Atomic-Orbital-Symmetry Based ␴-, ␲-,
and ␦-Decomposition Analysis of
Bond Orders
OLGA V. SIZOVA, LEONID V. SKRIPNIKOV,
ALEXANDER Yu. SOKOLOV, VLADIMIR V. SIZOV
Department of Chemistry, St. Petersburg State University, Universitetskii pr., 26,
198504 St. Petersburg, Russia
Received 29 September 2008; accepted 27 October 2008
Published online 10 April 2009 in Wiley InterScience (www.interscience.wiley.com).
DOI 10.1002/qua.21978
ABSTRACT: The atomic-orbital-symmetry based (AOSB) scheme for the
decomposition of Mayer and Wiberg bond orders into ␴-, ␲-, and ␦-components is used
to investigate different types of covalent bonds. Four series of compounds are studied:
simple molecules with homonuclear bonds, inorganic molecules with polar
heteronuclear bonds, [Ru(CN)5(XY)]q transition metal complexes with ␲-acceptor
ligands, and dimetal complexes with multiple metal–metal bonds. © 2009 Wiley
Periodicals, Inc. Int J Quantum Chem 109: 2581–2590, 2009
Key words: bond orders; ␴-, ␲-, and ␦-bonds; ab initio; DFT
Introduction
T
he concept of ␴-, ␲-, and ␦-bonds plays a
crucial role in theoretical chemistry. Because
the notion of ␴-, ␲-, and ␦-bonds is closely related to
the concept of chemical bond multiplicity, it is reasonable to break up the quantum chemical bond
order into the corresponding components. According to the qualitative molecular orbital theory, the
formal bond order (or “chemist’s bond order”[1]) in
a diatomic molecule is
BO ⫽
Nbonding ⫺ Nantibonding
,
2
(1)
Correspondence to: O. V. Sizova; e-mail: [email protected]
where Nbonding and Nantibonding are the number of
electrons on the bonding and antibonding MOs
respectively. This bond order can be separated into
␴-, ␲-, and ␦-contributions according to the symmetry of the occupied MOs. However, the notions of
␴-, ␲-, and ␦-bonds are widely used by chemists
even for the molecules, which do not belong to the
D⬀h or C⬀v symmetry. This practice implies the use
of ␴, ␲, and ␦ designations in a local sense to denote
the bond components of the specified diatomic
fragment of a molecule.
The widely used density-matrix-based quantum
chemical descriptors of bond orders known as
Wiberg bond indices (WBI) and Mayer bond indices (MBI) were developed in the 1960s/1980s by
Wiberg, Borisova, Giambiagi, and Mayer [1–9].
Bridgeman et al. [10] showed that the Mayer bond
International Journal of Quantum Chemistry, Vol 109, 2581–2590 (2009)
© 2009 Wiley Periodicals, Inc.
SIZOVA ET AL.
order can be fully or partially decomposed into the
contributions corresponding to each irreducible
representation of the point group of the molecule; a
similar decomposition technique was later incorporated into the aomix program [11]. This molecularorbital-symmetry-based (MOSB) decomposition directly gives the ␴-, ␲-, and ␦-components of the
bond order only for linear molecules, though these
components can also be obtained for some of the
more complex species by grouping the contributions with appropriate symmetry [10, 12]. It is also
possible to quantitatively estimate the contributions
from each irreducible representation into the energy of orbital interactions by the energy decomposition analysis (EDA) [13, 14], though the orbital
energies of ␴-, ␲-, and ␦-bonds can be evaluated
only if the molecule has suitable symmetry.
Taking into account the local nature of the bond
indices we proposed the atomic-orbital-symmetry
based (AOSB) bond order decomposition scheme, a
tool for chemical interpretation of the results of
quantum chemical calculations of the complex organic and inorganic molecules using the notions
traditionally accepted by chemists [12, 15]. This
decomposition technique, which is valid for any
bond index based on the density matrix, was applied to the well-known Mayer covalent bond order
indices (MBI) [5– 8] and WBI [2]. The main purpose
of the present work is to show the ability of the
AOSB bond order decomposition to effectively analyze the electronic structure of several types of
compounds: diatomics and organic species with
homonuclear XOX bonds, inorganic molecules
with heteronuclear XOY bonds, transition metal
complexes with ␲-acceptor ligands, and complexes
with metal-metal bonds.
The Mayer, BAB, and the Wiberg, WAB, bond
order indices of the bond between A and B atoms
for closed-shell systems were calculated as [2, 5– 8]
冘 冘共DS兲
B AB ⫽
␮␯
共DS兲␯␮ ,
(2)
␮僆A␯僆B
W AB ⫽
冘 冘共D
␮␯
兲2,
(3)
␮僆A␯僆B
where S is the overlap matrix, D is the total density
matrix:
冘C C .
occ.
D ␮␯ ⫽ 2
␮i
(4)
␯i
i
For the WBI calculation a transformation from
AO to the orthogonal natural atomic orbitals (NAO)
was performed [22–24]
P NAO⫽T †(SDS)T,
(5)
P NAO,␣⫽T †(SP ␣S)T, PNAO,␤⫽T†(SP␤S)T.
(6)
To obtain reasonable bond orders for the H⫹
2
cation and other systems with unpaired electrons
the BAB and WAB values for the open-shell systems
were calculated using the UHF method as [6, 9]
B AB ⫽
冘 冘 关共DS兲
␮␯
共DS兲 ␯␮ ⫹ 共QS兲 ␮␯ 共QS兲 ␯␮ 兴,
(7)
␮ 僆A ␯ 僆B
W AB ⫽ 2
冘 冘 关共P
␣ 2
␮␯
␤ 2
兲 ⫹ 共P ␮␯
兲 兴,
(8)
␮ 僆A ␯ 僆B
where
D⫽P ␣⫹P ␤, Q⫽P ␣⫺P ␤,
Computational Details
The electronic structure calculations were performed using the Gaussian 03 [16] package. The
geometries of the compounds were optimized and
the matrices of the atomic orbital coefficients in the
molecular orbitals (C) were obtained at the B3LYP/
cc-pVDZ levels of theory [17–19] for the molecules
of main group elements and at the B3LYP/
LanL2DZ levels of theory [20, 21] for the transition
metal complexes. The spin-unrestricted UB3LYP
method was used for the open-shell systems. The
atomic orbital populations were calculated by the
natural population analysis (NPA) [22, 23].
冘C C , P ⫽ 冘C C .
occ.
P
␣
␮␯
⫽
(9)
occ.
␮i
␯i
␤
␮␯
i僆 ␣
␮i
␯i
(10)
i僆␤
AOSB-A SCHEME
The common procedure of the partitioning of the
bond order into individual ␴-, ␲-, and ␦-contributions is based on the symmetry of the occupied
MOs and this is rather evident for the diatomic
molecules. The AOSB-A decomposition scheme
considers the diatomic A–B fragments of a polyatomic molecule and uses the C⬀v symmetry of the
2582 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
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ATOMIC ORBITAL SYMMETRY
A–B diatomic fragment for the sorting of (DS)␮␯ or
P␮␯ according to ␮ and ␯ orbital symmetry relative
to the A–B axis. For this purpose the coordinate
system of the molecule is transformed so that atoms
A and B are placed on the z-axis for each specified
A–B pair in the molecule. The C and S matrices are
transformed correspondingly and are used subsequently for the calculation of BAB indices and their
components, for the construction of NAO and for
the calculation and decomposition of WAB. As A
and B atoms are placed on the z-axis, their orbitals
with quantum number m ⫽ 0 can be assigned to ␴-,
those with m ⫽ ⫾1 to ␲- and with m ⫽ ⫾2 to
␦-orbitals. So, for the linear molecules the expressions for BAB and WAB can be written as
B AB ⫽
冘 冘 共DS兲 共DS兲 ⫹ 冘 冘 共DS兲 共DS兲
⫹ 冘 冘 共DS兲 共DS兲 ⫽ B ⫹ B ⫹ B
␮␯
␮僆A␮僆␴
␯僆B ␯僆␴
␮␯
␮僆A␮僆␦
␯僆B ␯僆␦
W AB ⫽
冘 冘 共P
␮僆A␮僆␴
␯僆B ␯僆␴
␯␮
␯␮
兲 ⫹
NAO 2
␮␯
␮␯
␮僆A␮僆␲
␯僆B ␯僆␲
␴
AB
冘 冘 共P
␮僆A␮僆␲
␯僆B ␯僆␲
␲
AB
兲 ⫹
NAO 2
␮␯
␯␮
␦
AB
(11)
冘 冘共P
␮僆A␮僆␦
␯僆B ␯僆␦
␴
␲
␦
⫽ WAB
⫹ WAB
⫹ WAB
兲
NAO 2
␮␯
(12)
For nonlinear molecules Eqs. (11) and (12) are
not exact due to the “cross-terms” arising if ␮ and ␯
orbitals do not belong to the same symmetry type.
Defining the sum of such “cross-term” contribuCTC
CTC
tions (CTC) as BAB
and WAB
one can rewrite the
expressions for MBI and WBI in the following form:
␴
␲
␦
CTC
B AB ⫽ BAB
⫹ BAB
⫹ BAB
⫹ BAB
(13)
nificantly distorted and the large CTC values may cast
doubt on the results of AOSB-A decomposition for
such molecular fragments. The largest CTC values
were observed in the following cases [12]:
i. organic molecules with small cycles;
ii. complexes with ␩2-linkage ligands;
iii. ␲-complexes with ␩k-cyclic (k ⬎ 2) ligands.
In the same molecule there can be atom pairs for
which the standard definition of ␴-, ␲-, and ␦-bonds
is suitable, and pairs, for which this definition does
not work. In the latter case the AOSB-A technique
fails to provide an adequate description of the nature of bonding. For the molecules with large CTC
the MBI and WBI values for non-adjacent atom
pairs can be noticeable, suggesting the use of multicenter bond indices [25–29].
If all CTC are related to the orbitals in the plane
of the molecule or fragment, the AOSB-P approach
can be useful. The AOSB-P scheme is based on the
assignment of the A and B AOs to symmetric or
antisymmetric orbitals with respect to the reflection
in this plane using the local Cs symmetry. In this
case it is necessary to specify three atoms, which
will define the (xy) plane in the new molecular
coordinate system. Then the pz, dxz, and dyz orbitals
of A and B atoms belong to the irreducible representation a⬙ (“out-of-plane” AOs)1 of Cs point group
and all other AOs belong to a⬘ (“in-plane” AO’s). As
a consequence, the A–B bond index breaks down
into two components
B AB ⫽
and
␴
␲
␦
CTC
⫹ WAB
⫹ WAB
⫹ WAB
.
W AB ⫽ WAB
冘 冘 共DS兲 共DS兲
⫹ 冘 冘 共DS兲
␮僆A ␯僆B
␮僆a⬘␯僆a⬘
␮␯
␮僆A ␯僆B
␮僆a⬙␯僆a⬙
␯␮
␮␯
a⬘
a⬙
共DS兲␯␮ ⫽ BAB
⫹ BAB
.
(15)
(14)
and
For the open-shell systems the technique is modified appropriately. In all calculations, the validity
of the AOSB-A procedure for any chosen bond can
be monitored through the CTC values: if CTC is not
negligible then B ⫽ B␴⫹ B␲⫹ B␦ or W ⫽ W␴⫹ W␲⫹
W␦, and the results of the AOSB-A decomposition
are controversial.
AOSB-P SCHEME
If two adjacent atoms A and B participate in the
bonding with the same X atom, the symmetry of
electron distribution around the A–B line can be sig-
VOL. 109, NO. 11
DOI 10.1002/qua
a⬘
a⬙
W AB ⫽ WAB
⫹ WAB
.
(16)
For the open-shell systems the technique is modified appropriately.
Calculations of MBI and WBI and their symmetry decomposition for both AOSB schemes were
carried out using the MWBI-AOSBD program [12],
for which atomic coordinates, C matrix, and numbers of atoms for each specified A–B pair were only
1
AO’s, which correspond to ␲-electron orbitals within the
Hückel molecular orbital approximation.
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 2583
SIZOVA ET AL.
TABLE I ______________________________________
Bond lengths, r, bond order indices, WXX, their
components, W␴ and W␲, computed at B3LYP/ccpVDZ level of theory for diatomics and simple
organic molecules.
Molecule
Bond
r (Å)
WXX
W␴
W␲
H2
H2⫹
H2⫺
F2
Cl2
O2
N2
C2H6
C6H12
C2H4
C2H2
C6H6
HOH
HOH
HOH
FOF
ClOCl
OOO
NON
COC
COC
COC
COC
COC
0.762
1.114
1.524
1.410
2.049
1.209
1.104
1.530
1.536
1.333
1.210
1.399
1.00
0.50
0.53
1.00
1.02
2.01
3.00
1.05
1.02
2.04
2.99
1.44
1.00
0.50
0.53
1.00
1.00
1.00
1.00
1.00
0.97
1.00
1.00
0.96
0.00
0.00
0.00
0.00
0.02
1.01
2.00
0.05
0.05
1.04
2.00
0.47
taken as the input data. The S matrix was obtained
via inversion of the C matrix. Thus, the results of
AOSB decomposition for all bonds in the molecule
can easily be obtained from one HF or DFT quantum chemical calculation. The detailed testing of
AOSB technique was carried out in [12]. In Tables
I–IV, VI–VII the values of WBI and their components are presented since WBI are more stable than
MBI with respect to basis set variation.
Results and Discussion
HOMONUCLEAR XOX BONDS IN DIATOMICS
AND ORGANIC MOLECULES
Table I presents the AOSB-A decomposition results for diatomic inorganic and simple organic
molecules with different multiplicities of the homonuclear XOX bond. For all molecules zero or negligible CTC values were obtained. The results of the
proposed technique are in total accordance with the
classical understanding of the nature of the chemical bond. Most of the molecules considered, including ethane (C2H6) and F2, ethylene (C2H4) and O2,
acetylene (C2H2) and N2, are well-known examples
of the variety of chemical bonds and can serve as
reliable reference compounds with single, double,
and triple bonds. The AOSB-A analysis of COC
bond in benzene gives reasonable results comparing with classical considerations about the (1␴ ⫹
1⁄2␲) nature of this bond. From the Table I one can
see that equilibrium COC distances and the total
bond orders are generally determined by the
␲-component of these bonds.
Tables I and II show the results of AOSB analysis
of cyclic organic molecules. For n ⱖ 5 (where n is
the number of carbon atoms in the cycle) the CTC
values are negligibly small, and the analysis is fully
justified. For n ⫽ 4 (cyclobutane) the CTC, while
being relatively small by absolute value, is already
comparable with the W␲ index. Finally, for n ⫽ 3
(cyclopropane and cyclopropene) the CTC values
become very significant, with the CTC values for
the double CAC bond being noticeably smaller
than for the single COC bond, implying a correspondingly smaller distortion of the cylindrical
symmetry of electron density distribution. As the
results of the AOSB-A analysis for three-linked cycles are apparently unreliable, the alternative
AOSB-P technique should be used for these systems. The calculated “in-plane” and “out-of-plane”
components (Table II) give a clear picture of bonding in C3H6 and C3H4, which is in agreement with
the classical treatment of such systems.
TABLE II _____________________________________________________________________________________________
COC bond lengths, r, bond order indices, WCC, and bond order components (AOSB-A and AOSB-P
decomposition schemes) computed at B3LYP/cc-pVDZ level of theory for cyclic organic molecules.
AOSB-A
C5H10
C4H8
C3H6
C3H4
␴
AOSB-P
␲
r (Å)
WCC
CTC
W
W
in-plane
out-of-plane
1.535–1.557
1.554
1.510
1.300
1.514
1.01–1.02
1.01
1.00
1.96
0.98
0–0.01
0.05
0.14
0.10
0.18
0.96–0.97
0.93
0.77
0.82
0.68
0.03–0.04
0.03
0.09
1.04
0.12
0.98
1.03
0.95
0.02
0.93
0.03
If COC bonds are not equivalent the range of the changes of bond orders and their components is presented.
2584 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
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ATOMIC ORBITAL SYMMETRY
TABLE III ____________________________________
Bond lengths, r, bond order indices, WXY, bond
order components, W␴ and W␲, for the diatomic XY
molecules computed at B3LYP/cc-pVDZ level of
theory.
Molecule
ne
r (Å)
WXY
W␴
W␲
CO
CN⫺
NO⫹
NO
O2⫹
NO⫺
F2⫹
O2⫺
FO
14
14
14
15
15
16
16
17
17
1.135
1.184
1.067
1.154
1.111
1.277
1.306
1.354
1.354
2.27
2.85
2.80
2.32
2.51
1.80
1.50
1.51
1.20
0.83
0.97
0.97
0.97
1.00
0.96
1.00
1.00
0.94
1.44
1.88
1.83
1.35
1.51
0.84
0.50
0.51
0.26
atoms and does not reflect the changes in the chemical environment of these atoms. However, bond
order reflects the covalent strength of the bond,
which is also affected by composition and structure
of the molecule. This dependence can be regarded
as the range of the bond order changes for any
given ⌬␹. AOSB-A analysis helps to unravel the
nature of the bond order dependence on the polarity of the bond. The correlation between ⌬␹ and W␴
shows that the ␴-component of the bond order is
mainly determined by the nature of X and Y atoms.
The range of W␲ changes for the specified pair of X
and Y atoms is much wider than in the case of W␴
Results for isoelectronic molecules are grouped together
(ne ⫺ number of electrons).
HETERONUCLEAR XOY BONDS IN
INORGANIC MOLECULES
Bond orders in several diatomic molecules have
been investigated using the AOSB-A technique (Table III). For all heteronuclear species W␴ is noticeably lower than in homonuclear compounds. This
component of the bond order decreases as the polarity of the bond increases and is close to 1 only in
the case of non-polar covalent bonds. As it could be
expected, in the row of isoelectronic molecules the
homonuclear X2q species have the largest ␴- and
␲-bond indices regardless of the charge q (Table III).
For the further study we supplemented the list of
molecules considered by several sets of main group
polyatomic compounds with different polarity and
multiplicity of the XOY bond: HX (X ⫽ H, F, Cl,
Br), H2X (X ⫽ O, S), XH3 (X ⫽ B, Al, N, P), XF3 (X ⫽
B, Al, N, P), XCl3 (X ⫽ B, N, P), XBr3 (X ⫽ N, P), SiY4
(Y ⫽ H, F, Cl, Br), F3XY (XY ⫽ NO, PO, PS), F3CO⫺,
BeO, SF4, and SF6. For all molecules the CTC values
were found to be negligible. The polarity of XOY
bond for these molecules changes significantly.
These changes can be characterized in terms of the
difference of Pauling’s electronegativities for the
elements X and Y: ⌬␹ ⫽ ␹ (X) ⫺ ␹ (Y). In Figure 1
W␴ (Fig. 1(a)) and W␲ (Fig. 1(b)) are plotted against
⌬␹ [30] for a total of 48 compounds. As one can see
from this Figure, W␴ decreases with increasing polarity of the bond. The dependence of the total XOY
bond order and its W␲ component on ⌬␹ is rather
complicated.
Since ⌬␹ is constant for each pair of elements, its
value is totally determined by the nature of X and Y
VOL. 109, NO. 11
DOI 10.1002/qua
W␴ (a) and W␲ (b) indices for the XOY
bond plotted against the absolute values of the difference of Pauling’s electronegativities, ⌬␹ ⫽ ␹ (X) ⫺ ␹
(Y) .
FIGURE 1.
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TABLE IV ____________________________________________________________________________________________
The indices W, W␴ and W␲ calculated at the B3LYP/LanL2DZ level of theory for the free ligands XY ⴝ CN, N2,
CO, NO and for the [Ru(XY)(CN)5]q complexes (W␴/W and W␲/W ratios are given in parenthesis).
XY
CN⫺
N2
CO
NO
[Ru(XY)(CN)5]q
Bond
index
Free ligand
XOY
RuOX
XOY
RuOC (cis)
RuOC (trans)
Total
␴
␲
Total
␴
␲
Total
␴
␲
Total
␴
␲
2.84
0.97 (34%)
1.86 (66%)
3.02
1.02 (34%)
2.00 (66%)
2.26
0.84 (37%)
1.42 (63%)
2.30
0.96 (42%)
1.34 (58%)
0.53
0.44 (83%)
0.09 (17%)
0.61
0.30 (49%)
0.31 (51%)
1.00
0.48(47%)
0.53(53%)
1.26
0.34 (27%)
0.92 (73%)
2.75
0.99 (36%)
1.76 (64%)
2.63
1.02 (39%)
1.61 (61%)
1.91
0.86 (45%)
1.05 (55%)
1.81
0.98 (54%)
0.83 (46%)
0.53
0.44 (83%)
0.09 (17%)
0.50
0.44 (87%)
0.06 (13%)
0.50
0.43 (88%)
0.06 (12%)
0.48
0.43 (90%)
0.04 (9%)
0.53
0.44 (83%)
0.09 (17%)
0.61
0.55 (89%)
0.07 (11%)
0.43
0.39 (91%)
0.04 (9%)
0.53
0.48 (92%)
0.04 (8%)
indicating that the ␲-component is generally determined by the chemical environment of the bond in
the molecule, i.e. composition and structure of the
molecule, valent, and oxidation states of the atoms.
METAL–LIGAND AND INTRALIGAND BONDS
IN THE [M(CN)5(XY)]q COMPLEXES
The WBI values and their ␴- and ␲-components
for the MOX and XOY bonds in the series of
[Ru(XY)(CN)5]q complexes (XY ⫽ CN⫺(q ⫽ ⫺4), CO
(q ⫽ ⫺3), N2 (q ⫽ ⫺3), and NO (q ⫽ ⫺2)), as well as
for the free XY ligands, are given in Table IV.2 From
the Table it follows that:
(i) the RuOX ␴-bonding is stronger for X ⫽ C
(XY ⫽ CN⫺ and CO) than for X ⫽ N (XY ⫽ N2 and
NO) as suggested by the lower electronegativity of
C atom as compared with N atom;
(ii) the ␲-bonding between ruthenium and X
atom of the XY ligand increases sharply in the order
CN⫺ ⬍ N2 ⬍ CO ⬍ NO;
(iii) in the complexes with the highest RuOX
bond orders (XY ⫽ CO, NO) the XOY bond order is
the lowest due to the decrease of its ␲-component,
i.e. coordination of XY to the metal atom results in
the noticeable weakening of the XOY ␲-bond as
compared with the free ligands;
(iv) the ␴-WBI for XOY bonds is almost identical
for free and coordinated ligands;
2
Note that results for free XY ligands computed with
LanL2DZ (Table IV) and cc-pVDZ (Table III) basis sets are very
close.
(v) the ␲-component of RuOCN bonds is diminished in all complexes under consideration, especially for XY ⫽ CO and NO;
(vi) trans-influence of XY in these complexes affects the ␴-components of RuOCN bonds: as the
XY ligand is varied, the most significant changes
are found for ␴-WBI(RuOCNtrans), while these indices remain constant for the RuOCNcis bonds.
It was found [31] that for similar [Fe(XY)(CN)5]q
complexes WFeX indices correlate with the Eint and
Eorb values calculated by EDA. Both AOSB-A and
EDA approaches suggest that the ␲-component of
the metal-XY bond rises in the CN⫺, CO, NO series
and for XY ⫽ NO the ␲-back-donation is the dominating component. One should keep in mind, however, that (i) Eint and its components describe the
interactions between the whole fragments while the
bond orders have local two-centered nature, and (ii)
the energies of ␴- and ␲-bonds are not the same:
␴-bond is usually estimated to be stronger than the
␲-bond; so, it is not surprising when such correlation is not found.
The charge redistribution between the XY ligand
and {Ru(CN)3⫺
5 } fragment via ␴- and ␲-bonds can
be estimated as:
⌬n ␴ ⫽ n共 ␴ 兲 free ⫺ n共␴兲coord.
and
⌬n␲ ⫽ n共␲兲coord. ⫺ n共␲兲free.
Here n(␴)coord. and n(␲)coord are the total populations of ␴- and ␲-orbitals of the coordinated ligands
calculated by NPA. If all XY are considered as
2586 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
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VOL. 109, NO. 11
ATOMIC ORBITAL SYMMETRY
TABLE V _____________________________________________________________________________________________
The metal–metal distances, r(Å), MOM and MOX bond indices, calculated at the B3LYP/LanL2DZ level of
theory for M2X8q complexes (all complexes have closed-shell ground state). The ␦-components are also
presented as sums of the dx2ⴚy2 (the first item) and dxy (the second item) contributions (in parenthesis).
MBI
Bond
total
[Re2Cl8]2⫺ (r ⫽ 2.209)
ReO Re
2.80
Re⫺Cl
0.96
[Re2H8]2⫺ (r ⫽ 2.172)
ReORe
3.37
ReOH
0.94
[Mo2Cl8]4⫺ (r ⫽ 2.153)
MoOMo
3.47
MoOCl
0.78
[Cr2Cl8]4⫺ (r ⫽ 1.754)
CrOCr
3.60
CrOCl
0.80
WBI
␴
␲
␦
total
␴
␲
␦
0.77
0.67
1.49
0.29
0.55 (⫺0.01 ⫹ 0.56)
2.51
0.62
0.75
0.48
1.26
0.13
0.50 (0.01 ⫹ 0.49)
0.82
0.94
1.55
1.00 (0.00 ⫹ 1.00)
3.04
0.58
0.99
0.58
1.03
1.01 (0.01 ⫹ 1.00)
0.86
0.52
1.77
0.25
0.84 (0.00 ⫹ 0.84)
3.26
0.46
0.90
0.36
1.57
0.08
0.79 (0.01 ⫹ 0.78)
0.99
0.52
1.75
0.26
0.87 (0.01 ⫹ 0.85)
3.44
0.45
0.96
0.35
1.63
0.07
0.85 (0.01 ⫹ 0.83)
closed-shell species (CN⫺, N2, CO, and NO⫹) then
n(␴)free ⫽ 10, n(␲)free ⫽ 4, and ⌬n values are
The AOSB-A analysis was performed for the
2⫺
4⫺
Re2Cl2⫺
(M ⫽ Mo and Cr)
8 , Re2H8 , and M2Cl8
complexes, for three paddlewheel dimetal carboxylates M2(BL)4 (M ⫽ Mo and Rh, BL ⫽ O2CH⫺; M ⫽
Ru, BL ⫽ O2CH⫺ and dpf, dpf ⫽ N,N⬘-diphenylformamidinate ion), and three M2(BL)4(NO)2 complexes (M ⫽ Ru, BL ⫽ O2CH⫺, dpf; M ⫽ Rh, BL ⫽
O2CH⫺), in which NO is axially coordinated to the
M2(BL)4 core (Tables V–VII).
CN ⫺ N 2 CO NO ⫹
⌬n : 0.51 0.30 0.54 0.34
⌬n ␲: 0.11 0.41 0.50 1.26
␴
These data show that in agreement with the W␴XY
values the XY ligands coordinated via C atom are
better ␴-donors than the ligands coordinated via N
atom. The ␲-acceptor properties of the XY ligands
increase sharply in the series CN⫺ ⬍ N2 ⬍ CO ⬍
NO and correlate with ␲-WBI for RuOX bonds. The
remarkable property of nitric oxide as a ligand is its
ability to form metal-NO bonds with extraordinarily high ␲-bond order, and this gives the basis to
treat RuONO bonding as covalent and NO ligand
as a radical NO0 with one unpaired electron on the
␲* MO (n(␲)free ⫽ 5 and ⌬n␲ ⫽ 0.26). Unlike the
charge transfer analysis or EDA, the bond orders
are independent from any presupposed scheme of
fragmentation of the molecule, so their values can
be useful for unbiased interpretation of the chemical bond.
VOL. 109, NO. 11
DOI 10.1002/qua
MOM BONDS IN DIMETAL TRANSITION
METAL COMPLEXES
M2Xq8 and M2(BL)4 Complexes. Placing both
metal atoms on the z-axis, which is the usual practice, one can see that the MOM ␴-bonds are formed
by dz2 AOs of the metal atoms, ␲-bonds — by dxz
and dyz orbitals, and ␦-bonds — by dxy and dx2-y2
orbitals. The dx2-y2 orbitals are responsible for the
formation of ␴-bonds with the X or BL ligands and
usually are not considered in the context of metal–
metal bonding. The formal metal-metal bond order
in M2Xq8 and M2(BL)4 is determined by the number
of electrons on the dimetal core (ne) and by the
sequence of MOs occupation and can be obtained
using Eq. (1). In the dimetal cores considered in this
work all bonding MOs {␴2␲4␦2} are occupied.
The Re2Cl2⫺
8 molecule (ne ⫽ 8, r(ReORe) ⫽ 2.24
Å) was originally considered as a molecule with
quadruple metal–metal bond [32], however, in the
recent theoretical studies [33–36] it was found that
the ␦-bond is rather weak and ReORe bond should
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 2587
SIZOVA ET AL.
TABLE VI ____________________________________________________________________________________________
Number of electrons on the dimetal core (ne), spin multiplicity (2Sⴙ1) and electronic configuration of the
ground state, formal bond order (BO, Eq.(1)), metal-metal distances (r, Å), and bond indices WMM, W␴, W␲, and
W␦ for MOM bonds in the M2(BL)4 complexes calculated at the B3LYP/LanL2DZ level.
Complex
ne
2S⫹1
Configuration
BO
r
WMM
W␴
W␲
W␦
Mo2(O2CÍ)4
Ru2(O2CH)4
Ru2(dpf)4
Rh2(O2CÍ)4
8
12
12
14
1
3
3
1
{␴2␲4␦2}
{␴2␲4␦2}(␲*)2(␦*)2
{␴2␲4␦2}(␲*)3(␦*)1
{␴2␲4␦2}(␲*)4(␦*)2
4
2
2
1
2.150
2.309
2.426
2.436
3.19
1.71
1.40
0.80
0.87
0.81
0.76
0.79
1.83
0.89
0.48
0.00
0.49
0.01
0.16
0.01
cupied by four electrons (double metal–metal bond;
spin-triplet ground state), whereas the {Rh2}4⫹ core
is populated by six electrons (single metal–metal
bond; closed-shell ground state) (Table V). In
{Ru2}4⫹ compounds these highest occupied orbitals
are likely to be close in energy and the sequence of
their population depends on the ligand environment [37, 38].
Since the ␦*-orbital interacts with the lower lying
orbitals of BL, while the ␲*-orbital does not, the
replacement of BL ⫽ O2CH⫺ by the stronger electron donor dpf⫺ leads to the changes in the electronic configuration of the dimetal core from
(␲*)2(␦*)2 (for Ru2(O2CH)4) to (␲*)3(␦*)2 (for
Ru2(dpf)4). Because of the higher efficiency of ␲-interactions, the diminishing of the ␦*-antibonding
contribution is overridden by the increasing population of ␲*-orbitals. As a result, the total metal–
metal bond order decreases and the RuORu distance increases (Table VI).
in fact be regarded as a triple bond. Gagliardi and
Roos using CASPT2 method obtained the effective
bond orders equal to 0.92, 1.74, and 0.54 for ␴-, ␲-,
and ␦-components respectively [34]. Michalak et al.
[36] calculated the Nalewajski–Mrozek multiplicities for the [Re2Cl8]2⫺ bond multiplicity (2.9),
[Re2H8]2⫺ (3.4), [Mo2Cl8]4⫺ (3.6), and [Cr2Cl8]4⫺
(3.7) complexes and came to the conclusion that the
weakening of the ReORe bond in [Re2Cl8]2⫺ results
from a competition between Cl ligands and the
other Re atom for the metal orbitals. According to
AOSB-A analysis, all three, ␴-, ␲-, and ␦-components, take part in the ReORe bond, however, the
corresponding bond orders, especially for ␦-bond,
do not get the full amount of the classical values 1
(for ␴), 2 (for ␲), and 1 (for ␦). The data of Table V
indicate that the decrease of total ReORe bond
order in M2Clq8 as compared with CrOCr and
MoOMo bond orders is determined by all components of this bond and is accompanied by significant increase of the MOCl bond orders. These results are valid both for WBI and MBI analysis,
although the MBI values are larger than those of
WBI.
If ne is greater than 8, the antibonding MOs are
being populated, the bond order decreases and the
metal–metal distance increases. In the case of
{Ru2}4⫹ the antibonding ␲*- and ␦*-orbitals are oc-
M2(BL)4(NO)2 Complexes. The coordination of
NO molecules in trans-position to the covalent metal–metal bond leads to the drastic increase of the
metal–metal distance [39 – 42]. Unlike the Ru2(BL)4
species, the Ru2(BL)4(NO)2 complexes are diamagnetic [40, 42]. All dinitrosyl complexes considered
in this section have closed-shell ground states,
TABLE VII ____________________________________________________________________________________________
The metal–metal distances r(Å), and Wiberg bond indices WMM and WMN calculated at the B3LYP/LanL2DZ
level for M2(BL)4(NO)2 complexes (all complexes have closed-shell ground state).
MOM bond
Complex
Ru2(O2CH)4(NO)2
Ru2(dpf)4(NO)2
Rh2(O2CH)4(NO)2
␴
␲
r
WMM
W
W
2.613
2.667
2.633
0.52
0.42
0.15
0.46
0.34
0.13
0.04
0.04
0.01
MON bond
␦
W
CTC
⬔MNO
WMN
W␴
W␲
CTC
0.01
0.00
0.09
0.01
0.03
0.00
149.8
156.9
121.1
1.11
1.19
0.79
0.34
0.32
0.49
0.67
0.79
0.22
0.05
0.08
0.08
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VOL. 109, NO. 11
ATOMIC ORBITAL SYMMETRY
though for Ru2(dpf)4(NO)2 the singlet and triplet
states are nearly degenerate.
For the M2(BL)4 complexes the metal–metal CTC
values are equal to zero, and for the M2(BL)4(NO)2
complexes these values are negligible. For the
MON bonds in M2(BL)4(NO)2 the CTCs are nonzero (Table VI) due to the electronic density delocalization in the bent {MNO} triatomic fragments
and to the non-zero MOO bond indices [15].
The RhORh WBI value in Rh2(O2CH)4(NO)2 is
so small, that one has no reason to recognize a
chemical bond between the metal atoms, which are
held together by four rigid carboxylate bridging
ligands. The destruction of the covalent RhORh
␴-bond (Table VII) is accompanied by the reorientation of two unpaired electrons from inside the
dimetal core towards the axial ligands, leading to
the formation of covalent RhONO bonds. The absence of the direct RhORh bond and the formation
of strong covalent RhONO bonds explain the atypically long RhORh distance and short RhON distances in Rh(II) tetracarboxylates with axially coordinated NO [39]. In the case of Ru2(BL)4 the
coordination of nitric oxide leads to the destruction
of metal–metal ␲-bonds and to the reorientation of
d␲ AO towards the formation of covalent RuONO
bonds (Table VII). The RuORu distances (experimental values are ca. 2.5–2.6 Å [40, 41]) are characteristic of single metal–metal bonds. So, the AOSB
analysis provides a compact representation of the
metal–metal bond destruction in the paddlewheel
dimetal complexes due to the coordination of nitric
oxide.
Conclusions
The works of Mayer [1, 5–9] have clearly demonstrated the validity of density matrix-based indices as quantum chemical analogues of the bond
multiplicities used by the classical structural theory. Quantum chemical bond orders cannot be replaced by empirical relations of the “bond length/
bond order” type, which do not depend on the
chemical environment [43] and do not reveal the
electronic nature of the chemical bond. In the
present work, we attempted to further develop the
computational tools linking together the classical
and quantum chemical concepts of bond order. As
compared to the MOSB approach [10], AOSB analysis has a wider range of potential applications,
being available for both MBI and WBI, enjoying
freedom from molecular symmetry limitations and
VOL. 109, NO. 11
DOI 10.1002/qua
yielding chemically justified results which are
closer to the universally accepted notions of ␴- and
␲-bonds.
ACKNOWLEDGMENTS
This work was supported by the Russian Foundation for Basic Research (Project No. 07– 03-01016).
The helpful discussions with Prof. V. I. Baranovski
are gratefully acknowledged.
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