Page 1
Accepted Manuscript
Halogen-halogen interaction in view of many-body approach
Justyna Dominikowska, Marcin Palusiak
PII: S0009-2614(13)00946-9
DOI: http://dx.doi.org/10.1016/j.cplett.2013.07.055
Reference: CPLETT 31425
To appear in: Chemical Physics Letters
Please cite this article as: J. Dominikowska, M. Palusiak, Halogen-halogen interaction in view of many-body
approach, Chemical Physics Letters (2013), doi: http://dx.doi.org/10.1016/j.cplett.2013.07.055
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Halogen-halogen interaction in view of many-body approach.
by
Justyna Dominikowska and Marcin Palusiak*
Department of Theoretical and Structural Chemistry, University of Łódź,
Pomorska 163/165, 90-236 Łódź, Poland
e-mail: [email protected]
Keywords: halogen bond, hydrogen bond, cooperativity, interaction energy decomposition,
many-body theory
Graphical abstract:
Abstract:
The many-body theory was applied in order to estimate the character of interaction in
quadruple complex consisting of four bromomethane molecules. The tetrameric complex used
as a model system was found in the crystal structure in which halogen bonds were stabilizing
the solid state structure. Decomposition of interaction energy on two-, three- and four-body
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terms allowed to conclude that individual halogen bonds in tetramer rather does not
cooperate forming the complex. The nonadditive contribution to interaction energy isnegative,
but of very small value in respect to total interaction energy (less than 0.01%), indicating very
weak cooperativity of halogen bridges.
1. Introduction
Due to its dual character resulting from anisotropic distribution of atomic charge[1-4]
(see Fig. 1) the halogen atom may act as both electron-donating and electron-accepting center
[5-7] and, therefore, one may expect that the same halogen center may in some specific
conditions form two X-bonds in which in one case it acts as a Lewis acid and in the second
case as a Lewis base.
Figure 1. Electrostatic potential mapped on electron density isosurface of H3CBr molecule,
illustrating anisotropic charge distribution on halogen atom. (See also ref. [4] for related
material.)
The simple search through CSD system [8] allows finding several examples of such
interactions. Therefore, one may ask whether such situation favours the formation of both
interactions, or, in other words, whether the two (or more) X-bonds cooperates forming more
complex molecular synthons. In this paper we investigate this problem using methodology
which involves the many-body approach. As it was recently shown [7], the cooperativity in
X-bond formation may strongly depend on the properties of molecular systems used as a
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model. For this reason it would be worth to eliminate the effect of differences in
acidity/basicity of the interacting fragments in the complex. Also, since the regions of acidic
and basic properties are clearly oriented perpendicularly in respect to halogen nuclei and R-X
bond (see Fig. 2a), the orientation of the interacting fragments must also be rather
perpendicular. Thus, a quadruple complex consisting of four fragments, each with halogen
atom is involved in two X-bonds, seem to be a perfect choice. Even better if the symmetry
would restrain interacting fragments such that each fragment have reflected the same Lewis
acidity/basicity. Figure 1a shows schematically the orientation of molecular fragments in such
model complex. The more detailed search through CSD allowed to find the example of such
complex in which four symmetrically identical X-bonds stabilizes the quadruple complex
shown in Figure 1b. The crystal structure of anti-α-bromoacetophenone oxime stabilized by
this molecular synthon was published by Wetherington and Moncrief already in 1973 [9]
(CSD refcode: ABACOX10 [8]). Interestingly, no X-bond term was used in order to classify
this interaction, which is obvious, since in that time the nature of such interactions was not
clear yet.
(a)
(b)
Figure 2. Schematic representation of fragment distribution in quadruple X-bonded complex
used as a model, (a), and the same complex formed by four anti-α-bromoacetophene oxime
molecules in crystal state, (b).
Instead, authors mention that “a cluster of four bromine atoms make van der Waals contacts
of 3.681Å about an S4 axis at 0,0,z”. Therefore, the stabilizing character of this interaction
was clearly pointed by Wetherington and Moncrief, [9] even if the nature of this interaction
could not be precisely recognized in that time.
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In this paper we investigate the structural pattern shown in Figure 2, using advanced
quantum-chemistry models. The nonadditive contribution to the interaction energy is
estimated in the framework of the many-body theory. In this way the cooperativity effect in
purely X-bonded quadruple complex (of C4 symmetry) is treated explicitly by the well-
defined and physically justified interaction energy decomposition.
2. Methods
2.1. Many-body approach for four-body system
According to Hankins et al. [10] interaction energy of n-body system [11] ( intE ) can
be decomposed to a sum of k-body terms:
,...
...
)1(
1
)2()(
...
2
1
1)3(
1
1
)2(
int
1 12 1
21
1 12 23
321
1 12
21
nn
i
nn
ii
n
ii
n
iii
n
i
n
ii
n
ii
iii
n
i
n
ii
ii
nn
n
E
two-body
three-body
n-body
(1)
where )(
...21
k
iii k are the k-body terms obtained from energies of interacting k-tuples and can be
expressed with the following equation:
,
...
1211
121
211
21
11
1
2121
...
)1(
...
)2(
...
)(
...
k
k
k
k
k
kk
i
jjij
k
jjj
i
jij
jj
i
ij
j
iii
k
iii
E
E
(2)
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The equation (1), for four-body system with C4 symmetry and consecutive circular monomer
numbering, can be written as:
.
4
24
)4(
1234
)3(
123
)2(
13
)2(
12
4
E
(3)
2.2 Basis set superposition error for four-body system
When the many-body terms are to be calculated, the problem of basis set superposition
error (BSSE) occurs. To avoid it, the counterpoise correction (CP) method was introduced by
Boys and Bernardi.[12] The CP method was initially designed for two-body systems only.
Considering the two-body system AB, let EX(Y) be the energy of system X calculated in the Y
basis set. The two-body interactions can be expressed in two different ways:
)()()()2( ABEABEABE BAABAB (4a)
)()()()2( BEAEABE BAAB
u
AB (4b)
The (4a) equation is based on the energies of subsystems calculated in the basis set of the
whole AB system while in (4b) “uncorrected” equation, the energy of each subsystem is
calculated in its own basis set. Defining BSSE as the differenc between corrected and
uncorrected interaction energies, BSSE for the AB system can be written as:
u
ABAB
)2()2(BSSE (5)
Since the CP method was originally created solely for two-body systems, a few
schemes of its generalization were proposed and all of them are used in this study. It was
suggested [13] that the only scheme which gives always the right sign of BSSE for n-body
systems is the one designed by Valiron and Mayer (VMFC). [14] However, as it will be
shown for here investigated system, both VMFC and other BSSE schemes give negative
correction energies for some of n-body interaction energy contributions, thus, indicating
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rather artificial results of BSSE correction for many-body interaction energy decomposition.
This aspect will be discussed in detail in the next section.
According to the scheme BSSE should be calculated as a difference between the
corresponding corrected and uncorrected (u) energy terms. The expansion of formulas (4a)
and (4b) to k-body interactions results in equations (6a) and (6b):
k
k
k
k
k
kk
i
jjij
k
k
jjj
k
i
jij
jj
k
i
ij
j
kiii
k
iii
iii
iii
iiiE
iiiE
1211
121
211
21
11
1
2121
...
21
)1(
...
21
)2(
21
21...
)(
...
)...(
...
)...(
)...(
)...(
(6a)
k
k
k
k
k
kk
i
jjij
uk
jjj
i
jij
u
jj
i
ij
j
kiii
uk
iii
jE
iiiE
1211
121
211
21
11
1
2121
...
)1(
...
)2(
1
21...
)(
...
...
)(
)...(
(6b)
Thus, the sum over all k-body terms of BSSE for n-body system can be expressed with the
equation:
n
k
kk
i
jjij
uk
jjj
k
jjj
k
..
)(
...
)(
..
211
2121][BSSE
(7)
The total BSSE is:
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n
k
k
2
VMFC BSSEBSSE
(8)
Such procedure of bringing together the equations (2a), (6a), (6b), (7) and (8) leads to the
same final formula as that given by Valiron and Mayer [14]:
n
iii
n
n
iiin
n
iii
n
ii
niiii
n
i
nii
n
nniiiiii
iiiii
iiiEiE
121
121121
21
2121
1
11
...
21
)1(
..121
)1(
..
21
)2(
21
)2(
211
VMFC
)]..()..([
...
)]..()([
)]..()([
BSSE
(9)
It is worth noting that in the equation (11) the terms are grouped in a different way than in the
procedure with k-body terms of BSSE.
For comparison with the VMFC scheme, the following schemes were applied in the
study: site-site function counterpoise (SSFC) [15] and the pairwise-additive function
counterpoise (PAFC). [15] SSFC can be obtained in the procedure similar to the one used for
VMFC scheme. The only difference lies in the corrected energy values where each k-body
term should be calculated in the basis set of the whole system. This leads to the following
BSSE formula:
)]...(-)([EBSSE 211
SSFC
1
1
1 ni
n
i
i iiiEi
(10)
The PAFC correction can not be partitioned into k-body terms since solely two-body
corrections (resulting from the interaction of pairs) are included in this scheme. The total
BSSE in this case equals:
)](-)([EBSSE 211
PAFC
1
21
1iiEi i
n
ii
i
(11)
Results of various above-described BSSE schemes will be discussed in the Results and
discussion section.
2.3 Computational details – different chemistry models
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The Hartree-Fock SCF method, post-SCF MP2 and three DFT functionals, namely,
B3LYP [16], M06-2X [17] and ω-B97XD [18], were initially chosen for the study. All the
calculations were performed with the use of Gaussian09 program.[19] In all cases the aug-cc-
pVTZ basis set was used. The geometry close to the one from the crystal structure (with the
C4 symmetry constrained) was chosen as a starting geometry for the optimization procedure.
Only MP2 and ω-B97XD calculations reproduced the quadruple complex being analogous to
that from Fig. 1. In all other cases the complex was not a stable structure. Since both MP2 and
ω-B97XD chemistry models include dispersion (explicitly in MP2 and by means of empirical
correction in the case of DFT functional), the dispersion interaction must be crucial in
stabilizing the complex itself. This statement is in line with recent report by Riley et al. on
importance of dispersion in halogen bond formation.[20] For ω-B97XD calculations, when
the starting geometry is optimized in one step with tightened optimization criteria
(Opt=VeryTight, Int=UltraFine in Gaussian) the calculation leads to the X-bonded quadruple
complex (as in the case of MP2 optimization). However, interestingly, when the two step
procedure is applied (first step with default Gaussian criteria and the second with the
geometry from the first step, being a starting geometry optimized with tightened Gaussian
criteria) the other stable quadruple complex is found for ω-B97XD method. This time,
however, it is stabilized by four symmetrically equivalent H-bonds. Graphical representation
of both types of complexes is shown in Fig. 3.
Since, at it was mentioned above, the methods that do not include dispersion effects
and which do not allow to reproduce experimentally observable interaction, are not suitable
for the analysis of halogen bonding within the system chosen for the study. This conclusion
can be extrapolated to other possible model systems in which nondispersive components of
interaction can be sufficient to stabilize the system through X-bonding. In such a case the use
of methods with no dispersion included can possibly lead to underestimated interaction
energies. Thus, the more general conclusion can be made according to which the methods
which do not include dispersion effects, even if they allow in some cases reproducing the
stabilizing character of X-bonding, are rather not suitable for analysis of that interaction.
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(a)
(b)
Figure 3. Geometries of X-bonded, (a), and H-bonded, (b), tetramers of bromomethane
optimized at MP2, (a), and ω-B97XD, (b), levels of theory.
3. Results and discussion
In Fig. 3 the graphical representation of geometries of X-bonded and H-bonded
complexes is given. There are some important differences in X-bonded geometries obtained
with the use of MP2 and ω-B97XD models. First of all the Br...Br distance is shorter in the
MP2 structure by about 0.2Å, being 3.503Å and 3.785Å for MP2 and ω-B97XD, respectively.
Another important difference is connected with the fact that in MP2 structure the line that is
the C-Br bond elongation is not directed exactly on adjacent halogen nuclei, but is slightly
moved into direction of carbon atom covalently attached to Br. This shift can be seen in Fig.
2. This geometrical feature is important taking into account the fact that the sigma-hole
actually lays on the elongation C-Br bond line. In the case of ω-B97XD structure the halogen
bond lays almost exactly on elongation of the C-Br bond. In the consequence the Br...Br-C
angle is of 81.476° in MP2 structure and 84.939° in DFT structure. There are other minor
differences between MP2 and DFT optimized structures. For instance the C-Br distance is
slightly shorter in MP2 structure (1.928Å) than in DFT structure (1.939Å). It is worth
mentioning that in the case of both methods the slight elongation of C-Br bond due to the
complexation was observed (by about 0.003Å and less than 0.001Å respectively for MP2 and
ω-B97XD). The longer distance found in DFT structure is in line with the fact that for DFT
calculations the absolute value of interaction energy is also smaller, when compared to the
one obtained with the MP2 method. For interaction energies and their components see
Table 1.
In general the total interaction energy is negative, indicating the stabilizing character
of interactions in tetramer. However, the absolute value of this energy is rather small being of
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around 5-7 kcal/mol for MP2 calculations, depending on the basis set and the BSSE scheme
applied, and about 60% of that for DFT calculations. (Note that this interaction energy
corresponds to four individual X-bonds in the complex and that it includes the nonadditive
contribution to interaction energy.) The system is mainly stabilized due to the two-body
interactions. The most significant is the stabilization resulting from 1...2 pairing (assuming
cyclic fragment notation in tetramer). However, also the 1...3 interaction corresponds to
slightly stabilizing effect. In that case the small attractive nature of interaction can be
connected with +...- attraction between oppositely charged local charges on Br surface.
(See Fig. 2a) Both three- and four-body interactions are also stabilizing but the effect is much
weaker than in the case of two-body terms. As the consequence the summarized nonaddtive
contribution to interaction energy is of about –0.25 kcal/mol, varying slightly for different
basis sets and BSSE schemes. Thus, similarly to hydrogen bond for which it was shown that
many-body terms usually corresponds to significantly stabilizing effect, [21-23] the
nonadditive terms stabilize also the halogen-bonded tetramer investigated here. However, the
effect is much weaker when compared to e.g. H-bond in water complexes. [21-23] (Note that
in the case of H-bond the cooperativity effect can also be interpreted as a polarization effect
which assists the H-bond. In such a case the term polarization-assisted hydrogen bonding –
PAHB - is often used for classification of this type interaction. PAHBs are observed mostly
when the proton donor acts simultaneously as a proton acceptor. [24])
It is also worth taking a closer look on H-bonded complex found with the use of ω-
B97XD functional. This structure was obtained with the procedure described in the
Computational Details section. Actually, the total interaction energy in H-bonded complex is
about twice of that found for X-bonded complex obtained on the same level of theory. Its
absolute value is still rather small, being of around 4.5 kcal/mol (see Table 1). The weak
stabilizing character of interactions in H-bonded complex are confirmed with structural
parameters of H-bridges. The (C)H...Br distance is of 2.984Å, which is comparable with the
sum of vdW radii (2.94Å). Also the angle in H-bridge is relatively far from linearity, adopting
value of 159.776°.
It is also worth mentioning that regardless of BSSE scheme and the basis set used the
SCF interaction energies estimated from MP2 calculations are always significantly positive
(see Table S1 in Supporting Information file for data) suggesting insufficiency of SCF model
in description of X-bonding. This conclusion is in line with the fact that HF calculations do
not allow reproduce experimentally observed [9] tetrameric structure stabilized via X-bond.
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The most significant component of the total interaction energy is the two-body term being of
the value close to 10 kcal/mol. Interestingly, the nonadditive contribution to SCF interaction
energy, although relatively small, is systematically negative, thus, suggesting cooperative
effect within the SCF model. What is more, the nonadditive contribution to interaction energy
is connected with cooperative character of three-body interaction. The four-body interaction
corresponds to weak anticooperative effect.
As far as considering the BSSE corrections, the values obtained with the use of
different BSSE schemes are rather similar. The maximum difference in values of the total
BSSE correction is of about 2 kcal/mol, being largest for PAFC scheme (almost 6 kcal/mol).
In the case of SSFC scheme the value of total BSSE correction is smallest and equals about 5
kcal/mol. This is also the value which can be obtained with counterpoise correction (for four
fragments of the complex) implemented as a standard procedure in Gaussian09. Since PAFC
does not give BSSE corrections to n-body components, only VMFC and SSFC schemes can
be compared when considering many-body interaction energy components. As expected, two-
body corrections are positive for both schemes and do not exceed the two-body interation
energy values. Their values are also the highest among all BSSE correction terms because the
two-body contribution to the interaction energy is also the greatest from all many-body
contributions. Apparently, the three-body BSSE corrections are systematically negative,
which must be an artifact resulting from BSSE generalization on interaction energy n-body
terms. What is more, the absolute values of BSSE corrections are even larger than three-body
interaction energies, thus the cooperative effect of three-body interactions, reflected by values
of three-body components collected in Table 1, in large part results from inconsistency in
BSSE generalization on many-body approach. It is worth mentioning, that the negative values
of BSSE corrections to three-body components of interaction energies were already reported
for some H-bonded complexes. [13,25] Thus, it seems that the generalization of BSSE
correction on many-body interaction energy decomposition still needs some more attention
and the more appropriate scheme is still needed. Nevertheless, in the case of here investigated
model system the artifact of three-body BSSE corrections does not change the possibilities of
data interpretation. The individual X-bonds very slightly cooperate between themselves when
forming tetramer, but this cooperative effect is rather insignificant. Moreover, the difference
between the non-additive component of the interaction energy and the artificially negative
three-body BSSE term enlarges with the increasing basis set. Such a result also supports the
conclusion that the very weak cooperation exists in the case of investigated tetramer. Also, the
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BSSE corrections values decreases with increasing basis set, which is expected and related
explicitly with definition of BSSE itself. This statement regards not only the whole BSSE
value but also the absolute values of its many-body contributions.
Finally, it is worth checking the procedure available in Gaussian and based on
Dannenberg et al. scheme [26], which introduces BSSE correction along optimization
process. We performed such calculation at MP2/aug-cc-pVTZ level of theory for our halogen
bonded tetramer. Geometrical and energetic parameters are given in Table S2. As it can be
seen, the changes in geometry are relatively small, with largest change observed for Br...Br
contact (0.117Å). Changes for covalent bonds lengths are practically meaningless. As far as
considering interaction energy, its value obtained with BSSE correction implemented during
optimization equals -6.46kcal/mol, being slightly larger than it was obtained with the use of
standard optimization procedure.
4. Conclusions
It was shown that the quadruple complex consisting of four bromomethane molecules
is stabilized by four halogen bonds in which each Br atom plays the dual role of Lewis acid
and Lewis base. The two-body contribution to interaction energy is of around –5_ -6 kcal/mol
and corresponds to stabilizing interaction between molecules in pairs, . The three- and four-
body components of interaction energy are also negative, but their values are very small
comparing the total interaction energy. Thus, the cooperative effect of halogen bonding is
very weak. As a consequence, the total interaction energy in complexes is also relatively
small, with its absolute value being of around 6-7kcal/mol or, depending on the computational
level.
The BSSE correction contributes significantly to estimated interaction energies. It
occurs that the three-body corrections are possessing negative values, which certainly
indicates inconsistency in BSSE generalization on n-body interaction energy decomposition.
This conclusions concerns both these BSSE correction schemes which allow estimate
corrections to n-body terms, that is SSFC and VMFC. However, these artificial results of
BSSE correction does not influence the final conclusion regarding very weak cooperative
character of interaction between individual X-bonds in tetramer. The values of BSSE
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corrections to total interaction energy are always of the proper sign and relatively similar
values, regardless of the scheme used.
Also, as shows the comparison of several different chemistry models used for
optimization of the tetrameric complex, the methods which do not include dispersion effects
are rather not suitable for calculations on X-bonded systems, even if such methods possibly
reproduce stabilizing character of that interaction.
Acknowledgments
Calculations using the Gaussian09 set of codes were carried out in Wroclaw Center for
Networking and Supercomputing (http://www.wcss.wroc.pl). Access to HPC machines and
licensed software is gratefully acknowledged. The financial support from National Science
Centre of Poland (grant no. 2011/03/B/ST4/01351) is also gratefully acknowledged. JD
acknowledges the financial support from University of Łódź Foundation (University of Łódź
Foundation Award).
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Table 1. Uncorrected and CP- corrected energies, BSSE values, their many-body contributions and
the sum of non-additive terms (kcal/mol).
MP2/aug-cc-pVTZ
uncorr. SSFC BSSEk VMFC BSSEk PAFC
(2) 12 -2.556 -1.379 1.177 -1.184 1.371 -
13 -0.372 -0.179 0.192 -0.169 0.203 -
(2) -10.966 -5.873 5.093 -5.076 5.891 -
(3) 123 -0.021 -0.076 -0.055 -0.081 -0.060 -
(3) -0.084 -0.304 -0.220 -0.324 -0.240 -
(4) 1234 -0.060 0.047 0.107 0.047 0.107 -
Eint -11.110 -6.131 - -5.353 - -5.220
BSSE - 4.980 - 5.757 - 5.891
non-add - -0.257 - -0.277 - -
MP2/aug-cc-pVQZ//MP2/aug-cc-pVTZ
uncorr. SSFC BSSEk VMFC BSSEk PAFC
(2) 12 -2.631 -1.621 1.009 -1.546 1.084 -
13 -0.377 -0.186 0.191 -0.180 0.196 -
(2) -11.275 -6.856 4.419 -6.546 4.730 -
(3) 123 -0.018 -0.072 -0.054 -0.073 -0.055 -
(3) -0.072 -0.288 -0.216 -0.294 -0.222 -
(4) 1234 -0.011 0.043 0.054 0.043 0.054 -
Eint -11.358 -7.100 - -6.796 - -6.628
BSSE - 4.258 - 4.562 - 4.730
non-add - -0.244 - -0.251 - -
MP2/aug-cc-pV5Z//MP2/aug-cc-pVTZ
uncorr. SSFC BSSEk VMFC BSSEk PAFC
(2) 12 -2.484 -1.700 0.784 -1.589 0.895 -
13 -0.329 -0.187 0.142 -0.184 0.145 -
(2) -10.595 -7.175 3.420 -6.723 3.871 -
(3) 123 -0.031 -0.070 -0.039 -0.071 -0.039 -
(3) -0.126 -0.280 -0.155 -0.283 -0.157 -
(4) 1234 0.001 0.041 0.040 0.041 0.040 -
Eint -10.720 -7.414 - -6.965 - -6.848
BSSE - 3.306 - 3.755 - 3.871
non-add - -0.239 - -0.242 -
-B97XD/aug-cc-pVTZ (halogen bonding)
uncorr. SSFC BSSEk VMFC BSSEk PAFC
(2) 12 -0.948 -0.896 0.052 -0.903 0.045 -
13 -0.058 -0.049 0.009 -0.050 0.008 -
(2) -3.906 -3.681 0.225 -3.712 0.194 -
(3) 123 -0.022 -0.026 -0.004 -0.028 -0.006 -
(3) -0.086 -0.103 -0.017 -0.111 -0.025 -
(4) 1234 -0.001 -0.004 -0.003 -0.004 -0.003 -
Eint -3.993 -3.789 - -3.828 - -3.799
BSSE - 0.204 - 0.165 - 0.194
non-add - -0.107 - -0.115 -
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-B97XD/aug-cc-pVTZ (hydrogen bonding)
uncorr. SSFC BSSEk VMFC BSSEk PAFC
(2) 12 -1.097 -1.034 0.063 -1.011 0.086 -
13 -0.099 -0.091 0.008 -0.092 0.007 -
(2) -4.586 -4.317 0.269 -4.227 0.359 -
(3) 123 -0.076 -0.079 -0.003 -0.078 -0.002 -
(3) -0.304 -0.316 -0.012 -0.313 -0.008 -
(4) 1234 -0.021 -0.020 0.001 -0.020 0.001 -
Eint -4.911 -4.654 - -4.560 - -4.553
BSSE - 0.258 - 0.351 - 0.359
non-add - -0.337 - -0.333 - -
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Graphical abstract:
*Graphical Abstract (pictogram) (for review)
