THE JOURNAL OF CHEMICAL PHYSICS 128, 124310 共2008兲
An ab initio investigation on „CO2…n and CO2„Ar…m clusters:
Geometries and IR spectra
K. V. Jovan Jose and Shridhar R. Gadrea兲
Department of Chemistry, University of Pune, Ganeshkhind 411007, India
共Received 9 November 2007; accepted 7 January 2008; published online 28 March 2008兲
An ab initio investigation on CO2 homoclusters is done at MPWB1K / 6-31+ + G共2d兲 level of
theory. Electrostatic guidelines are found to be useful for generating initial structures of 共CO2兲n
clusters. The ab initio minimum energy geometries of 共CO2兲n with n = 2 – 8 are T shaped, cyclic,
trigonal pyramidal, tetragonal pyramidal, tetragonal bipyramidal, pentagonal bipyramidal, and
pentagonal bipyramid with one CO2 molecule attached to it. A test calculation on 共CO2兲20 cluster is
also reported. The geometric parameters of the energetically most favored 共CO2兲n clusters match
quite well their experimental counterparts 共wherever available兲 as well as those derived from
molecular dynamics studies. The effect of clustering is quantified through the asymmetric C–O
stretching frequency shift relative to the single CO2 molecule. 共CO2兲n clusters show an increasing
blueshift from 1.8 to 9.6 cm−1 on increasing number of CO2 molecules from n = 2 to 8. The
energetics and geometries of CO2共Ar兲m clusters have also been explored at the same level of theory.
The geometries for m = 1 – 6 show a predominant T type of the argon-CO2 molecule interaction.
Higher clusters with m = 7 – 12 show that the argon atoms cluster around the oxygen atom after the
saturation of the central carbon atom. The CO2共Ar兲m clusters exhibit an increasing redshift in the
C–O asymmetric stretch relative to CO2 molecule of 0.7– 5.6 cm−1 with increasing number of argon
atoms through m = 1 – 8. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2838202兴
I. INTRODUCTION
Understanding the cluster to bulk transition point behavior and the underlying governing forces is an unexplored
area in chemistry and physics.1 Two exclusive properties of
clusters are their unique packing and the cluster size dependent properties, which arise from the large surface to volume
ratio. Specific cluster sizes exhibiting unusual stability are
called the “magic numbers.” The turning point from bulk to
crystal property is called the “critical size” and this point is
chemically diverse. The nature of interaction plays a predominant role in deciding the critical size of a species. Ionic
systems at least require few tens of atoms, while thousands
of atoms are needed for atomic clusters to reach bulk crystalline state.2
Some of the experimental techniques to study these large
clusters are molecular beam electron diffraction, highresolution microwave, and infrared spectroscopies. There is
an experimental limitation in generating and studying large
neutral clusters. Even though the molecular beam electron
diffraction spectroscopic technique enables generation of
large clusters, the method is not accurate enough to study
size dependent properties. Mass spectroscopy is another
method used to shed light on the size of clusters. Through it
one can readily explore the stability of ionic species rather
than that of a neutral entity. This makes one rely on popular
experimental techniques such as matrix isolation3 and supersonic jet expansion4 to generate clusters and probe the thus
prepared cluster samples by any of the spectroscopic techa兲
Author to whom correspondence should be addressed. Electronic mail:
[email protected].
0021-9606/2008/128共12兲/124310/9/$23.00
niques such as infrared spectroscopy, UV-visible absorption,
etc.5 Though matrix isolation spectroscopy affords a number
of advantages in the study of isolated molecules, one of the
major problems it often poses is the non-negligible interaction of the solid matrix medium with the probe molecules.
This is referred to as the matrix effect and has been addressed in our earlier study.6
The systems of interest in the present study are 共CO2兲n
clusters. The current work is pursued on these clusters because of the fascinating chemistry and importance of this
system in various fields.7 On the experimental side, generation of 共CO2兲n clusters through matrix isolation or through
supersonic jet expansion inevitably requires inert matrix. The
interaction of these matrixes with the probe molecule cannot
be neglected. The weak matrix-probe molecule interactions
drag the spectral peak either to red or blue regions relative to
free molecular frequency mode depending on the nature and
strength of interaction. Smaller 共CO2兲n clusters 共up to n = 3兲
are one of the classic clusters explored theoretically and
experimentally.8–12 The nature of the spectra makes these
clusters an ideal species for understanding the mode specificity of vibrational perturbation and vibrational predissociation.
IR investigations on CO2 clusters began with the works
of Walsh et al.8 and Jucks et al.9 The CO2 dimer has been
studied in the sub-Doppler resolution in the vicinity of the
upper member of the Fermi diad.9 The experimental geometry corresponds to the C2h point group, in agreement with
previous Raman spectroscopic studies.10 Meanwhile, the
high-resolution IR and Raman spectroscopic studies confirmed the predominant existence of shifted parallel configu-
128, 124310-1
© 2008 American Institute of Physics
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124310-2
J. Chem. Phys. 128, 124310 共2008兲
K. V. Jovan Jose and S. R. Gadre
ration over T-kind CO2 dimer species.8,9 The trimer species
shows experimentally two equivalent in energy geometries,
the cyclic trimer 共C3h兲 reported by Fraser et al.11 and the
Z-trimer configuration by Weida and Nesbitt.12 Recently in
the literature, an understanding of the larger 共CO2兲n clusters
and their spectral behavior have gained importance.13–16 The
investigations on 共CO2兲n mer clusters via the molecular
beam electron diffraction technique has begun with the
works of Torchet et al.13 A critical size of n = 30 is found to
be the cross over regime from icosahedral clusters to large
cubic morphology. The slow electron attachment time of
flight mass spectroscopic analysis combined with geometrical shell closing analysis by Negishi et al.14 led to a cuboctahedral motif for clusters containing more than
80 molecules, whereas a distorted face centered cubic packing was assumed by Ingolfsson and Wodtke15 for same sized
clusters.
On the contrary, direct absorption spectroscopy can provide wealth of information in this regime. Bonnamy et al.16
studied the signature of 共CO2兲n with cluster size in the range
of 30⬍ n ⬍ 14 500, shape, and structural effects through direct IR supersonic jet expansion spectroscopy in the asymmetric stretching region. Small clusters of specific sizes were
generated through supersonic expansion. As the cluster size
increases, the band peak was seen to shift towards blue region. Maillet et al.17 studied the changes in the thermodynamic parameter with temperature for 共CO2兲n small sized
clusters 共n = 2 – 13兲 using molecular dynamics simulations
and found a coexistence of liquid and solid phases at the
cluster size of 13.
Spectroscopic studies on CO2 – Ar complex began with
the microwave work of Steed et al.18 and the geometric parameters were in well agreement with the T-shaped CO2 – Rg
clusters. Later Sharpe et al.19 and Randall et al.20 studied the
periodic trends in the v4 rovibrational band of the complex,
still preserving the T-shaped geometry for these kinds of
complexes of CO2 with Ne, Ar, Kr, and Xe. Fraser et al.21
studied the sub-Doppler infrared spectra of CO2 – Ar in the
Fermi-diad region of CO2, viz., 3613– 3715 cm−1 using optothermal molecular beam color-center laser spectrometer.
Microwave spectra of CO2 – Rg 共Rg= Ne, Ar, Kr, and Xe兲
complexes show that the lower Fermi components are free of
perturbation, whereas the upper components are perturbed
and the reported geometry is consistent with T-shaped complex. CO2 – Ar clusters show a redshift 共1.1 cm−1兲 relative to
the free CO2 in the band origin. Studies on these systems
enabled development of empirical potential for clusters, viz.,
Hough and Howard potential.22 Theoretical and experimental
investigation on the vibrational relaxation and excitation in
CO2 – Ar has also been reported and the results are in agreement with the earlier studies, validating the predominant existence of T kind of CO2 – Ar complexes.21
A preliminary qualitative and quantitative understanding
on these smaller CO2 – Ar clusters has directed the theoretical
as well as experimental studies towards larger clusters. Xu et
al.23 reported a microwave study on larger clusters showing a
T-shaped CO2共Ar兲2. Sperhac et al.24 studied CO2共Ar兲2 complexes using direct absorption supersonic infrared spectroscopy and also with dynamics simulations. The calculations
show that these complexes show a redshift in the v3 region of
the CO2 molecule. CO2共Ar兲n complexes preferentially form
a T-shaped geometry to bind the central carbon atom and the
corresponding site gets saturated with five argon atoms
around it as also shown by the frequency shifts. The calculations show that the primary solvation sphere will be completed with 17 Ar atoms surrounding CO2 molecules.
Recently Vigasin et al.25 tried to comprehend all the
Raman and infrared spectroscopic experimental works on
Rg– CO2 complexes and tried to offer an explanation of the
frequency shifts on the basis of the matrix critical temperature. It was reported that the CO2 – Ar, CO2 – Kr, and
CO2 – Xe complexes show bathochromic 共red兲 shifts, while
the CO2 – He, CO2 – Ne, CO2 – N2, and CO2 – CO2 complexes
show hypsochromic 共blue兲 shift.
Considering the above, an ab initio investigation on the
effect of CO2 clustering and the effect of inert argon matrix
seems worthwhile. In view of this, the present work first
focuses on such a study of the energetics and geometries of
共CO2兲n clusters along with a comparison with experiment.
Second, ab initio studies on the geometry and energetics of
CO2共Ar兲m clusters along with the effect of inert argon matrix
on the CO2 probe molecule via the asymmetric stretching
frequency shifts are explored.
II. METHODOLOGY
Ab initio calculations were performed using the GAUSSprogram.26 Geometry optimizations were done at the
MPWB1K / 6-31+ + G共2d兲 level to obtain energy minima
corresponding to a variety of complexes. This level of theory
is recommended in the literature and found to give good
results for thermochemistry, thermochemical kinetics, hydrogen bonding, and weak interactions.27 Also it is reported that
the magnitude of interaction energy at this level of theory is
comparable in magnitude with the MP2 basis set superposition error 共BSSE兲 corrected interaction energy. The use of
density functional theory enables treatment of large clusters
and achieves good balance between size of clusters and computational time. All geometric parameters were optimized
with no constraints imposed on the molecular geometry during the optimization process. The vibrational frequencies
were calculated at the same level and basis set, viz.,
MPWB1K / 6-31+ + G共2d兲 level. None of the computed harmonic vibrational frequencies were found to be imaginary
for all the clusters reported in this work. The normal frequencies were visualized by using the in-house developed
package28 UNIVIS-2000. The BSSE 共Ref. 29兲 was estimated
for a few test geometries by the Boys-Bernadi counterpoise
method. For benchmarking purposes, smaller 共CO2兲n clusters
were also optimized at MP2 / 6-31+ + G共d兲 level of theory.
The energies and geometries obtained thereby were found to
be in good agreement with the corresponding density functional theory results. However, in view of substantially large
computational resources and time needed for MP2 level calculations, the larger systems were treated at MPWB1K level
of theory.
The molecular electrostatic potential30 共MESP兲; a physically observable property of a molecule, plays a key role in
IAN03
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124310-3
共CO2兲n and CO2共Ar兲m: Geometries and IR spectra
J. Chem. Phys. 128, 124310 共2008兲
FIG. 1. MESP isosurfaces of 共I兲 CO2 molecule, 共IIA兲
T-shaped 共CO2兲2, 共IIB兲 shifted parallel 共CO2兲2, 共IIIA兲
cyclic trimer, and 共IIIB兲 Z-trimer motifs at −0.018 a.u.
The 共3 , −3兲 CPs are shown as black dots and the respective function values are reported in a.u. and the distances are in Å.
weak intermolecular interactions. The MESP, V共r兲 at a point
r, due to nuclear charges 兵ZA其 at 兵RA其 and electron density,
␳共r兲 of the molecule is defined as
V共r兲 = 兺
A
ZA
−
兩r − RA兩
冕
␳共r⬘兲 3
d r⬘ .
兩r − r⬘兩
共1兲
The first and second terms on the right hand side of Eq.
共1兲 represent the bare nuclear and electronic potentials, respectively. The sign of the V共r兲 in any particular region depends on whether the nuclear or electronic effects are dominant there. The most negative site in a cluster acts as a
harbinger for the attack of the most positive site for the interacting molecule. In the present work, MESP is employed
in generating the large initial CO2 clusters by minimizing the
interaction energy within the electrostatic potential for intermolecular complexation 共EPIC兲 model.31 The Eint is defined
as
Eint = 1/2
再兺
i
冎
VA,iqB,i + 兺 VB,jqA,j .
j
共2兲
Here, the summation is over all the atoms of the species
A and B. V is the MESP defined by Eq. 共1兲 and q, the MESP
driven charges. The factor of 1 / 2 is incorporated in the equation to nullify the double counting. The local minima were
located through a genetic algorithm based energy minimization algorithm, under the constraint that the individual molecules are rigid. The molecules were docked over the van der
Waals surface of the other molecules to minimize the interaction energy. The geometries generated by EPIC were subjected to ab initio calculations. In order to gauge the strength
of two-, three-, four-, etc. body contribution to the total interaction in each of the clusters, a many-body analysis for
clusters 共MBAC兲 scheme of Xantheas32 calculation is performed on the ab initio level minimum geometries. These
calculations were performed via the MBAC code developed in
our laboratory.33
Larger 共CO2兲20 clusters were also generated from the
information of the smaller motifs and optimized at
MPWB1K level of theory using molecular tailoring
approach.34 In this method the whole cluster is divided into
two main fragments, typically 14 CO2 molecules each and
with an overlapping fragment of typically 8 CO2 molecules.
The respective energies of two main fragments and one overlapping fragments are Efi, Efj, and Efi艚fj. Now the total energy and gradients for the whole cluster is calculated using
the cardinality expression
Etotal = Efi + Efj − Efi艚fj + ¯ + 共− 1兲k−1 兺 Efi艚fj艚¯艚fk .
共3兲
Similar to the evaluation of the total cluster energy, the
total atomic gradients can be obtained by patching the fragment energy gradients via the general expression34
⳵EA
⳵Efi
⳵E fi艚fj
= 兺 fi − 兺 fi艚fj
⳵X␮
⳵X␮
⳵X␮
+ ¯ + 共− 1兲k−1 兺
⳵Efi艚fj艚¯艚fk
.
⳵X␮fi艚fj艚¯艚fk
共4兲
Once the atomic energy gradients are obtained for the
cluster, the new geometry is obtained from the old geometry
by moving each atomic coordinate in the opposite direction
to that of the energy gradient. The magnitude of the atomic
gradient is taken as the step size in each optimization cycle.34
The optimization procedure is continued until the energy and
gradients converge below a cutoff value. Finally these geometries are optimized using GAUSSIAN03. The geometry and
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124310-4
J. Chem. Phys. 128, 124310 共2008兲
K. V. Jovan Jose and S. R. Gadre
TABLE I. The most stable conformers of 共CO2兲n clusters with n = 1 – 8 at MPWB1K / 6-31+ + G共2d兲 and
MP2 / 6-31+ + G共d兲 levels of theory, geometry number, energy 共in a.u.兲, interaction energy 共in kcal/mol兲 at
MPWB1K 关MP2兴 levels, ⌬E, unscaled C–O frequencies 共in cm−1兲, and the corresponding intensities
共in km mol−1兲 are given in parentheses. See text for more details. 共The energies at MPWB1K 关MP2兴 level are
as follows 共in a.u.兲: I, −188.522 99 关−188.117 96兴; IIA, −377.047 49 关−376.238 40兴; IIB, −377.047 37
关−376.238 60兴; IIIA, −565.573 57 关−564.361 70兴; IIIB, −565.571 95 关−564.361 60兴; IVA, −754.099 78
关−752.486 80兴; IVB, −754.098 56 关−752.484 11兴; IVC, −754.098 17 关−752.483 39兴; V, −942.626 72
关−940.612 20兴; VI, −1131.154 26; VII, −1319.680 38; VIII, −1508.207 98.兲
Geometry
Energy 共⌬E兲
MPWB1K 关MP2兴
CO2 关I兴
共CO2兲2 关IIA兴
共CO2兲2 关IIB兴
共CO2兲3 关IIIA兴
0.00
−0.95
−0.87
−2.89
关0.00兴
关−1.57兴
关−1.68兴
关−4.89兴
共CO2兲3 关IIIB兴
−1.87 关−4.83兴
共CO2兲4 关IVA兴
−4.91 关−9.39兴
共CO2兲5 关V兴
−7.39 关−14.03兴
共CO2兲6 关VI兴
−10.24
共CO2兲7 关VII兴
−12.21
共CO2兲8 关VIII兴
−15.10
Frequency 共intensity兲
MPWB1K / 6-31+ + G共2d兲
2496.0 共780.2兲
2497.9 共743.3兲
2495.2 共3.2兲
2497.8 共390.7兲
2500.0 共1166.0兲
2495.5 共283.1兲
2496.0 共1440.7兲
2492.4 共62.6兲
2498.4 共1694.6兲
2492.4 共223.3兲
2497.8 共1280.1兲
2501.1 共1459.1兲
2495.3 共176.0兲
2500.6 共2183.6兲
2489.7 共230.4兲
2491.7 共194.6兲
2498.6 共1097.5兲
2501.9 共1496.7兲
2488.5 共465.5兲
2492.4 共428.2兲
2495.5 共202.6兲
2501.6 共653.8兲
energy of the most stable 共CO2兲20 cluster from the several
geometries explored by us is also reported.
III. RESULTS AND DISCUSSION
Results of our ab initio level investigation into the energetics and geometries of CO2 homoclusters at
MPWB1K / 6-31+ + G共2d兲 and at MP2 / 6-31+ + G共d兲 levels
of theory are discussed first. The effect of clustering is quantified through the C–O asymmetric stretching frequency
shifts and a comparison with experimental geometries 共wherever available兲 and frequencies is also made. Second, the
energetics and geometrical parameters of CO2共Ar兲m clusters
are studied at MPWB1K level and the effect of inert argon
matrix on the probe CO2 molecule is quantified through the
CO2 molecule asymmetric stretching frequency shifts.
A. CO2 homoclusters and the effect on asymmetric
stretching frequency
The CO2 molecule is optimized at MPWB1K and MP2
levels of theory and the MESP of single molecule are shown
in Fig. 1共I兲. The latter exhibits two negative valued regions
near the oxygen atoms and a positive region near the carbon
atom. The MESP predicts the existence of two kinds of CO2
dimer motifs, the first, T-shaped dimer 共C2v兲, with an electrostatic interaction between the negative MESP region of
oxygen with the positive carbon region of the other CO2
molecule and the second, a shifted parallel geometry 共C2h兲.
2500.2 共843.2兲
2495.5 共1521.5兲
2498.1 共811.3兲
2500.3 共638.3兲
2495.5 共123.1兲
2498.9 共1144.7兲
2495.1 共334.1兲
2499.9 共474.3兲
2502.7 共2186.7兲
MP2 / 6-31+ + G共d兲
2413.0 共13.4兲
2415.0 共12.9兲
2413.1 共0.0兲
2416.5 共20.3兲
2417.8 共0.3兲
2410.3 共2.9兲
2415.5 共1.8兲
2415.0 共10.2兲
2417.9 共9.6兲
2416.2 共11.4兲
2418.4 共9.1兲
2419.4 共17.7兲
—
2416.3 共14.5兲
2415.0 共25.7兲
2416.8 共20.5兲
2418.0 共31.9兲
2416.5 共23.9兲
2419.3 共8.8兲
2417.1 共3.1兲
2419.1 共24.8兲
—
2490.8 共387.9兲
2493.1 共235.7兲
2500.0 共1461.8兲
—
—
2491.3 共257.3兲
2494.7 共2435.3兲
2498.0 共989.4兲
2506.23 共736.4兲
—
—
The initial geometries were generated by maximizing the
magnitude of electrostatic interaction energy defined in Eq.
共2兲, using EPIC program.31 The shifted parallel geometry has
EPIC interaction energy of −0.97 kcal/ mol, while for
T-shaped it is −0.71 kcal/ mol. These initial geometries were
optimized at MPWB1K 关MP2兴 levels of theory. Both the
optimized geometries in Figs. 1共IIA兲 and 1共IIB兲 correspond
to a local minimum. Minimal nature of these is confirmed by
the absence of imaginary frequencies. The optimized C–O
bond length in CO2 molecule is 1.015 Å. It is found that this
value is not perturbed much in both the shifted parallel and
in T-shaped geometries. Table I reports the optimized energies 共in a.u.兲 of 共CO2兲n homoclusters along with the interaction energies 共in kcal/mol兲. MP2 / 6-31+ + G共d兲 energies and
unscaled IR frequencies are also shown up to n = 5 for a
comparison. It shows that both the CO2 dimers are very close
in interaction energies, shifted parallel with 共−0.87
关−1.68兴 kcal/ mol兲 and T-shaped conformer with 共−0.95
关−1.57兴 kcal/ mol兲 at MPWB1K 关MP2兴 levels of theory, respectively. In what follows, the reported parameters at
MPWB1K are reported outside the bracket and MP2 level is
given inside the square brackets. The BSSE corrections
within MPWB1K 关MP2兴 level of theory for shifted parallel
and
T-shaped
dimers
are
0.16
关0.94兴
and
0.08关0.88兴 kcal/ mol, respectively, and the trend in total interaction energy is not altered after counterpoise correction.
Figures 1共IIA兲 and 共IIB兲 show the corresponding optimized
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124310-5
共CO2兲n and CO2共Ar兲m: Geometries and IR spectra
FIG. 2. 共a兲 The geometrical parameters R 共the C–C distance兲, d 共perpendicular distance between CO2 molecules兲, x 共distance of shift between both
CO2 molecules兲, and ␪ 共⬔C – C – O兲 for the slipped parallel dimer and 共b兲
the RC–C and ␤ 共90-⬔C – C – C兲, the angle subtended between the C–C below CO2 molecules and the C–O bond of the third CO2 molecule. See text
for more details.
dimer geometries along with the MESP isosurface at
−0.018 a.u. The MESP minimum value indicates that the
T-shaped dimer has more affinity towards a neighboring molecule and hence a higher tendency to grow.
For a comparison of the shifted parallel CO2 dimer with
experiment, the geometric parameters as described in Fig.
2共a兲 are employed. Here, R is the vector connecting carbon
atoms of CO2 molecules, d is the perpendicular distance between two parallel CO2 molecular planes, x the distance of
shift of the second CO2 molecule relative to the first molecule, and ␪ the C–C–O angle. The corresponding calculated
structural parameters at MPWB1K 关MP2兴 levels of
theory respectively are R = 3.7关3.6兴 Å, ␪ = 55.0° 关57.9° 兴, x
= 2.0关1.9兴 Å, and d = 3.0关3.1兴 Å. These are in good agreement with those reported by Walsh et al.8 and Jucks et al.,9
viz., R = 3.6 Å, ␪ = 58.2°, x = 1.9 Å, and d = 3.1 Å. The experimental asymmetric stretching frequency of CO2 is
2350.8 cm−1. Hence the scaling factor of 0.9418 is selected
for MPWB1K level of theory for matching the theoretical
C–O frequency with its experimental counterpart. Theoretically calculated shifted parallel dimer conformer, reported in
Table I, shows a peak at 2352.6 cm−1 using the above scale
factor, which is blueshifted 1.8 cm−1 with respect to free
CO2 asymmetric stretch and this value is in agreement with
experimental shift reported by Walsh et al.8 The T-shaped
optimized geometry shown in Fig. 1共IIA兲 reports the distance
of 3.04 Å between the oxygen and carbon atoms of CO2
molecules. High-resolution spectroscopy confirmed the presence of only shifted parallel dimer motifs. However, the experimental geometrical parameters for T-shaped dimer are
not reported in the literature. The most intense asymmetric
stretching frequency reported in Table I shows a blueshift of
4.2 cm−1.
The MESP of the T-shaped dimer motifs, shown in Fig.
J. Chem. Phys. 128, 124310 共2008兲
1共IIA兲, also predicts cyclic trimer geometry, while the MESP
of the shifted parallel CO2, shown in Fig. 1共IIB兲 engenders a
Z type of trimer geometry. The latter is generated via EPIC
program by interacting another CO2 molecule with the
shifted parallel CO2 moieties in Fig. 1共IIB兲. The cyclic trimer and Z conformers have EPIC interaction energy of −2.04
and −1.57 kcal/ mol, respectively. These initial geometries
were optimized at ab initio level and are reported in Figs.
1共IIIA兲 and 1共IIIB兲 and which are the two predominant, experimentally reported trimer geometries in the literature.11,12
Essential structural parameters for comparison of these geometries with experimental counterparts are shown in Fig.
2共b兲, where the Rc–c is the distance between two adjacent
carbon atoms and ␤ 共90-⬔C – C – C兲 the angle. Theoretical
calculations at MPWBIK and MP2 levels could locate the
cyclic trimer conformer, which is the most stable trimer
conformation having interaction energy of −2.89
关−4.89兴 kcal/ mol at corresponding levels of theory. This is
again a local minimum as confirmed by the absence of
imaginary frequencies and the respective parameters at
MPWB1K 关MP2兴 levels are Rc–c = 4.07关4.04兴 Å and ␤
= 33.4° 关33.8° 兴. Weida et al.11 reported the existence of
stable cyclic trimer conformer obtained by fitting with model
potentials and the respective experimental parameters for the
cyclic trimer conformer are Rc–c = 4.04 Å and ␤ = 33.8°. The
cyclic trimer shows a blue frequency shift relative to the CO2
molecule.
A probable justification for the relative frequency shifts
can be given on the basis of he molecular electron density
共MED兲 value at the C–O bond critical point 共BCP兲 in the
共CO2兲n and CO2共Ar兲m clusters. The calculations were performed both at MPWB1K and MP2 levels of theory with
6-31+ + G共2d兲 basis set. The density function value at the
BCP is a signature of the strength of the respective bond. In
T-shaped CO2 dimer, three of BCP’s show a relative increase
共⬃0.001 a.u.兲 in density function value and a slight decrease
共⬃0.0006兲 at the fourth BCP. The cyclic trimer shows a predominant increase at three BCP value 共by 0.0015 a.u.兲 and a
relatively smaller decrease at three BCP’s 共by 0.0008 a.u.兲.
Thus one can say that there is an overall increase in the MED
value at C–O BCP’s in 共CO2兲n clusters, resulting into a blueshift of C–O asymmetric stretch.
Weida et al.12 have discussed the Z-trimer conformer
from the experiments performed in stagnation mixture of 2%
CO2 either in He, Ar, or 70% Ne/30% He. Calculations at
MPWBIK 关MP2兴 levels could locate the Z-trimer conformer
with stabilization energy of −2.83关−4.83兴 kcal mol−1, which
is very close in interaction energy to the cyclic trimer. This
conformation is again a local minimum. Weida et al.12 reported the spectrum of Z-conformer showing a 共red兲 shifted
band of −5.9 cm−1 and another 共blue兲 shifted band of
+3.6 cm−1 in the above mentioned matrix mixture, with respect to free CO2. The theoretical frequency of Z conformer
reported in Table I shows the same trend, but the magnitude
of shifts is less. One of the frequencies is redshifted by
−0.5关2.7兴 while the other is blueshifted by +4.3关4.9兴 cm−1.
Figure 1共IIIA兲 shows the MESP isosurface of cyclic CO2
trimer, which qualitatively predicts two directions of approach of the fourth molecule. The first is a sideways ap-
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124310-6
K. V. Jovan Jose and S. R. Gadre
J. Chem. Phys. 128, 124310 共2008兲
FIG. 3. Geometries of 共CO2兲n with n
= 4 – 8 at MPWB1K / 6-31+ + G共2d兲
level of theory. The geometry numbers
are given below the corresponding figure and the distances are in Å. See text
for more details and Table I for interaction energies and asymmetric
stretching frequencies.
proach in which the positive region of the ensuing CO2 molecule interacts with the negative region of the trimer motif
and the second is a perpendicular approach of the fourth
molecule with two negative regions of the oxygen atoms
interacting with the positive regions of the cyclic trimer and
the carbon atoms near the central negative region. These initial geometries were generated via the EPIC program and the
corresponding EPIC interaction energies were −3.65 and
−4.18 kcal/ mol. Many other EPIC tetramer geometries were
also generated based on the information of smaller motifs
and optimized at MPWB1K 关MP2兴 levels of theory. Three
most stable geometries of tetramers are shown in Figs.
3共IVA兲, 3共IVB兲, and 3共IVC兲. The most stable conformer corresponds to a trigonal pyramidal geometry with interaction
energies of −4.91关−9.39兴 kcal/ mol. Table I reports the respective asymmetric stretching frequencies. The most intense
peak is at 2353.1关2352.6兴 cm−1 after scaling and is blueshifted by 2.9关3.4兴 cm−1 relative to the CO2 molecule. The
most stable tetramer geometry is not in agreement with the
earlier ab initio studies by Knözinger and Beichert35 which is
a square 共C2h兲 geometry, shown in Fig. 3共IVC兲 and this geometry at MPWB1K level of theory is higher in interaction
energy compared to the pyramidal geometry by
0.81 kcal/ mol.
A number of initial CO2 pentamer geometries were generated by minimizing the EPIC interaction energy. The most
stable tetramer geometries direct the ensuing molecule to
form a tetragonal pyramidal geometry and which has the
highest EPIC interaction energy of −5.10 kcal/ mol. These initial geometries were optimized at ab intio level and depicted
in Fig. 3共V兲. The energetically lowest geometry is tetragonal
pyramidal one. The MPWB1K 关MP2兴 interaction energies
are −7.39关−14.03兴 kcal/ mol. This geometry shows that four
carbon atoms of CO2 molecules are in one plane interlocked
with T kinds of interactions and fifth CO2 molecule is above
the plane containing the four molecule and is again stabilized
by T-type dimer interactions with the molecules in the plane.
The intense asymmetric stretching frequencies of the most
stable conformer are reported in Table I. The most intense
frequency is 2355.6关2355.1兴 cm−1 after scaling and this peak
is blueshifted by 2.4关5.9兴 cm−1 relative to the CO2 molecule
asymmetric stretching frequency.
The MESP insights are used for generating around 20
initial hexamer geometries with smaller motifs. The most
favored structure on the basis of electrostatics is a square
bipyramidal structure. These geometries were optimized at
ab initio level with most stable CO2 hexamer being square
bipyramidal one, shown in Fig. 3共VI兲 with an interaction
energy of −10.24 kcal/ mol. This geometry shows four carbon atoms of CO2 molecules in one plane and the remaining
two are above and below the plane. The CO2 molecules in
the plane are mainly stabilized with T-type interactions. The
molecules above and below the plane have a Z-type interaction with the two molecules in the plane. The most intense
peak value is at 2357.1 cm−1 after scaling, which is blueshifted by 6.3 cm−1 relative to single molecule.
Fifteen initial heptamer geometries were generated on
the basis of MESP insights of smaller motifs and subjected to
EPIC geometry optimization. Subsequently, these geometries
were optimized at ab initio level. The most stable geometry
obtained on optimization at MPWB1K level of theory shows
that it prefers pentagonal bipyramidal geometry 共Fig. 3共VII兲
with an interaction energy of −12.21 kcal/ mol. This figure
shows that the five carbon atoms of CO2 molecules are in
one plane and two molecules are above and below the pentamer plane. The pentamers in the plane are stabilized with
major T type of interactions. Since the monomers in pentamer are located at larger distances, the molecules above
and below the planes are also stabilized by T type of interactions. The respective frequencies are reported in Table I.
The most intense frequency of the most stable configuration
shows a blueshift relative to the single molecule by 5.6 cm−1.
A series of initial octamer geometries were generated
with electrostatic guidelines and optimized at MPWB1K
level. It is found that the most stable geometry still preserves
pentagonal bipyramidal geometry at the core part and the
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124310-7
共CO2兲n and CO2共Ar兲m: Geometries and IR spectra
J. Chem. Phys. 128, 124310 共2008兲
FIG. 4. Geometry of optimized 共CO2兲20 cluster at MPWB1K / 6-31+
+ G共2d兲 level of theory. See text for more details.
eighth CO2 molecule causes slight distortion of the pentagonal bipyramidal geometry. The geometry of the most stable
octamer is shown in Fig. 3共VIII兲. The most intense peak
shows a redshift relative to single CO2 molecule, while the
second most intense frequency shows a blueshift of
1.8 cm−1. The ab initio frequency evaluation indeed shows
that the reported geometry is a local minimum. This observation is in agreement with Osber36 that the pure solid spectrum exhibits a redshift relative to the single CO2 molecule,
hence shows a frequency shift trend opposite to that of
smaller clusters.
Ten different 共CO2兲20 clusters were generated from the
MESP “lock and key” complementary information of smaller
clusters. These geometries were subsequently optimized using CG-MTA algorithm developed in our laboratory at
MPWB1K / 6-31+ + g共2d兲 level of theory. The best of these
clusters was subjected to Gaussian-based26 optimization.
This cluster with binding energy of −38.83 kcal/ mol is depicted in Fig. 4. The most stable 20 mer is seen to incorporate the maximum number of pentagonal bipyramids out of
all other energetically less favored clusters. This observation
is in agreement with the earlier studies37 and justifies that the
larger CO2 clusters prefer to have maximum number of pentagonal bipyramidal motifs. More detailed investigations on
large 共CO2兲n clusters are underway in our laboratory. The
variation of the MESP minimum with the cluster size is reported in Table II.
B. Effect of argon matrix on the asymmetric stretching
frequency of CO2 molecules
It has been observed that due to an increase in the percentage of argon in the matrix, the asymmetric stretching
TABLE II. The MESP minimum 共in a.u.兲 of 共CO2兲n clusters with n = 1 – 8 at
MPWB1K / 6-31+ + G共2d兲 level of theory. See text for more details.
Geometry
MESP minimum
CO2
共CO2兲2
共CO2兲3
共CO2兲4
共CO2兲5
共CO2兲6
共CO2兲7
共CO2兲8
−0.0217
−0.0252
−0.0239
−0.0244
−0.0216
−0.0242
−0.0208
−0.0237
FIG. 5. Geometries of CO2共Ar兲m with m = 1, 2, 3, 4, 5, 6, 8, and 12 at
MPWB1K / 6-31+ + G共2d兲 level of theory with the distances in Å. See text
for more details.
mode frequency shifts to a lower wave number region.38 In
order to quantify the effect of inert argon matrix on the probe
CO2 molecules, the frequency shifts in the asymmetric
stretching region of probe CO2 are explored. Hence a large
number of CO2共Ar兲m clusters with m = 1 – 12 were generated
and optimized at MPWB1K / 6-31+ + G共2d兲 level of theory.
We have performed these calculations only at this level of
theory to tackle large CO2共Ar兲m clusters and its frequency
evaluations. The most stable complexes are depicted in Fig.
5; typical distances between C¯Ar is 3.7 Å. The most stable
CO2共Ar兲1–5 complex geometry at the MPWB1K level is in
agreement with the dynamics simulation calculations reported by Severson et al.38 The interaction energy at
MPWB1K level of theory shows that in the CO2共Ar兲m clusters, argon atoms preferably show a predominant interaction
with carbon over oxygen atom of CO2 molecule. This observation is in agreement with experimental observation of
Steed et al.18 and dynamics simulation calculation reported
by Severson et al.38 The reported CO2 – Ar structural parameters of Steed et al.18 were R = 3.5 and ␪ = 82.5° and the geometry belongs to C2v point group. While the theoretical
structural parameters are R = 3.7 and ␪ = 90.0°. Even though
the geometries of smaller CO2共Ar兲m clusters with m = 1 and 2
are in agreement with the current observations at MPWB1K
level of theory, the higher clusters reported by Böyukata39
with m = 3 – 7 show a preferential interaction of the argon
atoms with the oxygen atom of CO2. This disagreement in
the geometries may be due to the empirical model potential
employed in the dynamics calculations. The BSSE counterpoise correction does not seem to alter the relative trends in
interaction energy of CO2共Ar兲m clusters. The BSSE correction for CO2共Ar兲2 and CO2共Ar兲4, respectively, are 0.09 and
0.60 kcal/ mol, which does not alter the trends in the interaction energies. To check the consistency of these results, we
have
performed
these
calculations
again
at
MPW1B95/ 6-31+ + G共2d兲 level of theory. Even though the
interaction energies at this level of theory are also small
compared to that of MP2 level, the interaction energy is
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124310-8
J. Chem. Phys. 128, 124310 共2008兲
K. V. Jovan Jose and S. R. Gadre
TABLE III. The CO2共Ar兲m clusters with m = 1 – 12 at MPWB1K / 6-31+
+ G共2d兲 level of theory, geometry, number of CO2 in the cluster, energy 共in
a.u.兲, interaction energy 共⌬E兲 共in kcal/mol兲, unscaled C–O frequencies 共in
cm−1兲, and the corresponding intensities 共in km mol−1兲 are given in parentheses. See text for more details. 共CO2 and Ar energies at MPWB1K / 6-31
+ + G共2d兲 level are −188.522 99 and −527.579 58 a.u., respectively.兲
Geometry
CO2Ar 关M-I兴
CO2共Ar兲2 关M-II兴
CO2共Ar兲3 关M-III兴
CO2共Ar兲4 关M-IV兴
CO2共Ar兲5 关M-V兴
CO2共Ar兲6 关M-VI兴
CO2共Ar兲8 关M-VII兴
CO2共Ar兲8 关M-VIII兴
Energy 共⌬E兲
Frequencies 共intensity兲
−716.102 88共−0.20兲
−1243.682 80共−0.41兲
−1771.262 80共−0.67兲
−2298.842 89共−0.99兲
−2826.423 30共−1.51兲
−3354.003 00共−1.59兲
−4409.164 17共−2.85兲
−4409.163 24共−2.27兲
2495.3共742.6兲
2494.5共705.8兲
2492.2共678.4兲
2494.2共626.8兲
2495.5共614.2兲
2490.3共591.4兲
2490.1共676.4兲
2493.7共850.7兲
that beyond the saturation of the central carbon atom, the rest
of the argon atoms cluster around the oxygen atoms. Dynamics modeling shows that the 17 argon atoms will saturate the
first interaction or solvation sphere of the probe CO2
molecule.38,39 This observation is indeed in corroboration
calculations at MPWB1K level of theory.
C. Many-body interaction energy analysis of „CO2…n
and CO2„Ar…m clusters
comparable with the BSSE corrected MP2 interaction energies. The trends in interaction energies and frequency shifts
are consistent with MPWB1K level of theory.
The graph of number of Ar atom versus frequency shift38
of CO2共Ar兲m clusters reaches a saturation region beyond m
艋 5; it corresponds to a saturation of the central carbon atom
of CO2 molecule. The interaction energy at MPWB1K level
for CO2共Ar兲5 is −1.51 kcal/ mol and that of CO2共Ar兲6 is
−1.59 kcal/ mol. Calculated frequencies reported in Table III
also shows a saturation in frequency shift in CO2共Ar兲m clusters with m 艌 5. Figure 5 displays the geometries of most
stable CO2共Ar兲m with m = 1 – 12. Table III reports the frequencies of the CO2共Ar兲m clusters through m = 1 – 8 and the
values show an increasing red frequency shifts from 0.7
through 5.6 cm−1 with number of argon atoms. This direction
of shifts are in agreement with the earlier dynamics studies
of Severson38 and Böyukata et al.39 and direct supersonic
fourier transform infrared spectroscopy 共FTIR兲 experimental
studies of Thiévin et al.40 The present ab initio studies show
a good agreement in geometry and frequency shifts with the
most stable higher CO2共Ar兲m clusters with m = 6 – 8, obtained
via molecular dynamics simulation.38,39 This red asymmetric
C–O stretching frequency in CO2共Ar兲m clusters can be justified on the basis of MED values at the C–O BCP. It is observed that the MED value shows a slight decrease in
CO2共Ar兲m clusters, viz., with m = 6 and 8, respectively, are
0.47520 and 0.47519 a.u. compared to CO2 molecule with
0.47524 a.u. This decease 共⬃0.000 04 a.u.兲 in MED, however small, indicates a weakening of both C–O bond strength
and hence a redshift in frequency. These geometries show
Many-body analyses on 共CO2兲n and CO2共Ar兲m clusters
were carried with a view to partition the total interaction
energy into two-, three-, etc. body contributions, based on
the partitioning scheme of Xantheas32 and to get a better
molecular level understanding in terms of many-body interactions. Table IV gives a glimpse of the MBAC calculations
on 共CO2兲n and CO2共Ar兲m clusters. It shows that the total
interaction energies of 共CO2兲n is higher than in CO2共Ar兲m
clusters. In all the complexes, the two-body contribution to
interaction energy is higher compared to that of three-body
energy. Also the two-body interaction between CO2 – CO2 in
共CO2兲n is higher compared to that of Ar– CO2 interactions in
CO2共Ar兲m. The CO2 – CO2 interaction in CO2 clusters is
nearly −0.95 kcal/ mol, which is approximately 4.8 times
that for Ar– CO2 共−0.20 a.u.兲. This may be the reason for a
higher frequency shift in CO2 clusters than in CO2共Ar兲m
clusters. The magnitude of total two-, three-, and four-body
interaction energy increases with the number of CO2 molecules in the cluster and this is due to the increase in the
number of respective body interactions in the cluster. But the
average contribution of each body to the total interaction
energy goes on decreasing with the number of molecules,
meaning that the individual body in higher clusters looses
interaction energy compared to that in the lower one. Even
though the interaction between the argon and CO2 probe
molecule is negligibly small, there is definite shift in frequency in the CO2共Ar兲m clusters, with respect to free CO2
molecule.
IV. CONCLUSIONS
One of the aims of this work is to study the geometric
patterns of CO2 clusters and also how argon atoms cluster
around a CO2 molecule. The MESP critical points 共CPs兲 are
seen to serve as a harbinger for probing the addition of CO2
molecules to a given cluster and EPIC model is seen to be
useful for generating energetically favorable initial structures
for the CO2 clusters. This model generally brings out correct
TABLE IV. Many-body analysis of clusters 共MBAC兲 partitioning of the interaction energy of 共CO2兲m and
CO2共Ar兲m clusters at MPWB1K / 6-31+ + g共2d兲 level of theory. The reported values are in kcal/mol. See text for
details.
Cluster
共CO2兲3
共CO2兲4
共CO2兲5
共CO2兲6
CO2共Ar兲
CO2共Ar兲2
Total ⌬E
Max two body
Total two body
Max three body
Total three body
−2.89
−4.91
−7.39
−10.24
−0.20
−0.41
−0.95
−0.94
−0.94
−0.92
−0.20
−0.19
−2.84
−4.64
−6.84
−9.86
−0.20
−0.38
−0.97
−0.09
−0.12
−0.12
¯
−0.01
−0.97
−0.15
−0.30
−0.44
¯
−0.01
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124310-9
共CO2兲n and CO2共Ar兲m: Geometries and IR spectra
trends in the interaction energy of these clusters as compared
with those exhibited by the ab initio level calculations. These
initial geometries generated are found to be good starting
points for subsequent calculations at MP2 and MPWB1K
level of theory. The optimized geometries at MPWB1K level
of theory will be furnished on request. The present calculations could locate all the experimentally reported geometries
of 共CO2兲2,3 and ab initio level n = 4 clusters and which are
energetically favorable. Critical comparisons of these geometries and frequency shifts have been done with the experimental counterparts and are reported in the current work. The
effect of CO2 clustering on the frequency shifts are quantified and the clusters show an increasing amount of blue frequency shifts from 4.2 to 5.9, n = 2 – 7, while the most intense
peak of 共CO2兲8 shows a slight redshift. These directions of
shifts are in agreement with the earlier experimental frequency shifts. A test calculations on 共CO2兲20 cluster employing molecular tailoring approach followed by GAUSSIAN
共Ref. 26兲 based geometry optimization is also presented. The
geometry of this cluster shows pentagonal bipyramidal patterns. We have also reported a study of the energetics and
geometrical parameters of the CO2共Ar兲m clusters. The effect
of inert matrix on the CO2 molecule asymmetric stretching
frequency is quantified and the frequency shifts shows a redshift of 0.73– 5.9 cm−1 on increasing the number of argon
atoms in the cluster from m = 1 to 8 and this observation is
indeed in good agreement with experiments. The investigations on large CO2 共⬃25– 30兲 clusters are in progress employing molecular tailoring approach.34 It is hoped that further such calculations on large clusters will be able to shed
light on cluster to bulk cross over regime and frequency
shifts.
ACKNOWLEDGMENTS
One of the authors 共J.J.K.V.兲 is thankful to the Council
of Scientific and industrial Research 共CSIR兲, New Delhi for
followship and CAS program of the UGC to the Department
of Chemistry, University of Pune for computational facilities.
Another author 共S.R.G.兲 is thankful to DST for the award of
J. C. Bose fellowship and Naval Research Board 共NRB兲 for
financial support. The authors 共J.J.K.V. and S.R.G.兲 thank R.
Georges for valuable discussion. One of the authors
共J.J.K.V.兲 thank Subodh and Ritwik for their valuable help.
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