QUANTUM CHEMICAL PREDICTIONS ON THE LINEAR CONFORMER S [610247]

1
QUANTUM CHEMICAL PREDICTIONS ON THE LINEAR CONFORMER S
OF THE HYPOTHETICAL (CO 2)2H2 AND (CO 2)3H2

Mihai MEDELEANU ,a,* Mihai -Cosmin PASCARIU ,b Maria MRACECc and Mircea MRACECc

a Department of Organic Chemistry, Faculty of Industrial Chemistry and Environmen tal
Engineering, University Politehnica Timișoara, 6 Carol Telbisz , RO -300006 Timișoara, Roumania
b Department of Pharmaceutical Sciences, Faculty of Pharmacy, “Vasile Goldiș” Western
University of Arad, 86 Liviu Rebreanu, RO -310414 Arad, Romania
c Department of Computational Chemistry, Institute o f Chemistry Timișoara of Roumanian
Academy, 24 Mihai Viteazul Av., RO -300223 Timișoara, Roumania
Received …………………
A search for distinct en ergy and geometric conformers, within the limits of the PM3 approximation ,
was performed for the hypothetical molecules (CO 2)2H2 dimer and, respectively, (CO 2)3H2 trimer,
by using the Conformational Search module in the HyperChem 7.5 (Hy) software . The located
conformers were reoptimized with the PM3, PM6 and PM7 semi -empirical methods using version
16.299L of MOPAC 2016 (M16) software . For the dimer we obtained 11 PM 3-M16 conformers, 11
PM6 -M16 conformers and 11 PM7 -M16 conformers, while for the trimer we obtained 57 PM3 –
M16 conformers, 33 PM6 -M16 conformers and 38 PM7 -M16 conformers. After overlapping the
geometries generated for the resulting conformations , a symmet rical distribution over a central
plane was obtained . In the case of the trimer, an unstable cyclical conformer appear ed due to the
flexibility of the bonds . It was found that there is no regularity between the standard enthalpy of
formation (Δ fH0) of conf ormers and the HOMO, the LUMO, the dipole moment (μ), the first
vibration (υ 0) or the zero point vibration energy (E ZVE). For all conformers and for each method , the
dependence between the formation enthalpy and temperature on the T[100 K, 600 K] range wa s
calculated with a 10 K step. With the calculated data , the interpolation functions of a three -order
polynomial type (y = a 0 + a 1T + a 2T2 + a 3T3; y = Δ fH) were established. Since for the dimer the free
term dispersion is below 5 % and the other parameters do not vary significantly with the calculation
method, an average polynomial is proposed for calculating the temperature dependence of the
enthalpy of formation. For the trimer, this dispersion is dependent on the calculation method and
varies between 10 and 16 %. It is demonstrated that there is a linear dependence between the free
term a 0 and the standard formation enthalpy (Δ fH0).

*corresponding author. E -mail: [anonimizat] , [anonimizat]

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INTRODUCTION

Carbon dioxide (CO 2) is one of the most used compounds in the extraction technology of
temperature sensitive organic compounds under supercritical conditions.1-5 It is also o ne of the main
causes of global warming.3,6-11 For these reasons it continues to be one of the main simple
molecules being studied, both experimentally and theoretically through various methods of
quantum chemistry.3-14 The main experimental method of stud ying CO 2 in gaseous or solid state is
high resolution infrared (IR) spectroscopy.3-14 Starting from experimentally proven facts that CO 2
molecules can form clusters with different symmetrical structures through van der Waals forces ,3-14
we prove in this pa per by semi -empirical methods of quantum chemistry and thermodynamics that
they can also form polymer structures through linear chains O = C – O – C – … – O – C = O. To
this end, w e study here the conformation of dimeric and trimeric molecules by semi -empirical
methods. We use the PM3 Hamiltonian from the HyperChem7.515 software , which allows the
performing of a conformational analysis and establishment of a number of conformers, while the
PM3, PM6 and PM7 Hamiltonians from MOPAC1616 allow us to calculate the thermodynamic
properties of those conformers. To demonstra te conformers’ stability we have first consider ed the
formation enthalpy under standard conditions (Δ fH0) and its dependence on temperature in the
range T [100 K, 600 K], for which we have establish the third order interpolation functions (y = a 0
+ a1T + a 2T2 + a 3T3; y = Δ fH). We have also analyze d some structural properties for the found
conformers .

METHOD

General methods

The PM3 method from the HyperChem 7.52 (Hy) software15 was used, along the PM3, PM6
and PM7 methods implemented in MOPAC16 , version 1 6.299L (M16 ) software.16 Because the
PM6 and PM7 methods have not been implemented in programs that search for conformers, we
have used the Conformational Search module from the Hy software pack and the PM3 method for
the optimiz ation of the conformers' ge ometries.15 The Polak -Ribière (conjugate gradient) method
was used to optimize the geometry in Hy, with a RMS gradient of 0.001 kcal Ǻ-1 mol-1.15 For the
computation of Δ fH(T) properties with the M16 program it was necessary to work in two stages. In
the first stage , the geometries of the conformers were optimized with one of the PM3, PM6 or PM7

3
methods, while in the second stage the thermodynamic properties were calculated.16 In the M16
software, the Eigenvector method was used for geometry optimization , with a RMS gradient of
0.001 k cal Ǻ -1 mol-1.16 For the conversion of “.hin” files from Hy software into “.mop” files, the
AVOGADRO17 program was used, applying the “MOPAC ” and “Geometry Optimization ”
keywords . For the geometry optimization with the M16 program, in the first stage the foll owing
sets of keywords were used: PM3 (PM6 or PM7), SCFCRT=1.D -10, GEO -OK, PRECISE,
GNORM=0.001, CYCLES=5000 , T=345600, LET. The obtained geometries were converted into
M16 input files for computing the thermodynamic properties. For the calculation of Δ fH(T)
thermodynamic property in the second stage , the following keywords were used : PM3 (PM6 or
PM7), SCFCRT=1.D -10, GEO -OK, PRECISE, GNORM=0.001, CYCLES= 5000, T=345600,
LARGE, CHARGE=0, SINGLE T, SYMMETRY, FORCE, THERMO(100,600,10), LET. For
conversion of the M16 geometry in to Hy geometry , the Open Babel softwar e was used .18

Molecul ar model s and conformational search

In order to perform the calculations and the conformational analysis, two starting models of
molecular geometries were constructed:
For the dimer (Fig. 1a), in order to compensate for the bonds from C1 and O6 atoms, one
hydrogen (H7 and H8) was added to each atom, a formic and , respectively, carbonic acid
((CO 2)2H2) structure being thus formed.

a. b.
Fig. 1. Atom numbering in dimer (a.) and trimer (b.)

For the trimer (Fig . 1b), a hydrogen (H10 and H11) was added to compensate for the bonds
at the C1 and O9 atoms, a formic and, respectively , a carbonic acid structure ( (CO2)3H2) in the
ending part being thus formed.
In the search for conformers with the Conformational Search module in Hy, in the case of
the dimer molecule, rotation s were performed around the C1 -O3 and O3 -C4 bonds, while in the
case of the trimer molecule, rotation s were performed around the C1 -O3, O3 -C4 , C4 -C6 and C6 –

4
C7 bonds . To p erform these rotations, the dihedral angles O2-C1-C3-C4 and C1 -C3-C4-O5 were
selected for the dimer , while in the case of the trimer the dihedral s O2-C1-C3-C4, C1-O3-C4-O6,
O3-C4-C6-C7 and C4 -C6-C7-O8 were chosen . These dihedral angles were randomly varied on the
0 ± 180 ° range , with a ±15 ° step. The starting geometry was taken at random, and the maximum
range of energy in which the conformers were found was 50 kcal/mol. Duplicate conformers were
considered to be those for which the optimal conformations differ in energy with values less than
0.05 kcal/mol , while the dihedral angles were less than 5 °. The g enerat ion of random numbers was
done according to the computer clo ck. As we have shown in previous papers ,19-23 the ratio between
the number of s tarting geometries and the number of optimal conformers that are found should be
greater than 20:1, a ratio that guarantee s that at least 95% of the conformers were found. By using
the PM3 Hamiltonian and by applying these constraints, 12 energ y and geomet rically distinct
conformers were found in the case of the dimer , while 57 conformers were found in the case of the
trimer. The results of the calculations for these conformers , along with the data for formic acid,
carbon dioxide and water (compared to expe rimental data taken from Wikipedia, the free
encyclopedia) are presented in the Supplementary Material. The geometries of these 12 and,
respectively, 57 conformers obtained with PM3 from Hy were re -optimized with PM3 -, PM6 – and,
respectively, PM7 -M16. T he distinct geometries of the resulting conformers were ordered in
descen ding order of standard formation enthalpies . Through the reoptimization of the PM3 -M16
geometries of the 12 PM3 -Hy conformers and the calculation of Δ fH(T) , a large deviation between
the free term (a 0) of the interpolation function and Δ fH0 was revealed, thus this conformer was
discarded and only 11 conformers remained (Supplementary Material). After the PM3 -M16
reoptimization of the 57 PM3 -Hy conformers and the calculation of Δ fH(T), it was found that two
conformers (PM3 -15 and PM3 -22) had negative  vibration, but the differences between the Δ fH0
values obtained at the geometries ’ optimization and the ones obtained during calculati on of the
thermodynamic properties are not that large (less than 2 ‰) , so these conformations have been
taken into account when calculating the ΔfH(T) interpolation functions, alth ough they are transition
states.
After reoptimization of geometries and computation of the PM6 -M16 therm odynamic
properties f or the 12 conformers, of which 11 remaining PM3 -M16, it was found that a conformer
(PM6 -08) has four negative vibrations (Δ fH0 is very different from the value obtained during
geometry optimization) , so it was thus discarded, being con sidered a transition state. By
reoptimizing the geometries and by calculating the thermodynamic properties with the PM6 -M16 of
the 57 conformers, it was found that several conformers has equal energies ( Supplementary
Material) . As a result of discarding the geometrically identical conformers , there were only 33

5
geometrically and e nergy distinct conformers, of which one (PM6 -20) has negative vibration, but
because between the Δ fH0 values obtained after the geometries ’ optimization and those obtained in
the calculation of the thermodynamic properties no large differences exist (being less than 2 ‰) ,
this conformer was also taken into accou nt in the calculation of the ΔfH(T) interpolation functions
(Supplementary M aterial).
After reoptimiz ation of geometries and c omputation of the PM6-M16 thermodynamic
properties of the 11 PM3 conformers , it was found that a conformer ( PM6 -05) has negative
vibrations (Δ fH0 is very different from the geometry optimization value) , so it was discarded, being
considered a transition state. By reoptimizing the geometries and calculating the thermodynamic
properties with PM7 -M16 f or the 57 PM3 conformers, several conformers with identical geometries
and energies were produced (Supplementary Material). As a result of the elimination of conformers
with identical geometries, only 38 geometrically and energ y distinct conformers remained , of which
one (PM7 -31) has negativ e vibration, but because the differences between the Δ fH0 values
obtained after optimizing the geometries and those obtained in the calculation of the
thermodynamic properties are not large ( being less than 2 ‰) , this conformer was taken into
account in t he calculation of ΔfH(T) interpolation functions (Supplementary M aterial).

RESULTS AND DISCUSSION

Analysis of geometric and electronic properties

Following the selection and removal of suspect conformers , 11 PM3 -M16 conformers, 10
PM6 -M16 and 11 PM7 -M16 conformers remained for the dimer, while 57 PM3 -M16, 33 PM6 -M16
and 38 PM7 -M16 conformers remained for the trimer, and these were used for the geometry
analysis and for computation of interpolation functions Δ fH(T).
To overlay the geometries, the M16 “.out” files have been converted to “.hin ” files usin g the
Open Babel 2.4.1 program .18 By overlapping the distinct geometr ies of the PM3 -M16 conformers ,
taking as overlapping centers the C1O3C4 atoms for the dimer and , respectively, the O3C4O6
atoms for the trimer , almost symmetrical images with respect to the plane of the overlapping centers
are obtained (Figs. 2a, 2b).

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a. b. c. d. e.
Fig. 2. Dimer and trimer PM3 -M16 c onformers ’ superposition

These symmetrical arrangements are also explanations for obtaining pairs of conformers
with almost identical proper ties, but positioned spatially distinct, symmetric al with respect to a
central plane. There are, within each calculation method, a case where there are three conformers
with identical properties. In this situation of energy degeneration, two conformers are symmetrical
to the overlap plane (Fig . 2c), and for the third degenerate d energy conformer , the terminal
functions of the dimer or trimer are oriented in different directions (Fig. 2d). In this way the
degeneration term g ij must be interpreted when calcul ating the partition function , Zij = g ijexp(-
Eij/kT).24-26 Similar overlapping s (Figs. 2a and 2b) are also obtain ed for the 33 PM6 -M16 and 38
PM7 -M16 conformers, but these are more tightly arranged . To highlight the difference between the
degenerate d energy conformer pairs, the angles between the planes containing the O3 -C4 bond for
the dimer and , respectively, the C4-O5 bond for the trimer were measured. In the case of the dimer,
this angle between plane s has a maximum value of 12.63 ° for the PM3 -03/PM3 -04 conformer pair,
of 3.04 ° for the PM6 -07/PM6 -08 conform er pair and of 6.07 ° for the PM7 -08/PM7 -09 conformer
pair. For the other pairs of conformers, the angles between these planes have lower values. In the
case of the trimer, this angle between plane s has the maximum value of 22.91 ° for the PM3 –
45/PM3 -46 conformer pair, of 12.45 ° for the PM6 -17/PM6 -18 conformer pair and of 13.24 ° for
the PM7 -08/PM7 -09 conformer pair. For the other pairs of conformers, the angles between these
planes have lower values.
Because the C4 -O6 bonds, in case of the dimer, and, respectively, the C7 -O9 bonds in the
case of the trimer, are able to freely rotate, the hydrogen atoms bonded to O6 and, respectively, O9,
verify the distance conditions and can form internal hydrogen bonds with oxygen atoms. In the case
of the dimer, these internal hydrogen bonds take place between H8· ··O2, in PM3 -03 and PM3 -05,
PM6 -03 and PM6 -04, and, respectively, PM7 -03 and PM7 -11 conformers. In the case of the trimer,
these internal hydrogen bonds take place between H11···O5, in PM3 -01, PM3 -09, PM3 -10, PM3 –
11, PM6 -04, PM6 -06, PM6 -07, PM6 -12, PM7 -05, PM7 -06 and PM7 -14 conformers, between
H11···O3 in PM3 -26, PM3 -50, PM3 -51, PM6 -05, PM6 -10, PM6 -11, PM7 -10, PM7 -11 and PM7 –

7
15 conformers and, respectively, between H11···O2 in the PM3 -48, PM3 -49 and PM3 -52
conformers. The numbering system for the conformers is given in the Supplementary Material . The
PM6 and PM7 methods do not predict c onformers having hydrogen bonds between H11···O2.
Since there are more flexible bonds in the trimer, after rotation it becomes possible that the
O2 and O8 oxygen atoms , which possess double bonds (C1=O2 and C7=O8) , to get close to a
distance of 2 Ǻ. In this case, by optimizing the geometry, a bond is broken and a four -atom cycle is
formed, generating a new conformer (Fig . 2e). Through all methods (PM3, PM6 and PM7), these
conform ers have the lowest stability ( Supplementary Material) and have standard formation
enthalp ies slightly different fro m the other conformers.
By comparing the HOMO level , calculated with the PM6 method , with the first ionization
potential for formic acid, it was noticed that the result was very close to the one obtained through
experiment , the other method s also giving comparable results. Also , for water , the calculated data
are comparable to the experimental ones, the best estimation being given by the PM3 method. For
CO 2, the calculated data underestimates the experimental value. The electron affinity is much
underestimated by LUMO for all the test molecules , in the case of all three methods being used.
The dipole moment for formic acid is over estimated by all three methods, however, for water, the
PM3 method underestimates the value relatively slight , while PM6 and PM7 overestimate this
value . The order of the  vibration is correctly estimated , although the values are quite different.
For these comments on exper imental data see Table 1 in the Supplementary M aterial.
Because the considered mole cules (dimer and trimer) are only hypothetical molecules, no
experimental data are available. By following the computed data for the dimer and the trimer ’s
conformers, no regularity is observed in the case of EHOMO and ELUMO , and also for μTOT, ν0, EZVP.
For this reason only the mean values were considered (Supplementary M aterial , Table s 2, 9 and 15
for the dimer , and, respectively Table s 6, 12b and 19 for the trimer ). For the dimer , the mean
calculated values for the HOMO levels using PM3 -M16 are EHOMO = -11.7588 ±0.0683 eV, using
PM6 they are EHOMO = -11.9883 ±0.1335 eV, while using PM7 they are EHOMO = –
11.7659 ±0.1155 eV, while the LUMO levels calculate d with PM3 -M16 are ELUMO =
0.1682 ±0.2114 eV, with PM6 they are ELUMO = -0.7094 ±0.1646 eV, and, respectiv ely, with PM7
they are ELUMO = -0.1665 ±0.1155 eV. For the trimer , the calculated mean values for the HOMO
levels with PM3 -M16 are EHOMO = -11.9555 ±0.0971 eV, with PM6 they are EHOMO = –
12.2695 ±0.0971 eV, respectiv ely with PM7 they are EHOMO = -11.9742 ±0.0883 eV, while the LUMO
levels calculate d with PM3 -M16 are ELUMO = 0.1859 ±0.1718 eV, with PM6 they are ELUMO = –
1.2821 ±0.1305 eV), and, respectiv ely, with PM7 they are ELUMO = -0.8566 ±0.1460 eV. Similar ly,
the mean values for the dipole moment were computed for the dim er: μTOT(PM3 -M16) =

8
3.4025 ±1.4308 D; μ TOT(PM6) = 2.7270 ±1.5981 D and μTOT(PM7) = 2.8931 ±1.5543 D, while for the
trimer: μTOT(PM3 -M16) = 3.4981 ±1.3582 D; μ TOT(PM6) = 3.3477 ±1.2092 D and μTOT(PM7) =
3.1137 ±1.1772 D, and also for the first vibration (in calcul ating the mean values, the negative values
were not considered ) for the dimer: υ 0(PM3 -M16) = 59.1172 ±25.3950 cm-1; υ 0(PM6) =
64.8498 ±36.1174 cm-1 and υ0(PM7) = 60.6323 ±20.9727 cm-1, while for the trimer: υ 0(PM3 -M16) =
33.1727 ±16.7312 cm-1; υ 0(PM6) = 32.6212 ±18.3862 cm-1 and υ0(PM7) = 32.6770 ±15.8285 cm-1.
Also, the mean values for the zero vibration point energy (EZVP) were calculated for the dimer:
EZVP(PM3 -M16) = 29.5550 ±0.4257k cal·mol-1; E ZVP(PM6) = 26.3892 ±0.0637k cal·mol-1 and
EZVP(PM7) = 27.1256 ±0.0996k cal·mol-1, wh ile for the trimer: E ZVP(PM3 -M16) =
38.4396 ±0.3466k cal·mol-1; E ZVP(PM6) = 34.5353 ±0.2701k cal·mol-1 and EZVP(PM7) =
35.5838 ±0.2702k cal·mol-1 (Supplementary M aterial , Table s 2, 9 and 15 for the dimer , and Table s 6,
12b and 19 for the trimer ). By analyzing these mean values, some observations can be made :
– For both dimer and trimer , there ar e no significant differences in the case of the
HOMO levels. However, we can observe that the PM3 values are quite similar to the PM7
values, while the PM6 values are slightly higher. The data dispersion is relatively narrow,
which means that these level s depend in a relatively small degree on the geometry of the
conformers. For the LUMO level, the PM3 method leads to positive values, while the PM6
and PM7 methods lead to similar negative values. The data dispersion is relatively large,
which means that t hese levels depend on the conformers ’ geometry.
– In the case of the dipole moment , for the dimer, the PM3 method provides a
different value when compared to the PM6 and PM7 methods. This shows that the PM3
method provides a charge distribution which is different from the one obtained by using the
other two methods. For the trimer , the dipole moment values are relatively similar, which
shows that the charge distributions are similar.
– In the case of the first vibration, the average values predicted by the three methods
(PM3, PM6 and PM7) are highly dispersed for both the dimer and the trimer, which proves
the dependence of vibrations on the geometry of the conformers.
– For average values of the zero point vibration energies (E ZVP), which correspond t o
equilibrium R 0 values, the dispersion values are very small, which shows that the
equilibrium states are relatively constant and do not significantly depend on the conformers ’
geometries .

9
Analysis of formation enthalpies

For each energy and geometric ally distinct conformer, the temperature depe ndence of the
enthalpy of formation (Δ fH) on the T [100 K, 600 K] interval , with a 10 K step , was calculated
with the three methods (PM3 -M16, PM6 -M16 and PM7 -M16). The primary data from the
calculation are shown in the Supplementary Material (Table s 4, 10 a nd 16 for the dimer , and Table s
7, 13 and 19 for the t rimer ). With these primary data, we established the third -order interpolation
functions (y = a 0 + a 1T + a 2T2 + a 3T3; y = ΔfH) for each conformer and for each method (PM3 –
M16, PM6 -M16 and PM7 -M16) . The in terpolation functions for the dimer ’s conformers are given
in Table s 5, 11 and 17 from the Supplementary Material , while for the trimer ’s conformers are
given in Table s 8, 14 and 20 from the Supplementary Material. The graphical representations of
these interpolation functions for the dimer are shown in Fig s. 3a, 3b and 3c, while for the trimer in
Figs. 5a, 5b and 5c. For the dimer, since the a0 free term range is located between 3 and 5 %, the a1,
a2 and a 3 parameters of the polynomial are relatively ide ntical, the standard deviation is of the order
of 10-3, and the F statistical test is of the order of 107, a set of mean interpolation functions (1 -3)
was build using the mean values :

 for PM3
ΔfH(T) = -177.7094 (2.2168) + 0.0075 (0.0010) T + 2.8051 (0.1396 )·10-5T2 – 1.0406 (0.0843)·10-8T3 (1)
T[100K,600K] SD = 8.8323 (4.2720)·10-4 F = 4.3086 (2.1669)·108
Δa0 = 8.4748 kcal·mol-1 (4.77%)
 for PM6
ΔfH(T) = -166.0397 (1.5083) + 0.0066 (0.0006) T + 2.9602 (0.0896)·10-5T2 – 1.0557 (0.0526)·10-8T3 (2)
T[100K,600K] SD = 12.5669 (3.2534)·10-4 F = 1.9659 (1.0962)·108
Δa0 = 4.8415kcal·mol-1 (2.92%)
 for PM7
ΔfH(T) = -165.8631 (1.9254) + 0.0070 (0.0007) T + 2.8259 (0.0853)·10-5T2 – 0.9745 (0.0423)·10-8T3 (3)
T[100K,600K] SD = 0.0010 (0.0003 ) F = 16.771 6(6.3002 )·107
Δa0 = 6.3663 kcal·mol-1 (3.84%)

Within the limits of reasonable statistical errors, these mean interpolation functions can be
used to compute the formation enthalpies on the T [100 K, 600 K] domain for any of the
conformers. The interpolation functions in Tabl es 5, 11 and 17 (Supplementary M aterial) must be
used for the computation of the correct value s, according to the working method which is being
used.

10

100 200 300 400 500 600-180-178-176-174-172-170-168-166-164-162-160-158fH(kcal mol-1)
T(K) PM3-01
PM3-02
PM3-03
PM3-04
PM3-05
PM3-06
PM3-07
PM3-08
PM3-09
PM3-10
PM3-11
PM3-12
100 200 300 400 500 600-168-166-164-162-160-158-156-154-152-150fH(kcal mol-1)
T(K) PM6-01
PM6-02
PM6-03
PM6-04
PM6-05
PM6-06
PM6-07
PM6-09
PM6-10
PM6-11
100 200 300 400 500 600-168-166-164-162-160-158-156-154-152-150fH(kcal mol-1)
T(K) PM7-01
PM7-02
PM7-03
PM7-04
PM7-05
PM7-06
PM7-07
PM7-08
PM7-09
PM7-10
PM7-11
a. b. c,
Fig. 3. Plot of f unctional dependencies of ΔfH (kcal·mol-1) in the temperature range T[100K, 600K] with a step 10K resulted f rom the PM3 -M16 (a.),
PM6 -M16 (b.) and PM7 -M16 (c.) calculati ons of (CO 2)2H2
-176 -174 -172 -170 -168 -166-176-174-172-170-168a0(kcal mol-1)
fH0(kcal mol-1) a0 = f(fH0)
Linear fit of data

-164 -163 -162 -161 -160 -159 -158-168-167-166-165-164-163a0(kcla mol-1)
fH0(kcal mol-1) a0 = f(fH0)
Linear fit of data
-165 -164 -163 -162 -161 -160 -159 -158 -157-169-168-167-166-165-164-163-162a0(kcal mol-1)
fH(kcal mol-1) a0 = f(fH0)
Linear fit of data
a0(PM3) * = -0.2997 + 0.9984 ·ΔH (PM3) a0(PM6) = -12.5009 + 1.0504 ·ΔH(PM6) a0(PM7) = -12.6469 + 0.9486 ·ΔH(PM7)
(r2 = 0.9988 ; SD = 0.0779 ; F = 8.604 9·103) (r2 = 0.9980 ; SD = 0.0667 ; F = 0.4596 ·104) (r2 = 0.9985 ; SD = 0.0740; F = 0.6762 ·104)
a. b. c.
Fig. 4. Linear dependence of a 0 from ΔfH0 for PM3 -M16 (a. *plot of the d ata when the PM3 -11 was excluded ), for PM6 -M16 (b.) respectively for
PM7-M16 (c.) of (CO 2)2H2

11

100 200 300 400 500 600-262-260-258-256-254-252-250-248-246-244-242-240
12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152
ABCDEFGHIJKLMNOPQRSTUVWXYZAAABACADAEAFAGAHAIAJAKALAMANAOAPAQARASATAUAVAWAXAYAZ
abcdefghijklmnopqrstuvwxyzaaabacadaeafagahaiajakalamanaoapaqarasatauavawaxayazfH(kcal mol-1)
T(K) PM3-01
PM3-02
PM3-03
PM3-04
PM3-05
PM3-06
PM3-07
PM3-08
PM3-09
PM3-10
PM3-11
PM3-12
PM3-13
PM3-14
PM3-15
PM3-16
1 PM3-17
A PM3-18
a PM3-19
PM3-20
PM3-21
PM3-22
PM3-23
PM3-24
100 200 300 400 500 600-260-255-250-245-240-235-230-225-220-215
12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152
ABCDEFGHIJKLMNOPQRSTUVWXYZAAABACADAEAFAGAHAIAJAKALAMANAOAPAQARASATAUAVAWAXAYAZ
abcdefghijklmnopqrstuvwxyzaaabacadaeafagahaiajakalamanaoapaqarasatauavawaxayazfH(kcal mol-1)
T(K) PM3-25
PM3-26
PM3-27
PM3-28
PM3-29
PM3-30
PM3-31
PM3-32
PM3-33
PM3-34
PM3-35
PM3-36
PM3-37
PM3-38
PM3-39
PM3-40
1 PM3-41
A PM3-42
a PM3-43
PM3-44
PM3-45
PM3-46
PM3-47
PM3-48
PM3-49
PM3-50
PM3-51
PM3-52
PM3-53
PM3-54
PM3-55
PM3-56
PM3-57
100 200 300 400 500 600-240-230-220-210-200-190
12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152
ABCDEFGHIJKLMNOPQRSTUVWXYZAAABACADAEAFAGAHAIAJAKALAMANAOAPAQARASATAUAVAWAXAYAZ
abcdefghijklmnopqrstuvwxyzaaabacadaeafagahaiajakalamanaoapaqarasatauavawaxayazfH(kcal mol-1)
T(K) PM6-01
PM6-02
PM6-03
PM6-04
PM6-05
PM6-06
PM6-07
PM6-08
PM6-09
PM6-10
PM6-11
PM6-12
PM6-13
PM6-14
PM6-15
PM6-16
1 PM6-17
A PM6-18
a PM6-19
PM6-20
PM6-21
PM6-22
PM6-23
PM6-24
PM6-25
PM6-26
PM6-27
PM6-28
PM6-29
PM6-30
PM6-31
PM6-32
PM6-33
100 200 300 400 500 600-240-230-220-210-200-190
12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152
ABCDEFGHIJKLMNOPQRSTUVWXYZAAABACADAEAFAGAHAIAJAKALAMANAOAPAQARASATAUAVAWAXAYAZ
abcdefghijklmnopqrstuvwxyzaaabacadaeafagahaiajakalamanaoapaqarasatauavawaxayaz
12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152
ABCDEFGHIJKLMNOPQRSTUVWXYZAAABACADAEAFAGAHAIAJAKALAMANAOAPAQARASATAUAVAWAXAYAZabcdefghijklmnopqrstuvwxyzaaabacadaeafagahaiajakalamanaoapaqarasatauavawaxayazfH(kcal mol-1)
T(K) PM7-01
PM7-02
PM7-03
PM7-04
PM7-05
PM7-06
PM7-07
PM7-08
PM7-09
PM7-10
PM7-11
PM7-12
PM7-13
PM7-14
PM7-15
PM7-16
1 PM7-17
A PM7-18
a PM7-19
PM7-20
PM7-21
PM7-22
PM7-23
PM7-24
PM7-25
PM7-26
PM7-27
PM7-28
PM7-29
PM7-30
PM7-31
PM7-32
PM7-33
PM7-34
PM7-35
1 PM7-36
A PM7-37
a PM7-38
PM3 -01 to PM3 -24 PM3 -25 to PM3 -57
a. b. c.
Fig. 5. Plot of f unctional dependencies of ΔfH(kcal·mol-1) in the temperature range T[100K, 600K] with a step 10K resulted from the PM3 -M16 (a.),
PM6 -M16 (b.) and PM7 -M16 (c.) calculations of (CO 2)3H2

-255 -250 -245 -240 -235 -230 -225-265-260-255-250-245-240-235-230a0(kcal mol-1)
fH(kcal mol-1) a0 = f(fH0)
Linear fit of data
-240 -235 -230 -225 -220 -215 -210 -205-245-240-235-230-225-220-215-210a0(kcal mol-1)
fH0(kcak mol-1) a0 = f(fH0)
Linear fit of data
-240 -235 -230 -225 -220 -215 -210 -205 -200 -195-245-240-235-230-225-220-215-210-205-200a0(kcal mol-1)
fH0(kcal mol-1) a0 = f(fH0)
Linear fit of data
a0(PM3) = 6.193 4 + 1.0504 ·ΔfH(PM3) a0(PM6)* = 4.4895 + 1.0467 ·ΔfH0(PM6) a0(PM7) = 2.6523 + 1.0388· ΔfH0(PM7)
(r2 = 0.9978 ; SD = 0.1909 ; F = 2.5192 ·104) (r2 0.9988; SD = 0.1889 ; F = 2.4904 ·104) (r2 0.9989; SD = 0.2001; F = 3.279·104)
a. b. c.
Fig. 6. Linear dependence of a 0 from ΔfH0 for PM3 -M16 (a.), for PM6 -M16 (b. *plot of the d ata when the PM6 -20 was excluded) respectively for
PM7 -M16 (c.) of (CO 2)3H2

12
For the trimer, because the variations of the a0 free term are between 10 and 16 %, although
the a1, a2 and a3 polynomial parameters are relatively alike , the standard deviation is of the order of
10-3, and the F statistical test is of the order of 107, the mean interpolation functions (4-6)
constructed with the mean values :
 for PM3
ΔfH(T) = -257.1786 (4.0541) + 0.0103 (0.0014) T + 4.3194 (0.1894)·10-5T2 – 1.7377 (0.1156)·10-8T3 (4)
T[100K,600K] SD = 0.00287 (0.00089) F = 4.80898 (2.43867)·107
Δa0 = 27.6058kcal•mol-1 (10.73%)

 for PM6
ΔfH(T) = -238.2712 (5.4470) + 0.0093 (0.0015) T + 4.5688 (0.2202)·10-5T2 – 1.6876 (0.6399)·10-8T3 (5)
T[100K,600K] SD = 0.00137 (0.00044) F = 2.79351 (0.94751)·107
Δa0 = 31.9535kcal•mol-1 (13.41%)
 for PM7
ΔfH(T) = -237.2321 (5.9604) + 0.0096 (0.0015) T + 4.3860 (0.1926)·10-5T2 – 1.6728 (0.0833)·10-8T3 (6)
T[100K,600K] SD = 0.0011 (0.0007) F = 5.6696 (2.6043)·107
Δa0 = 37.9536kcal•mol-1 (16.00%)

cannot be used to compute the formation enthalpies for the conformers in the T[100 K, 600 K]
domain , because the errors are too great . In this case, t he use of the each conformer’s own function
from Tables 8, 14 and 20 (Supplementary Material ) is recommended.
By perfo rming linear correlations between the free term of the interpolation polynomial (a0)
and the standard formation enthalpy Δ fH0, we find a very good correlation. The free term follows
the same direction as the standard enthalpy of formation . In the case of t he trimer, which is much
more flexible, Figs . 6a, 6b and 6c show an isolated point corresponding to the most unstable
conformer. This conformer involves the overlap of O2 and O8 oxygen atoms , which have double
bonds (C1=O2 and C7=O8) and which approach to a distance of less than 2 Å, which leads to the
dissociation of the double bond and the formation of a cyclic conformer with a four atoms ring (Fig.
2e).

CONCLUSIONS

From t he analysis of electronic data it is found that there is no regularity between the
standard formation enthalp ies and the HOMO or LUMO levels, the dipole moment (μ), the first
vibration (υ 0) or the zero point vibration energy (E ZVE).
For the geometrically and energ y distinct conformers of the (CO 2)2H2 dimer and of the
(CO 2)3H2 trimer obtained with the PM3, PM6 and , respectively , PM7 Hamiltonian , the cubic
interpolation relations of the functional dependence of the enthalpy of formation from temperatu re

13
(ΔfH = a0 + a 1T + a 2T2 + a 3T3) were established . It is shown that the a1, a2, a3, r2, SD and F values
do not significantly depend on the calculation method and on the geometry of the conformer. The α0
free term is the only one that depends on geometry and decreases with the decreas e in the
conformer ’s stability .
For the dimer, with a certain degree of confidence, an average polynomial function can be
used, but for the trimer these mean functions are no longer usable.
In the case of the trimer, due to the flexibility of the bonds, intern al hydrogen bonds or an
overlap of atoms may occur, which may lead to the formation of cyclic structures.

Acknowledgments : Part of this work was supported by the Romanian National Authority for
Scientific Research (CNCS -UEFISCDI) through project PN -II-PCCA-2011 -142. We are gratefully
acknowledging the generous support of J. J. P. Stewart for providing an academic license for the
MOPAC1 6 software.

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