non-Lewis” bonds,
for example, multi-center two-electron bonds,[78–80] are also
now a pivotal part of chemical discourse. Therefore, there is no
general theoretical Scheme that may claim to encompass the
description of all known types of chemical bonds. Furthermore,
new types of non-Lewis chemical bonds are discovered from
time to time, particularly in the realm of “weak interactions”
demonstrating temporary nature of current “bond-type” classi-
fications. The recently proposed dihydrogen bond is an exam-
ple that helps us to comprehend the “real procedure” of intro-
ducing new chemical bonds.[81–85] Detailed reading of the origi-
nal papers reveals the fact that no “single” criterion has been
used to introduce the dihydrogen bond. But, the reasoning
was based on geometrical (X-ray and neutron diffraction data),
spectroscopic (e.g., IR, NMR spectroscopies), thermochemical
criteria (e.g., melting points, HH interaction energies), theo-
retical analysis of the electronic structures (e.g. , qualitative mo-
lecular orbital models, atomic charges) as well as considering
reaction mechanisms and tracing relevant intermediates. Clear-
ly, the dihydrogen bonds deserve to be named and introduced
as independent “chemical phenomena” since their fingerprints
are seen in all abovementioned properties. Based on the case
of the dihydrogen bond as a typical example, one may con-
clude that the identification of a chemical bond is a delicate
task that is performed by combination/synthesis of various
chemical and physical properties. Assuming that a type of
chemical bond is well-established, that is, the list of relevant
criteria to be checked for its presence is well-known, the afore-
mentioned combinatorial/synthetic approach must be used to
identify the relevant chemical bond in a new molecule. Of
course there will be always “gray” examples, that is, at the
Table 4. Some geometrical parameters (inter-nuclear distance are given
in ), relative energies [kcal mol1], the electron delocalization index (d),
atomic charges and the magnitude of total basin dipole moments com-
puted at both the equilibrium (denoted as Opt) and one selected non-
equilibrium geometry of [(F3C)FSiONMe2].
SiN Si-O-N DE[a] d(SiN) Si charge N charge Si dipole N dipole
Opt 2.06 81.3 0 0.10 3.19 0.74 0.48 1.17
A 1.84 70.5 2.11 0.16 3.20 0.85 0.50 0.96
[a] Relative to the energy of the equilibrium geometry. This is the lowest
energy discovered non-equilibrium geometry that yet contains a LCP be-
tween Si and N atoms.
Chem. Eur. J. 2014,20, 10140 – 10152 www.chemeurj.org  2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim10148
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edge of bond/no-bond, which are hard to categorize because
they satisfy some criteria of the list but not others, demonstrat-
ing the inherent ambiguity/fuzziness of the bond classifica-
tions. In fact, in a high “magnification”, the concept of chemi-
cal bond has an intrinsic floppiness. The same combinatorial
approach must be used when tracing chemical bonds within
the context of the QTAIM while the gray examples also inevita-
bly emerge. It is worth reemphasizing that this ambiguity
points to the intrinsic floppiness of the concept of chemical
bond rather than a deficiency of the QTAIM (probably it is
a solace for chemists to know that biologists also encounter
the same kind of “classification obstacles”).[86, 87]
The QTAIM analysis yields point, surface, and basin proper-
ties. The point properties include the amount of the one-elec-
tron and the property densities at LCPs, usually called topolog-
ical indices (in principle, any other type of CPs or even an arbi-
trary point in space can be used for “density sampling”).[2–4]
The surface properties similarly include surface integrals of the
one-electron and the property densities over the inter-atomic
surfaces.[88–90] Finally, the basin properties include volume inte-
grals of the one-electron and the property densities over the
atomic basins.[2–4] Although none of these properties uniquely
determines the presence/absence of a chemical bond, the
combinatorial approach must be used for the recognition and
classification of chemical bonds. In this QTAIM-based combina-
torial approach the list of criteria is composed of a set of
QTAIM derived indices. Indeed, beyond Bader’s classics on co-
valent and ionic bonds in general,[2] and a dedicated study on
transition-metal-carbonyl bonds in particular,[91] in the case of
the dihydrogen,[81–85] hydrogen and agostic bonds,[92–94] Popeli-
er has introduced and elaborated such combinatorial method-
ology.[95–97] Also, in the case of metal–metal bonds from transi-
tion-metal block, the same methodology has been employed
masterfully by Macchi and Sironi (see also section 13 in
ref. [4]).[6] In the combinatorial methodology, the QTAIM analy-
sis yields a wealth of “raw properties”, which can be used to
construct correspondence rules.[6] Accordingly, a set of mole-
cules with well-established target bond must be analyzed as
a first step, seeking for correspondence between the bond and
the QTAIM-derived properties of the atomic basins involved in
the bonding. Subsequently, the derived correspondence rules
should be used to establish the presence/absence of a target
bond in a larger set of molecules. There is no unique recipe
guiding how to choose the primary set of typical molecules
containing a certain type of bond, or what type of the QTAIM
indices should be used as proper criteria to decipher the bond,
or how to tune the range of continuous indices determining
the bond/no-bond boundary. Thus, such a combinatorial ap-
proach does not yield the “final” set of correspondence rules
so it is prone to revision and constant evolution. Particularly,
since the gray examples are also conceivable within the
QTAIM-based correspondence rules, one must always be pre-
pared to discover two AIMs, in which it is hard to ascertain the
presence/absence of a certain kind of chemical bond in be-
tween them. In other words, this observation points to the fact
that the QTAIM does not make chemical bond classification
procedure less fuzzy or more definite.
Based on all previous discussions on LCPs and LPs, the em-
phasis on the point and surface properties in such a list of cri-
teria does not seem to be legitimate. Such emphasis dismisses,
unreasonably, the possibility of considering the presence of
bonds between atomic basins that are not neighbors. Conse-
quently, in designing correspondence rules, the presence of
LPs as a necessary and/or sufficient criterion for a certain kind
of chemical bond must be eliminated and the emphasis has to
be shifted toward the basin properties as well as general quali-
tative features of various property densities at the region of
bonding; the Laplacian of one-electron density, as a typical
density, has been used to recognize various types of bonds.[98]
Atomic electron population (atomic charge),[2–4] atomic polari-
zation dipole and higher order multipoles,[2–4] atomic polariza-
bility,[2–4] atomic energies,[2,99] localization and delocalization in-
dices,[2,26–38] and the source function,[100–104] are just examples of
proper basin properties that have been utilized in previous
high-quality QTAIM analyses.
Conclusion
Prospects beyond the orthodox QTAIM: The nature of AIMs
interactions
Inherent in all previous discussions on various types of chemi-
cal bonding was the strength of the interactions of AIMs, for
example, covalent bonds are typical strong interactions, where-
as the dihydrogen bonds are examples of weakly interacting
AIMs. Since each atom in a molecule always interacts with all
other atoms of the system,[44] the key point in using the QTAIM
analysis is not distinguishing the interacting and non-interact-
ing AIMs, but “gauging” the strength and nature of interac-
tions. Currently, gauging the interactions of neighbors is done
“indirectly”, that is, without direct evaluation of the interaction
energy. This is done by considering topological indices at LCPs
and/or surface integrals over inter-atomic surfaces or more
generally, through combinatorial strategy, employing various
basin properties thus encompassing the case of the interaction
of non-neighbors. Such an indirect approach may trigger con-
troversies on the true nature of interacting AIMs, of which the
case of the “HH bonding controversy” is a vivid but not exclu-
sive example.[105–114] Such controversies clearly point to the fact
that a “direct” gauging approach attributing interaction energy
to each pair of AIMs is a missing ingredient within the formal-
ism of the QTAIM. The Interacting Quantum Atoms (IQA) meth-
odology, developed by Pends and co-workers,[115–125] is
a proper example of such a direct approach that is used cur-
rently in parallel with the QTAIM analysis.[126–129] However, in
long term, the QTAIM needs its own direct gauging approach
to answer delicate questions regarding the nature of interac-
tions of AIMs that are currently debated usually without clear-
cut conclusions.[130–171] One may hope that upon introduction
of direct gauging, a truly consistent and comprehensive chemi-
cal interpretation will emerge relegating most of the current
ongoing controversies around the QTAIM.
Chem. Eur. J. 2014,20, 10140 – 10152 www.chemeurj.org  2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim10149
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Acknowledgements
The authors are grateful to Masumeh Gharabaghi and Shahin
Sowlati for their detailed reading of a draft of this paper and
helpful suggestions. This work was supported by the Czech
Science Foundation (P206/12/0539 to R.M.) and carried out at
CEITEC-Central European Institute of Technology with research
infrastructure supported by the project CZ.1.05/1.1.00/02.0068
financed from the European Regional Development Fund as
well as by the Program of “Employment of Newly Graduated
Doctors of Science for Scientific Excellence” (grant number
CZ.1.07/2.3.00/30.009) co-financed from European Social Fund
and the state budget of the Czech Republic (C.F.-N.). Access to
the computing and storage facilities owned by parties and
projects contributing to the National Grid Infrastructure Meta-
Centrum provided under the program “Projects of Large Infra-
structure for Research, Development, and Innovations”
(LM2010005) and the CERIT-SC computing and storage facilities
provided under the program Center CERIT Scientific Cloud,
part of the Operational Program Research and Development
for Innovations, reg. no. CZ. 1.05/3.2.00/08.0144 is appreciated.
Keywords: bond theory ·chemical bonds ·computer
chemistry ·noncovalent interactions ·quantum chemistry
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Received: February 13, 2014
Published online on July 2, 2014
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