Self archive of H. Torii, J. Chem. Phys. 119 (4), 2192 (2003)

The role of atomic quadrupoles in intermolecular electrostaticinteractions of polar and nonpolar molecules
Hajime Torii
Department of Chemistry, School of Education, Shizuoka University,
836 Ohya, Shizuoka 422-8529, JapanFor the purpose of getting insight into the reason for the anomalous vibrational frequency shifts
observed in some usually used solvents for a mode that has a large dipole derivative, the role of atomic
quadrupoles in intermolecular electrostatic interactions is studied for some halogen-containing molecules(CX4, HX, and X2 with X = F, Cl, and Br), CH4, CO2, and CS2. From the fitting to the electrostatic
potentials around the molecules, large atomic quadrupoles are obtained for the chlorine, bromine, andsulfur atoms, suggesting that the atomic quadrupolar effect is important for electrostatic interactions aroundcovalently bonded atoms on the third and higher rows in the periodic table. Taking the case of the chlorine
atoms as an example, the electron densities inside the atoms in CCl4, HCl, and Cl2 are examined. It is foundthat these electron densities are highly anisotropic. This anisotropy in electron densities is reasonably
explained by the forms of the occupied molecular orbitals, and is considered to be the electronic structural
origin of the large atomic quadrupoles.
where kp is the vibrational quadratic force constant.
When the vibrational mode is not strictly harmonicin a mechanical sense, this structural displacement
gives rise to a shift in the vibrational frequency(νp):
fp
f p ∂μδν p=
δq p = 2
E
(3)
νp
2k p
2k p ∂q p

I. INTRODUCTION
It is well recognized that various molecularproperties are affected more or less by intermolecular electrostatic interactions. By analyzing
those properties, the nature of the intermolecularelectrostatic interaction in a particular system may
be examined. From this viewpoint, vibrationalmodes with large dipole derivatives deserve
detailed analysis in vibrational spectroscopic
studies of condensed phase systems. The reason
for the high sensitivity of these modes tointermolecular electrostatic interactions is that thescalar product between electric field and dipole
derivative of a vibrational mode provides theleading order term for electrostatic force along themode:1
∂V
∂μ
=−
E
(1)
∂q p
∂q p

where fp is the cubic force constant of this mode.
In fact, electric field induces an additional shift inthe vibrational frequency because of electronicanharmonicity,3,4 so that we obtainδν p∂ 2 μ ⎞⎟
1 ⎛⎜ f p ∂μ
=
−
E
(4)
2k p ⎜ k p ∂q p ∂q 2p ⎟νp⎝⎠
In many cases, however, the first term isconsidered to be dominant.5 Therefore, studies on
the behavior of the vibrational modes with large
dipole derivatives [more precisely, those with large
values of (fp/kp) (∂μ/∂qp) – ∂2μ/∂qp2] are expected
to be useful to clarify the nature of the intermolecular electrostatic interactions in condensed
phases.
The C=O stretching mode of carbonylcompounds are known to have large dipole derivatives, and to exhibit significant low-frequency
shifts in polar solvents. In a recent study, Kollingwhere E is the electric field and ∂μ/∂qp is the
dipole derivative of the pth vibrational mode (qp).
This force induces a displacement of the potential
energy minimum along this mode:1,2
1 ∂μ(2)
δq p =
Ek p ∂q p

1

Self archive of H. Torii, J. Chem. Phys. 119 (4), 2192 (2003)

has analyzed the vibrational frequencies of the
C=O stretching mode of 2-butanone in various
solvents.6 It has been shown that the vibrationalfrequencies observed in many polar and nonpolar
solvents are in a good linear relation with the
Kirkwood–Bauer function K(ε) defined as K(ε) ≡(ε – 1)/(2ε + 1), where ε is the dielectric constantof solvent. This result is reasonably explained by adielectric continuum theory, where electric field E
in Eqs. (3) and (4) is described as the reaction fieldof solvent generated by the permanent dipole
moment of the solute molecule. However, it has
also been clarified that there are some widely usedsolvents in which the C=O stretching band shows
anomalous low-frequency shifts. These solvents
include chloroalkanes (carbon tetrachloride,
chloroform, and dichloromethane), carbon disulfide, and aromatic hydrocarbons (benzene andtoluene), many of which are generally classified
into nonpolar solvents.
In our previous study, the solvent effect ofcarbon tetrachloride on the C=O stretching modeof acetone has been analyzed by carrying out ab
initio MO calculations for an acetone–(CCl4)4cluster.7 It has been shown that, in the optimized
structure of the cluster, one C–Cl bond of eachCCl4 molecule points to the carbonyl oxygen, andstrong electric field is operating on the C=O bondin the direction that the chlorine atoms look as if
they had large positive charges, in contrast to the
expectation from the values of electronegativity incommon electronegativity scales.8–11 This electricfield is significantly stronger that that expected forthe reaction field, and is described approximately
as the sum of the electric field present aroundindividual CCl4 molecules. Because of this strong
electric field, the C=O stretching mode shows asignificant low-frequency shift, in accord with theexperimental result.6 It has been clarified from thefitting to the electrostatic potential around an
isolated CCl4 molecule that the atomic quadrupoles of the chlorine atoms, with the magnitude of
Θ(Cl) ≅ 1.5 au,12–14 give rise to this effect. Thismagnitude of atomic quadrupole is also found forthe chlorine atom of hydrogen chloride. It has been
therefore suggested that this may be a generalfeature of chlorine atoms bonded covalently toanother atom.
The strong electric field around CCl4 may berelated to the apparently peculiar structures of the
Cl–..CCl4 and CCl4..Cl–..H2O clusters obtained in
the previous studies.15,16 In those structures, one of
the C–Cl bond of CCl4 points to the chloride ion,just like the structure obtained for the acetone–(CCl4)4 cluster7 where one C–Cl bond of each CCl4
molecule points to the carbonyl oxygen that has apartial negative charge. The interaction between anegative charge and the electric field in the C→Cl
direction, in addition to the charge transfer effectas discussed in the previous studies,15,16 seems tocontribute to the stability of those structures. This
result may indicate that atomic quadrupoles havesignificant effects on the interaction energies
between polar and nonpolar molecules.
From the fitting to the electrostatic potential
around a CCl4 molecule,7 a negative atomic chargeof q(Cl) ≅ –0.1 e is obtained for the chlorine atomwhen we take into account the atomic quadrupole,in accord with the expectation from electronegativity scales,8–11 but a positive atomic chargeof q(Cl) ≅ 0.2 e is obtained when we do not. This
result is suggested to indicate the limitation of an
atomic charge model. The limitation of an atomic
charge model is also evident from the difference inthe signs of the values of q(Cl) estimated by theMK [q(Cl) = 0.13 e],17,18 CHelp [q(Cl) = –0.09e],19 and CHelpG [q(Cl) = 0.09 e]20 methods at theMP3/6-31+G(2df) level.7
Taking all these points into account, it isinteresting to compare the results obtained7 for
CCl4 and HCl with those of other related
molecules, and to study the electronic structural
origin of atomic quadrupoles. The present study isdevoted to these subjects. We first analyze theelectric field around a Cl2 molecule, where theuncertainty between bond dipole and atomic
dipole7 does not exist. The electronic structure ofthis molecule is studied to get insight into thereason for the existence of atomic quadrupoles.
Then we study the situation of the F- andBr-substituted molecules (CF4, HF, F2, CBr4, HBr,
and Br2) and CH4, and compare the results with2

Self archive of H. Torii, J. Chem. Phys. 119 (4), 2192 (2003)

those of the Cl-containing compounds. We also
compare the situation of CO2 and CS2 molecules tostudy the reason for the anomalous low-frequency
shift of the C=O stretching mode observed in CS2
solvent, and to see the generality of the tendency
of atomic quadrupolar effect found for halogencontaining molecules.

III. RESULTS AND DISCUSSION
A. CCl4, HCl, and Cl2
The electric fields present around isolated
molecules of CCl4, HCl, and Cl2 are shown in
Fig. 1. As discussed in our previous study,7 theelectric field around CCl4 is going out of thechlorine atom on the line extended from the C–Cl
bond, and is going toward the chlorine atom on theside of the bond. This form of electric field isrationalized by taking into account the existence of
atomic quadrupoles on the chlorine atoms. In thecase of HCl, the electric field around the chlorine
atom on the line extended from the H–Cl bond ismuch weaker because of the cancellation with theeffect of molecular dipole moment, but the vectorof electric field is also going out of the chlorine
atom on that line in the vicinity (within ∼3.8 Å) of
the atom. As shown in Fig. 1 (c), the same form ofelectric field is also found around a Cl2 molecule.
Because Cl2 is a diatomic molecule consistingof the same atomic species, the atomic charge isexactly zero and there is no bond dipole moment.
As a result, there is no uncertainty between bonddipole and atomic dipole in the fitting to theelectrostatic potential. This point is considered to
be a merit for examining the atomic multipole
moments obtained from electrostatic potential andelectric field around the molecule.
The atomic charges, dipoles, and quadrupoles
obtained from the fitting to the electrostatic
potentials are shown in Table I.12 The electricfields in the X→Cl direction on the line extended

II. COMPUTATIONAL METHODS
Ab initio MO calculations were carried out at
the MP3 level with the 6-31+G(2df,p) basis set byusing the Gaussian 98 program.21 The electrostatic
potential and the electric field were evaluatedaround molecules at the optimized structures. For
part of the molecules treated in the present study,
calculations were also carried out at the Hartree–
Fock and MP2 levels and with various basis sets.
The results were found to be insensitive to theselection of theoretical level and basis set.
Atomic partial charges, atomic dipoles, andatomic quadrupoles were obtained from the fitting
to the electrostatic potential around molecules byusing our original program. In the fitting procedure,
the atomic charges were constrained so that the
total charge of the molecule is zero. The
evaluation points were taken with the interval of
0.4 Å in a box with L ≥ 16 Å, but those within 1.6
Å from hydrogen or within ∼2.8 Å (precise values
depending on the atomic species in the range of
2.5–3.1 Å) from any other atoms were excluded.
The total number of independent evaluation points
varied with the molecular symmetry. It was ∼1400
for linear molecules with D∞h symmetry, ∼2400 for
those with C∞v symmetry, and ≥ 16000 for othermolecules.
All the above calculations were performed on
Fujitsu VPP5000 and NEC SX-5 supercomputersat the Research Center for Computational Scienceof the Okazaki National Research Institutes, andon Compaq XP1000 and DS20E workstations in
our laboratory.

——————————————————————
FIG. 1 (next page). The electric fields around (a) CCl4,(b) HCl, and (c) Cl2 molecules calculated at theMP3/6-31+G(2df,p) level. The filled circles represent
the carbon and chlorine atoms located on the sheet. The
open circle in a represents two chlorine atoms locatedabove and below the sheet. The hydrogen atom of HClis located at one end of the thick straight line without afilled circle in b. The points within 1.6 Å from thehydrogen atom or within 2.8 Å from any other atoms
are excluded. In b and c, the electric field on the left
side of the molecule is symmetric to the one on the rightside, and is omitted in this figure.

3

(a)

(b)

H. Torii, Figure 1(c)Self archive of H. Torii, J. Chem. Phys. 119 (4), 2192 (2003)

TABLE I. Values Atomic charges (q), dipoles (μ), and quadrupoles (Θ) of CCl4, HCl, and Cl2 molecules
obtained from the fitting to the electrostatic potentiala==========================================================================q/e
μ / auΘ / aumolecule fitting procedure
residual deviationb / au atom——————————————————————————————————————————
with q and Θ
5.6 × 10-3
C
0.422
–
0.000
CCl4
Cl
–0.105–1.413
1.1 × 10-2
C
3.358
0.000
–
with q and μCl–0.8401.232–-2C–0.894––
with q
4.4 × 10
Cl
0.224
–
–
7.6 × 10-4
H
0.213
–
0.154
HCl
with q and ΘCl–0.213–1.535-34.5 × 10H
0.770
–0.310–with q and μCl–0.770–1.039
–
H
0.236
–
–
with q
4.3 × 10-2Cl
–0.236––-3
with Θ
4.1 × 10
Cl
0.000
–
1.249
Cl2
-2
1.0 × 10
Cl
0.000
0.326
–
with μ==========================================================================ab

The electrostatic potentials are calculated at the MP3/6-31+G(2df,p) level.
Calculated as the square root of Σ (ρMO – ρfitted)2, where ρMO and ρfitted are the electrostatic potentials obtained fromthe ab initio MO calculations and the fitting, respectively. Comparison of the values is meaningful only within thesame molecule.

or in the wrong direction (for HCl).
At this point, it is interesting to see how theatomic charges and quadrupoles obtained from thefitting contribute to the molecular multipole
moments. In the case of CCl4, the molecularoctopole moment is observed as Ωmol = 15 ± 3 DÅ2or 21 ± 4 au (Ref. 23) and calculated as Ωmol =20.32 au at the MP3/6-31+G(2df) level. The
atomic quadrupole of Θ(Cl) = 1.413 au (with the
C–Cl bond length of d = 3.345 au) corresponds toΩmol = 10Θ d / 3 = 27.29 au. This is partiallycanceled by the contribution of atomic charges
q(Cl) = –0.105 e that corresponds to Ωmol = 10 qd3/ 3 3 = –7.59 au. The sum of these values (19.70au) is consistent with the molecular octopole
moment. If this molecular octopole moment ofΩmol ≅ 20 au is to be reproduced only with atomic
charges, q(Cl) should be as large as ~0.3 e,24,25 but
such a set of atomic charges cannot correctly
reproduce the electric field around the molecule asdiscussed in our previous study.7 In the case ofHCl, the molecular quadrupole moment isfrom the X–Cl bond (where X = C for CCl4, H for
HCl, and Cl for Cl2) calculated with these atomic
multipole moments as well as those obtained
directly from the MO calculations are shown in
Fig. 2. The atomic charges obtained from thefitting with atomic charges and dipoles shown in
Table I (as well as those obtained in a similar way
in a previous study22 for CCl4) seem to be out ofan acceptable range in the case of CCl4 and a little
too exaggerated in the case of HCl, probably
because the effect of an atomic dipole is quite
similar to that of a bond dipole, which isrepresented by the partial charges of the two atoms
that are connected by the relevant bond. This is not
the case for Cl2. Even in this case, however, thefit with atomic quadrupoles is much better thanthat with atomic dipoles, indicating the importanceof atomic quadrupoles. For the other two
molecules also, a better fit is obtained by taking
into account atomic quadrupoles. If the fit is done
only with atomic charges, we obtain electric fieldthat is too weak (for CCl4), totally zero (for Cl2),

4

Self archive of H. Torii, J. Chem. Phys. 119 (4), 2192 (2003)

8

About half of this value is obtained for each of thechlorine atoms from the fitting [Θ(Cl) = 1.249 au]
as shown in Table I.
In the potential functions developed previously
for HCl, the third interaction points were placed on
the H–Cl bond28 or on the line extended from thisbond,29,30 and a negative charge is assigned to thepoint in the middle and positive charges to theother two points, to get good agreement in themolecular dipole and quadrupole moments.26 In aprevious study based on a distributed multipole
model,31 this kind of charge distribution was also
assumed by taking the bond center as the thirdinteraction point. The results obtained in thepresent study show that the electrostatic potentialand electric field around the molecule and themolecular quadrupole moment are reasonably
explained by introducing atomic quadrupoles on
the chlorine atoms, and suggest that this is not
special to HCl but is a general feature of chlorine
atoms bonded covalently to another atom. The
vectors of electric field shown in Fig. 1 indicatethat the atomic sites are the primary interaction
points. Inclusion of atomic quadrupoles issuggested to be the simplest but reasonably correct
way of representing the electrostatic interactions ofthese molecules.
To see the electronic structural origin of thelarge atomic quadrupoles calculated for thechlorine atoms, the electron densities inside thechlorine atoms are examined. The result obtained
for Cl2 is shown in Fig. 3. Those obtained for theother two molecules are very similar. It is clearlyseen that the electron density is substantiallyhigher in the x direction (perpendicular to thebond) than in the z direction (on the molecular axis
along the bond). This highly anisotropic electron
density in the hemisphere in the z ≥ 0 region (withthe origin at the center of the atom) contributes tothe atomic quadrupole by 0.874 au. Consideringthat the electron density in the other hemisphere is
expected to contribute also to the atomic quadrupole to a similar extent, this value is considered tobe consistent with the atomic quadrupole of Θ(Cl)= 1.249 au obtained from the fitting to theelectrostatic potential.

(a)

6
4
2

-3

Electric field / 10 au

0

4

(b)

2
0
-2
-4
-6

10

(c)

8
6
4
2
0
3
4
5
Distance from Cl / ÅFIG. 2. The electric fields in the X→Cl direction on theline extended from the X–Cl bond calculated for (a)CCl4, (X = C), (b) HCl (X = H), and (c) Cl2 (X = Cl).
The values calculated with the atomic multipole
moments obtained from the fitting (Table I) are shownin broken lines (fitted with atomic charges only),
dot-dashed lines (atomic charges and dipoles), anddotted lines (atomic charges and quadrupoles). The
solid lines show the values obtained directly from theMO calculations at the MP3/6-31+G(2df,p) level.

observed as Θmol = 3.74 DÅ or 2.78 au (Ref. 26)
and calculated as Θmol = 2.86 au at the MP3/
6-31+G(2df,p) level or 2.67 au at the CCSD(T)level with an extended basis set.27 The atomic
charges of q(H) = –q(Cl) = 0.213 e (with the H–Cl
bond length of d = 2.412 au and the center of mass
taken as the origin) correspond to Θmol = 1.17 au.
The sum of this value and the atomic quadrupoles[Θ(H) = 0.154 au and Θ(Cl) = 1.535 au], whichamounts to 2.86 au, is consistent with the molecular quadrupole moment. In the case of Cl2, themolecular quadrupole moment is calculated asΘmol = 2.41 au at the MP3/6-31+G(2df) level.

5Self archive of H. Torii, J. Chem. Phys. 119 (4), 2192 (2003)
1.0

0.3

(a)

4

0.10

(a)

(b)

(b)

0.8
0.08

3

0.6

0.2

2

Wave function squaredElectron density / au

5

0.1
1
0

0.0
0.0

0.1

0.2

0.3

0.2

0.6

1.0

1.4

Distance from the center of the Cl atom / Å

0.06

0.4

0.04

0.2

0.02

0.0

0.00

1.0
0.10

(c)

0.08
0.6

z
(c)

(d)

0.8

0.06

0.4

0.04

0.2

0.02

0.0

0.00
0.0

0.1

0.2

0.3

0.2

0.6

1.0

1.4

Distance from the center of the Cl atom / ÅxFIG. 4. The values of the electronic wave functions
squared in one of the chlorine atoms of Cl2 plottedagainst the distance (0.0–0.3 Å in a and c, 0.2–1.5 Å inb and d) from the center of the atom. Solid and brokenlines in a and b: |Ψ11(r)|2 + |Ψ12(r)|2 with r on the z andx axes [defined in Fig. 3 (c)], respectively; solid andbroken lines in c and d: |Ψ13(r)|2 with r on the z and xaxes, respectively; dot-dashed lines in c and d: the sumof |Ψm(r)|2 for m = 14–17 with r on the x axis.
respectively.

FIG. 3. The electron density in one of the chlorine
atoms of Cl2 calculated at the MP3/6-31+G(2df) level,plotted against the distance (0.0–0.3 Å in a and 0.2–1.5
Å in b) from the center of the atom. The values
calculated for the points on the z and x axes (defined inc) are shown in solid and broken lines, respectively.

There are 17 occupied molecular orbitals in Cl2,
out of which the highest seven are the valenceorbitals. The 11th and 12th molecular orbitals areapproximately described as linear combinations of
the 3s orbitals of the two chlorine atoms. The
values of |Ψ11(r)|2 + |Ψ12(r)|2, with r on the z and xaxes defined in Fig. 3 (c), are shown in solid andbroken lines, respectively, in Fig. 4 (a) and (b).
The plotted values are almost isotropic, as
expected from the 3s character of these orbitals.
The 13th molecular orbital mainly consists of the
3pz orbitals. The value of |Ψ13(r)|2, plotted in solidlines for r on the z axis and in broken lines for r on
the x axis in Fig. 4 (c) and (d), clearly shows the
3pz character of this orbital, although it also hassome sp hybrid character because the value of|Ψ13(r)|2 at the origin is nonzero. The 14th to 17th
molecular orbitals are made of the 3px and 3py
orbitals. The sum of the values of |Ψm(r)|2 of theseorbitals for r on the x axis is shown in dot-dashedlines in Fig. 4 (c) and (d).
It is clear from these plots that the sum of thevalues of |Ψm(r)|2 on the x axis is significantly
larger than that on the z axis mainly because of thecontributions from the 13th to 17th molecularorbitals. There is only one occupied molecularorbital with the 3pz character (the σ orbital),because one of the linear combinations of 3pz isunoccupied (the σ* orbital). In contrast, there are
two occupied molecular orbitals each for 3px and3py. This is considered to be the reason for thehighly anisotropic electron density in the chlorine
atom shown in Fig. 3, and hence for the large
atomic quadrupole obtained from the fitting to theelectrostatic potential.
B. F- and Br-containing molecules and CH4
The atomic charges and quadrupoles obtained
for CH4, CF4, HF, F2, CBr4, HBr, and Br2 from thefitting to the electrostatic potentials around themolecules are shown in Table II. We have notincluded atomic dipoles in the fitting, because

6Self archive of H. Torii, J. Chem. Phys. 119 (4), 2192 (2003)

TABLE II. Atomic charges (q) and quadrupoles(Θ) of CH4, CF4, HF, F2, CBr4, HBr, and Br2
molecules obtained from the fitting to theelectrostatic potentiala

and HBr on the line extended from the H–X bond[shown for HCl in Fig. 2 (b), not shown for HF
and HBr], it is recognized that the direction of theelectric field is correctly represented with atomic
charges only in the case of HF. In the other twocases, the atomic quadrupoles are essential. It istherefore suggested that atomic quadrupoles areimportant for representing the electrostatic interactions around covalently bonded atoms on thethird and higher rows.

================================
moleculeatomq/eΘ / au——————————————————
C–0.289
0.000
CH4
H
0.072
0.131
C
1.204
0.000
CF4
F
–0.301
0.373
HF
H
0.428
0.109
F
–0.428
0.495
F
0.000
0.400
F2
C
0.290
0.000
CBr4
Br
–0.072
1.935
HBr
H
0.137
0.213
Br
–0.137
2.113
Br
0.000
1.843
Br2

C. CO2 and CS2
The electric fields present around isolated
molecules of CO2 and CS2 are shown in Fig. 5.
The solid lines in Fig. 6 (a)–(d) are the plots of
those fields along the z and x axes [defined in Fig.
6 (e)]. In the case of CO2, the field on the z axis isgoing toward the oxygen atom, and the field on thex axis is going out of the carbon atom. This formof electric field is reasonably explained by considering that the carbon atom is positively charged
and the oxygen atoms are negatively charged.
However, in the case of CS2, the field around thesulfur atom is going out of the atom on the z axis
and is going toward the atom on the side of thebond. On the x axis, the field is going out of the
carbon atom in the vicinity (within ∼3.3 Å) of theatom.
The atomic charges and quadrupoles obtained
from the fitting to the electrostatic potentials
around these molecules are shown in Table III.
The electric fields along the z and x axes
calculated with these values are also shown in Fig.
6 (a)–(d). It is seen that, in the case of CO2, areasonable fit to the electric field around themolecule is obtained with atomic charges only,
and inclusion of atomic quadrupoles in the fitting================================a

The electrostatic potentials are calculated at
the MP3/6-31+G(2df,p) level.

atomic dipoles are not so effective as comparedwith atomic quadrupoles in the correct representation of electrostatic potentials, and inclusion ofthem may result in atomic charges out of an
acceptable range, as shown above in the previous
section.
We can see some general tendencies bycomparing the results shown in Tables I and II.
(1) Similar values of atomic quadrupoles are
obtained for the same atomic species: Θ(H) is in
the range of 0.1–0.3 au, Θ(F) in the range of
0.3–0.5 au, Θ(Cl) in the range of 1.2–1.6 au, andΘ(Br) in the range of 1.8–2.2 au. Within the
halogen group (group 17) in the periodic table, thevalue of Θ(X) gets larger down the table. (2) The
signs (and qualitatively the relative magnitudes) of
the atomic charges are in complete accord with the
expectation from electronegativity scales.8–11 This
is in contrast to the cases of some of the potentialfunctions used for Cl-containing molecules,25,32–35
where a positive charge is assigned to the chlorine
atom bonded to a less electronegative atom.
Inspecting the electric field around HF, HCl,——————————————————————
FIG. 5 (next page). The electric fields around (a) CO2
and (b) CS2 molecules calculated at the MP3/
6-31+G(2df) level. The filled circles represent theatoms, which are located on the sheet. The points within
2.8 Å from any of the atoms are excluded. The electricfield on the left side of each molecule is symmetric tothe one on the right side, and is omitted in this figure.

7

(a)

(b)

H. Torii, Figure 5

Self archive of H. Torii, J. Chem. Phys. 119 (4), 2192 (2003)

(c)

6
-3

4

(a)

-3

Electric field / 10 auvalue and the atomic quadrupoles of Θ(C) = 0.537
au and Θ(S) = 1.166 au, which amounts to 2.114
au, is consistent with the molecular quadrupole
moment of Θmol = 2.127 au obtained directly from
the MO calculation at the MP3/6-31+G(2df) level.
This result indicates that the atomic quadrupoleson the sulfur atoms are the primary origin of theelectric field on the z axis of this molecule.
This electric field is considered to induce ananomalous low-frequency shift observed6 for the
C=O stretching band of a carbonyl compound dissolved in CS2 solvent. An optimized structure of
the acetone–(CS2)6 cluster has three CS2 moleculeswith one of the C=S bonds pointing toward thecarbonyl oxygen,36 just like the structure obtained
for the acetone–(CC4)4 cluster.7 The electric fieldoriginating from the CS2 molecules is operating on
the C=O bond in the direction that it induces a
low-frequency shift. It is therefore suggested thatthe atomic quadrupolar effect is important in thesolvent effect of some widely used solvents on
vibrational modes that are sensitive to intermolecular electrostatic interactions.

8

-1

2
-5
0
2
9

(b)

0

6
-2
3

(d)
-4

0
3

4

5

3

4

5

Distance from the atom center / Å

z

(e)

x

FIG. 6. The electric fields around CO2 (a and b) andCS2 (c and d) molecules, in the z direction (a and c)plotted against the distance from the oxygen or sulfur
atom, and in the x direction (b and d) plotted against the
distance from the carbon atom. The values calculatedwith the atomic charges and quadrupoles obtained fromthe fitting (Table III) are shown in broken lines (fittedwith atomic charges only) and dotted lines (fitted withatomic charges and quadrupoles). The solid lines show
the values obtained directly from the MO calculations at
the MP3/6-31+G(2df) level.

IV. SUMMARY
In the present study, we have examined theeffects of atomic quadrupoles in some halogencontaining molecules (CX4, HX, and X2 with X =F, Cl, and Br), CH4, CO2, and CS2 in their intermolecular electrostatic interactions. The results are
summarized as follows. (1) A reasonably correct
representation of intermolecular electrostatic interactions is obtained by taking into account atomic
quadrupoles. From the fitting to the electrostatic
potentials around molecules, similar values ofatomic quadrupoles are obtained for the sameatomic species [Θ(H) = 0.1–0.3 au, Θ(F) = 0.3–0.5
au, Θ(Cl) = 1.2–1.6 au, and Θ(Br) = 1.8–2.2 au].
Within the halogen group, the value of Θ(X) getslarger down the periodic table. From the comparison of the results of the fitting for CO2 and CS2,
it is seen that a larger atomic quadrupole isobtained for the sulfur atom in CS2 than for theoxygen atom in CO2. As a result, it is suggested
that the atomic quadrupolar effect is important for

improves the agreement. In the case of CS2,
however, it is clear from the large difference
between the solid and broken lines in Fig. 6 (c)and (d) that the atomic charge model is notsatisfactory. A reasonable fit is obtained only by
including atomic quadrupoles. Comparing theatomic quadrupoles of the oxygen in CO2 and thatof the sulfur in CS2, the latter is significantly larger.
These results support the suggestion made in theprevious section that the atomic quadrupolar effectis important for electrostatic interactions aroundcovalently bonded atoms on the third and higherrows.
The atomic charges of q(S) = –0.044 e for thetwo sulfur atoms in CS2 (with the C=S bond lengthof 2.930 au) contribute to the molecular quadrupole moment by Θmol = –0.755 au. The sum of this8

Self archive of H. Torii, J. Chem. Phys. 119 (4), 2192 (2003)

TABLE III. Atomic charges (q) and quadrupoles (Θ) of CO2 and CS2 molecules obtained fromthe fitting to the electrostatic potentiala=================================================================
molecule fitting procedure
residual deviationb / au atomq/e
Θ / au—————————————————————————————————————
with q and Θ
1.2 × 10-4C
0.575
–1.039CO2
O–0.287
0.177
-3
C
0.719
–
with q
2.3 × 10O
–0.360–-4
with q and Θ
7.9 × 10
C
0.088
0.537
CS2
S
–0.0441.166-2
C–0.235–
with q
1.2 × 10S
0.118–=================================================================ab

The electrostatic potentials are calculated at the MP3/6-31+G(2df,p) level.
Calculated as the square root of Σ (ρMO – ρfitted)2, where ρMO and ρfitted are the electrostatic potentials
obtained from the ab initio MO calculations and the fitting, respectively. Comparison of the values ismeaningful only within the same molecule.

electrostatic interactions around covalently bonded
atoms on the third and higher rows. (2) Atomic
dipoles are not so effective as compared withatomic quadrupoles in the correct representation ofelectrostatic interactions. If the fitting is done onlywith atomic charges, we may obtain electric fieldin the wrong (opposite) direction, as in the cases of
HCl and CS2. (3) The electron densities inside thechlorine atoms of CCl4, HCl, and Cl2 are found tobe highly anisotropic. Inspecting the molecularorbitals of Cl2, it is noticed that there is only one
occupied molecular orbital with the 3pz character[with the z axis defined in Fig. 3 (c)] while there
are two occupied molecular orbitals each for 3px
and 3py. This is considered to be the electronic
structural origin of the large atomic quadrupoles
obtained for the chlorine atoms.
The results in the present study and those in
our previous study7 suggest that the atomic
quadrupolar effect is important in the solvent
effect of some widely used solvents on vibrationalmodes that are sensitive to intermolecular electrostatic interactions. The present study has dealt withmolecules with simple structures, so that all theatomic quadrupoles are uniaxial (except the carbon
atoms in CX4 for which the atomic quadrupoles
are exactly zero for symmetry reason). In
molecules with more complex structures, atomic

quadrupoles have more number of independent
elements. The role of atomic quadrupoles of thosemolecules in intermolecular electrostatic interactions may deserve further studies.
Acknowledgement
This study was supported by a Grant-in-Aid for
Scientific Research from the Ministry of Education,
Culture, Sports, Science, and Technology.

References and Notes
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