Journal of Molecular Structure 556 (2000) 303–313
www.elsevier.nl/locate/molstruc
B3LYP/6-31G ⴱ, RHF/6-31G ⴱ and MM3 heats of formation of
disaccharide analogs
A.D. French a,*, A.-M. Kelterer b, G.P. Johnson a, M.K. Dowd a
a
Southern Regional Research Center, Agricultural Research Service, US Department of Agriculture, P.O. Box 19687, New Orleans,
LA 70179-0687, USA
b
Institut für Physikalische und Theoretische Chemie, Technische Universität Graz, Technikerstrasse 4, A-8010 Graz, Austria
Dedicated to Professor N.L. Allinger on the occasion of this special issue
Received 11 October 1999; accepted 27 October 1999
Abstract
DHf values were calculated with the bond- and group-enthalpy method for analogs of 13 disaccharides to learn the relative
stabilities for various configurations about the carbon atoms connected to the linkage oxygen atom. The analogs were based on
tetrahydropyran and, in the case of the sucrose analogs, tetrahydrofuran. The molecular mechanics program MM3 calculates
these values as an option. The method was also used with RHF/6-31G ⴱ and B3LYP/6-31G ⴱ quantum mechanics (QM) theory.
DHf values had ranges of 15–17 kcal (all energies herein are molar) by the different methods, with the analogs of the three
trehaloses and sucrose having the lowest values. DHf values for the isomeric analogs of non-reducing sugars with two anomeric
centers are about ⫺150 kcal, about 8 kcal lower than for the analogs of the reducing dimers (nigerose, maltose, laminarabiose,
cellobiose and galabiose (a-d-galactosyl-d-galactose)) by all three types of calculation. The analogs of the di-axial, axial–
equatorial and di-equatorial, non-glycosidic pseudo-disaccharides had DHf values within 1.0 kcal by each of the three methods.
They were about 8 kcal higher than for the molecules that contain one glycosidic sequence. The relative DHf values by QM
were within 1.0 kcal of their corresponding relative electronic energies, except for the methylated sucrose analog, for which the
discrepancy was about 1.85 kcal at both QM levels. Compared to our B3LYP results, the RHF-based calculations overestimated
the stability of all molecules by about 2.85 kcal. The MM3 values were close to the B3LYP numbers, with the largest
discrepancy for the cellobiose analog, 1.76 kcal. The stabilization embodied in the group enthalpy increment for anomeric
centers is another manifestation of the anomeric effect. 䉷 2000 Elsevier Science B.V. All rights reserved.
Keywords: Carbohydrates; Cellobiose; Exo-anomeric effect; Galabiose; Laminarabiose; Maltose; Nigerose; Sucrose; Tetrahydropyran; Tetrahydrofuran; Trehalose
1. Introduction
Once standard states are defined, the heat of formation, DHf, is a quantity that can be calculated by a
* Corresponding author. Tel.: ⫹1-504-286-4410; fax: ⫹1-504286-4217.
E-mail address: [email protected] (A.D. French).
variety of methods as well as determined experimentally. Thus it provides a useful connection between
methods of predicting relative energies and structures
and observable reality. We have previously found it
useful to calculate DHf in our studies of carbohydrates. For example, lattice energies give useful information on the forces that might be distorting
molecules in crystals compared to their shapes in an
0022-2860/00/$ - see front matter 䉷 2000 Elsevier Science B.V. All rights reserved.
PII: S0022-286 0(00)00648-7
304
A.D. French et al. / Journal of Molecular Structure 556 (2000) 303–313
Fig. 1. The compounds used in this paper, with the conformations determined by B3LYP/6-31G ⴱ theory. The oxygen atoms are shown in gray,
the hydrogen atoms are not shown. All carbon atoms, shown in black are nominally tetrahedral. The atoms are numbered according to the
corresponding numbers for the disaccharides with standard carbohydrate nomenclature.
A.D. French et al. / Journal of Molecular Structure 556 (2000) 303–313
isolated state [1]. In that work, the lattice energy is the
difference between the calculated DHf of the isolated
molecule and the DHf determined by the experimental
heat of combustion for the crystalline solid. Another
example of our use of DHf calculations concerns the
relative stabilities of monosaccharide isomers such as
fructofuranose and fructopyranose that have different
sizes of rings [2]. DHf values give information on the
relative stability that is not inherent in the simple
steric energies. However, calculations of correct DHf
values for molecules such as carbohydrates that have
anomeric centers are more difficult than for other
molecules [3], and further verification of those calculated values is especially important. In some recent
work [4,5], we used both RHF/6-31G ⴱ quantum
mechanics (QM) theory and MM3 molecular
mechanics (MM) theory to construct hybrid energy
surfaces for a number of disaccharides. Those results,
which covered the full range of linkage conformations, depend on compatibility between the QM and
MM methods. One aspect of that compatibility would
be demonstrated if the calculated DHf values for those
molecules were comparable when calculated by both
methods. Determination of the degree of compatibility of both the QM and MM theory was also a
prime motivation for carrying out the present work.
This paper presents the DHf values for analogs of 13
disaccharides (Fig. 1) that were calculated with
MM3(96) 1 molecular mechanics software and with
RHF/6-31G ⴱ and B3LYP/6-31G ⴱ levels of QM
theory. As far as we know, these molecules exist at
present only as computer models. They have various
configurations of the linkages between the tetrahydropyran (THP) rings (or, in two cases, tetrahydrofuran
(THF) rings) that are used to represent the participating monosaccharides. Addition of a methyl group
to the C2 0 of the THP–O–THF sucrose analog gave a
second, more representative analog of sucrose that has
the same number of atoms and bonds of each type as
the analogs of the other disaccharides that are based
on THP. These THP–O–THP and THP–O–THF
models have most of the types of two-bond, interring linkages found in disaccharides. The configura1
Academic users may obtain MM3 from the Quantum Chemistry
Program Exchange, Indiana University, Bloomington, IN 47405
USA. All others may obtain it from Tripos Associates, 1699
South Hanley Road St. Louis, MO 63144.
305
tions include di-axial, axial-equatorial, and di-equatorial variations of typical glycosidic linkages (such as
cellobiose or maltose), head-to-head glycosidic
linkages (such as sucrose and the trehaloses) and
non-glycosidic linkages (for our tail-to-tail, 3,3-linked
pseudo-disaccharide analogs). These last three molecules are analogs of disaccharide molecules that do
not themselves exist, either.
The conformations and energies of disaccharide
linkages are thought to be affected substantially by
the general anomeric effect [6,7]. Anomeric carbon
atoms are attached to both endo- and exo-cyclic
oxygen atoms, and the bond between the anomeric
carbon and the exo-cyclic oxygen atom is called a
glycosidic bond. At our current level of understanding, the stereoelectronic, general anomeric effect
causes C–O–C–O sequences to tend to take gauche
conformations, rather than the trans conformation that
is preferred for the C–C–C–C sequences. In structures such as 2-hydroxytetrahydropyran, the gauche
conformation of the C–O–C–O sequence results in an
axial disposition of the glycosidic bond (the a-gluco
configuration), while the trans conformation corresponds to an equatorial orientation (the b-gluco
configuration). However, the configurational energy
is also governed by other torsional energies, such as
C–C–C–O or C–C–O–C, as well as other factors,
and the gauche conformation may not predominate.
Therefore, the anomeric effect is, in practice, an unexpected degree of preference for gauche conformations, if our expectations were based on the behavior
of C–C–C–C sequences. When the gauche conformation refers to the favored orientation of the C1–
O1 glycosidic bond to the O5–C5 bond (Fig. 1), the
cause is said to be the endo-anomeric effect. When it
concerns the preferred orientation of the bond to the
aglycon, e.g. the O5–C1–O1–C4 0 sequence in
maltose, the cause is described as the exo-anomeric
effect.
2. Methods
Both MM and QM (Hartree–Fock and Density
Functional Theory) can use the bond- and groupenthalpy increment scheme to calculate DHf. Even
when economical levels of theory are employed, the
method has demonstrated accuracy within
306
A.D. French et al. / Journal of Molecular Structure 556 (2000) 303–313
experimental error for a wide variety of typical
organic compounds. This method, championed by
Prof. Allinger for both MM and QM [3,8–10] is
based on the premise that the change in enthalpy of
the elements in their standard states to the enthalpy
of a compound (i.e. DHf) can be reduced to a summary
of the enthalpy of formation of the individual bonds,
along with a measure of the strain induced by
constructing the molecule. Certain additional enthalpies for formation of rings, for terminal methyl groups
etc., which are systematically not calculated correctly
are added to this information. In the most correct
calculations of DHf, there are also compensations for
conformational flexibility and the overall translational
and rotational degrees of freedom. The bond- and
group-enthalpy effects are determined for various
classes of molecules by optimizing their values so
as to best reproduce the known DHf values for a
number of different molecules. Typically, the molecules used for these calibrations are small, and
systematic errors can be magnified for compounds
as large as studied in the present work.
As an option, the MM3 program [11], draws upon
built in tabulations of such bond- and group-enthalpies to calculate the DHf for the compound in question. The remaining term is the strain in the molecule
induced by its formation. In the formalism of MM3,
this strain corresponds to the “final steric energy”
term, composed of the bond stretching, the angle
bending, the van der Waals terms, the torsional
term, the dipole–dipole energy and a few smaller
contributions. As a practical note for users of MM3,
the steric energy values for isomers with different ring
sizes do not correctly indicate their relative enthalpies
but their DHf values should. This is precisely because
the group effects for different ring sizes must also be
included in the comparison.
The QM calculations are quite similar, but are not
available in the usual QM software. Instead, they may
be carried out expeditiously with a spreadsheet
program. Here the computed quantity that is modified
by the sum of the bond- and group-enthalpy increments is the total electronic energy [8]. That total
contains the calculated DHf information, and the
task for the bond enthalpy increments is to convert
the total electronic energy to the DHf by removing
the contributions arising from the electronic energy
not only for the formation of bonds between the
atoms in their standard states but also the formation
of the atoms themselves from the electrons and nuclei.
This makes the bond-enthalpy increments for the QM
calculations very large. The total calculated electronic
energy of a molecule depends very much on the particular level of QM theory that was used, and so the
individual bond- and group-enthalpy contributions
will be applicable to only one level of theory. For
example, the RHF/6-31G ⴱ and B3LYP/6-31G ⴱ total
electronic energies for our a,a-trehalose analog differ
by 3.82 Hartrees (almost 2400 kcal, all energies
herein are molar), and the C–C bond-enthalpy increments for the two levels of theory differ by 74 kcal/
mol of bonds.
Ideally, the group-enthalpy increments would not
be required with QM methods, but the limitations of
affordable levels of theory require these additional
empirical corrections. Besides the ring-size effects,
our molecules require group increments for anomeric
centers with various substitutions, O–C–C–O
sequences, and secondary oxygen atoms such as
those that are part of the ring when they are not adjacent to an anomeric center. The QM calculations of
DHf values now also include group enthalpy increments for six-membered rings [10]. Besides values
for C–C and C–H bonds and applicable groups
from that work on alkanes, our QM-derived values
used preliminary, unpublished new parameters for
the bond and group enthalpies of alcohols and ethers
from the Allinger group [12]. 2
MM3 values of DHf were calculated for isolated
molecules with its optional routine and the default
dielectric constant of 1.5, after constructing the
complete f , c energy surfaces and determining the
global minimum energy conformations. The RHF/631G ⴱ values were for the global minima from the
relaxed conformational surfaces calculated with that
level of theory using Gaussian 94 [13] or gamess
[14]. The locations of the global minima for the laminarabiose and cellobiose analogs are different on the
RHF/6-31G ⴱ and MM3 surfaces. However, these
minima correspond to positions that are just slightly
higher in relative energy on the surfaces calculated by
the other method. The B3LYP/6-31G ⴱ values,
computed using Gaussian 94, were from
2
These provisional parameters are believed to be final but are still
subject to last-minute corrections.
A.D. French et al. / Journal of Molecular Structure 556 (2000) 303–313
307
Table 1
Linkage torsion angles and energies of the global minima computed with B3LYP/6-31G ⴱ, RHF/6-31G ⴱ and MM3(96) for the disaccharide
analogs (kcal)
fa
Sugar analog
ca
DHf (kcal)
Rel. DHf (kcal)
QM Rel. b Eelect. (kcal)
a,a-Trehalose
Axial–axial
1,1-Linked
B3LYP/6-31G ⴱ
RHF/6-31G ⴱ
MM3
70.14
71.39
74.50
70.14
71.38
76.50
⫺149.86
⫺153.40
⫺150.03
0.36
0.00
0.35
0.00
0.00
–-
a,b-Trehalose
Axial–equatorial
1,1-Linked
B3LYP/6-31G ⴱ
RHF/6-31G ⴱ
MM3
58.85
61.90
76.91
⫺96.11
⫺95.33
⫺79.53
⫺147.38
⫺150.43
⫺147.79
2.84
2.95
2.59
2.48
2.95
–-
b,b-Trehalose
Equatorial–equatorial
1,1-Linked
B3LYP/6-31G ⴱ
RHF/6-31G ⴱ
MM3
⫺71.50
⫺70.09
⫺76.87
⫺71.50
⫺70.09
⫺76.87
⫺147.45
⫺150.89
⫺148.60
2.77
2.50
1.78
2.41
2.50
–-
Sucrose (THF–O–THP)
Axial–pseudo-axial
1,2-Linked (28 atoms)
B3LYP/6-31G ⴱ
RHF/6-31G ⴱ
MM3
58.38
60.77
78.70
⫺97.22
⫺98.16
⫺78.03
⫺140.18
⫺143.57
⫺141.67
10.04
9.82
8.71
Sucrose (Methyl–THF–O–
THP)
Axial–pseudo-axial
1,2-Linked (31 atoms)
B3LYP/6-31G ⴱ
98.67
⫺36.51
⫺150.22
0.00
1.88
100.72
80.05
⫺40.16
⫺63.67
⫺152.76
⫺150.38
0.63
0.00
2.44
–
Nigerose
Axial–equatorial
1,3-Linked
B3LYP/6-31G ⴱ
RHF/6-31G ⴱ
MM3
⫺48.53
⫺46.77
⫺41.77
⫺36.46
⫺33.28
⫺40.52
⫺142.59
⫺145.16
⫺141.97
7.64
8.22
8.41
7.29
8.25
–
Laminarabiose
Equatorial–equatorial
1,3-Linked
B3LYP/6-31G ⴱ
RHF/6-31G ⴱ
MM3
51.12
54.33
35.66
⫺20.83
⫺14.43
⫺51.21
⫺141.69
⫺145.08
⫺140.57
8.54
8.31
9.81
8.19
8.33
–
Maltose
Axial–equatorial
1,4-Linked
B3LYP/6-31G ⴱ
RHF/6-31G ⴱ
MM3
⫺47.15
⫺47.00
⫺45.46
⫺30.70
⫺32.91
⫺40.41
⫺141.31
⫺143.79
⫺140.06
8.92
9.60
10.32
7.99
9.37
–
Cellobiose
Equatorial–equatorial
1,4-Linked
B3LYP/6-31G ⴱ
RHF/6-31G ⴱ
MM3
50.94
53.84
34.74
⫺19.33
⫺13.59
⫺51.72
⫺140.62
⫺143.94
⫺138.86
9.60
9.45
11.52
8.67
9.21
–
Galabiose
Axial–axial
1,4-Linked
B3LYP/6-31G ⴱ
RHF/6-31G ⴱ
MM3
⫺48.87
⫺47.97
⫺44.66
⫺37.12
⫺35.83
⫺38.49
⫺140.44
⫺143.64
⫺140.06
9.78
9.75
10.32
8.85
9.51
–
Pseudo dimer d
Di-axial
3,3-Linked
B3LYP/6-31G ⴱ
HF/6-31 ⴱ
MM3
35.26
32.91
33.36
35.26
32.92
35.65
⫺134.14
⫺136.63
⫺133.59
16.08
16.76
16.79
15.75
16.80
–
Pseudo dimer d
Axial–equatorial
3,3-Linked
B3LYP/6-31G ⴱ
HF/6-31 ⴱ
MM3
35.56
32.34
42.14
35.47
33.87
41.24
⫺134.47
⫺136.54
⫺133.55
15.75
16.84
16.83
15.42
16.89
–
Pseudo dimer d
Di-equaotrial
3,3-Linked
B3LYP/6-31G ⴱ
HF/6-31 ⴱ
MM3
35.91
37.29
38.88
35.91
37.29
38.38
⫺134.76
⫺136.32
⫺133.57
15.46
17.07
16.81
15.14
17.11
–
RHF/6-31G ⴱ
MM3
c
c
–
a
For the trehalose and sucrose analogs, f is defined as O5–C1–O1–C1 0 (O5–C1–O1–C2 0 for sucrose) and c is defined by O5 0 –C1 0 –O1–
C1 (O5 0 –C2 0 –O1–C1 for sucrose). For nigerose, laminarabiosse, maltose and cellobiose analogs, f is defined by H1–C1–O1–C4 0 (H1–C1–
O1–C3 0 for nigerose and laminarabiose), and c is defined by H4 0 –C4 0 –O1–C1 (H3 0 –C3 0 –O1–C1 for nigerose and laminarabiose).
b
Energies calculated with gamess and Gaussian 94 were the same to seven decimal places. The lowest electronic energies were for the a,atrehalose analog, at ⫺613.7542486 a.u. (RHF/6-31G ⴱ) and ⫺617.5741626 a.u. (B3LYP/6-31G ⴱ).
c
Relative energies are affected because the sucrose analog without the methyl group has fewer atoms than the other analogs. Its electronic
energies are ⫺574.7118895 (RHF) and ⫺578.25309698 (B3LYP) a.u. Therefore, these relative values are not included.
d
These molecules have an equivalent minimum at ⫺f , ⫺c .
308
A.D. French et al. / Journal of Molecular Structure 556 (2000) 303–313
minimizations of the RHF/6-31G ⴱ structures. No
other basis sets were used. In some calculations with
the B3LYP level of theory, satisfactory termination of
the minimization required the use of a step size of 0.05
[IOP(1/8 ˆ 5)]. This was used with the SCF ˆ Tight
keyword. All DHf values are for 298⬚C.
For the molecules with head-to-head glycosidic
bonds, the f torsion angle was defined by O5–C1–
O1–C1 0 (O5–C1–O1–C2 0 for the sucrose analogs),
and the c torsion angle was defined by C1–O1–C1 0 –
O5 0 (C1–O1–C2 0 –O5 0 for the sucrose analogs). See
Fig. 1 for the atom numbering. Torsion angles for the
other structures were defined with the hydrogen atoms
attached to the carbon atoms that participate in the
inter-ring linkages (e.g. f ˆ H1–C1–O1–C4 0 and
c ˆ C1–O1–C4 0 –H4 0 for galabiose (a-galactosyl(1,4)-galactose), maltose and cellobiose).
3. Results and discussion
DHf values and f , c coordinates of the global
minima for the 13 analogs are in Table 1. Fits of the
QM DHf values to the relative electronic energies are
shown in Fig. 2, and the fits of the RHF/6-31G ⴱ and
MM3 DHf results to the B3LYP/6-31G ⴱ DHf data are
shown in Fig. 3. The THF–O–THP analog of sucrose
is not included in Fig. 2 and its relative electronic
energy is not in Table 1. Its electronic energy is
much smaller than for the other molecules because
it has three fewer atoms. Alternative treatments of
the data based on subtracted averages instead of relative energies give essentially the same results.
The most important results are the large, 15–
17 kcal ranges for these isomers, and the fact that
the DHf values computed with all three levels of
theory and the relative electronic energies are roughly
parallel. Also, the energies are in three distinct clusters. The lowest values of about ⫺150 kcal are for the
analogs of the trehaloses and the methylated THF–O–
THP analog of sucrose. Those molecules have two
glycosidic bonds that connect to the same oxygen
atom. In a second cluster, about 8 kcal higher, are
the analogs with a single glycosidic bond (nigerose,
laminarabiose, maltose, cellobiose and galabiose).
The THF–O–THP sucrose analog, with two
anomeric centers and three fewer atoms and
bonds, coincidentally falls in that group, too.
Highest, by another 7–8 kcal, are the molecules
having simple ether linkages with no glycosidic bonds.
The a,a-trehalose analog with di-axial linkages had
the lowest electronic energy by either QM method.
This molecule, wherein all of its O–C–O–C
sequences are in the preferred gauche conformations,
is stabilized by both endo- and exo-anomeric effects.
The lowest DHf values for the B3LYP and MM3
levels were found for the methylated sucrose analog.
It has an axial linkage bond from the pyranosyl ring,
and a pseudo-axial linkage bond from the furanosyl
ring. Along with its f and c values, the O1–C2 0 –
O5 0 –C5 0 torsion angle deviates substantially from the
theoretically preferred gauche conformations because
of the furanosyl ring. These non-ideal conformations,
as well as slightly higher electronic energies, suggest
that the MM3 and B3LYP DHf values might be somewhat lower than the correct value for Me–THF–O–
THP.
Labanowski et al. found that the B3LYP/6-31G ⴱ
calculations gave statistically more accurate calculations of DHf than did the RHF calculations with the 6/
31G ⴱ basis set [10]. Also, the magnitude of the groupenthalpy increments is generally considerably smaller
with B3LYP theory than for the RHF theory. This is
especially so for the group-enthalpies for the ring size,
with the value for six-membered rings being 5.97 kcal
by RHF/6-31G ⴱ theory and only 1.37 kcal by B3LYP
theory. The five-membered ring enthalpy is similarly
reduced, from 4.60 to 0.70 kcal. The reduction in
group-enthalpy increment size is probably due to the
accounting for more of the electron correlation in
B3LYP theory. The main exception is for the O–C–
C–O group (see below). The other B3LYP group
enthalpies are smaller than their RHF counterparts
by about one third. This suggests that the B3LYP
theory has less need of supplemental information to
predict DHf values [10]. However, a recent study of
monosaccharides indicates that B3LYP/6-31G ⴱ
theory should not be used on compounds with many
hydroxyl groups such as sugars [15].
The average difference between the B3LYP relative
electronic energies and the relative DHf values
(0.33 kcal) is slightly larger than the comparable
difference for the RHF calculations (⫺0.10 kcal).
Table 1 and Fig. 2 show that for both QM methods,
the biggest deviation between the relative electronic
energy and the relative DHf is for the methylated
A.D. French et al. / Journal of Molecular Structure 556 (2000) 303–313
309
Fig. 2. Agreement between the relative electronic energy and the relative DHf for our compounds, as calculated by the RHF and B3LYP levels
of theory. The points corresponding to the methylated sucrose analog (Me–THF–O–THP) are indicated, showing the relative electronic energy
to be 1.85 kcal/mol greater than the relative DHf while the other compounds agree quite well. Relative DHf and electronic energies are relative
to the lowest values found for any of the compounds—see Table 1.
sucrose analog (B3LYP ˆ 1.88 kcal, RHF ˆ 1.81
kcal). This suggests that methyl groups attached to
the anomeric carbon may require a different groupenthalpy increment, compared to the methyl group
attached to non-anomeric carbon atoms. Table 1 and
Fig. 3 show that the disagreement between the B3LYP
and MM3 DHf values is strongest for the cellobiose
analog (1.76 kcal) and for THF–O–THP analog of
sucrose (1.49 kcal). Other than the offset shown by
Fig. 3, there is no large discrepancy between the
RHF and the B3LYP DHf values.
The preference for the axial glycosidic-bonded
members of the nigerose/laminarabiose and maltose/
cellobiose pairs is small by the B3LYP method and
smaller by RHF theory. Either on the basis of the
relative electronic energies or the relative DHf values,
the axial–equatorial analog of nigerose is preferred by
0.90 kcal (B3LYP) or 0.08 kcal (RHF) over the diequatorial laminarabiose analog. The axial–equatorial
maltose analog is preferred by 0.68 kcal (B3LYP) or
0.15 kcal (RHF) over the di-equatorial cellobiose
analog. The axial–equatorial members of these pairs
are favored more with MM3, by 1.40 kcal and
1.20 kcal, respectively. By MM3, the DHf and f
and c values for the galabiose analog are roughly
comparable to those for maltose. However, the
Table 2
Relative values of the most variable MM3 energy components for the trehaloses at the respective minimum-energy conformations
Trehalose
Total
Bending
Dipole–dipole
Torsion
Sum of major components
a,a
a,b
b,b
0.00
2.23
1.43
1.68
1.23
0.00
0.00
1.60
0.95
0.00
1.24
2.39
1.68 (0.00)
4.07 (2.39)
3.34 (1.66)
310
A.D. French et al. / Journal of Molecular Structure 556 (2000) 303–313
Fig. 3. Comparison of the MM3 and RHF heats of formation to the B3LYP DHf. The RHF values are systematically higher than the B3LYP
values by 2.85 kcal, while the MM3 values and the B3LYP values agree very well, except for the cellobiose analog and THF–O–THP
molecule, for which the agreement is only fair.
axial–axial configuration in the former has a less
stable B3LYP DHf value by 0.9 kcal, compared to
the DHf value for the axial–equatorial configuration
in maltose.
As shown in Fig. 3, the RHF values of DHf are
offset from the B3LYP values by ⫺2.85 kcal on the
average, a rather large systematic discrepancy. On the
other hand, the MM3 values are fairly similar to
the B3LYP values, being 0.34 kcal higher on average.
Indeed, differences between our RHF and MM3
calculated DHf values are relatively large, compared
to the mean errors of less than 1 kcal reported for
these methods [3,11]. Still, confidence in each method
is enhanced by the parallel results. The relative QM
electronic energies strongly support the ranking of the
DHf values for these isomeric analogs. Work to be
published elsewhere that involved hybrids of RHF/
6-31G ⴱ and MM3 energy surfaces [4,5] depends to
some extent on compatibility between these two
levels of theory. The comparable DHf values from
both the QM and MM3 methods imply the validity
of the assumption of comparable bond stretching,
angle bending and van der Waals forces to a first
approximation.
Table 2 shows the MM3 components of the energy
that differ the most among the minimum energy structures of the three trehaloses. (Our previous energies
for the methylated analogs of the trehaloses [16] were
calculated at a dielectric constant of 4.0, not appropriate for comparison with QM calculations.)
Although not expected to be quantitatively meaningful (because of coupling between terms during
parameterization), such component analysis offers
qualitative insight into dominant energetic interactions when differences are large. Di-axial a,a-trehalose has the most bending energy and di-equatorial
b,b-trehalose has the least. The dipole–dipole interactions are the highest (most destabilizing) for a,btrehalose. In its minimum-energy conformation, the
C1–O5 and C1 0 –O5 0 bonds are adjacent and roughly
parallel (Fig. 1) while the same bonds are more distant
and directed antiparallel in the other two trehalose
analogs. The relative torsion angle energies of 0.00,
1.24 and 2.38 kcal are consistent with the notion of
A.D. French et al. / Journal of Molecular Structure 556 (2000) 303–313
311
Table 3
Relative energy components of a,a-trehalose and cellobiose analogs from MM3 DHf calculations (kcal)
Component
Compression
Bending
Bend–bend
Stretch–bend
van der Waals 1,4
van der Waals Other
Torsion
Torsion–stretch
Dipole–dipole
Total steric energy
Bond enthalpy increments
C–C
C–O
C–H
Group enthalpy increments
O–CHR–O
O-Sec sp 3
O–C–C–O
Population, torsion
&translation/rotation
Total DHf and D DHf
a,a-Trehalose
Cellobiose
Difference
0.40
4.02
⫺0.07
0.25
13.59
⫺0.60
⫺2.17
⫺0.02
⫺5.62
9.76
0.52
2.97
⫺0.11
0.20
13.20
⫺0.22
2.76
⫺0.03
⫺1.80
17.50
0.12
⫺1.05
⫺0.04
⫺0.05
⫺0.39
0.38
4.93
⫺0.01
3.82
7.74
8 × 2:45 ˆ 19:58
6 × ⫺14:32 ˆ ⫺85:91
18 × ⫺4:59 ˆ ⫺82:62
8 × 2:45 ˆ 19:58
6 × ⫺14:32 ˆ ⫺85:91
18 × ⫺4:59 ˆ ⫺82:62
0
0
0
2 × ⫺6:618 ˆ ⫺13:236
–
–
2.40
1 × ⫺6:618 ˆ ⫺6:62
1 × ⫺5:03 ˆ ⫺5:03
1 × 1:84 ˆ 1:84
2.40
6.62
⫺5.03
1.84
0
⫺150.03
⫺138.86
11.17
two, one and zero anomeric stabilizations for the a,a-,
a,b- and b,b-trehaloses, respectively, while the exoanomeric stabilizations remain constant. These exoanomeric stabilization energies may not actually be
constant, however. For one thing, the c torsion for
the a,b-trehalose analog deviates substantially from
the gauche conformation. Another consequence of the
preferred conformation of a,b-trehalose affects the
ability of the lone pairs of electrons on the linking
oxygen atom to delocalize into the s ⴱ orbitals for
the C1–O5 and C1 0 –O5 0 bonds. For a,a- and b,btrehalose analogs in their minimum-energy conformations, each of the lone pairs of electrons on the linkage
oxygen is positioned to stabilize the molecule by
being trans to a different C1–O5 bond. In the a,btrehalose analog, one of the lone pairs is trans to both
C1–O5 bonds and the other lone pair is trans to
neither. This analysis is based on the very rough
approximation that the lone pairs on oxygen have
sp 3 orientations. Still, less stabilization would be
expected under this situation. Research continues in
this area.
Table 3 shows the different components of the
MM3 DHf calculation for the analogs of a,a-trehalose
and cellobiose which, except for the analogs of the
imaginary pseudo disaccharides, gave the low and
high MM3 DHf values. The differences between the
two molecules include 7.8 kcal of steric energy
(MM3’s standard “Total Steric Energy”) and
3.4 kcal of group enthalpy additions. The main contributions to the steric energy difference include
1.05 kcal for bending, favoring the di-equatorial
cellobiose analog; 4.93 kcal of torsional energy
favoring the trehalose analog (two anomeric effects
and one extra exo-anomeric effect); and 3.82 kcal of
dipole–dipole interactions, also favoring the trehalose
analog. The extra anomeric group effect in trehalose is
offset to some extent by the presence of a group effect
for the secondary sp 3 oxygen atom in the cellobiose
analog.
The nigerose analog has a considerably lower MM3
DHf (Table 1, 1.91 kcal) than the maltose analog at least
partly because of a group increment of 1.83 kcal for the
O5 0 –C5 0 –C4 0 –O1 sequence in the maltose analog
model. The nigerose/maltose difference of 1.91 kcal
by MM3 can be compared with the smaller 1.37 and
1.28 kcal differences calculated by RHF and B3LYP
theory for the same molecules. The O–C–C–O group
312
A.D. French et al. / Journal of Molecular Structure 556 (2000) 303–313
enthalpy increments for the RHF and B3LYP calculations are only 0.26 and 0.58 kcal, respectively (the only
group-enthalpy increment that is bigger for B3LYP
theory than for RHF theory). A special O–C–C–O
group enthalpy increment was not developed for when
one of the oxygen atoms was part of an anomeric center.
The laminarabiose analog also has a lower MM3 DHf
(by 1.71 kcal) than the cellobiose analog. The laminarabiose analog is favored by the RHF and B3LYP DHf
values by the smaller amounts of 1.14 and 1.07 kcal.
Because our compounds are isomeric (except for
the THP–O–THF analog of sucrose), there are 18
H–C, 8 C–C and 6 C–O bonds in all of the molecules.
Despite the different group effects that apply, their
DHf values track their electronic energies fairly well
as shown in Fig. 2. That property makes the present
compounds a valuable and interesting set, even
though they have not yet been synthesized and no
experimental DHf values exist. Of course, synthesis
and DHf experiments on the different members of
this set would be interesting to confirm the very
large differences in DHf. Limiting the study to only
the tetrahydropyran and tetrahydrofuran models has
several advantages over the studying the actual sugars
when QM is used. For one, the computer time is
reduced. Secondly, the hydrogen bonding, a serious
matter for true carbohydrates, is avoided in these
molecules. In particular, basis set superposition errors
are fairly large (e.g. 2.3 kcal per hydrogen bond) at the
B3LYP/6-31G ⴱ level for hydrogen-bonded water and
methanol clusters [15], so molecular energies of
actual disaccharides which have 8 or so hydroxyl
groups might be badly exaggerated. In compensation
for ignoring the hydrogen bonding, we were able to
focus on the impact of various sorts of glycosidic
linkages on molecular energies. While these calculations suggest that it is feasible to calculate heats of
formation for these compounds and thus the disaccharides themselves with some expectation of accuracy, our analog DHf values cannot be extrapolated to
the parent disaccharides themselves. For example, the
maltose molecule would not be expected to have a
DHf value some 8 kcal above that of a,a-trehalose.
The real disaccharide has two anomeric centers, but
the analog has only one. The additional group effect
plus the intrinsically calculated enthalpy for the
second anomeric center would increase the calculated
stability of the maltose molecule.
4. Conclusions
DHf values for disaccharide analogs based on tetrahydrofuran and tetrahydropyran were calculated at the
B3LYP/6-31G ⴱ, RHF/6-31G ⴱ and MM3 levels.
Insight was gained by studying five molecules having
various configurations of glycosidic linkages with
overlapping exo-anomeric sequences and five others
having ordinary glycosidic linkages. Three other
isomers with simple ether linkages were also studied.
There was a large (15–17 kcal) range of electronic
energies and DHf for the conformations at the global
minima of the different isomeric analogs, and there
was reasonable correspondence between the DHf
values from both MM3 and QM calculations. The
mostly excellent agreement between the MM3 and
B3LYP/6-31G ⴱ values for these moderately large,
problematic compounds having anomeric sequences
is especially remarkable because the parameterization
for MM3 was established more than a decade ago.
Even the fit to the RHF/6-31G ⴱ results, after compensating for a 2.85 kcal offset, was quite good. Of
course, synthesis of these compounds and experimental verification of the large range in their heats
of formation is a good example of actual experiment
suggested by computations.
Acknowledgements
We thank Professor N.L. Allinger for valuable
discussion along with Dr K.-H. Chen for furnishing
the prepublication parameters. Dr Chen also assisted
with the QM DHf calculations. We also thank
Professor Lothar Schäfer for suggesting the collaboration between ADF and AMK. Besides Professor
Allinger, Drs J. Labanowski and W.E. Franklin read
presubmission versions of this paper.
References
[1] A.D. French, M.K. Dowd, S.K. Cousins, R.M. Brown, D.P.
Miller, ACS Symp. Ser. 618 (1995) 13–37.
[2] A.D. French, M.K. Dowd, P.J. Reilly, J. Mol. Struct. (Theochem) 395/396 (1997) 271–287.
[3] N.L. Allinger, L.R. Schmitz, I. Motoc, C. Bender, J.K. Labanowski, J. Phys. Org. Chem. 3 (1990) 732–736.
[4] A.D. French, A.-M. Kelterer, C.J. Cramer, G.P. Johnson, M.K.
Dowd, Carbohydr. Res. 326 (2000) 305–322.
A.D. French et al. / Journal of Molecular Structure 556 (2000) 303–313
[5] A.D. French, A.-M. Kelterer, G.P., Johnson, M.K. Dowd, C.J.
Cramer. Constructing and evaluating energy surfaces of crystalline disaccharides. J. Mol. Graphics and Modelling 18
(2000) in press.
[6] G.R.J. Thatcher (Ed.), The Anomeric Effect and Associated
Stereoelectronic Effects ACS Symposium Series, vol. 539,
ACC, Washington, DC, 1994.
[7] I. Tvaroška, T. Bleha, Adv. Carbohydr. Chem. Biochem. 47
(1989) 45–123.
[8] N.L. Allinger, L.R. Schmitz, I. Motoc, C. Bender, J.K. Labanowski, J. Am. Chem. Soc. 114 (1992) 2880–2883 (See the
supplemental information for this paper for many of the
details of the heat of formation calculations. Copies are available from Professor Allinger).
[9] N.L. Allinger, K. Sakakibara, J. Labanowski, J. Phys. Chem.
99 (1995) 9603–9610.
[10] J.K. Labanowski, L. Schmitz, K.-H. Chen, N.L. Allinger, J.
Comput. Chem. 19 (1998) 1421–1430.
[11] N.L. Allinger, M. Rahman, J.-H. Lii, J. Am. Chem. Soc. 112
(1990) 8293–8307.
313
[12] K.-H. Chen, N.L. Allinger, personal communication.
[13] M.J. Frisch, G.W. Trucks, H.B. Schlegel, P.M.W. Gill, B.G.
Johnson, M.A. Robb, J.R. Cheeseman, T. Keith, G.A.
Petersson, J.A. Montgomery, K. Raghavachari, M.A. AlLaham, V.G. Zakrzewski, J.V. Oritz, J.B. Foresman, J.
Cioslowski, B.B. Stefanov, A. Nanayakkara, M. Challacombe,
C.Y. Peng, P.Y. Ayala, W. Chen, M.W. Wong, J.L. Andres,
E.S. Replogle, R. Bomperts, R.L. Martin, D.J. Fox, J.S.
Binkley, D.J. DeFrees, J. Baker, J.P. Stewart, M.L. HeadGordon, C. Gonzalez, J.A. Pople, Gaussian 94. Gaussian,
Inc., Pittsburgh, PA, 1995.
[14] M.W. Schmidt, K.K. Baldridge, J.A. Boatz, S.T. Elbert, M.S.
Gordon, J.H. Jensen, S. Koseki, N. Matsunaga, K.A. Nguyen,
S.J. Su, T.L. Windus, M. Dupuis, J.A. Montgomery, J.
Comput. Chem. 14 (1993) 1347–1363.
[15] J.-H. Lii, B. Ma, N.L. Allinger, J. Comput. Chem. 20 (1999)
1593–1603.
[16] M.K. Dowd, P.J. Reilly, A.D. French, J. Comput. Chem. 13
(1992) 102–114.