(BOOK CHAPTER) - 2015 - The Covalent Bond Classification Method and Its Application To Compounds That Feature 3-Center 2-Electron Bonds - Green
(BOOK CHAPTER) - 2015 - The Covalent Bond Classification Method and Its Application To Compounds That Feature 3-Center 2-Electron Bonds - Green
(BOOK CHAPTER) - 2015 - The Covalent Bond Classification Method and Its Application To Compounds That Feature 3-Center 2-Electron Bonds - Green
DOI: 10.1007/430_2015_206
# Springer International Publishing Switzerland 2016
Abstract This article provides a means to classify and represent compounds that
feature 3-center 2-electron (3c–2e) interactions according to whether (1) the two
electrons are provided by one or by two atoms; (2) the central bridging atom
provides two, one, or zero electrons; and (3) the interaction is open or closed.
Class I 3c–2e bonds are defined as those in which two atoms each contribute one
electron to the 3-center orbital, while Class II 3c–2e bonds are defined as systems in
which the pair of electrons are provided by a single atom. The use of appropriate
structure-bonding representations enables the [MLlXxZz] covalent bond classifica-
tion of the element of interest to be evaluated. This approach is of considerable
benefit in predicting metal–metal bond orders that are in accord with theory for
dimetallic compounds that feature bridging hydride and carbonyl ligands.
Contents
1 Introduction
2 The Covalent Bond Classification Method: A Synopsis
3 Classification of 3-Center 2-Electron Bonds and Their Representations
4 Examples of Compounds with 3c–2e Bonds
4.1 Class I: Closed μc–Z 3c–2e Bonds
4.2 Class I: Open μo–Z 3c–2e Bonds
1 Introduction
Following the classic 1916 paper by Lewis on “The Atom and the Molecule” [1–3],
the ability to represent molecules by so-called Lewis structures [4], in which a
2-center 2-electron (2c–2e) bond is illustrated as a solid black line between two
atoms, must be regarded as one of the most important developments in molecular
chemistry over the past 100 years. Despite its importance and the elegance of its
simplicity, however, the limitations of the 2c–2e bond as a bonding model are well
known, such that many molecules, as exemplified by B2H6,1 must be represented in
terms of multicenter bonding. Although the nature of such compounds is best
analyzed by the application of theoretical methods, this approach lacks the sim-
plicity of allowing one to evaluate the chemical reasonableness of a molecule by
employing simple electron counting procedures, such as the octet [2, 3, 8, 9]2 and
18-electron [8, 11, 12] rules. However, despite the fact that the bonding in such
compounds may be highly delocalized, it can often be expressed in terms of a
combination of 2-center 2-electron and 3-center 2-electron (3c–2e) interactions.
Therefore, we describe herein ways to represent various classes of 3-center 2-elec-
tron (3c–2e) interactions such that they can be used in conjunction with the covalent
bond classification (CBC) method for analyzing molecules [13–16].
Although covalent molecules are often classified in terms of the oxidation states of
the atoms of interest, it is evident that this approach has a number of shortcomings.
For example, since the oxidation state corresponds to the charge on an isolated
atom, with no ligands attached, the assigned values often either convey no useful
chemical information or can result in misleading interpretations [13–17]. In con-
trast to the oxidation state approach, however, which focuses on the charge on an
isolated atom, the covalent bond classification (CBC) method evaluates a molecule
1
It is interesting to note that the bridged structure of B2H6 was first proposed in 1943, 27 years
after Lewis’ introduction of the 2c–2e bond and only 3 years prior to his death; see [5–7].
2
Kossel also recognized the tendency for atoms to form ions with the adjacent noble gas
configuration but did not extend this concept to the formation of molecules; see [10].
The Covalent Bond Classification Method and Its Application to Compounds. . .
Fig. 1 The covalent bond classification (CBC) of L, X, and Z ligands. Note that the ligands are
always classified in their neutral forms. The metal, ligand, and metal–ligand orbitals are arbitrarily
placed at the same respective energies
by identifying the number and types of bonds that surround the element of interest
(M). As such, by evaluating the intact molecule, the classification provides a more
comprehensive description of the molecule than that which is provided by the
oxidation state.
The basic premise of the CBC method is that many covalent molecules can be
satisfactorily represented in terms of 2-center 2-electron bonding interactions in
which the neutral monodentate ligands may contribute either two (L), one (X), or
zero (Z) electrons to the bonding orbitals. The molecular orbital representations of
these interactions are illustrated in Fig. 1, while representative examples of L-, X-,
and Z-type ligands are listed in Table 1.3 Thus, (1) L-type ligands (electron pair
donors) are neutral molecules that have available lone pairs and are Lewis bases
(e.g., H2O, H3N, and R3P), (2) Z-type ligands (electron pair acceptors) are neutral
molecules that exist as Lewis acids (e.g., BF3) [25, 26], and (3) X-type ligands
(one-electron donors) are neutral molecules that are radicals (e.g., H•, Cl•, and
H3C•).
Interactions involving X-type ligands correspond to M–X normal covalent
bonds, whereas those involving L- and Z-type ligands correspond to dative covalent
bonds [13, 27] and are represented with the use of arrows, as either M L or M!Z.
In addition to representing a dative bond with an arrow, it can also be represented as
a line with formal charges,4 i.e., M L+ and M+ Z (Fig. 2); however, it must
be emphasized that, despite these different appearances, they correspond to exactly
the same electronic structure and are not resonance structures [28]. For example,
3
For examples of textbooks that employ the classification of ligands as L, X, or Z type, see
[18–24].
4
The formal charge is the charge remaining on an atom when all ligands are removed
homolytically. See [17].
M.L.H. Green and G. Parkin
Fig. 3 [LlXxZz]
classifications of some
common ligands as derived
by summing the individual
components that correspond
to the valence bond
representation
Multidentate and multiply bonded ligands that are attached by more than one
covalent bond may be classified as [LlXxZz], where l, x, and z are the respective
number of L, X, and Z functionalities that are associated with the frontier orbitals
of the ligand in the geometry that corresponds to its binding mode. Common
examples of such ligands include allyl, cyclopentadienyl, and benzene, as illus-
trated in Fig. 3 and Table 1.
In many cases, the classification can be simply derived by summing the
individual bonding components that are implied by their valence bond represen-
tations, but, in certain cases, consideration of the frontier orbitals is essential to
obtaining the correct representation. As an illustration, the η7-cycloheptatrienyl
ligand is classified as L2X3 rather than L3X because of the availability of Z
functions (Fig. 4). Another illustration of a multifunctional ligand that features
a Z function is provided by linear NO, which results in a classification of
X3 [29].
Structure-bonding representations for some of these cyclic ligands are provided
in Fig. 5, which also includes the familiar forms that use a single line between the
metal center and the ring centroid to indicate connectivity.
After the ligands attached to the element of interest have been identified
according to the CBC method, the molecule itself is classified in the form
[MLlXxZz]Q by summing all the L, X, and Z functionalities, where Q is the
charge on the molecule. Comparison of molecules that have different charges,
however, require the [MLlXxZz]Q assignment to be reduced to its “equivalent
neutral class” by formally localizing the Q charge on the ligands, rather than on
M. The equivalent neutral class is thus the classification that results if the
M.L.H. Green and G. Parkin
Fig. 5 Structure-bonding
representations for benzene
and cyclopentadienyl
ligands (top) and
representations that are
commonly used for clarity
to indicate connectivity
(bottom)
Q charge were to be localized on the ligand rather than on the metal center. The
reduction of [MLlXxZz]Q to its equivalent neutral class is described in detail
elsewhere [14–16] and is readily achieved by the application of some simple
transformations, the most essential of which are:
(1) For cations, L+ ! X and, if no L ligand is present, X+ ! Z
(2) For anions, X ! L and, if no X ligand is present, L ! LX
If the derived classification after performing these transformations contains both
an L and a Z function, the classification is reduced further by using the transfor-
mation LZ ! X2.
Illustrations of these procedures are provided by (1) [Cp2WH3]+, which is
classified as [ML4X5]+, transforming to [ML3X6] by application of the rule L+
! X, and (2) [Cp2ZrR3], which is classified as [ML4X5], transforming to
[ML5X4] by application of the rule X ! L (Fig. 6).
Not only does the CBC method provide a simple classification of a covalent
molecule, but the MLlXxZz description also contains useful information pertaining
to the nature of a molecule, such as the electron number (EN), valence number
(VN), number of nonbonding electrons (vn, i.e., dn for transition metals), and ligand
bond number (LBN), as summarized in Table 2.
The Covalent Bond Classification Method and Its Application to Compounds. . .
Fig. 6 [MLlXxZz]
classifications of some
metallocene compounds
Table 2 Definitions pertaining to the CBC method and the equivalent neutral class
Symbol Definition
L Two-electron donor function
l Number of L functions
X One-electron donor function
x Number of X functions
Z Zero-electron donor function
z Number of Z functions
m Number of valence electrons on neutral M atom
VN Valence number
VN ¼ x + 2z
LBN Ligand bond number
LBN ¼ l + x + z
EN Electron number (or electron count)
EN ¼ m + 2 l + x
vn Number of electrons in “nonbonding” M orbitalsa:
n ¼ m – x – 2z ¼ m – VN
a n
v corresponds to dn for transition metal compounds.
By analogy to the fact that 2c–2e bonds can be classified according to the
number of electrons that each partner contributes, i.e., [XX] (normal covalent) or
[LZ] (dative covalent), 3c–2e interactions can likewise be classified according to
the number of electrons that each center contributes, with the only possibilities
being [X2Z] (Class I) and [LZ2] (Class II). Thus, a Class I interaction corresponds to
a situation in which two atoms each contribute one electron to the bonding
molecular orbital, whereas a Class II interaction corresponds to a situation in
which a single atom contributes both electrons. These interactions can also be
subclassified according to (1) the identity of the bridging atom, i.e., μ–L, μ–X, or
μ–Z, and (2) whether the interaction is open (i.e., little overlap between one pair of
orbitals) or closed (i.e., substantial overlap of all three orbitals) [30], as distin-
guished by the symbols μo- and μc-, respectively. Unless one is dealing with
symmetric [AB2] 3c–2e interactions where the identity of the bridge (A) is self-
evident, more than one reasonable interpretation is possible for an asymmetric
[ABC] arrangement. Therefore, for clarity, the identity of the bridging atom should
always be specified when using the μ–Z, μ–X, and μ–L classification to describe the
bonding within an asymmetric [ABC] arrangement. With respect to the issue of
whether the interaction is open or closed, appropriate consideration needs to be
given to ascertain whether there is an interaction between all pairs of atoms,
recognizing that such differentiation may not be possible for borderline situations,
in which case it is preferable to just use μ- rather than μo- or μc-.
A summary of the structure-bonding representations of the various 3c–2e inter-
actions is presented in Fig. 7. For example, with respect to μ–Z interactions, the μc–
Z closed form can be conveniently represented by drawing an arrow from the
midpoint of the X–X bond to Z, which clearly illustrates how the X–X bonding
pair of electrons serves as a “dative bond” to Z and thus contributes two electrons to
its electron count (Fig. 8).5 This class of bonding is illustrated well by dihydrogen
compounds, in which coordination by H2 to a metal center (Z) increases the electron
count by two.
An open μo–Z interaction may be depicted by using a “dot-dashed-line” repre-
sentation, i.e., “•----,” in which each dot is intended to convey the electron that is
provided by each X to the 3c–2e interaction, while the two dashed lines attached to
each Z are intended to indicate that the electron number of Z has increased by two
units. Conceptually, the open μo–Z interaction may be considered to emerge from
the closed form by breaking the X–X bond in a homolytic manner, thereby
localizing, in a formal sense, an unpaired electron on X (Fig. 9).
Note that the two dashed lines attached to Z do not modify the valence of Z
since, by definition, Z does not contribute any electrons to the bond; as such, the two
X atoms of an open μo–Z interaction may be viewed as serving as an L donor
toward Z, in a similar way that the X–X bond serves as an L donor in a closed μc–Z
interaction. With respect to the X groups, although not connected directly, the
5
The notion of using an arrow to represent donation of electron density from a bond to another
atom was first introduced by Walsh to describe the bonding in B2H6; see [31].
The Covalent Bond Classification Method and Its Application to Compounds. . .
Fig. 7 Classification of 3-center 2-electron (3c–2e) interactions according to the nature of the
specified bridging atom. Although the orbitals shown are depicted as s-orbitals, the classification
also applies to p-, d-, and spn-hybrid orbitals. Note that the μc–X description is an alternative
setting for μc–Z. Two structure-bonding representations are shown for μc–X and μo–X; of these,
the upper ones are preferred but the lower ones are also acceptable. Class II [ZZL] interactions are
not included in this table because (1) we are not aware of any Class II μo–Z interactions and (2) the
closed Class II [ZZL] μc–Z interaction is formally equivalent to the closed Class II [ZLZ] μc–L
Fig. 8 Illustration of how three atoms, one with a Z function and two with X functions, may
combine to form open and closed 3c–2e interactions in which Z is assigned as the bridge. In each
case, the electron count of the Z atom increases by two, while that for the X atoms increases by
one. The closed μc–Z form can be conveniently represented by drawing an arrow from the
midpoint of the X–X bond to Z, while the open μo–Z interaction may be depicted by using a
“dot-dashed-line” representation, i.e., “•----”
M.L.H. Green and G. Parkin
Fig. 9 Illustration of how the open μo–Z interaction is derived from the closed μc–Z interaction
via a formal cleavage of the X–X bond
Fig. 10 Illustration of how three atoms, one with a Z function and two with X functions, may
combine to form open and closed 3c–2e interactions in which X is assigned as the bridge. In each
case, the electron count of the Z atom increases by two, while that for the X atoms increases by
one. The closed μc–Z form can be conveniently represented by drawing an arrow from the
midpoint of the X–X bond to Z, while the open μo–Z interaction may be depicted by using a
“half-arrow” representation, i.e., “X–X⇀Z”
electron count of each X increases by one because the electrons are shared with its
partner by virtue of the 3c–2e orbital. In essence, the empty orbital on Z provides a
mechanism for the two electrons on X to couple even though there is no direct
interaction between the atoms.
Similar to a closed μc–Z interaction, a closed μc–X interaction can also be
represented with an arrow drawn from the midpoint of the X–X bond to Z
(Fig. 10) because these two descriptions are formally equivalent and merely differ
by the identity of the atom that is ascribed to being the bridge. For example, in the
case of η2-dihydrogen compounds, the symmetry of the situation (i.e., two equiv-
alent X substituents) is such that the metal (i.e., Z) can be chosen as the bridge,
whereas in, for example, η2-silane compounds, [M–H–Si], the hydrogen atom (i.e.,
X) could be selected as the bridge on the basis that it has the two shortest bonds.
Thus, as noted above, it is essential for one to specify the identity of the bridging
atom when discussing a closed Class I system. Although the situation is less
ambiguous, it is also prudent to assign the bridging atom in an open μo–X system
in order to remove any uncertainty.
The Covalent Bond Classification Method and Its Application to Compounds. . .
Since there is no interaction between the outer X and Z atoms in an open μo–X
interaction, the bonding is more conveniently represented by using the X–X⇀Z
“half-arrow” notation in which the “half arrow” is drawn from the central X atom to
the outer Z atom (Figs. 7 and 10) [32]. Whereas a closed μc–X interaction typically
requires an angle at the central X atom that is distinctly nonlinear, an open μo–X
interaction is characterized by a large bond angle, as illustrated by the bridging
hydride complex, {[(CO)5Cr]2(μ-H)} [30]. The purpose of using a half arrow
(rather than a full arrow) from the atom is to emphasize that the arrow does not
correspond to donation of a lone pair but rather donation of the X–X bond pair. In
this regard, both the representation of a full arrow from the center of the X–X bond
in a closed μc–X system [31] and a half arrow from the central X in a μo–X system
convey the same information with respect to the impact on Z in terms of electron
counting purposes, i.e., donation of a pair of electrons. In many cases, the precise
nature (i.e., open versus closed) of the bonding of an asymmetric [XXZ] system
may not be known, either experimentally or theoretically. However, since both
bonding representations convey the same information with respect to electron
counting purposes, the form that is used may be a matter of convenience and should
not necessarily be used as a criterion to indicate whether an author considers that
the interaction is open or closed, unless it is explicitly stated.
For both open and closed μ–L interactions, the bridging L atom contributes a pair
of electrons to the 3-center orbital, while the outer Z atoms provide no electrons.
Since all three atoms share the pair of electrons, the donor L atom may be regarded
as contributing two electrons to the electron count of both outer Z atoms (Fig. 11).
This type of interaction may be represented by a pair of half arrows that are drawn
from the central atom (L) to each outer atom (Z); half arrows, rather than full
Fig. 11 Illustration of how three atoms, one with an L function and two with Z functions, may
combine to form open and closed 3c–2e interactions in which L is assigned as the bridge. In each
case, the electron counts of the Z atoms increase by two, while that for the L atom is unchanged.
Both open and closed interactions are represented with a pair of half arrows that are drawn from L
to Z to indicate that a single electron pair is being shared simultaneously with both Z atoms. A
dashed line is shown between the two Z atoms of the closed form to indicate that these atoms are
within bonding distance. Unlike a 2c–2e bond, however, none of the electron density between
these atoms derives from the Z atoms but derives only from the L atom
M.L.H. Green and G. Parkin
arrows, are used to indicate that a single electron pair is being shared simulta-
neously with both Z atoms. A dashed line is shown between the two Z atoms of the
closed form to indicate that these atoms are within bonding distance. However, it is
possible that the close proximity of these atoms may not solely be a consequence of
the 3c–2e interaction because closed-shell metallophilic interactions can also pro-
vide a means to bring two atoms into proximity [33–37].
Examples of metal-containing moieties that feature these types of interactions
and the corresponding CBC designation with respect to the metal are summarized in
Table 3.
Table 3 Summary of CBC designations for coordination of various ligands that feature 3c–2e
interactions
Description Example CBC classification with respect to M
Class I: μc–Z L
Class I: μo–Z L
Class I: μo–Z X
Class I: μ–X L
Class I: μ–X LX
6
For other examples in which 3c–2e bonding is represented as donation of a M–M bonding pair of
electrons to Au+, see [39].
M.L.H. Green and G. Parkin
7
Another closely related species is the hydride derivative, [([IPr]Au)2H]+, in which a gold center is
formally replaced by hydrogen and which possesses an Au–Au distance of 2.701 Å. See [41, 42].
The Covalent Bond Classification Method and Its Application to Compounds. . .
Fig. 17 Qualitative
molecular orbital diagram
for dihydrogen complexes
Fig. 19 An example of a compound which features coordination of a C–C bond to a metal center
Fig. 20 Comparison of closed μc–Z interactions in dihydrogen and olefin adducts. Note that the
olefin interaction is supplemented by a μc–L backbonding interaction from the metal
is that the accompanying π-backbonding interactions for the latter do not cleave the
C–C bond since the carbon atoms are also attached by a σ-bond; in contrast,
π-backbonding cleaves the bond in dihydrogen. The π-backbonding interaction for
olefins is classified as a μc–L interaction (vide infra), and thus coordination of an
olefin may be considered to be comprised of both μc–Z and μc–L interactions.
More common than σ-complexes that feature coordination of C–C bonds are those
that feature Si–Si bonds [66]. In this regard, it is interesting to note that the first
example of a transition metal compound that is now recognized to involve coordi-
nation of Si–Si bonds to a transition metal [67, 68] was originally proposed to be a
hexasilyl derivative (Fig. 21) [69].8
Another example of an intermolecular coordination of a Si–Si bond is provided
by the copper complex, {[Ph2P(C6H4)SiMe2–SiMe2(C6H4)PPh2]Cu}+, as illus-
trated in Fig. 22 [71]. Evidence for the interaction is provided by the observation
that the Cu–Si distances [2.7196(14) and 2.7212(14) Å] are only 12% longer than
the sum of the covalent radii.
An interesting example of an intermolecular disilane adduct is provided by
the N-heterocyclic carbene platinum complex, [IPr]Pt(η2-Me2PhSi–SiMe2Ph), as
illustrated in Fig. 23, although it was originally formulated as a disilyl derivative,
[IPr]Pt(SiMe2Ph)2 [72, 73]. However, key data to support the formulation as a
disilane σ-complex are provided by a very acute Si–Pt–Si bond angle [80.9(1) ]
and a Si–Si distance [2.980(5) ] that is much shorter than that in four-coordinate
8
For a highlight of this article, see [70].
M.L.H. Green and G. Parkin
Fig. 21 A compound that was originally proposed to be a hexasilyl derivative (right) but was
subsequently identified as a compound that features coordination of Si–Si bonds
Fig. 22 An example of a copper compound which features coordination of a Si–Si bond to a metal
center
Fig. 27 An open μo–Z interaction involving boron. (a) Overlap of the boron p-orbital with the
σ-symmetry orbitals of two X atoms results in a 3c–2e interaction that results in the electron count
of boron increasing by two (but not its valence), while the electron count of each X increases by
one. (b) Structure-bonding representation of the interaction
Fig. 28 (a) Structure of B5H9 showing connectivity. (b) Structure-bonding representation of B5H9
in the absence of an open μo–Z interaction, which indicates that B5 has a sextet configuration,
while B1 and B3 have septet configurations. (c) Structure-bonding representation of B5H9 showing
that each boron has an octet MLX3 configuration in the presence of an open μo–Z interaction
9
For a discussion of the bonding in B5H9, see [90–92].
M.L.H. Green and G. Parkin
in Fig. 28c. It should be noted that Fig. 28c represents only one resonance structure.
Specifically, all four equatorial boron atoms (B1–B4) are equivalent by virtue of a
C4 axis, and thus an equivalent resonance structure in which there is a μo–Z
interaction involving B2, B4, and B5 also exists. For comparison, the structure-
bonding representation for the form without the μo–Z interaction is illustrated in
Fig. 28b. This structure features an axial boron (B5) that possesses a sextet config-
uration and two equatorial boron atoms (B1 and B3) that have septet configurations;
the remaining two equatorial boron atoms (B2 and B4) possess octet configurations.
Thus, comparison of these two structure-bonding representations (Fig. 28a and b)
makes it evident how the presence of the μo–Z interaction allows the axial boron
(B5) and the two of the equatorial boron atoms (B1 and B3) to achieve an octet
configuration.
The axial [BH] moiety of B5H9 may be formally replaced by metal centers to give
(B4H8)[M] derivatives. Two illustrative examples are provided by the iron and
cobalt compounds, (B4H8)Fe(CO)3 [93] and (B4H8)CoCp [94–96], which, by
analogy with B5H9, can be illustrated by the structure-bonding representations in
Fig. 29. As with B5H9, it is worth noting that equivalent resonance structures can
also be drawn with bonds between the other two boron atoms and the metal centers.
Consideration of the structure-bonding representations indicates that the iron com-
pound belongs to the 18-electron class of ML4X2, while the cobalt compound
belongs to the 18-electron class of ML3X3. Both of these descriptions correspond
to well-known CBC designations for the respective metals.10
10
It is pertinent to note that, in contrast to B5H9, the iron and cobalt compounds have additional
valence electrons such that they could support additional 2c–2e bonds. For such a situation, the
iron and cobalt compounds would be, respectively, categorized as ML3X4 and ML2X5, which are
much less common than ML4X2 and ML3X3 for these elements.
The Covalent Bond Classification Method and Its Application to Compounds. . .
Compounds with Class I interactions that feature a μ–X bridge are very common, as
illustrated by (1) hydrocarbon and silane σ-complexes, (2) bridging hydride com-
plexes, and (3) agostic complexes.11 In many of these complexes, however, there
may be insufficient experimental or theoretical evidence to distinguish whether the
interaction is best described as closed or open. Therefore, compounds
corresponding to both classes of interactions are included in this section, with it
being recognized that the choice of representation, i.e., a full arrow from the center
of the X–X or a half arrow from the central X, is not necessarily being used to
distinguish between closed and open situations.
Transition metal hydrocarbon [54, 98–100] and silane [101–111] adducts corre-
spond to a class of σ-complexes [51–54] in which C–H and Si–H bonds interact
with a metal center (Fig. 30).12 Such molecules are closely related to η2-dihydrogen
complexes but are classified as possessing μ–X rather than μ–Z interactions because
they lack the symmetry that is present in η2-dihydrogen complexes. Thus, the
hydrogen atom of hydrocarbon and silane adducts is more appropriately designated
as the bridge on the basis that it exhibits the two shortest bond distances. Two
representations to describe these interactions are provided in Fig. 30, recognizing
that they can be used equivalently to determine the electron count and CBC
designation of a metal center.
By comparison to η2-dihydrogen complexes, hydrocarbon σ-complexes
are particularly unstable, with there being no crystal structures of adducts that
persist in solution. Interactions between alkanes and a metal center have, never-
theless, been observed in the solid state by X-ray diffraction, although their
Fig. 30 Two
representations of
hydrocarbon (top) and
silane (bottom)
σ-complexes. Both
representations are intended
to convey the same
information with respect to
electron counting
11
It should be emphasized that not all M–H–Y interactions must be described as 3c–2e bonds
because some are better represented as 3c–4e “hydrogen bond” interactions. See, e.g., [97].
12
Compounds that feature coordination of Ge–H and Sn–H bonds have also been investigated; see,
e.g., [112].
M.L.H. Green and G. Parkin
Fig. 33 Bridging silane compounds, including one that features a six-coordinate hypervalent
silicon center
Lewis base adducts of boranes, alanes, and gallanes (LEH3; E ¼ B, Al, Ga) are
isoelectronic with methane and silanes, and so the interaction of the E–H bonds in
these molecules with metal centers can likewise be represented by the use of either
the full-arrow or half-arrow notations (Fig. 34).
Examples of borane compounds that feature such interactions are provided
by the chromium, molybdenum, and tungsten complexes, (CO)5M(κ1-H3BL)
13
Note that the hypervalent representation of the silicon of {[PhBCH2CH2PPh2]Ru}2(μ-SiH6) is
drawn for convenience.
M.L.H. Green and G. Parkin
(L ¼ NMe3, PMe3, PPh3), which represent early examples of this class of com-
pounds (Fig. 35) [147–149].
A variety of other adducts of four-coordinate neutral boranes have also been
synthesized [150–152], including compounds in which the borane coordinates
via two of the hydrogen atoms, e.g., [Rh(PBui3)2H2(κ2–H3BNMe3)]+ [153] and
[NacnacAr2]Cu(κ2–H3BL) (L ¼ Me3N, lutidine) [154], as illustrated in Fig. 36.
σ-Complexes of Lewis base adducts of alanes [155, 156] and gallanes [157, 158]
have also been reported, as illustrated in Figs. 37 and 38.
The B–H bond of electronically unsaturated three-coordinate boranes, R2BH,
can also interact with a metal center. However, an important distinction with
respect to the coordination of the B–H bond of tetrahedral LBH3 is that the boron
of R2BH has an additional empty orbital that enhances the backbonding interaction
The Covalent Bond Classification Method and Its Application to Compounds. . .
Fig. 36 B–H σ-complexes of Lewis base adducts of boranes involving bidentate coordination
[105, 159, 160]. The coordination of R2BH can, therefore, be described in terms
of two interactions, namely, (1) a 3c–2e interaction that involves donation from the
B–H bond to the metal (i.e., L) and (2) a 2c–2e backdonation interaction from the
metal to boron (i.e., Z), as illustrated in Fig. 39 (left).
M.L.H. Green and G. Parkin
A large variety of bridging hydride compounds are known [30, 164–166], and the
angle at hydrogen may be very obtuse. For example, the average Cr–H–Cr bond
angle of [(Ph3P)2N]{[(CO)5Cr]2(μ-H)} is 155.8 [167]. As such, the half-arrow
representation to describe the bonding is particularly appropriate (Fig. 41).
In addition to [MHM’] bridges involving transition metals, variants in which
one of the metals is a main group element are also known. For example,
[NacnacAr2]Cu(η2-toluene) reacts with aluminum and zinc hydride compounds to
afford [NacnacAr2]Cu(σ-HMLn) (e.g., MLn ¼ [NacnacAr’2]Zn, [NacnacAr”2]AlH)
[168]. However, the formation of [NacnacAr2]Cu(σ-HMLn) is reversible, such that
these complexes have been described as weak copper(I) σ-complexes [168].
Compounds that feature double, triple, and quadruple hydride bridges are also
known (Figs. 42 and 43) [164] and can likewise be represented by the half-arrow
representation.
Fig. 41 An example of a bridging hydride complex with a large M–H–M bond angle
Fig. 42 Different descriptions of the metal–metal bond orders in some dinuclear complexes with
bridging hydride ligands according to the electron counting method. The “half-arrow” method
(left) predicts M–M bond orders that are in accord with theory, in contrast to the “half-electron”
method (right)
M.L.H. Green and G. Parkin
14
Although the hapto “ηx” notation [178] is often used to describe the coordination mode of
borohydride ligands [175–177], such notation is strictly inappropriate because ηx refers to the
number of contiguous atoms that are attached to a specific element [179]. If the atoms are not
contiguous, as in borohydride, the kappa “κx” notation [180] should be used instead.
15
Note that alternative structure-bonding representations involving donation of electron density
from a B–H bond of anionic [BH4] to cationic M+ can also be drawn. Such representations,
however, are equivalent to those shown in Fig. 44, which do not portray formal charges.
M.L.H. Green and G. Parkin
Fig. 46 Early examples of compounds that feature /- and β-agostic interactions (top) and an
example of an agostic phenyl ligand (bottom)
Agostic compounds, i.e., those which feature 3c–2e M–H–C interactions (Fig. 45), are
now recognized to be an important feature of organometallic chemistry [181–183].
Particularly notable examples of compounds that possess agostic interactions are
provided by the titanium alkyl compounds, (dmpe)TiMeCl3 and (dmpe)TiEtCl3, in
which the titanium centers interact, respectively, with the /- and β-C–H bonds
(Fig. 46) [184–186]. Agostic interactions are also observed in aryl compounds, as
illustrated in Fig. 46 [187]. In each case, an agostic ligand is considered to be an LX
three-electron donor.16
Bridging alkyl groups adopt a variety of coordination modes, some of which may
feature agostic interactions [13, 192–195], although the energetic preferences may
16
This view of the bonding in agostic compounds is necessarily simplistic. For more detailed
discussions, see [188–191].
The Covalent Bond Classification Method and Its Application to Compounds. . .
Fig. 47 Coordination modes of bridging methyl ligands (the lines between atoms are only to
indicate connectivity and are not intended to be structure-bonding representations)
Compounds that possess Class II interactions with μ–L bridges, in which a single
atom provides both electrons for the 3-center interaction, are much less common
than those with Class I interactions in which two of the atoms each provide one
electron. As with Class I compounds, there may be insufficient experimental or
theoretical data to identify whether the interaction is best described as closed or
open. Therefore, for expediency, the compounds described herein are represented
with the open form, but it must be emphasized that this representation is not being
used to infer that the interaction is not closed.
While PR3, AsR3, and SbR3 ligands almost exclusively coordinate in a terminal
manner, examples in which these ligands bridge two metals are also known, as
illustrated by the rhodium compounds shown in Fig. 49 [201, 202].17 The migration
of PR3 ligands between two metal centers has also been postulated to occur via
intermediates that possess μ-PR3 ligands [212]. In addition, μ-phosphole derivatives
of Pd [213–215], Pt [213, 216], Cu [213, 217, 218], and Ag [219] have also been
synthesized (Fig. 50).
Fig. 49 Examples of
compounds that feature
bridging PR3, SbR3, and
AsR3 ligands
17
Compounds with triply bridging PF3 ligands are also known; see, e.g., [203–205]; for an early
speculation of a complex with a bridging PR3 ligand, see [206]; for compounds with asymmetri-
cally bridging PR3 ligands, see [207–210]; for calculations on hypothetical bridging PF3 com-
plexes, see [211].
The Covalent Bond Classification Method and Its Application to Compounds. . .
Fig. 50 Examples of
compounds that feature
bridging phosphole ligands
(the dotted lines indicate
metallophilic interactions)
The nitrogen atom of acetonitrile has also been shown to bind in a similar manner
and bridge two metals in a variety of compounds [220–227], as illustrated by the
dinuclear copper compound, [(dpen)Cu2(μ-NCMe)]2+ (Fig. 51) [221]. The bonding
in this complex involves overlap of the nitrogen lone pair with an empty in-phase
combination of spn hybrids on each copper (Fig. 52). Density functional theory
calculations indicate that there is no formal bond between the two copper centers of
[(dpen)Cu2(μ-NCMe)]2+ because all pairs of Cu–Cu bonding and antibonding
orbitals are filled [221], while the quantum theory of atoms in molecules predicts
a bond critical point that has characteristics which are consistent with a cuprophilic
closed-shell interaction, rather than that of a formal single bond. Consistent with
this description, the structure-bonding representation of this molecule (Fig. 51)
indicates that each copper belongs to the 18-electron class ML3X, such that a direct
Cu–Cu bond would not be expected.
M.L.H. Green and G. Parkin
Fig. 52 Qualitative molecular orbital diagram for a bridging acetonitrile compound adapted
from [221]; the primary interaction involves donation of the nitrogen lone pair into the empty
in-phase combination of spn-hybrid orbitals on copper
The Covalent Bond Classification Method and Its Application to Compounds. . .
18
In addition to bridging in a symmetric manner, carbonyl ligands are also known to adopt bent
semibridging and linear semibridging coordination modes; the bonding in these complexes is
highly varied and is not part of the scope of the present article; for key references, see [228–237].
19
A subsequent higher-quality structure revealed an Fe–Fe distance of 2.523(1) Å; see [242].
20
Although some theoretical articles have suggested the possibility of a weak direct Fe–Fe
attractive interaction in Fe2(CO)9, the interpretation has been questioned [271].
21
We are aware of only two textbooks that discuss the absence of a direct Fe–Fe bond in Fe2(CO)9.
Of these, one does not include a drawing of the molecule [272], while the other draws the molecule
both without an Fe–Fe bond and also with a dotted FeFe bond [273]; however, there is no
discussion as to how the latter description should be employed with respect to electron counting
purposes. Also of note, [CpM(CO)(μ-CO)]2 has been represented on the cover of a textbook,
without including a M–M bond, but the bonding was not discussed [274].
M.L.H. Green and G. Parkin
Fig. 53 The two bonding molecular orbitals derived from interaction of the 5σ HOMO and one of
the 2π* C–O antibonding orbitals with the in-phase and out-of-phase combination of metal
d-orbitals
22
Co2(CO)8 exists as an equilibrium between bridged, (CO)3Co(μ-CO)2Co(CO)3, and non-bridged
isomers, (CO)4Co–Co(CO)4, of which the former is the major isomer.
23
Experimental charge-density studies are also in accord with the absence of a direct Co–Co bond
in the bridged form of Co2(CO)8; see [276].
24
For other articles that describe the absence of M–M bonds in other carbonyl bridged compounds,
for a similar reason, see [287].
The Covalent Bond Classification Method and Its Application to Compounds. . .
Fig. 54 MOs for a M(μ-CO)M interaction (a) occupied by four electrons, giving two 2c–2e
bonds, and (b) occupied by two electrons, giving a 3c–2e bond because the orbital corresponding
to π-backdonation is empty. For (a), the CO is classified as a μ–X2 donor and contributes one
electron to each metal, whereas for (b), the CO is classified as a μ–L donor and contributes a pair of
electrons to both metals
The bonding within Fe2(CO)9 has received considerable attention, and a simple
molecular orbital description that focuses on the bridging interactions (Fig. 54) in
D3h symmetry is provided in Fig. 55 [259, 260]. Thus, each [Fe(CO)3] fragment has
three orbitals and two electrons available for bonding, while each bridging carbonyl
ligand contributes a 5σ orbital, the 2πz* orbital that lies parallel to Fe–Fe axis (z),25
and a pair of electrons. The component of the molecular orbital diagram that
pertains to the Fe(μ-CO)3Fe moiety thus contains ten electrons, with six being
contributed by the three CO ligands and four being contributed by the two
[Fe(CO)3] fragments (Fig. 55).
The most important aspect of this molecular orbital diagram is that there are
three occupied bonding orbitals (a1’ and e’) that correspond to donation by the
three 5σ orbitals (Fig. 55), but only two occupied bonding orbitals (e”) that
correspond to backbonding into the 2πz* orbitals (Fig. 55). Since each μ–X2
“ketonic” description of a carbonyl ligand requires occupation of both the σ- and
π-type orbitals, it is evident that the bonding within Fe2(CO)9 cannot be described
as possessing three “ketonic” bridging carbonyl ligands. The bonding within
Fe2(CO)9 is thus more appropriately described as a resonance hybrid in which
each structure possesses two μ–X2 “ketonic” carbonyl ligands and one μ–L car-
bonyl ligand (Fig. 56). This resonance description of the bridge bonding is in accord
with other theoretical studies. For example, Ponec et al. state that “. . . 4 out of the
5 bonding electron pairs are involved in localized 2c–2e bonding of two bridging
ligands, while the remaining ligand is bonded via a delocalized electron pair with
the character of [a] 3c–2e bond” [271].
If the three bridging carbonyl ligands in Fe2(CO)9 are described by one μ–L and
two μ–X2 interactions, each iron center may achieve an electron count of 18 without
the necessity of forming an Fe–Fe bond, as illustrated by the structure-bonding
representation in Fig. 57.
Electron counting viewing carbonyl ligands as μ–X2 “ketonic” moieties is often
invoked for assigning M–M bond orders, with the outcome that it often results in
bond orders that are not in accord with theory. In contrast, consideration of
the possibility that a carbonyl ligand can serve as a μ–L ligand results in M–M
bond orders that are in better accord with theory. For example, structure-bonding
representations of the bridged form of Co2(CO)8 and [CpFe(CO)(μ-CO)]226 [294],
which depict the absence of M–M bonds that are in accord with theory [265, 275,
278–286], are illustrated in Fig. 58.
Thus, whereas a symmetrically bridging carbonyl is commonly represented as a
μ–X2 “ketonic” ligand, it is evident that its alternative description as a μ–L donor
can provide a more accurate representation of the bonding, which thereby allows
the prediction of M–M bonds that are in better accord with theory.
25
The 2π* orbitals of CO which are perpendicular to the Fe–Fe axis are neglected from the
analysis on the basis that there is little overlap with the metal d-orbitals.
26
For discussion pertaining to bonding in [CpFe(CO)(μ-CO)]2, see [293].
The Covalent Bond Classification Method and Its Application to Compounds. . .
Fig. 55 Qualitative molecular orbital diagram showing the interaction of fragment orbitals
involved in bridge bonding in Fe2(CO)9 with D3h symmetry (the z axis is coincident with the
Fe Fe vector). For clarity, orbitals associated with bonding to the terminal carbonyl ligands are
not illustrated, while only the carbon 2pz orbital is used to represent the 2π* orbital of CO (adapted
from [13])
M.L.H. Green and G. Parkin
Fig. 56 Three resonance structures that correspond to the bonding illustrated in the molecular
orbital diagram of Fig. 55. Each structure corresponds to three donor and two backbonding
interactions
Fig. 57 Structure-bonding representation of Fe2(CO)9 illustrating that each iron can achieve an
18-electron configuration without an Fe–Fe bond
experimental studies [295, 296]. The experimental and computational studies may,
however, be reconciled by viewing the RB: ligand as a μ–L donor (Fig. 59, right). In
this way, the manganese centers can achieve an 18-electron configuration without
having to form a Mn–Mn bond.
27
For example, the B. . .B distance of 2.11 Å is ca. 0.4 Å longer than the value in catBBcat
derivatives; see [298].
M.L.H. Green and G. Parkin
Fig. 60 Structural
representation (top), in
which lines are used merely
to indicate connectivity, and
structure-bonding
representations (bottom) for
Cp2Ti(η2-HBcat)2. The
structure on the bottom left
views the compound as an
adduct between [Cp2Ti] and
two HBcat moieties,
whereas the structure on the
bottom right views the
compound as an adduct
between Cp2TiH2 and
catBBcat
center, in which a dn metal center becomes dn–2 [28, 301–303]. As such, the
titanium center of Cp2Ti(η2-HBcat)2 is classified as tetravalent d0 18-electron
ML5X4 (Fig. 60), rather than as a Ti(II) derivative [161, 297]. Correspondingly,
the boron is classified as MLX3 with an octet configuration.
While the view of Cp2Ti(η2-HBcat)2 as an adduct between [Cp2Ti] and two
molecules of HBcat (Fig. 60, bottom left) is useful because the compound is
obtained via a reaction with HBcat, other bonding descriptions are also possible.
In particular, Cp2Ti(η2-HBcat)2 can be regarded as an adduct between Cp2TiH2 and
the diborane, catBBcat (Fig. 60, bottom right). Such a description is of use because
it emphasizes the B–B bonding interaction. Regardless of which bonding descrip-
tion one prefers to adopt, however, the CBC descriptions of the molecule (ML5X4)
are the same. In contrast, formalisms based on oxidation state concepts result in
different descriptions, namely, Ti(II) and Ti(IV) [161, 297].
Fig. 61 3c–2e interaction for Cp2Ti[η2-HB(OH)2]2. Note that while the configuration of the
titanium in the [Cp2Ti] moiety is d2, it becomes d0 upon coordination of the two borane ligands
and formation of the 3c–2e bond
28
The structure-bonding representation for the zirconium compound is one in which only one of
the oxygen lone pairs of each aryloxide ligand participates in π-donation; additional π-donation
would result in an 18-electron ML6X2 classification.
M.L.H. Green and G. Parkin
metal (M’) and as an L3 ligand to the other (M), and (3) a bridging borole ligand
serves as an L2X ligand to each metal.
Furthermore, these motifs may be elaborated into multi-decker structures. For example,
triple-decker arene complexes are common [313, 314], of which a classic example
is provided by the mesitylene complex (η6-MesH)Cr(μ-η6,η6-MesH)Cr(η6-MesH)
[315, 316] (Fig. 66, left). Since bridging arene ligands serve as L3 donors to each
The Covalent Bond Classification Method and Its Application to Compounds. . .
Fig. 66 Triple-decker compounds that feature bridging benzene, cyclopentadienyl, and borole
ligands
Fig. 68 Anti- and syn-coordination of two metals to a pentalene ligand, indicating only
connectivity
The Covalent Bond Classification Method and Its Application to Compounds. . .
Fig. 70 M–M bond orders for some dinuclear bis(pentalene) derivatives (substituents are not
shown for clarity) predicted by assuming that the bridging pentalene is either (i) a four-electron L2
donor to each metal (top) or (ii) a five-electron L2X donor to each metal (bottom). The M–M bond
orders predicted by the latter method are in accord with theory, whereas those predicted by the
former approach are in disagreement
29
For other calculations on this system which propose a Mn–Mn bond, see [335].
The Covalent Bond Classification Method and Its Application to Compounds. . .
DFT calculations also indicate the absence of an Fe–Fe bond in the iron
compound syn-(μ-η5,η5-Pn*)[Fe(CO)2]2(μ-CO) [331], which is of note because
the less substituted derivative, syn-(μ-η5,η5-Pn)[Fe(CO)2]2(μ-CO), has been
described as possessing a formal Fe–Fe single bond on the basis that the pentalene
ligand donates four π-electrons to each iron (Fig. 73, top) [336, 337]. However, the
absence of an Fe–Fe bond is predicted by adopting the view that the pentalene
ligand is an L2X donor to each iron center (Fig. 73, bottom).
The ruthenium compound syn-(μ-η5,η5-Pn)[Ru(CO)2(GeMe3)]2 has also been
represented with a metal–metal bond (Fig. 74, top) [338, 339], but subsequent
calculations indicate that there is no significant interaction between the metal
centers [340]. As such, the calculations are in agreement with the structure-bonding
representation that invokes the pentalene ligand as an L2X donor to each ruthenium
(Fig. 74, bottom).
Two metal centers may also coordinate via opposite faces of the pentalene ligand
in an anti-manner. The bonding in complexes with this arrangement can likewise
be viewed in terms of the double bond of the ring junction acting as a μ–L donor,
such that each ring is a five-electron L2X donor to each metal. In this manner,
the iron and manganese complexes, anti-(μ-η5,η5-Pn)[FeCp*2] [341, 342] and anti-
(μ-η5,η5-Pn)[Mn(CO)3]2, are predicted to have 18-electron configurations (Fig. 75)
rather than the 17-electron configurations that would otherwise be predicted if the
pentalene ligands were simply considered to be four-electron donors to each metal.
The metal centers in these compounds are, therefore, formally analogous to those in
the well-known cyclopentadienyl compounds, CpMn(CO)3 and Cp2Fe, which
thereby reiterates the validity of classifying the pentalene ligand as an L2X donor
to each metal.
The Covalent Bond Classification Method and Its Application to Compounds. . .
5 Summary
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