Dedicated to Professor Dumitru Oancea on the occasion on his 75th anniversary ENTROPIC EFFECT AND THERMODYNAMIC PROPERTIES FOR THE CONFORMERS OF… [610271]

ACADEMIA ROMÂNĂ
Revue Roumaine de Chimie
http://web.icf.ro/rrch/
Rev. Roum. Chim. ,
2016 , 61(6-7), 597-607

Dedicated to Professor Dumitru Oancea
on the occasion on his 75th anniversary
ENTROPIC EFFECT AND THERMODYNAMIC PROPERTIES
FOR THE CONFORMERS OF (3R,5S,6 R)-6-ACETYLAMIDOPENICILLANIC
ACID CALCULATED WITH THE PM3, PM6
AND PM7 SEMIEMPIRICAL MO METHODS**
Mihai MEDELEANU,a Mihai-Cosmin PASCARIU,b Eugen ȘIȘU,b Tudor OLARIU,b
Doina GEORGESCU,b Maria MRACECc and Mircea MRACECc,*
a University Politehnica Timișoara, Faculty of Industrial Chemis try and Environmental Engineering, 6 Carol Telbisz, RO-300001
Timișoara, Roumania
b “Victor Babeș” University of Medicine and Pharmacy of Timișoar a, 2 Eftimie Murgu Sq., RO-300041 Timișoara, Romania
c Department of Computational Ch emistry, Institute of Chemistry Timișoara of Roumanian Academy, 24 Mihai Viteazul Blvd.,
RO-300223 Timișoara, Roumania
Received December 21, 2015
Thirty conformers of (3R,5S,6R)-6-acetylamidopenicillanic acid have been obtained by using
the PM3 Conformational Search implemented in the HyperChem 7.5 software. After performing
the energy minimization with the PM3, PM6 and PM7 semiempirical MO methods, which are
included in the MOPAC12 software, thirty, eight, and, respectiv ely, nine conformers were
obtained. The following cubic interpolation relations of functi onal dependence of enthalpy (ΔH),
entropy (ΔS) and C P on temperature (T ∈[100K,1000K]) were established:
• for PM3:
ΔH(T)=0.0613( ±0.0241)+0.0079( ±0.0004)·T+1.1621( ±0.0055)·10-4T2–3.4367(±0.0270)·10-8T3
ΔS(T)=60.4738( ±1.2434)+0.2981( ±0.0029)·T–1.2594( ±0.0411)·10-4T2+3.2563(±0.1969)·10-8T3
CP(T)=7.8283(±0.6499)+0.2364(±0.0029)·T–1.0955(±0.4350)·10-4T2+1.2023(±0.2133)·10-8T3
• for PM6:
ΔH(T)=0.1015( ±0.0188)+0.0084( ±0.0005)·T+1.1741( ±0.0069)·10-4T2–3.3992(±0.0324)·10-8T3
ΔS(T)=61.4428( ±1.4661)+0.3043( ±0.0033)·T–1.3004( ±0.0524)·10-4T2+3.4746(±0.2617)·10-8T3
CP(T)=9.3563( ±0.7470)+0.2312( ±0.0033)·T–10.0693( ±0.4757)·10-T2+24.3696(±22.4091)·1010T3
• for PM7:
ΔH(T)=0.0400( ±0.0218)+0.0089( ±0.0006)·T+1.1498( ±0.0082)·10-4T2–3.2727(±0.0367)·10-8T3
ΔS(T)=61.3072( ±1.2728)+0.3049( ±0.0039)·T–1.3469( ±0.0589)·10-4T2+3.7687(±0.2888)·10-8T3
CP(T)=9.4670( ±0.8905)+0.2286( ±0.0036)·T–10.0442( ±0.5069)·10-5T2+39.8522(±23.6941)·10-10T3
Considering the ΔG = ΔH – T ⋅ΔS relation, we can state that the thermodynamic factor, which determines the variation of the conformers’
biological activity, is the entropic factor, which represents t he quantity of bound energy. Based on this observation, the fol lowing theorem can be
stated: The dispersion of biological activities of all conformers of a flexible molecule at a temperature T is determined by their entropy .

INTRODUCTION*
About sixty years ago one hypothesis was
developed which stated that between the
thermodynamic potentials, namely the ΔF free

energy, but especially the ΔG Gibbs free enthalpy
(Gibbs free energy) and various descriptors of the chemical structure should exist some relations of
linear dependence.
1-16
The structural descriptors of molecules have two
sources: experimentally obtained descriptors, and
0 200 400 600 800 100080100120140160180200220240260280ΔS(cal K-1 mol-1)
T(K) PM7-01
PM7-02
PM7-03
PM7-04
PM7-05
PM7-06
PM7-07
PM7-08
PM7-09
* Corresponding author: [anonimizat];
** Supporting information on http: //web.icf.ro/rrch or http: //re vroum.lew.ro

598 Mihai Medeleanu et al.
calculated descriptors obtai ned by using different
methods of computational chemistry.17-22 Starting
from old ideas (Wiener-Hansch-Charlton),1-16 today’s
field of quantitative chemical structure – properties
relationships (QSPR) or, in particular, the quantitative chemical structure – biological activity
relationships (QSAR) were developed.
17,18
QSPR/QSAR relationships rely on the fact that
for ΔG i (Gibbs free energies) of a M = {m i} set of
molecules with similar structures a functional
relation can exist. This can be expressed by a
functional dependence of the type f(D) → A ,
where:
− f represents the different analytical
functions of the Gibbs free energy;
− D is the set of real values of the {x i}
descriptors, which is a subset of the set of
real numbers D ⊂ ℜ;
− A ⊂ ℜ for the molecules in the M set, is a
set of real values of Gibbs free energies, or equivalent forms: equilibrium or
complexation constants, biological
activities, etc.
Because usually the f function is not known, it
is replaced by an interpolation function of type
f(D
1) → A1, where D1 ⊂ D ⊂ ℜ, and A1 ⊂ A ⊂ ℜ. It is
considered that the f interpolation function on the
D1 domain is a good estimate for the known A1
values if the root-mean-square deviation between
the values calculated through f and the
experimental values is minimal. From analytical point of view, any f interpolation function is
strictly true only on the D
1 definition domain and
any extrapolation of its use outside of this domain
can lead to serious interpretation errors. Because it is admitted that the biological activities or different
chemical properties of the molecules from the M
set (A
1 set) are linear functions of free energy, then
the f function is proposed as being a linear or
linearizable dependence (but not only) of the D =
{xi} set of descriptor variables.23-28 T h e
determination of the analytical expression of this f
function constitutes the whole strategy of
QSPR/QSAR relationshi ps or of molecular
modeling, and applied mathematical method is the least squares method, with a series of validation statistical methods.
23-38 In the case of compounds
of biological interest (QSAR) the majority of the
molecules from the M s e t a r e n o t r i g i d m o l e c u l e s
and have a multitude of {c ij} = C conformers, and
each of them could be characterized through its
own ΔG ij Gibbs free energy. If the m i ∈ M
molecule has j conformers (the Ci ⊂ C s e t o f conformers of the m i molecule), then a certain
conformer can be characterized through a Pi set of
its structural and energetic properties, one of them being its Gibbs free energy, or ΔG
ij. At a
temperature T at which the m i ∈ M molecule exists
in solution or in gaseous state, the x ij m o l a r
fraction of the c ij conformer is proportional with
the partition function: Z ij o r Q ij = g ij⋅exp(-
Eij/kT)/Σigij⋅exp(-E ij/kT), where g ij i s t h e
degeneration degree of the E ij energy state of the c ij
conformer, and E ij = E ij,rot+ E ij,vib+ E ij,el + E ij,nuc
represents the total energy of the c ij conformer
located in the E ij energy state.39-40 T h e
experimental determination of the Pi s e t o f
properties of a certain conformer is virtually
impossible, because the separation of a certain conformer from the mixture is in most of the cases
impossible. In principle only properties in the
fundamental state, when the geometry is frozen, can be known. When a molecule of biological interest binds to a protein, then through the
protein’s isolation and crystallization just some
properties of the bound conformer can be known.
The estimation of the P
i properties for Ci
conformers of the m i molecule can be done mainly
through computational chemistry.
The knowledge of conformers for a molecule
which presents a biological interest (ligand,
substrate, effector etc.) is important because,
during the interaction with proteins the involved
conformer is not the global minimum conformer.
This specific conformer determines the biological activity, a fact that was proven through the
different docking techniques.
41-49 For now, the only
experimental method for showing the existence of
conformers is the high resolution nuclear magnetic
resonance (NMR), and 1H NMR for the organic
compounds, through the jj coupling constants.50
The objectives of this paper are by using different
quantochemical methods (the PM3, PM6 and PM7
MO semiempirical methods), the computation of
their ΔH, ΔS and C P thermodynamic properties, and
also their temperature dependence on the
T∈[100K,1000K] interval.
METHOD
The PM3 method from the HyperChem7.52 –
Hy51 software was used, and also the PM3, PM6
and PM7 from the MOPAC12 Version 13.004W –
M12 software.52 B e c a u s e t h e P M 6 a n d P M 7
methods were not implemented in programs that

Semiempirical MO methods 599
search for conformers, we have used the
Conformational Search module from the Hy
package and the PM3 method for the optimization
of the conformers’ geometries.53 F o r t h e
computation of the thermodynamic properties: ΔH, C
P, and ΔS with the M12 program it was necessary
t o w o r k i n t w o s t a g e s .52 In the first stage the
geometries of the conformers were optimized with
one of the PM3, PM6 and PM7 methods, and in the
second stage the thermodynamic properties were calculated.
52 For the conversion of the “hin” files53
from the Hy software in “mop” files, the
AVOGADRO54 program was used, applying the
“MOPAC” and “Geometry Optimization”
keywords.
For the geometry optimization with the M12
program,52 in the first stage the following 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 in
M12 input files for the computing of the
thermodynamic properties. For computing the
thermodynamic properties ΔH(T), C
p(T) and ΔS(T)
i n t h e s e c o n d s t a g e t h e f o l l o w i n g k e y w o r d s w e r e
used: PM3, (PM6 or PM7), SCFCRT=1.D-10,
GEO-OK, PRECISE, GNORM=0.001, CYCLES=5000, T=345600, AUX, LARGE,
CHARGE=0, SINGLET, SYMMETRY, FORCE, THERMO(100,1000,50), LET. The atom
numbering in the (3R,5S,6R)-6-
acetylamidopenicillanic acid is given in Fig. 1.
RESULTS AND DISCUSSION
Thirty conformers of (3R,5S,6R)-6-
acetylamidopenicillanic acid were obtained by
using the PM3 Conformational Search module
from the Hy software.
51 The standard formation
enthalpies (Δ fH0 – PM3, PM6 and PM7
calculations) were used for ordering the conformers.
53
For each conformer of (3R,5S,6R)-6-
acetylamidopenicillanic acid obtained with the
PM3, PM6 and PM7 M12 Hamiltonians, we
computed the functional dependencies of the molar enthalpies, the molar entropies and the molar heats
at constant pressure as polynomial interpolation
functions depending on temperature until the third
degree (cubic interpolation functions y = a
0 + a 1T +
a2T2 + a 3T3, where y is ΔH, ΔS or C p). The chosen
temperature range was T ∈[100K,1000K] at a step
o f 5 0 ș . T h e p r i m a r y d a t a a r e p r e s e n t e d i n t h e
Supplementary material: Tables 1a, 1b, 3a, 3b, 5a,
5b for PM3, Tables 7, 9, 11 for PM6 and Tables
13, 15, 17 for PM7.

Fig. 1 – Atom numbering in (3R,5S,6R)-6-acetylami dopenicillanic acid.

600 Mihai Medeleanu et al.
0 200 400 600 800 1000020406080100
1234567891011121314151617181920
ABCDEFGHIJKLMNOPQRST
abcdefghijklmnopqrstΔH(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
PM3-25
PM3-26
PM3-27
PM3-28
PM3-29
PM3-30

a.
0 200 400 600 800 1000020406080100ΔH(kcal mol-1)
T(K) PM6-01
PM6-02
PM6-03
PM6-04
PM6-05
PM6-06
PM6-07
PM6-08
0 200 400 600 800 1000020406080100ΔH(kcal mol-1)
T(K) PM7-01
PM7-02
PM7-03
PM7-04
PM7-05
PM7-06
PM7-07
PM7-08
PM7-09

b. c .
Fig. 2 – Functional dependence ΔH(T) plots in the range T ∈[100K,1000K] resulted for: a. PM3; b.PM6; and c. PM7 – M12
conformers of (3R,5S,6R)-6-acetylami dopenicillanic acid.55

Enthalpy. With the primary data (Supplementary
material: Tables 1a and 1b) for ΔH(T) of the thirty
PM3 conformers of (3R,5S,6R)-6-acetylami-
dopenicillanic acid, we have calculated the cubic functions, ΔH
PM3(T) = a 0 + a 1T + a 2T2 + a 3T3, for the
T ∈ [100K,1000K] (Supplementary material, Table
2). The plots for these functions are shown in
Fig. 2a.55
Similarly, we have calculated the cubic functions
of correlation for the T ∈ [100K,1000K],
ΔH PM6orPM7 (T) = a 0 + a 1T + a 2T2 + a 3T3, for the eight
PM6 conformers, and, respectively, for the nine PM7 conformers. The diagrams for these functions, resulted from the primary data (Supplementary
material: Table 7 and, respectively, Table 13) are shown in Fig. 2b for the PM6 conformers, and,
respectively, in Fig. 2c for the PM7 conformers.
55
The analysis from Supplementary material data
(Table 2 and Figs. 2a – 2c for the PM3 conformers, Table 8 and Figs. 8a – 8c for the PM6 conformers,
and, respectively, Table 14 and Figs. 14a – 14c for
the PM7 conformers) shows that, between the a
0
(or a 1 o r a 2 o r a 3) values of the interpolation
functions and the Δ fH0 o r Δ H0 values there is no
functional dependency (correlation). 5 5 F o r t h e
PM3 conformers, the difference between min(a0)
and Max(a0) is only 0.0956 kcal·mol-1, the mean
value being 0.0613±0.0241. For the PM6
conformers, the difference between min(a0) and
Max(a0) is only 0.0475 kcal·mol-1 and the mean

Semiempirical MO methods 601
value is 0.1015±0.00188, while for the PM7
conformers this difference is only 0.0609 kcal·mol-1,
the mean value being 0.0400±0.0218. The
grouping on a tight interval of interpolation
functions proves that, in the case of the same molecule’s conformers, the ΔH enthalpy (the
internal energy content of the system) is relatively
unchanged and it does not depend significantly on
the conformers’ geometry.
Through the computation of the a 1/a2 ratio
between the mean values from the interpola-tion mean functions we obtain the following
values: 680 for the PM3 conformers, 716 for the
PM6 conformers, and, respectively, 774 for the
PM7 conformers. This means that, through
interpolation, we obtain three packages of parallel
functions which are almost overlapped, and which have the same concavity (Fig. 2a – Fig. 2c),
differing only by the a
0, free term.55 Owing to this
observation, the following three interpolation mean
relations can be introduced:

ΔH PM3(T) = 0.0613( ±0.0241) + 0.0079( ±0.0004)T + 1.162( ±0.006)·10-4T2- 3.437(±0.027)·10-8T3
T∈[100K,1000K] (SD = 0.0347( ±0.0010); F = 4.2154( ±0.2232)·106)
ΔH PM6(T) = 0.1015( ±0.0188) + 0.0084( ±0.0005)T + 1.174( ±0.007)·10-4T2– 3.399(±0.032)·10-8T3
T∈[100K,1000K] (SD = 0.0298( ±0.0003); F = 5.9632( ±0.1224)·106)
ΔH PM7(T) = 0.0400( ±0.0218) + 0.0089( ±0.0006)T + 1.150( ±0.008)·10-4T2 – 3. 273(±0.037)·10-8T3
T∈[100K,1000K] (SD = 0.0251( ±0.0008); F = 8.0891( ±0.2542)·106)

These mean relations can constitute a way of
computing the enthalpies for every single
conformer, without any significant error, keeping
in mind that the statistic parameters (SD, F) correspond to some relations with a confidence degree of over 99.9%. The interpolation relations from Supplementary material (Tables 2, 8 and 14)
allow the enthalpy computation for every single
conformer in the T ∈[100K,1000K] temperature
domain. The thus calculated enthalpy can be used
for the establishment of some QSAR/QSPR relationships. With the same confidence degree, the interpolation mean relation can also be used,
which allows the estimation of an enthalpy with a
high confidence degree.

Entropy. The primary data for the ΔS(T)
entropies of the thirty PM3 conformers of
(3R,5S,6R)-6-acetylamidopenicillanic acid,
resulted through the use of the PM3 M12 Hamiltonian, are presented in Supplementary material (Tables 3a and 3b). With these primary
data of each PM3 conformer, the ΔS
PM3(T) = a 0 +
a1T + a 2T2 + a 3T3 cubic interpolation function for
the T∈[100K,1000K] temperature domain were
calculated (Supplementary material, Table 4). The
d i a g r a m s f o r a l l f u n c t i o n s a r e p l o t t e d i n F i g . 3 a . The analysis of data from Supplementary material
(Table 4, and Figs. 4a and 4c) shows that, between
the a 0 (or a 1 or a 2 or a 3) values of the interpolation
functions and the Δ fH0 o r Δ H0 values there is no
correlation. From Supplementary material (Fig. 4b) it results that between the free term, a
0, of the
polynomial and ΔS 0, there is a somewhat good
correlation, and this suggests that the entropies in
standard conditions might be used as a criterion for ordering the conformers. The difference between min(a
0) and Max(a0) is of only 3.8031 cal·K-1·mol-
1, and the mean value is 60.4738±1.2434. The
dispersion on the 3.8031 cal·K-1·mol-1 interval of
the interpolation functions (Suplementary material, Fig. 4a and 4b) proves that, in the case of conformers of the same molecule, the ΔS(T)
entropy is not a constant and it depends on the
conformer geometry.
55
The a 1/a2 ration between the mean values from
the interpolation mean functions is -2367. This
means that, through interpolation, a parallel
polynomial package results, which have the same
concavity (Fig. 3a), and they only differ through
the free term, a 0.55 Owing to these observations, the
interpolation mean relation:

ΔS PM3(T) = 60.4738( ±1.2434) + 0.2981( ±0.0029)T – 1.2594( ±0.0411)·10-4T2 + 3.256(±0.197)·10-8T3
T∈[100K,1000K] (SD = 60.4738( ±1.2434); F = 2.17517(0.33015)·105)

cannot be used for computing the entropy of each
conformer, although the statistic parameters (SD, F) correspond to a relation with a confidence
degree of over 99.9%. This statement is based on

602 Mihai Medeleanu et al.
the high error degree resulted from the significant
dispersion of the interpolation function, and that
can be observed in Fig. 3a.55 The interpolation
relations from Supplementary material (Table 4)
allow the computation of entropy in the T ∈
[100K,1000K] temperature domain.
The interpolation relations from Supplementary
material (Tables 2, and 4) can be used for c o m p u t i n g t h e f r e e e n t h a l p y , Δ G ( T) = Δ H ( T) – T
ΔS(T), at any temperature on the T ∈[100K,
1000K] domain. According to the data obtained for the enthalpy in this temperature domain we can
state that only the entropy depends on the
conformers’ geometry. In conclusion, the Gibbs
free energies will depend on the conformers’
geometries for the same molecule through the entropy. As the Gibbs free energy is the measure
that determines the biological activity, we can
conclude that, in reality, the thermodynamic factor
that influences the variation of the biological
activity with the conformer’s nature is the entropic factor.

0 200 400 600 800 100080100120140160180200220240260280
1234567891011121314151617181920
ABCDEFGHIJKLMNOPQRST
abcdefghijklmnopqrstΔS(cal K-1 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
PM3-25
PM3-26
PM3-27
PM3-28
PM3-29
PM3-30

a.
0 200 400 600 800 100080100120140160180200220240260280ΔS(cal K-1 mol-1)
T(K) PM6-01
PM6-02
PM6-03
PM6-04
PM6-05
PM6-06
PM6-07
PM6-08
0 200 400 600 800 100080100120140160180200220240260280ΔS(cal K-1 mol-1)
T(K) PM7-01
PM7-02
PM7-03
PM7-04
PM7-05
PM7-06
PM7-07
PM7-08
PM7-09
b. c.
Fig. 3 – Functional dependence ΔS(T) plots in the range T ∈[100K,1000K] resulted for: a. PM3; b. PM6;
and c. PM7 – M12 conformers of (3R,5S,6R)-6-acetylami dopenicillanic acid.55

Semiempirical MO methods 603

The primary data for the ΔS(T) entropies of the
eight conformers of (3R,5S,6R)-6-
acetylamidopenicillanic acid, resulted from the
PM6 M12 Hamiltonian, are given in Supplementary material (Table 9). With this
primary data for each PM6 conformer the ΔS(T) =
a
0 + a 1T + a 2T2 + a 3T3, interpolation cubic functions
for the T∈[100K,1000K] temperature domain were
computed (Supplementary material, Table 10). The plots for these functions are given in Fig. 3b. The
analysis of data in Supplementary material (Table
10, and Fig.s 10a and 10c) shows that, between the
a
0 ( o r a 1 o r a 2 o r a 3) values of the interpolation
functions and the Δ fH0 o r Δ H0 values there is no
correlation. The difference between min(a0) and
Max(a0) is 3.5487 cal·K-1·mol-1, w h i l e t h e m e a n value is 61.4428±1.4661 cal·K-1·mol-1. The
dispersion of the a 0 values (3.5487 cal·K-1·mol-1
from Supplementary material, Figs. 10a and 10c)
and of interpolation functions (Fig. 3b) prove that, in the case of the same molecule the ΔS(T)
entropies of its conformers are not constant and
they depend of the conformer’ geometry.
55 T h e
computed a 1/a2 ratio between the mean values
becomes negative, having the value of -2333.59. This means that through interpolation a package of
parallel functions result, which have the same
concavity (Fig. 3b), and differ through the free
term, a
0. Owing to the dispersion ( ±1.4661) and to
the fact that Δa 0 = 3.5487cal·K-1·mol-1, the
interpolation mean relation:

ΔS PM6(T) = 61.443( ±1.4661) + 0.304 ( ±0.0033)T – 1.300( ±0.0524)·10-4T2 + 3.475(±0.2617)·10-8T3
T∈[100K,1000K] (SD = 0.3344 ±0.0291; F = 1.7692( ±1.4057)·105

cannot be used for computing the entropy for each
conformer, because of the high dispersion degree of the interpolation curves (Fig. 3b).
55
The interpolation relations from Supplementary
material (Table 10) allow to calculate the entropy
for each conformer in the T ∈[100K,1000K]
temperature domain. The interpolation relations from Supplementary material, Table 10) can also
be used for computing the free enthalpy, ΔG(T) =
ΔH(T) – T ·ΔS(T), at any temperature in T ∈[100K,
1000K] interval. In this temperature domain
(T∈[100K,1000K]) we can also state that only the
entropy depends on the conformer geometry.
Based on this observation we can also conclude
that the Gibbs free energy depends on conformer
geometries by means of their entropy.
The primary data for the entropies of the nine
conformers of (3R,5S,6R)-6-acetylamidopenicil-
lanic acid calculated with the PM7 M12
Hamiltonian are shown in Supplementary material (Table 15). With this primary data, for each PM7
conformer of (3R,5S,6R)-6-acetylamidopenicil-
lanic acid, we computed the Δ S ( T ) = a
0 + a 1T +
a2T2 + a 3T3, cubic interpolation functions for the T ∈ [100K,1000K] temperature domain (Supple-
mentary material, Table 16). The diagrams of these functions are shown in Fig. 3c. The analysis of
data from Supplementary material (Table 16, Fig.s.
16a and 16c) shows that,
55 between the a 0 (or a 1 or
a2 o r a 3) values of the interpolation functions
and the Δ fH0 or ΔH0 values there is no correlation.
The difference between min(a0) and Max(a0) is
3.3011 cal·K-1·mol-1, and the mean value is
61.3072±1.2728. The dispersion of the a 0
parameter 3.3011 cal·K-1·mol-1 from the interpola-
tion functions, (see Fig. 3c, and Supplementary material (Figs. 16a and 16c),
55 p r o v e s t h a t i n t h e
case of the conformers of a molecule the entropy is
not a constant and depends of the conformer
geometry. The computation of the a 1/a2 ratio
between the mean values gives a negative value of -2263.72. This means that through interpolation a
pack of parallel functions results, which have the
same concavity (Fig. 3c), and differ through the
free term, a
0. Owing to these observations, the
interpolation mean relation:

ΔS PM7(T) = 61. 307( ±1.278) + 0.305( ±0.004)T – 1.347( ±0.059)·10-4T2 + 3.769 (±0.289)·10-8T3
T∈[100K,1000K] (SD = 0.3448 ±0.0298; F = 1.6426( ±0.3056)·105

cannot be used for computing the entropy for each
conformer, because of the high error degree (the
dispersion of the interpolation curves). The interpolation relations from Supplementary
material (Table 16) allow the computation of the
entropy in the T ∈ [100K,1000K] temperature

604 Mihai Medeleanu et al.
domain for each conformer of a molecule. The
interpolation relations from Supplementary
material (Table 16) can be used for the calculation
of ΔG(T), Gibbs free energy for each conformer of
a molecule at any temperature in the
T∈[100K,1000K] domain.
Because this analysis proves that the conformer
entropies depend on their geometries, the following theorem can be stated: The dispersion of
the biological activities of conformers of a flexible
molecule at a T temperature is determined by the
conformer entropy .

Molar heat at constant pressure. The primary
data for the C
P(T) molar heat at constant pressure,
for the thirty conformers of (3R,5S,6R)-6-
acetylamidopenicillanic acid, obtained with the
PM3 M12 Hamiltonian, are presented in
Supplementary material (Tables 5a and 5b). With
t h i s p r i m a r y d a t a , f o r e a c h P M 3 c o n f o r m e r w e have computed the C
P(T) = a 0 + a 1T + a 2T2 + a 3T3
cubic interpolation functions for the T ∈[100K, 1000K] temperature interval (Supplementary
material, Table 6). The plots of these functions are
presented in Fig. 4a.55 The analysis of the data
from Supplementary material (Table 6, and Figs 6a
and 6c) shows that, between the a 0 (or a 1 or a 2 or
a3) values of the interpolation functions and the
ΔfH0 o r Δ H0 values there is no correlation. The
difference between min(a0) and Max(a0) is only
2.1923 cal·K-1·mol-1, and the mean value is
7.8283±0.6499. The grouping in a relatively tight interval of the interpolation functions (Fig. 4a)
proves that,
55 for the conformers of a molecule, the
CP(T) molar heat at constant pressure is relatively
constant, and does not significantly depend on the
conformer geometries. Through the computation of the a
1/a2 ratio between the mean values we obtain a
negative value of -2157.92. This means that
through interpolation, a package of overlapping
parallel functions results, which have the same
concavity (Fig. 4a),55 and do not significantly
differ through the free term, a 0. Owing to these
observations, the interpolation mean relation:

CP,PM3(T) = 7.828(±0.650) + 0.236(±0.003)T – 1.096(±0.435)·10-4T2 + 1.202(±0.213)·10-8T3
T∈[100K,1000K] (SD = 0.5360 ±0.0191; F = 25549.35 ±2042.21)

can be used for computing the molar heats at
constant pressure for each conformer. The statistic
parameters (SD, F) show that the mean cubic function of C
P(T) has a correlation with a
confidence degree of over 99.99%.
The interpolation relations from Supplementary
material (Table 6) allow the computation of the
molar heats at constant pressure in the
T∈[100K,1000K] temperature domain. With the
same confidence degree the interpolation mean relation can be used, which allows the estimation of an molar heat at constant pressure with a good confidence degree. The interpolation relations from
Supplementary material (Table 6) and the
interpolation mean relation can be used in the computation of the enthalpy, ΔH = ∫
TCP(T)dT. The
tight interval in which C P(T) varies as a function of
the conformer geometries justifies the tight interval
of enthalpy variation.
The primary data for the C P(T) molar heat at
constant pressure of the eight conformers of
(3R,5S,6R)-6-acetylamidopenicillanic acid, obtained
with the PM6 M12 Hamiltonian, are presented in Supplementary material (Table 11). With these
primary data for every PM6 conformer of (3R,5S,6R)-6-acetylamidopenicillanic acid, we
have calculated the C
P(T) = a 0 + a 1T + a 2T2 + a 3T3
interpolation cubic functions for the
T∈[100K,1000K] temperature domain (Supple-
mentary material, Table 12). The plots of these
functions are shown in Fig. 4b.
The analysis of data from Supplementary
material (Table 12, Figs. 12a and 12c) shows that,
between the a 0 ( o r a 1 o r a 2 o r a 3) values of the
interpolation functions and the Δ fH0 or ΔH0 values
there is no correlation. The difference between
min(a0) and Max(a0) is only Δa 0 = 1.9245 cal·K-1·mol-
1, while the mean value is 9.3563±0.7470. The
grouping on a relatively tight interval of the
interpolation functions (Fig. 4b) proves that, in the
case of the conformers of a certain molecule, the
CP(T) molar heat at constant pressure is relatively
constant and does not significantly depend on the
conformer geometries. Through the computing of
the a 1/a2 ratio between the mean values a negative
value of -2296.09 was obtained.
This means that, through interpolation, a
package of parallel functions results, which have
the same concavity (Fig. 4b), and do not
significantly differ through the free term, a 0. Thus,
the interpolation mean relation:

Semiempirical MO methods 605
CP,PM6(T) = 9.356( ±0.747) + 0.231( ±0.003)T – 10.070( ±0.476)·10-5T2 + 24.370( ±22.409)·10-10T3
T∈[100K,1000K] (SD = 0.5388 ±0.0138; F = 26654.93 ±1570.81)

can be used for computation of the molar heats at
constant pressure for each conformer, with very
small errors. The interpolation relations from
Supplementary material (Table 12) allow the
computation of molar heats at constant pressure in
the T∈[100K,1000K] temperature domain. With
the same confidence degree, the interpolation mean relation can be used for the estimation of an
enthalpy, PTHC ( T ) d T .Δ=∫ The tight interval in
which C P(T) varies, as a function of the conformer
geometries, justifies the tight interval of enthalpy
variation.

0 200 400 600 800 100020406080100120140
1234567891011121314151617181920
ABCDEFGHIJKLMNOPQRST
abcdefghijklmnopqrstCP(cal K-1 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
PM3-25
PM3-26
PM3-27
PM3-28
PM3-29
PM3-30

a.
0 200 400 600 800 100020406080100120140160CP(cal K-1 mol-1)
T(K) PM6-01
PM6-02
PM6-03
PM6-04
PM6-05
PM6-06
PM6-07
PM6-08
0 200 400 600 800 100020406080100120140160CP(cal K-1 mol-1)
T(K) PM7-01
PM7-02
PM7-03
PM7-04
PM7-05
PM7-06
PM7-07
PM7-08
PM7-09

b. c .
Fig. 4 – Functional dependence C P(T) plots in the range T ∈[100K,1000K] resulted for: a. PM3; b.PM6;
and c. PM7 – M12 conformers of (3R,5S,6R)-6-acetylami dopenicillanic acid.55

The primary data for the C P(T) molar heat at
constant pressure for the nine PM7 M12
conformers of (3R,5S,6R)-6-acetylamidopenicillanic acid are presented in Supplementary material
(Table 17). With this primary data, for each PM7
conformer we have calculated the C P(T) = a 0 + a 1T

606 Mihai Medeleanu et al.
+ a 2T2 + a 3T3 cubic interpolation functions for the
T∈[100K, 1000K] temperature interval (Supplemen-
tary material, Table 18). The plots for these functions are presented in Fig. 4c. The analysis of
the data from Supplementary material (Table 18,
and Figs 18a and 18c) shows that, between the a
0
(or a 1 o r a 2 o r a 3) values of the interpolation
functions and the Δ fH0 o r Δ H0 values there is no
correlation. The difference between min(a0) and
Max(a0) is only 2.3109 cal·K-1·mol-1, and the mean
value is 9.4670±0.8905. The grouping on a relatively tight interval of the interpolation
func tion s (F ig . 4c) p rov es tha t fo r th e con form ers
of a molecule, the C P(T) molar heat at constant
pressure is relatively constant and does not depend
on the conformer geometries. The calculated a 1/a2
ratio between the mean values is -2275.96. This
means that, through interpolation, a package of
parallel functions results, which have the same
concavity (Fig. 4c), and do not significantly differ
through the free term, a 0. Owing to these
observations, the interpolation mean relation:

CP,PM7(T) = 9.467( ±0.891) + 0.229( ±0.004)T – 10.044( ±0. 507)·10-5T2 + 39.852( ±23.694)·10-10T3
T∈[100K,1000K] (r2 = 0.9998; SD = 0.4453 ±0.0149; F = 38621.9 ±2969.1)

can be used in the computation of molar heats at
constant pressure for each conformer, without
having significant errors, especially because the statistic parameters (SD, F) present it as a
trustworthy relation. The interpolation relations
from Supplementary material (Table 18) allow the
computation of the molar h eats at constant pressure
in the T∈[100K,1000K] temperature interval.
With the same confidence degree, we can also use
de interpolation mean rel ation. The interpolation
relations from Supplementary material (Table 18) and the mean values relation can be used for
computing the enthalpy,
PTHC ( T ) d T .Δ=∫ T h e
tight interval in which C P(T) varies as a function of
the conformers’ geometries justifies the tight variation interval of enthalpy.
CONCLUSIONS
We have obtained the interpolation cubic
relations of the temperature dependence of
enthalpy, entropy and molar heat at constant
pressure (Y = a
0 + a 1T + a 2T2 + a 3T3, Y = ΔH(T),
ΔS(T) and C P(T)) for the PM3, PM6 and PM7
conformers of (3R,5S,6R)-6-acetylamidopenicillanic
acid. The a 1 a2 and a 3 values do not depend on the
computational method and the conformer
geometries. For the enthalpy and the molar heat at
constant pressure, the a 0, free term depends
relatively little on the conformer geometries in a tight interval, while the interpolation functions
form a pack of nearly identical parallel curves that
can be replaced with a mean interpolation function
on the T∈[100K, 1000K] temperature domain. For
the entropy, we have noticed that the interpolation
functions depend on the conformer geometries. The interpolation relations as temperature
functions for the ΔH(T), ΔS(T) and C
P(T),
thermodynamic properties of the (3R,5S,6R)-6-acetylamidopenicillanic acid’s conformers prove
that the entropy, through its dispersion, is the
factor that determines the variation of biological
activity of the conformers, because the biological
activity is dependent on the ΔG(T) free enthalpy. Because the free energy, ΔG, can be calculated as
Δ G = Δ H – T · Δ S , i t m e a n s t h a t Δ G h a s a
dispersion that is dependent on the entropy
dispersion, namely conformer geometries. Thus,
the following theorem can be stated: The
dispersion of both the chemical reactivities and
the biological activities of conformers of a
flexible molecule at a T temperature is
determined by the conformer entropy .

Acknowledgments: Part of this work was supported by the
Romanian National Authority for Scientific Research (CNCS-UEFISCDI) through project PN-II-PCCA-2011-142 and was also done at the Center of Genomic Medicine of the “Victor
Babes” University of Medicine and Pharmacy of Timisoara
through POSCCE 185/48749, Agreement 677/09.04.2015.We are gratefully acknowledging the generous support of J. J. P. Stewart for providing an academic license for the MOPAC12
software.
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608 Mihai Medeleanu et al.