Chemical Physics 328 (2006) 1–7
www.elsevier.com/locate/chemphys
Conformational topology of ribose: A computational study
Abraham F. Jalbout *, Ludwik Adamowicz, Lucy M. Ziurys
Department of Chemistry, The University of Arizona, Tucson, AZ 85721, USA
Received 17 January 2006; accepted 21 March 2006
Available online 27 March 2006
Abstract
This work concerns the theoretical study of the configurational topology of the ribose molecule. MP2/6-31G** geometry optimizations of the system have yielded several structures corresponding to local minima on the ground-state potential energy surface and
the lowest energy configuration was identified. The stability of a few lowest lying conformations has been recalculated at the
CCSD(T)/6-31G** level of theory.
Ó 2006 Published by Elsevier B.V.
Keywords: Ribose; MP2; Coupled cluster theory; Ab initio; Potential energy surface
1. Introduction
Increasing number of organic molecules are being identified in interstellar space. Some of those systems have
fairly complicated structures and include several different
functional groups. In the search for molecules involving
carbon, hydrogen, nitrogen, and oxygen a special attention
is directed towards finding molecular systems which are
fragments of molecules of life. Sugars, which form the
backbone of DNA and RNA, are among such systems.
Since the detection of interstellar species is mostly achieved
by the microwave spectroscopy, it is helpful if the equilibrium configuration(s) of the search for system is described
by precision quantum-mechanical calculations, the rotational constants are determined, and a computer simulation of the spectrum is performed.
In the present work, we perform calculations to characterize the configurational topology of R-ribose. As it will
be shown the structural flexibility of this system leads to several low-energy minima that are connected with shallow
pathways on the potential energy hypersurface. Previous
*
Corresponding author.
E-mail address: [email protected] (A.F. Jalbout).
0301-0104/$ - see front matter Ó 2006 Published by Elsevier B.V.
doi:10.1016/j.chemphys.2006.03.026
calculations on furanose [1] showed that the furanose ring
has two minimum energy conformations at the C3 0 -endo
and C2 0 -endo ring puckerings and that the barrier between
these minima is only 0.6 kcal/mol. As the authors stated this
result has a simple physical explanation. Their argument is
based on first considering a five-membered ring of equivalent atoms like the cyclopentane ring. Because the ring
closes on itself, the torsion angles about all five C–C single
bonds must be close to the unfavorable eclipsed position.
These unfavorable interactions make ring closure difficult,
but once the ring is formed the energy is equally unfavorable for all the conformations along a path in conformational space known as pseudorotation. In the ring of the
furanose system, as well as in ribose, one methylene group
is replaced by an oxygen. As the barrier to internal rotation
is lower about a C–O single bond than about a C–C single
bond, the ring will prefer a conformation where the C–O
torsion angles are most eclipsed. When these torsion angles
are zero, the ring atoms across the ring from them, C3 0 and
C2 0 , will be maximally puckered and these two conformations (C3 0 -endo and C2 0 -endo) become energy minima.
As the COC bond angle prefers a larger value than the
CCC bond angle, the ring will have a high energy when
the oxygen O1 0 is maximally puckered (O1 0 -endo and
O1 0 -exo). Levitt and Warshel [1] found the energy of a
2
A.F. Jalbout et al. / Chemical Physics 328 (2006) 1–7
furanose ring to be almost constant between the C3 0 -endo
and C2 0 -endo conformations. However, ring conformations
observed by X-ray crystallography do tend to cluster at
these two extremes. They suggested a possible explanation
for this in the influence of crystal packing forces acting on
the C5 0 and O3 0 atoms. They also concluded that the variation of the energy of the ring system is almost the same
for both the ribose and deoxyribose group since the extra
hydroxyl group should not affect the conformational preferences of the ring. As ribonucleic acid double helices have
only been observed in the C3 0 -endo conformation, while
deoxyribonucleic acids double helices have been observed
in both the C3 0 -endo and C2 0 -endo conformations [2], this
difference must be due to interactions of the O2 0 hydroxyl
group with the more distant parts of the sugar-phosphate
backbone. Single-stranded ribonucleic acid does adopt
both puckers [3].
In this work, we perform ab initio calculations to
describe the configurational topology of unsubstituted
dehydro R-ribose. The literature search has shown that
there has been no report on the structural investigation
of this system. The only similar study was recently performed on the abstraction reactions of ribose [4]. However,
no account up to now deals with the stability and structure
of dehydro R-ribose conformers. The variability of structural forms an isolated ribose molecule can assume results
from superposition of two different puckers of the ribose
five-member ring and several different intramolecular
hydrogen bonds that can be formed by four OH groups.
This work is important to the astrobiological study of the
precursors of life among other topics. There is some evidence that suggests that various biological sugars such as
ribose may be formed from simple interstellar precursors
[5]. Therefore, it is important to carefully analyze the conformational behavior of ribose in order to fully understand
the stable forms that may be observed. In addition, the conformational properties of ribose may be directly applied to
its ability to undergo chemical reactions and modifications.
2. Methods
All the calculations done in this work have been performed using the GAUSSIAN03 suit of programs [6]. The
geometry optimizations have been done using the MP2/631G** method with the frozen-core (FC) approximation.
That level of theory is sufficiently accurate to identify the
equilibrium points of the potential energy surface of ribose.
The relative energies of the different ribose configurational
isomers have been determined at the MP2/6-31G** level.
From our experiences [7,8] this method is sufficient for
large-scale biological systems. For several low-energy conformers the relative energies have also been calculated at
the CCSD (T)/6-31G** level of theory. For those conformers we also calculated the zero-point vibrational energies
using the MP2/6-31G** level of theory and the harmonic
approximation. The relative zero-point vibrational energies
were added to the electronic relative energies to determine
the final energy differences between the ribose conformational isomers. The zero-point energies were scaled by
0.9646 which is customary for this theory level used [7].
It is clear that hydrogen bonding plays an important
role in the stability of a conformer. Thus we have
attempted to characterize the hydrogen bonds using the
‘‘atoms in molecules’’ (AIM) methods [9]. However, this
was unsuccessful due to problems with convergence. As
an alternative to AIM we used natural bond orbital
(NBO) analysis [7] to determine the presence of hydrogen
bonds. In the Wiberg model of NBO the cutoff bond order
value for a hydrogen bond is 0.0002 and for the molecular
orbital (MO) model the cutoff bond order for a hydrogen
bond is 0.015. We have applied the MO model to analyze
the hydrogen bond structure, for each conformer and we
got results consistent with the information obtained from
the analysis HO–H distances.
3. Results
The R-ribose equilibrium structures obtained in the calculations are shown in Fig. 1. To aid in the discussion of
the structures, we will group the R-ribose isomers according to similarity in geometrical properties, which are shown
in Table 1. We have presented only the most important
parameters, in the interest of space. The supporting material for this manuscript contains MP2/6-31G** cartesian
coordinates for all species.
The data in Table 2 include the total MP2/6-31G** energies that correspond to the equilibrium structures obtained
from geometry optimizations performed at that level of
theory. For several lowest-energy structures we also show
the CCSD(T)/6-31G**//MP2/6-31G** energies, i.e., the
energies obtained using the coupled cluster method with
single and double excitations and with non-iterative inclusion of triple excitations calculated at the MP2/6-31G**
optimal geometries. The relative energies of the different
configurational isomers calculated at the MP2/6-31G**
and CCSD(T)/6-31G** levels are also shown in Table 2.
Table 2 also shows the final relative energies that include
the electronic contribution obtained at the CCSD(T)/631G** level and the zero-point vibrational contribution calculated at the MP2/6-31G* level using the harmonic
approximation approach. The last column of the table
shows dipole moments (l) in Debye (D).
For the structural analysis of the isomers in this work we
have followed all of the vibrational modes in order to
attempt to locate more structural isomers, however, we
have considered only those with positive vibrational frequencies. These 16 were the only low energy structures that
we were able to locate. This is after a thorough analysis of
all possible vibrational modes and configurations of this
molecular species.
In agreement with the previous investigation [4], we
have considered two puckered orientations of the ring
(the first orientation corresponding to C3 located above
the C1OC4 plane and C2 below the plane, and the second
A.F. Jalbout et al. / Chemical Physics 328 (2006) 1–7
3
Fig. 1. MP2/6-31G** equilibrium structures of the ribose molecule which are grouped based on similarities. Structures are organized from most to least
stable (relative to 6).
orientation corresponding to C3 below the plane and C2
above the plane). For each puckering orientation of the
ring, we also considered all possible configurations of internal hydrogen bonds formed by hydrogens attached to O1,
O2, O3, and O4. That produced the structures being the
starting point for geometry optimizations.
We will begin by identifying the lowest energy structure.
There are many isomers for R-ribose, however, Table 2
shows that the lowest energy structure is 6. Also, this structure has atom labels that will be used throughout this
work. For this isomer, two other alternatives exist, 6b
and 6ab. As we can see from Fig. 1, 6 has two intermolec-
ular hydrogen bond, whereas 6b has only one due to the
fact that two of the hydroxyl groups (OH) have their
hydrogen atoms facing each other, preventing this type of
interaction. Although, the isomers of 6 are higher in
energy, 6b is only 0.83 kcal/mol higher than 6 while 6ab
is around 2.07 kcal/mol higher in energy. This slight difference in energy has to do with the rotation of one of the OH
groups, in which the hydrogen atoms is pointing away from
the molecule (HO3), relieving it from intermolecular strain.
It is interesting to note that structure 6 in fact more stable
than the regular R-ribose molecule (0), which has an energy
that is 6.27 kcal/mol higher than it.
4
Table 1
Geometrical parameters for structures shown in Fig. 1
n
R(O2O3)b
\O1C1C2c
\O2C2C1c
\O3C3C4c
\O4C5C4c
\H(O1)O1C1C2c
\H(O2)O2C2C1c
147.7
150.2
80.9
57.1
56.6
Group 1
1
16
3
4
2
5.215
5.667
4.969
6.048
6.004
2.118
2.149
2.196
2.144
2.146
106.4
106.5
106.2
107.8
107.7
104.5
105.1
108.6
106.9
106.9
105.3
104.9
105.5
105.5
105.4
108.0
106.9
107.8
107.3
108.0
170.8
169.9
162.3
175.9
176.0
Group 2
0
13
15
14
5.016
4.926
5.037
5.122
1.958
1.957
2.047
1.950
106.7
106.7
106.3
106.8
104.9
104.9
108.5
104.6
108.2
108.1
104.9
108.2
107.8
107.2
107.3
107.2
177.0
176.9
168.7
176.5
Group 3
12
7
9
8
1.994
2.488
2.060
2.052
2.028
1.950
1.983
1.997
106.5
106.7
104.2
104.6
108.0
105.1
107.5
105.3
105.8
108.5
105.7
108.7
108.5
104.4
108.1
107.0
Group 4
6
6b
6ab
5
11
10
2.220
2.184
2.244
2.359
1.937
1.974
2.039
2.071
2.098
2.096
2.055
2.064
106.9
107.1
107.3
107.1
105.7
105.6
104.9
105.3
105.9
108.6
108.5
105.2
107.6
107.7
108.6
95.6
104.7
107.4
105.9
106.3
104.5
107.3
108.4
107.9
a
b
c
The distance (Å) between the interaction between O4 and O1 via an HO–HO bond.
The distance (Å) between the interaction between O2 and O3 via an HO–HO bond.
Bond angles are represented as \XYZ, and dihedral angles are shown as \XYWZ, both in degrees (°).
\H(O3)O3C3C4c
\H(O4)O4C5C4c
53.8
57.9
73.0
154.1
155.0
176.6
83.0
41.0
67.0
178.1
89.85
89.9
85.0
90.9
82.3
82.7
79.7
97.5
178.0
175.5
176.7
73.7
178.4
179.4
62.7
58.8
86.2
90.5
58.1
84.5
151.8
86.5
143.0
75.5
60.2
61.3
176.4
175.4
171.4
64.6
61.1
175.0
87.5
88.3
138.9
139.8
152.9
84.9
75.8
138.9
157.1
162.1
72.3
38.4
80.8
162.2
41.1
67.8
65.7
41.7
161.1
156.1
A.F. Jalbout et al. / Chemical Physics 328 (2006) 1–7
R(O4O1)a
A.F. Jalbout et al. / Chemical Physics 328 (2006) 1–7
5
Table 2
Total MP2/6-31G** and CCSD(T) energies of ribose conformers found to be local minima on the ribose potential energy surface with the use of the MP2/
6-31G** calculations
n
MP2/6-31G**
0
1
2
3
4
5
6b
7
8
9
10
11
12
13
14
15
16
571.0587076
571.0665235
571.0582391
571.0638051
571.0592873
571.0669134
571.0687070
571.0668144
571.0606408
571.0637682
571.0655256
571.0668089
571.0674584
571.0587073
571.0581445
571.0586860
571.0659204
CCSD(T)/6-31G**
571.1750194
571.1754961
571.1772456
571.1754046
571.1738721
571.1750684
571.1760640
571.1745569
DMP2/6-31G**
6.27
1.37
6.57
3.08
5.91
1.13
0.0
1.19
5.06
3.10
2.00
1.19
0.78
6.27
6.63
6.29
1.75
DMP2/6-31G**a
DCCSD(T)/6-31G**
DCCSD(T)/6-31G**a
ZPEc
1.06
1.40
1.08
0.16099
0.80
0.0
1.25
1.10
0.0
1.15
0.78
0.0
1.22
0.16098
0.16149
0.16159
1.78
0.84
0.82
2.12
1.37
0.74
1.90
1.02
0.77
0.16114
0.16093
0.16155
1.47
1.69
1.41
0.16105
Md
3.12
4.02
2.64
4.38
1.92
4.74
1.35
3.43
6.86
4.04
4.84
1.11
2.14
3.13
3.67
3.58
2.57
Total energies in hartrees and the relative energies in kcal/mol.
a
Corrected for zero-point energies.
b
Two other isomers exist shown in Fig. 1 as 6b, and 6ab which are higher in energy by 0.83 and 2.07 kcal/mol, respectively.
c
ZPE scaled by 0.9646.
d
Dipole moment of the molecules in Debye (D).
Therefore, from the results of the calculations in Table 2
we can see that there exist 7 low-lying (lowest energy) structures that are energetically similar to 6, these molecules are
12, 5, 7 = 11, 1, 16, 10. For these molecules MP2/6-31G**
frequencies as well as CCSD(T)/6-31G**//MP2/6-31G**
calculations have been carried out to verify their position
on the potential energy surface. The structural reasons
for this difference in energy will be described below based
on groups. It is important to classify the conformers
according to geometrical traits, in order to form general
conclusions about the structures. The following figure will
be used for the numbering scheme discussed in this work.
3.1. Group 1
The first set consists of molecules 1, 2, 3, 4, and 16,
which share close equilibrium geometries. From previous
nomenclature [4] these molecules belong to a twist configuration. We can further classify these molecules according to
the position of their OH groups, as either endo (up) or exo
(down). For structures 1, 3 and 2 we can say that O1, O2
(see structure 6 for numbering scheme) are endo, whereas
O4, O3 are exo. However, structures 16, 4 have their O1,
O3 as endo, and O2, O4 as exo. Therefore, the primary difference between 1, 3, 2 and 16, 4 is that the O2 has rotated
downward and the O3 has rotated upward. Table 1 shows
the geometrical parameters for the molecules shown in
Fig. 1. The hydrogen bond interactions between O4 and
O1 are labeled as R(O4O1) and for the interaction between
O2 and O3 it is labeled as R(O2O3). As we can see for all
the structures in this group, there is no hydrogen bond
interaction between O4 and O1 as can be seen by the large
R(O4O1) distances in the table. However, there is substantial bonding between the O2 and O3 groups as can be seen
for the R(O2O3) values that are about 2.1 Å which is typical for a hydrogen bond.
These structures are higher in energy than the majority
of the other isomers in this study as Table 2 shows. The relative energies of 1, 2, 3, 4, and 16 are 1.37, 6.27, 3.08, 5.91,
and 1.75 kcal/mol, respectively. Due to the low-lying nature of structures 1 and 16 we have performed further
CCSD(T)/6-31G**//MP2/6-31G** calculations to obtain
relative energy values of 1.08 and 1.41 kcal/mol, respectively. With ZPE corrections we obtain MP2/6-311G**//
6
A.F. Jalbout et al. / Chemical Physics 328 (2006) 1–7
MP2/6-311G** values of 1.04 and 1.45 kcal/mol, as well as
CCSD(T)/6-31G**//MP2/6-31G** of 1.06, and 1.47 kcal/
mol, for 1 and 16, respectively. As we can see 1 has the lowest energy in this group. The table also shows dipole
moments in Debye (D). As we can see from the table, 2,
3, 4, and 16, have dipole moments of 2.64, 4.38, 1.92 and
2.57 D, respectively. The highest dipole moment being for
3 due to the fact that the dipoles of the OH groups are parallel to the ring.
Overall, the stability of these molecules is greatly influenced by the orientation of the H in the OH groups. They
determine whether or not hydrogen bonding will form, as
well as the general interatomic interactions in the molecular
framework.
3.2. Group 2
The second group of molecules consists of 0, 13, 15, and
14, belong to an enveloped configuration. For 0, 13, 14 we
observe that the general trend is where the O1, O2, O4
groups are endo, and the O3 is exo. The only deviation is
for 15, where O1, O3, O4 are endo, and O2 is exo.
Structures 0, 13, 14, and 15 have relative energies of
6.27, 6.27, and 6.63, and 6.29 kcal/mol, respectively. As
we can see 0, 13, both have the lowest energy, which is also
verified by a single point CCSD(T)/6-31G**//MP2/6-31G**
calculation that yields a relative energy of 1.40 kcal/mol,
which is very similar to the MP2/6-31G** result.
The high stability of this group relative to the other three
groups can be attributed to the fact the outer ACH2(OH)
adopts an ‘L’ structure that should lower the energy [10].
In addition, the presence of one hydrogen bonding interaction, which is shown in Table 1 as R(O4O1), also stabilizes
the structure. This interaction occurs at about 2 Å which is
typical for a hydrogen bond. Structures 0 and 13 have are
almost isoenergetic, with their structural differences being
very small. These two structures along with 14 share a similar geometry, however 14 is partially destabilized by the
rotation of C5 that causes the OH group to move slightly
closer to the ring, causing minor steric interactions. Molecule 15 has a different OH configuration than then others,
but is stabilized over 14 since the dihedral angle is around
177°, which causes less interactions with the ring, due its larger distance from the other atoms as a result from this rotation. In addition, we have also calculated dipole moments in
Table 1 for 0, 13, 15, 14 are 3.12, 3.13, 3.58, and 3.67 D,
respectively. As we can see the dipole moment for 14 is
the highest which can be attributed to the orientation of
the O1, and O4 groups.
3.3. Group 3
The third group consists of 7, 8, 9, and 12, which belong
to a twist configuration. These molecules can be classified
as have their O1, O2, O4 atoms endo, and O3 as exo. Structures 7, 8, 9, and 12, have energies of 1.19, 5.06, 3.10, and
0.78 kcal/mol, respectively. The lowest energy structures in
this group are 7, and 12, for which single point CCSD(T)/
6-31G**//MP2/6-31G** calculations reveal a relative
energy of 1.15 and 0.74 kcal/mol, respectively. With ZPE
corrections we obtain CCSD(T)/6-31G**//MP2/6-31G**
values of 1.22 and 0.77 kcal/mol for 7, 12, respectively.
These structures are stabilized due to the fact that two
hydrogen bonding interactions can be observed, as can be
seen for the R(O4O1) and R(O2O3) distances of about
2 Å which is typical of hydrogen bonding. Structures 12
and 7 are very similar in structure except that the direction
of their hydrogen atoms in their OH groups differ (HO2,
HO3), leading to a slight destabilization of 7. Molecule 9
has a similar structure as 7, however, the rotation of H
(O4) and H (O1) causes it to have a higher energy since
these is more interaction of H (O4) with the hydrogen
atoms of C5. Structure 8, has a higher energy due to the
fact that all of its groups run parallel leading to a destabilization of energy also due to unfavorable interactions as
opposed to the most stable structure 12, of this group.
From Table 1 we can also see dipole moments for 12, 7,
9, 8 of 2.14 D, 3.43 D, 4.04 D, 6.84 D, the highest being
for 8 due again to the parallel orientation of the OH
groups.
3.4. Group 4
In the final group, 5, 6, 10, and 11 are in a twist configuration with the general trend of O1, O3, and O4 as endo,
and O2 as exo. Structures 5, 6, 10, and 11 have relative
energies of 1.13, 0.0, 2.00, 1.19, and 1.75 kcal/mol, respectively. Since all of these structures are low lying on the
potential energy surface, single point CCSD(T)/6-31G**//
MP2/6-31G** calculations have been carried out that yield
relative energies of 1.10, 2.12, 1.37 kcal/mol for molecules
5, 10, and 11, respectively. With ZPE corrections we get
CCSD(T)/6-31G**//MP2/6-31G** for 5, 10, 11 of 0.78,
1.90, and 1.02 kcal/mol, respectively. As already mentioned
structure 6 is the most stable, and is used as a reference
point for the energies of the other molecules in this study.
As we can see the CCSD(T)//MP2 are very similar to the
results of the MP2//MP2 calculations with an average difference of 0.11 kcal/mol for the above mentioned
molecules.
The stability of 6 over 5 has to do with the orientation of
the second H (O2) group. It is rotated away from the molecule leading to an energy that is 1.13 kcal/mol higher.
Structure 10, 11 show a slight destabilization as a result
of the rotation of the HO1, HO4 in an opposite direction
as 6, 5. Furthermore, rotation of H (O3) in 10 causes a
slight destabilization relative to 11. Table 1 shows that
the \H (O2) O2C2C1 of 11 is 75.8°, whereas the angle in
10 is 138.9° as a result of this effect. Table 1 also shows
that 6, 5, 11, 10 have dipole moments of 1.35, 4.75, 1.11,
and 4.84 D, respectively. The highest dipole moment is seen
for 10 which may be due to the orientation of the hydroxyl
groups as can be seen for the variations in the dihedral
angles from the table.
A.F. Jalbout et al. / Chemical Physics 328 (2006) 1–7
slight energy changes. We saw that for groups 1, 2 only one
hydrogen bond can form between O2 and O3 due to rotation of the C5 atom that prevents an interaction between
O1 and O4. In groups 3 and 4, however, this same rotation
now occurs in a favorable manner permitting a second
hydrogen bonding interaction that stabilizes the molecule.
This work has not only summarized our location of 16 conformers of the dehydro-R ribose molecule, but also
described energetic and structural differences in these structures. The located minimum molecules can be used as a reference for future experimental studies related to this
system.
6.5
6.0
5.5
-1
Relative Energy (kcal mol )
7
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
Acknowledgement
0.0
r6 r12 r5 r11 r7
r1 r16 r10 r9
r3
r8
r4 r13 r0 r15 r2 r14
Ribose Isomer
Fig. 2. Relative energies in kcal mol of the structures shown in Fig. 1.
We would like to thank The University of Arizona for
valuable resources.
Appendix A. Supporting information
4. Conclusions
In this work, we have explored several isomers of the
dehydro R-ribose molecule. We have identified 16 stable
molecules on the potential energy surface. Out of these
16 structures 6, 12, 5, 7 = 11, 1, 16, 10 (where 7 and 11
are isoenergetic) were found to be low lying. For these
structures not only were their frequencies calculated at
the MP2/6-31G** level, but their energies were also verified
at the CCSD(T)/6-31G**//MP2/6-31G** level of theory.
The relative energies of the structures in Fig. 1 are represented graphically in Fig. 2.
If we refer back to Fig. 2, we can see that most stable
structures are 6, 12, 5, in that order. These structures have
two hydrogen bonds, however, in their formation the
hydrogen atoms rotates away from the molecular structure.
Such an effect can be seen by observing that H (O1), H (O3)
are point away from the ring, whereas in 12 only H (O1) is
in this similar orientation, and in 5 only HO3 is pointing
away.
In general, we have discussed that the primary reason
associated with the differences in energy had to with the
fact that each of them had structural deviations that caused
Supplementary data associated with this article can be
found, in the online version, at doi:10.1016/
j.chemphys.2006.03.026.
References
[1] M. Levitt, A. Warshel, J. Am. Chem. Soc. 100 (1978) 2607.
[2] A. Arnott, S.D. Dover, A.J. Wonacott, Acta Crystallogr., Sect. B 25
(1969) 2192.
[3] C. Altona, M. Sundaralingam, J. Am. Chem. Soc. 94 (1972)
8205.
[4] N. Luo, A. Litvin, R. Osman, J. Phys. Chem. A. 103 (1999) 592.
[5] S.A. Benner, A. Ricardo, M.A. Carrigan, Curr. Opin. Chem. Biol. 8
(2004) 672.
[6] M.J. Frisch et al., GAUSSIAN03, Gaussian Inc., Pittsburgh, PA,
2003.
[7] J.B. Foresman, Æ Frisch, Exploring Chemistry with Electronic
Structure Methods, second ed., Gaussian, INC, Pittsburgh, PA, 1996;
A.F. Jalbout, F. Nazari, L. Turker, J. Mol. Struct. (Theochem) 1
(2004) 627.
[8] A.F. Jalbout, L. Adamowicz, Adv. Quant. Chem., in: J. Sabin (Ed.),
vol. 51 (in press).
[9] P.L.A. Popelier, J. Phys. Chem. A 102 (1998) 1873.
[10] S. Carles, C. Defrançois, J.P. Schermann, A.F. Jalbout, L. Adamowicz, Chem. Phys. Lett. 334 (2001) 374.