The conformational distribution in diphenylmethane determined by nuclear magnetic resonance spectroscopy of a sample dissolved in a nematic liquid crystalline solvent

Article abstract of DOI:10.1063/1.1555631

The application of the probabilistic approach, coupled with the AP model for the conformational dependence of the interaction with liquid crystal molecules, gives a distribution P_LC(τ₁, τ₂) which is similar to that found b

Full text of DOI:10.1063/1.1555631

The conformational distribution in diphenylmethane determined by nuclear magnetic
resonance spectroscopy of a sample dissolved in a nematic liquid crystalline solvent
G. Celebre, G. De Luca, J. W. Emsley, E. K. Foord, M. Longeri, F. Lucchesini, and G. Pileio
Citation: The Journal of Chemical Physics 118, 6417 (2003); doi: 10.1063/1.1555631
View online: http://dx.doi.org/10.1063/1.1555631
View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/118/14?ver=pdfcov
Published by the AIP Publishing
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JOURNAL OF CHEMICAL PHYSICS
VOLUME 118, NUMBER 14
8 APRIL 2003
The conformational distribution in diphenylmethane determined
by nuclear magnetic resonance spectroscopy of a sample dissolved
in a nematic liquid crystalline solvent
G. Celebre and G. De Luca
Dipartimento di Chimica, Universita della Calabria, 87030 Arcavacata di Rende (Cs), Italy
J. W. Emsley and E. K. Foord
Chemistry Department, University of Southampton, Southampton SO17 1BJ, United Kingdom
M. Longeri, F. Lucchesini,a) and G. Pileio
Dipartimento di Chimica, Universita della Calabria, 87030 Arcavacata di Rende (Cs), Italy
͑Received 10 October 2002; accepted 3 January 2003͒
The deuterium decoupled, proton nuclear magnetic resonance spectrum of a sample of
diphenylmethane-d3 dissolved in a nematic liquid crystalline solvent has been analyzed to yield a
set of dipolar couplings, Dij . These have been used to test models for the conformational
distribution generated by rotation about the two ring-CH2 bonds through angles ␶ and ␶2 .
1
Conformational distributions, particularly when obtained from a quantum chemistry calculation, are
usually described in terms of the potential energy surface, V(␶1 ,␶2), which is then used to define
a probability density distribution, P(␶1 ,␶2). It is shown here that when attempting to obtain
P(␶1 ,␶2) from experimental data it can be an advantage to do this directly without going through
the intermediate step of trying to characterize V(␶ ␶2). When applied to diphenylmethane this
1,
method shows that the dipolar couplings are consistent with a conformational distribution centered
on ␶1ϭ␶2ϭ56.5Ϯ0.5°, which is close to the values calculated for an isolated molecule of 57.0°,
and significantly different from the asymmetric structure found in the crystalline state. © 2003
American Institute of Physics. ͓DOI: 10.1063/1.1555631͔
I. INTRODUCTION
methods are either of limited applicability, or are at an early
stage of development. Here we present a study of diphenyl-
methane by the method which relies on being able to obtain
partially-averaged dipolar couplings, Dij , between pairs of
nuclei by analysing the nuclear magnetic resonance ͑NMR͒
spectrum of a sample dissolved in a liquid crystalline phase.
This so-called LXNMR method has a wide range of applica-
bility, and has been used for small, rigid molecules such as
benzene1 in order to make a comparison at high precision
between structures obtained in solid, liquid, and gaseous
phases, and for molecules as large as ubiquitin to obtain
coarse grained structures of this and similar proteins in their
folded states.2,3
A study of the conformation of diphenylmethane is im-
portant because it is the simplest case of linkage of two aro-
matic rings through a methylene group. Such a linkage ex-
ists, for example, in trimethoprim, an important therapeutic
agent, and in some recently synthesized liquid crystals.4
Diphenylmethane also presents several interesting chal-
lenges to the LXNMR method, which we will address here.
First, the proton spectrum of a sample dissolved in a liquid
crystalline solvent is that of 12 interacting spins, and it is too
complex to be analyzed without following some spectral
analysis simplification strategy. Also, the question has to be
asked of whether it is necessary to analyze the spectrum of
the fully protonated molecule or would a smaller data set,
obtained from the analysis of selectively deuteriated isoto-
pomers, be sufficiently large to test models for the confor-
mational distribution? The approach adopted here is to pre-
Molecules with internal bond rotational possibilities usu-
ally adopt a single conformation in the solid state, but have a
conformational distribution in liquids and gases. The possi-
bility exists that the global energy minimum changes in
structure between the three phases because of the influence
of intermolecular forces. To investigate this possibility it is
necessary to determine the structure and conformation in
each phase. The structure in the solid state can be obtained
with high precision, and relatively easily by x-ray or neutron
diffraction, if the compound forms good crystals at an acces-
sible temperature. These methods may also be applied to
molecules of large or small molar mass. For the gaseous state
the experimental methods are more limited in their range of
application, and there may also be severe experimental diffi-
culties. An alternative is to calculate an energy surface by
¨
solving Schrodinger’s equation for the electronic wave func-
tions of an isolated molecule. This approach always involves
some degree of approximation, which varies with the number
of atoms and electrons in the molecule, and at the present
time there is considerable interest in investigating the valid-
ity of the commonly available computational methods, to-
gether with their associated computer programs.
For the liquid state, which chemically and biologically is
generally the most interesting, the available experimental
a͒
Present address: Dipartimento di Chimica e Tecnologie Farmaceutiche e
`
Alimentari, Universita di Genova, 16147 Genova, Italy.
0021-9606/2003/118(14)/6417/10/$20.00
6417
© 2003 American Institute of Physics
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6418
J. Chem. Phys., Vol. 118, No. 14, 8 April 2003
Celebre et al.
and the deuterated benzene removed by rotary evaporation.
The crude product, which was about 80% deuterated, was
used without further purification.
2. Synthesis of compound 2
Benzyl chloride ͑3.80 g, 30.0 mmol͒ and benzene-d6 ͑15
ml͒ in a three-necked flask equipped with a calcium chloride
guard tube, a magnetic stirrer and a water-cooled condenser
were cooled with an ice bath under stirring and tin ͑IV͒ chlo-
ride ͑0.40 g, 1.53 mmol͒ was added slowly in five equal
amounts. After stirring for 15 min D2O was added (5
ϫ10 ml) to decompose the catalyst. The mixture was
washed with water (2ϫ10 ml) and 1 M HCl (2ϫ10 ml), the
organic layer was separated and dried over magnesium sul-
phate. Volatile impurities were removed at 100 °C/3 Torr,
leaving behind the desired diphenylmethane-d5 2.
FIG. 1. Structure of isotopomers and atomic labeling of diphenylmethane.
pare suitably deuteriated samples and to analyze the simpler,
1H– 2H spectra.
3. Synthesis of compound 3
͕
͖
a. Preparation of bromobenzene-2,4,6-d3. Aniline hy-
drochloride ͑6.00 g, 47.4 mmol͒ and D2O ͑12.00 g͒ ͑Sigma-
Aldrich Dϭ99.9%) were introduced into a flask equipped
with a stirrer and a water-cooled condenser. The solution was
heated at reflux under nitrogen for 24 h and the water was
removed under reduced pressure. The procedure was re-
peated four times, then the aniline hydrochloride-2,4,6-d3
was dissolved in 10 ml 10% NaOH solution, the base ex-
tracted into Et2O (3ϫ50 ml) and dried over sodium sul-
phate. Removal of the solvent left the aniline-2,4,6-d3 ͑3.23
Second, in order to derive a conformational distribution
from a set of Dij it is necessary to have a theoretical model
to treat the coupling between internal and reorientational mo-
tions, and a parameterized functional form for the conforma-
tional surface which is then brought into agreement with the
experiment data by varying the parameters in a system-
atic way.
It is usual to describe the conformational surface by a
potential energy function, most often a Fourier series, plus a
term to take into account steric, repulsive interactions. Such
an approach is possible for diphenylmethane, but we will
show that it is simpler to use a direct probability description
of the conformational distribution. This new probability ap-
proach is of a general nature, and has particular advantages
for cases where rotations about two bonds ͑such as diphenyl-
methane͒ or more are of a cooperative nature.
2
g, 52% yield, Hϭ98%).
The aniline-2,4,6-d3 ͑3.0 g, 31.2 mmol͒ was dissolved in
aqueous 8.83 M HBr ͑22 ml͒ and diazotized with 12.2 ml of
aqueous 2.5 M sodium nitrite at TϽ5 °C. This diazonium
salt solution was poured into a three-necked flask equipped
with a stirrer and a water-cooled condenser containing cop-
per ͑I͒ bromide ͑6.34 g, 44.2 mmol͒ and 18 ml conc. HBr.
The mixture was heated at reflux for 2 h, steam distilled, and
the product extracted into Et2O. The organic layer was
washed with 10% sodium hydroxide solution, then with wa-
ter up to neutrality, dried over sodium sulphate, and the sol-
vent removed by rotary evaporation. The crude product was
distilled (53 °C/23 Torr) to yield bromobenzene-2,4,6-d3
͑2.10 g, 50% yield͒.
b. Preparation of diphenylmethanol-2,4,6-d3. Bromo-
benzene-2,4,6-d3 ͑2.00 g, 12.6 mmol͒ in anhydrous Et2O ͑35
ml͒ was cooled to 0 °C under nitrogen. n-Butyl lithium in
hexanes ͑1.68 M, 8.3 ml͒ was added slowly, dropwise, keep-
ing the temperature below 2 °C. The mixture was stirred at
room temperature for 1 h after which was cooled again at
0 °C and benzaldehyde ͑1.48 g, 14.0 mmol͒ added. The cool-
ing bath was removed and the mixture left at room tempera-
ture for 2.5 h. A solution of NH4Cl ͑10 ml͒ was added, the
organic layer separated, washed with water and dried over
sodium sulphate. Removal of the solvent left the crude prod-
uct ͑2.3 g, 98% yield͒, which was used in the next stage
without purification.
Third, diphenylmethane has been studied by x-ray
diffraction,5 and by several quantum chemistry methods,6,7
and these results act as a useful set of criteria to judge the
success of the LXNMR method.
II. EXPERIMENT
A general way of extracting dipolar couplings between
1
protons is to simplify the spectrum by replacement of H by
2H, followed by deuterium decoupling. To apply this method
to diphenylmethane required the synthesis of compounds
1–3 ͑Fig. 1͒.
A. Synthesis of isotopically labeled samples
1. Synthesis of compound 1
Diphenylmethane ͑0.50 g, 2.97 mmol͒ and benzene-d6
͑10 ml͒ were introduced into a three-necked flask fitted with
a calcium chloride guard tube and a water-cooled condenser.
The flask was flushed with nitrogen; a 1 M solution of ethy-
laluminum dichloride in hexanes ͑2 ml͒ ͑Aldrich͒ was added
and the solution refluxed for 1 h under stirring. The solution
was allowed to cool, opened to the air, and D2O added until
the yellow color in the organic layer disappeared. The or-
ganic layer was separated, dried over magnesium sulphate,
c. Preparation of benzophenone-2,4,6-d3. Diphenyl-
methanol-2,4,6-d3 ͑2.35 g, 12.5 mmol͒ in dry dichlo-
romethane ͑17 ml͒ was added to a stirred mixture of pyri-
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J. Chem. Phys., Vol. 118, No. 14, 8 April 2003
The conformational distribution of diphenylmethane
6419
FIG. 3. 200 MHz 1H– 2H NMR spectrum of a sample of diphenyl-
͕
͖
methane-d3 ͑compound 3͒ dissolved in the nematic solvent ZLI 1132 at 300
K.
FIG. 2. 200 MHz 1H– 2H NMR spectrum of a sample of dipheny-
͕
͖
lmethane-d5 ͑compound 2͒ dissolved in the nematic solvent ZLI 1132 at 300
K.
observed. The first step in obtaining the ‘‘good’’ set of Dij
1
was to record a H– 2H spectrum of 1, which is a doublet
͕
͖
2
with a separation of 3
͉
D24,25͉. To proceed further, the H
dinium chlorochromate ͑4.13 g, 19.1 mmol͒ and sodium ac-
etate ͑0.31 g, 3.8 mmol͒ in dry dichloromethane ͑17 ml͒. The
mixture was stirred for 2 h at room temperature and then dry
Et2O ͑35 ml͒ was added. The organic phase was separated
and the solvent was removed using a rotary evaporator to
leave an oily product, which was distilled under reduced
pressure (105 °C/0.9 Torr) to yield the product ͑1.35 g, 58%
yield͒.
d. Preparation of diphenylmethane-2,4,6-d3. A solution
of benzophenone-2,4,6-d3 ͑1.34 g, 7.23 mmol͒, hydrazine
hydrate ͑1.24 ml, 0.0247 mol͒, and KOH ͑1.85 g, 0.0330
mol͒ in triethylene glycol ͑4 ml͒ was refluxed for 2 h with
vigorous stirring. Pentane ͑20 ml͒ was added at room tem-
perature and the organic layer separated, washed with water
and dried with sodium sulphate. The solvent was removed
and the residue distilled under reduced pressure
(69 °C/2 Torr) to give the desired product 3 ͑0.586 g, 47%
yield͒.
spectrum of 2 was recorded. This consists of just four lines:
an outer pair from the deuterium at position 21, with com-
bined intensity of unity, and a separation ⌬␯p , and an inner
pair, intensity 4, separation ⌬␯
from the deuteriums at
o,m
positions 19, 20, 22, and 23. The deuteriums at positions 19
and 23 are expected to be equivalent and to give one qua-
drupolar doublet, and the equivalent pair of deuteriums at
positions 20 and 22 should in principle give another, separate
doublet. The observation that only one doublet is observed
shows that the chemical shift difference between these four
deuteriums is negligible, and that the vectors r9,19 and r12,22
are collinear, and so too are r10,20 and r13,23
.
The quadrupolar splittings are related to local order
parameters,8,9 S␣R␤ , by
⌬␯ ϭ 3/2͒qCDSR
,
͑1͒
͑
p
cc
⌬␯ ϭ 3/4͒q SR 3 cos2 ␪o,mϪ1͒
͑
͓
CD
͑
cc
o,m
All the compounds have been characterized by NMR
and the isotopic purity of compounds 2 and 3 was deter-
mined to be Ͼ95%.
ϩ SRaaϪSRbb͒sin2 ␪
,
͔
͑2͒
͑
o,m
where it has been assumed that the quadrupolar tensors for
the deuteriums at the three ring positions are equal and are
each axially-symmetric about the bond directions. The axes
a and c are fixed in the ring plane, as shown in Fig. 1, and b
is the normal to the plane. The angle between the C–D di-
rections and the c axis, ␪o,m , is taken to be 60°. The qua-
drupolar coupling constant, qCD , is assumed to have the
value of 185 kHz. The signs of the quadrupolar splittings are
not determined by the experiments, and so there are four
possible solutions for the two order parameters. These are
used to predict the dipolar couplings between protons in the
ring of 2 from
B. NMR experiments
Samples of compounds 1–3 were prepared by dissolving
approximately 10% by weight in the nematic mixture ZLI
1132, obtained from Merck ͑UK͒ Ltd. The 1H– 2H , and 2H
͕
͖
spectra were obtained with a Bruker MSL 200 spectrometer
at 300 K on samples contained in 5 mm o.d. sample tubes
using a probe with a 7 mm horizontal solenoid coil. Deute-
rium decoupling was achieved using a single pulse at the
center of the deuterium spectrum and applied only during the
acquisition period. A delay of 5 s between pulses was al-
lowed in order to minimize the heating effects of the deute-
rium irradiation.
D ϭϪ ␮ ␥2 h/32␲3r3 ͒ SR 3 cos2 ␪ijcϪ1͒
͑
͓
͑
cc
ij
0
H
ij
ϩ SRaaϪSbRb͒ cos2 ␪ijaϪcos2 ␪ ͒ ,
͑3͒
͑
͑
͔
ijb
III. ANALYSIS OF SPECTRA
where ␮ is the permeability of free space and ␥H is the
0
1
The H– 2H spectra of 2 and 3 are complex, as shown
gyromagnetic ratio for the proton.
͕
͖
2
in Figs. 2 and 3, and to analyze them by the conventional
approach requires that a trial set of Dij are obtained which
can be used to simulate spectra which are close to those
The H spectrum given by 3 is shown in Fig. 4, and has
fine structure on the peaks from dipolar coupling to the pro-
tons as shown in the expanded section. The triplet structure
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6420
J. Chem. Phys., Vol. 118, No. 14, 8 April 2003
Celebre et al.
TABLE I. Chemical shifts ⌬␯ij , and dipolar couplings, Dij , obtained from
the analysis of the 200 MHz 1H– 2H spectrum of
a sample of
͕
͖
diphenylmethane-d5 ͑compound 2͒ dissolved in ZLI 1132.
Dij /Hzb
ij
J
ij /Hza
19,20
19,21
19,22
19,23
19,24
20,21
20,22
20,24
21,24
24,25
8.00
2.00
¯
Ϫ994
Ϫ146
Ϫ38
2.00
¯
Ϫ14
Ϫ429
Ϫ311
Ϫ15
Ϫ83
Ϫ54
8.00
2.00
¯
¯
¯
2068
⌬␯ij /Hz
19,21
20,21
24,21
Ϫ27
Ϫ44
Ϫ614
aW. Brugel, NMR Spectra and Chemical Structure ͑Academic, New York,
1967͒.
bThe spectrum was simulated with a total rms error of 7 Hz. The error on
each NMR parameter is estimated to be Ϯ0.5 Hz.
˜
DijϵDijZZ
ϭ
P
␶ ,␷ ͒D ␶ ,␷ ͒d␶ d␷ ,
͑4͒
͑
LC
͑
͵
k
m
ij
k
m
k
m
where PLC(␶k ,␷m) is the probability in the liquid crystal
phase that the molecule is in a conformation defined by the
bond rotation angles ␶k , and a vibrational state defined by
␷m . A vibrational state is defined here as involving only
small amplitude displacements of nuclei, and does not there-
fore include bond rotational motion. The definition of
Dij(␶k ,␷m) is
FIG. 4. 30.7 MHz spectrum from the deuteriums in
a sample of
diphenylmethane-d3 ͑compound 3͒ dissolved in the nematic solvent ZLI
1132 at 300 K: ͑top͒ normal spectrum and, ͑bottom͒ expanded sections
where the fine structure of multiplets is shown.
TABLE II. Chemical shifts, ⌬␯ij , and dipolar couplings, Dij , obtained
of the peak from deuterium at position 16 is from coupling to
the protons at positions 15 and 17, and this allows a value to
be obtained for ͉D15,16͉. The peaks from deuterium at posi-
tion 14 ͑and 18͒ show a doublet from coupling to proton at
position 15, and a triplet from coupling to protons at posi-
tions 24 and 25, and these allow values to be obtained of
from the analysis of the 200 MHz 1H– 2H spectrum of a sample of
͕
͖
diphenylmethane-d3 ͑compound 3͒ dissolved in ZLI 1132; the calculated
values were obtained by the probabilistic approach without ͑column A͒ and
with ͑column B͒ vibrational corrections to the dipolar couplings.
D
ij /Hz
i, j
Calculated
͉
D14,15
͉ and ͉D14,24͉.
Experiment
Ϫ15.0Ϯ0.1
A
B
These experiments enabled good estimates to be made of
15,17
15,19
15,20
15,21
15,24
19,20
19,21
19,22
19,23
19,24
20,21
21,24
24,25
R/Hz
Ϫ15.1
Ϫ70.0
Ϫ40.6
Ϫ36.4
Ϫ73.0
Ϫ897.0
Ϫ133.2
Ϫ35.2
Ϫ15.1
Ϫ391.9
Ϫ281.1
Ϫ50.1
1847.5
0.4
Ϫ14.1
Ϫ70.2
Ϫ41.1
Ϫ36.3
Ϫ73.6
Ϫ896.5
Ϫ135.0
Ϫ35.8
Ϫ14.1
Ϫ391.9
Ϫ281.2
Ϫ51.1
1847.5
1.0
all the dipolar couplings within the C6H5CH2-group, with
the exception of D15,24 , and this was estimated by compari-
son with the set of data obtained for benzyl chloride dis-
solved in the same liquid crystalline solvent.10 With this set
Ϫ70.0Ϯ0.1
Ϫ40.7Ϯ0.1
Ϫ35.7Ϯ0.1
Ϫ73.0Ϯ0.1
Ϫ897.1Ϯ0.1
Ϫ133.2Ϯ0.1
Ϫ35.4Ϯ0.1
Ϫ15.0Ϯ0.1
Ϫ391.9Ϯ0.1
Ϫ281.1Ϯ0.1
Ϫ48.6Ϯ0.1
1847.5Ϯ0.1
1
of starting values it was possible to analyze the H– 2H
͕
͖
spectrum of 2 shown in Fig. 2 by the method of Celebre
et al.11 to give the data in Table I. These Dij were then used
1
as starting parameters in the analysis of the H– 2H spec-
͕
͖
trum of compound 3, which is shown in Fig. 3, to give the
data in Table II.
IV. CONFORMATIONAL ANALYSIS: THEORY
A. General
⌬␯ij /Hz
15,21
19,21
24,21
Ϫ18.0Ϯ0.1
Ϫ50.0Ϯ0.1
Ϫ628.8Ϯ0.1
The Dij are averages over all the normal modes of the
molecule relative to the applied magnetic field, which in the
present case is parallel to the mesophase director, n. Thus,12
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J. Chem. Phys., Vol. 118, No. 14, 8 April 2003
The conformational distribution of diphenylmethane
6421
␮
␥i␥jh
ULC ␤,␥,␶ ͒ϭU ␤,␥,␶ ͒ϩU ␶ ͒.
͑10͒
͑
͑
ext
͑
0
k
k
int k
Dij ␶ ,␷ ͒ϭ
S
͓
3
␶ ,␷ ͒
͑
zz k m
͑
ͩ ͪ
k
m
␶
,␷
2
k
ij
m
4␲
8␲
r
͒
͑
The function PLC(␶k) is related to PLC(␤,␥,␶k),
PLC ␶ ͒ϭ ␤,␥,␶ ͒sin ␤d␤d␥
ϫ 3 cos2 ␪k,mϪ1͒ϩ S ␶ ,␷ ͒
͑
͑
͑
xx k
m
ijz
P
͑11͒
͑
͑
͵
k
k
LC
ϪSyy ␶ ,␷ ͒͒ cos2 ␪ikj,xmϪcos2 ␪ ͒ ,
k,m
͑
͑
͔
ijy
k
m
ϭZϪ1 exp ϪU ␶ ͒/k T
͔
B
͓
͑
int
k
͑5͒
where Szz(␶k ,␷m), etc. vary with the conformation and vi-
brational state. These order parameters are for principal axes,
and these axes will also change their orientation in different
conformational and vibrational states.
ϫ
exp ϪU ␤,␥,␶ ͒/k T sin ␤d␤d␥. ͑12͒
͓ ͑ ͔
ext k B
͵
The order parameters S␣␣(␶k) are obtained as
1
2
k
3 cos2 ␪␣nϪ1͒
It is usual to assume that the dependence of the order
parameters on the vibrational state13 is much smaller than
their dependence on the bond rotational angles, and can be
neglected to a good approximation. Similarly it is argued that
Ϫ1
S␣␣ ␶ ͒ϭW ␶ ͒
͑
͑
͑
͵
k
k
ϫexp ϪU ␤,␥,␶ ͒/k T sin ␤d␤d␥ ͑13͒
͓
͑
͔
ext
k
B
with
W ␶ ͒ϭ exp ϪU ␤,␥,␶ ͒/k T sin ␤d␤d␥, ͑14͒
the dependence of Dij(␶k ,␷m) on ␷ can also be neglected
m
when the main interest is in trying to characterize PLC(␶k),
the conformational distribution, from a set of experimental
dipolar couplings. To support this view it can be noted that
vibrational corrections to Dij in rigid molecules are usually
͑
͓
͑
ext
͔
B
͵
k
k
k
where ␪
is the angle between the director and principal
␣ n
axis ␣ when the molecule is in the conformation specified by
the values of the set ␶k .
Making the division shown in Eq. ͑10͒ allows a confor-
mational distribution, Piso(␶k), to be defined as
Ͻ5%, whereas the dependence of Dij(␶k ,␷m) on ␶ is often
k
Ͼ100%. Of course, the main reason for neglecting the ef-
fects of vibrations when large amplitude motion is also
present is the difficulty of their calculation. Here the effects
of vibrations will be neglected at first, but later an approxi-
mate method for their inclusion will be explored.
Neglecting vibrational motion reduces Eqs. ͑4͒ and ͑5͒
to
Piso ␶ ͒ϭQϪ1 exp ϪU ␶ ͒/k T
͔
B
͑15͒
͑16͒
͑
͓
͑
int
k
k
with
Qϭ exp ϪU ␶ ͒/k T d␶ .
͓
͑
͔
B
͵
int
k
k
D ϭ
ij
P
␶ ͒D ␶ ͒d␶
͑6͒
͑
LC
͑
͵
k
ij
k
k
This distribution is for the same temperature, and the same
solvent as PLC(␶k), but is for an isotropic phase. Note that
and
P
iso(␶k)ÞPLC(␶k), but that these distributions are related by
␮
␥i␥jh
0
k
Dij ␶ ͒ϭϪ
S
␶ ͒ 3 cos2 ␪ijzϪ1͒
͑ ͑
zz k
͑
͓
ͩ ͪ
Piso ␶ ͒ϭP ␶ ͒ Z/ W ␶ ͒Q ͑17͒
.
͔
͖
͑
͑
LC
͓
͑
k
k
͕
␶
2
3
k
k
k
ij
4␲
8␲ r ͒
͑
k
k
ϩ S ␶ ͒ϪS ␶ ͒͒ cos2 ␪ijxϪcos2 ␪ ͒ .
B. The additive potential „AP… model for Uext „␤,␥,␶k…
͑
͑
͑
k
͑
͔
ijy
xx
k
yy
There have been several suggestions as to how to model
Uext(␤,␥,␶k).14,15 For flexible molecules in particular there
are advantages in using a general expansion in spherical har-
monics, and for a uniaxial phase this is
͑7͒
It is tempting to assume that the dependence of S␣␣(␶k) on
␶ can also be neglected, but although this is usually done by
k
those using dipolar couplings to study protein structures it is
likely to lead to major errors in the function PLC(␶k) and the
structures derived when there is an appreciable degree of
intramolecular motion present in the molecule.
ϱ
Uext ␤,␥,␶ ͒ϭϪ
␧
␶ ͒C
k
␤,␥͒.
͑
L,m
͑18͒
͑
͑
͚
L,m
even
k
L,m
To proceed further, a probability, PLC(␤,␥,␶k), that the
molecule is at an orientation with the director defined by the
polar angles ␤ and ␥ whilst in a conformation described by
The CL,m(␤,␥) are modified spherical harmonics, and the
L,m(␶k) are conformationally-dependent coefficients whose
␧
magnitudes depend upon the strength of the anisotropic part
of the solute–solvent interaction. Computer simulations of
solutes in liquid crystalline solvents suggest16 that the expan-
sion in Eq. ͑18͒ can be restricted to Lϭ2, and mϭ0, Ϯ2
when the aim is to use it to obtain averages of second-rank
interactions, such as dipolar or quadrupolar couplings. In the
version of the AP method used here the ␧L,m(␶k) will be
approximated as sums of contributions ␧L,m(j) from rigid
molecular subunits,
the set ␶ is introduced. It is defined in terms of a mean
k
potential, ULC(␤,␥,␶k), such that
PLC ␤,␥,␶ ͒ϭZϪ1 exp ϪU ␤,␥,␶ ͒/k T
͔
B
͑8͒
͑
͓
͑
LC
k
k
with
Zϭ exp ϪU ␤,␥,␶ ͒/k T sin ␤d␤d␥d␶ .
͑9͒
͓
͑
LC
͔
B
͵
k
k
The mean potential can be divided into a part, Uext(␤,␥,␶k),
which vanishes for an isotropic phase, and a part, Uint(␶k),
which does not,
␧
␶ ͒ϭ
k
␧
j͒D2 ⍀ ␶ ͒͒,
͑19͒
͑
͑
͑
͑
j k
͚
2,m
2,p
p,m
p, j
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6422
J. Chem. Phys., Vol. 118, No. 14, 8 April 2003
Celebre et al.
This gives an energy surface of the correct periodicity and
symmetry, but it has the disadvantage shared by Fourier ex-
pansions for Uint(␶k) that it is not possible to vary the posi-
tion of the energy minima, their relative energies, and the
barriers between them in an independent way. This disadvan-
tage can be overcome by adding more terms to the expan-
sion, when fitting a functional form to the results from a
calculation of an energy surface. Adding extra terms to Eq.
͑21͒ is usually not feasible when fitting observed data, such
as averaged dipolar couplings, to a model for a probability
distribution as in Eqs. ͑6͒, ͑8͒, and ͑10͒, because of the lim-
ited amount of experimental data available.
D. The direct probability description
of conformational distributions
An alternative is to determine the probability PLC(␶k)
directly without going through the intermediate stage of first
characterizing an energy surface. Such a probabilistic ap-
proach was proposed initially by Zannoni17 for obtaining
P
LC(␤,␥,␶k) from averaged NMR data. The original
Zannoni method derived a form for PLC(␤,␥,␶k) by apply-
ing the maximum entropy ͑ME͒ principle. This gives
FIG. 5. Conformations of diphenylmethane and their symmetry point
groups.
ME
P
␤,␥,␶ ͒ϭZϪ1 exp ϪU
␤,␥,␶ ͒/k T
͔
B
͑22͒
͑
͓
͑
ME
k
k
LC
ME
with
ZME
where D2p,m(⍀j(␶k)) is the Wigner function describing the
orientation of fragment j in molecular reference axes.
ϭ
exp ϪU
␤,␥,␶ ͒/k T sin ␤d␤d␥d␶ .
͑ ͔
ME k B k
͓
͵
͑23͒
C. Modelling the internal potential Uint„␶k…
UME(␤,␥,␶k) has the same form as Uext(␤,␥,␶k) in Eq. ͑18͒
except that the coefficients depend directly on the experi-
mental data and hence contain information on both the ori-
entational order and the conformational distribution. This has
the advantage of not going beyond the data available, and
hence has been described as being the least biased approach.
The method has the disadvantage however that PLC(␤,␥,␶k)
in the isotropic phase becomes a constant independent of
both orientational order, as it should, and conformation,
which it should not. This disadvantage can be removed by
introducing the constraint that PLC(␤,␥,␶k) become identi-
cal with Piso(␶k) in the isotropic phase.18,19
The method proposed here offers a different vision of the
probabilistic approach. The distribution function PLC(␶k) is
modeled directly as a sum ͑for interdependent conforma-
tions͒ or product ͑for independent states͒ of Gaussian func-
tions. Thus, for rotation about the lth bond between Nl con-
formations,
as a truncated Fourier series
There are also advantages in expanding Uint(␶k) in a
general way, and the most commonly adopted choice is as a
truncated Fourier series, such as
Uint ␶ ͒ϭ
V cos n␶ ͒,
͑20͒
͑
͑
͚
k
n
l
l,n
with n taking values to give the correct periodicity of
Uint(␶k), and the series being kept as short as will represent
the correct overall shape of this bond rotational potential ͑at
least near to the expected potential minima͒. However, for
molecules like diphenylmethane this simple form for
Uint(␶k) is not able to describe correctly the energy surface.
Thus, with nϭ2 and 4, which allows for energy minima with
0°Ͻ␶1 , ␶2Ͻ90°, Eq. ͑20͒ predicts equal energies for struc-
tures with symmetry C2 and Cs ͑see Fig. 5͒. This energy
degeneracy can be removed by adding additional terms.
Thus, adding a term to account for short-range repulsion,
which depends on rij , differentiates between the structures
with C2 and Cs symmetries, but by only a small amount that
could be insufficient for diphenylmethane. A more flexible
N
l
0
0
2
PLC͑␶ ,N ,k ,␶ ͒ϭ
A exp͓Ϫ k ͑␶ Ϫ␶ ͒ ͔,
͕ ͖
l l l
l
j j
j
͚
l
l
l
l
j
j
jϭ1
͑24͒
0
l
approach is to add terms to Eq. ͑20͒ which depend on ␶
where the ␶ are the positions of maximum probability, Al
ϩ
j
j
ϭ␶1ϩ␶ and ␶Ϫϭ␶1Ϫ␶2 ,
2
are their relative amplitudes, and kl gives the width of the
j
distribution. The physical interpretation is that the molecule
Uint ␶ ,␶ ͒ϭV cos 2␶ ϩcos 2␶ ͒
͑
͑
1
2
2
1
2
exists in j conformations, centered on ␶0 , and that there is a
l
j
ϩV4 cos 4␶ ϩcos 4␶ ͒
͑
1
2
symmetric oscillation at each. The relative amounts of each
conformer are obtained by integration.
ϩVϩ cos 2␶ϩϩVϪ cos 2␶Ϫ .
͑21͒
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J. Chem. Phys., Vol. 118, No. 14, 8 April 2003
The conformational distribution of diphenylmethane
6423
and the HCH plane are ␶1ϭ63.9° and ␶2ϭ71.1°. The asym-
metry must arise because of the asymmetric environment in
the solid state, and is not expected either for an isolated
molecule, or for an achiral liquid phase. Quantum chemistry
calculations on an isolated diphenylmethane molecule6,7
suggest that only the structures with C2 symmetry are appre-
ciably populated. The probability function, PLC(␶1 ,␶2)
describing these conformations is conveniently expressed
by introducing the angular variables ␶ϩϭ␶1ϩ␶ and
2
␶Ϫϭ␶1Ϫ␶2 ,
PLC ␶ ,␶ ͒ϭP ␶ ͒P ␶ ͒.
͑25͒
͑
͑
͑
Ϫ Ϫ
ϩ
Ϫ
ϩ
ϩ
The correct periodicity of PLC(␶ϩ ,␶Ϫ) is obtained by intro-
ducing indices s, n, and j,
1ϩs 2Ϫs͒
͑
PLC ␶ ,␶ ͒ϭ
͑
͚
͚
͚
ϩ
Ϫ
2
sϭ0,1,2 nϭϪ 1/2 , 1/2 jϭϪ 1/2 , 1/2
• exp Ϫx2 /2h2 exp Ϫx2 /2h2
͑26͒
͓
͔
͓
͔
Ϫ
ϩ
ϩ
Ϫ
with
and
0
x ϭ␶ ϩ2j s␲Ϫ␶ ͒
͑27͒
͑28͒
͑
ϩ
ϩ
ϩ
x ϭ␶ ϩ2 sn ␲ 2Ϫs͒.
͑
Ϫ
Ϫ
The maximum probability is constrained to have ␶Ϫϭ0° by
the C2 symmetry requirement, and to be at positions along
␶
determined by ␶0ϩ.
ϩ
The value of hϩ gives the full width at half maximum
͑FWHM͒ of the Gaussian function ͑related to the amplitude
of conformational oscillation in the ␶ direction͒ whilst hϪ
ϩ
is the same along ␶Ϫ . The maps represented in Fig. 7 show
how the shape of the distribution PLC(␶ϩ ,␶Ϫ) depends on
the relative values of hϩ and hϪ .
The AP method was used to model Uext(␤,␥,␶1 ,␶2)
R
R
R
with identical contributions ␧ and ␧xxϪ␧ from each phe-
zz
yy
nyl ring. These were varied, together with hϩ , hϪ , ␶0ϩ, and
any appropriate geometrical parameters, to minimize the tar-
get function
1/2
2
Rϭ MϪ1
D
observed͒ϪD calculated͒
͑ ͑ ͔
ij ij
͓
͚
ͭ
ͮ
iϽj
͑29͒
FIG. 6. The three Gaussian functions used to describe the three conforma-
tions generated by rotation about the C–C bond in a compound of type
XCH2CH2Y.
with Mϭnumber of independent couplings.
VI. RESULTS AND DISCUSSION
A simple example is rotation about the C–C bond in
compounds of the type XCH2CH2Y, as shown in Fig. 6. The
variables used to describe this situation are ␶g , kg , kt , and
At ; these can be varied independently, but with the con-
straint that PLC(␶, j,kj ,␶j) is normalized. We will return to a
more detailed description of this case, with a comparison to
the conventional cosine expansion of a potential energy func-
tion, in the future.
The calculations proceeded first with the neglect of the
effect on the Dij of small amplitude vibrational motion. The
first step was to adjust the positions of protons in the aro-
matic rings to minimize R for the intraring Dij . The spectral
analysis produces D19,23ϭD20,22 (ϭD15,17), and so the dis-
tances r19,23 , r20,22 , and r15,17 must be equal. Fixing the ring
geometry to have:
all rCCϭ1.40 Å, except r12,13 (ϭr6,7ϭr9,10ϭr3,4);
all CCC angles 120° in the rings;
V. APPLICATION OF THE PROBABILISTIC
APPROACH TO DIPHENYLMETHANE
all rCHϭ1.085 Å except r11,21 (ϭr5,16);
The diphenylmethane molecules in the crystalline state
are asymmetric in that the angles between the ring normals
R
R
R
yy
the function R was minimized by varying ␧zz , ␧xxϪ␧ ͑or
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6424
J. Chem. Phys., Vol. 118, No. 14, 8 April 2003
Celebre et al.
FIG. 7. The effect of hϩ and hϪ on the distribution PLC(␶ϩ ,␶Ϫ); ͑a͒ has
0
h
ϩϭhϪ , and in ͑b͒ hϩϭ1.5 hϪ . The value of ␶ is the same in both cases.
ϩ
TABLE III. Optimized and assumed geometrical parameters for
diphenylmethane-d3 ͑compound 3͒ and local order parameters, SR␣␤ , for the
aromatic ring.
Parameters
Optimized values
FIG. 8. The distribution PLC(␶ϩ ,␶Ϫ) determined by the probabilistic ap-
proach: ͑a1͒ and ͑a2͒ without, and ͑b͒ with vibrational corrections to the
dipolar couplings ͑see text for explanation͒.
r
r
SRcc
9,10 /Å
1.398Ϯ0.002
1.080Ϯ0.001
0.1140Ϯ0.0001
0.1336Ϯ0.0001
Assumed valuesa
1.085
11,21 /Å
SRaaϪSRbb
equivalently SRzz and SRxxϪSRyy), r9,10 and r11,21 to give the
results shown in Table III. The ring geometries were then
kept fixed in all subsequent calculations.
The potential energy surface for diphenylmethane has
been calculated by a number of computational methods,6,7
and it was decided to investigate how well two such surfaces
fit the observed dipolar couplings. In both cases the first
CH ͑ring͒/Å
CC ͑ring͒/Å
1.400
Angles ͑ring͒/Å
120.0
r
1,2 /Å
1.521
aTaken from Ref. 22 and kept fixed.
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J. Chem. Phys., Vol. 118, No. 14, 8 April 2003
The conformational distribution of diphenylmethane
6425
TABLE IV. The values of the total rms error, R, and the geometrical parameters obtained by fitting the observed
*
dipolar couplings with the potential energy surface calculated from MNDO and B3LYP/6-31G .
*
B3LYP/6-31G
MNDO
␣/°
␤/°
R/Hz
114.7a
105.2a
84.7
126.3Ϯ0.5
105.2a
29.7
104.4Ϯ0.5
124.7Ϯ0.5
17.3
114.7a
105.2a
91.9
118.6Ϯ0.5
105.2a
5.1
119.5Ϯ0.5
118.3Ϯ0.5
4.7
aTaken from Ref. 22 and kept fixed.
A. The effect of allowing for vibrational averaging
of the dipolar couplings
calculations used the same geometry as the quantum chem-
R
R
istry calculations, and the only variables were ␧ and ␧
zz
xx
R
Ϫ␧yy . The values of R obtained, given in Table IV, are
unacceptably large, and so the effect of retaining the calcu-
lated energy surface but varying the geometry was explored.
The results are also shown in Table IV, and again are unac-
ceptable. Note that when the R value is not very high the
value of the ␤ or ␣ angles are unacceptable.
The inclusion of vibrational averaging into the deriva-
tion of PLC(␶ϩ ,␶Ϫ) is not straightforward. The main prob-
lem is the inclusion of small-amplitude vibrational motion in
addition to the large amplitude bond rotational motion. This
is a general problem, but it is compounded in the case of
averaging of dipolar or quadrupolar couplings by having to
include the variation of the orientational order as the mol-
ecules changes shape during these two kinds of motion. An
approximate method is used for diphenylmethane, which is
described more fully by Lesot et al.20 This method consists
in calculating a wave function for small amplitude, harmonic
vibrational motion for the molecule in the conformation with
Fitting the Dij by the probabilistic approach, varying
R
R
R
␧
zz , ␧xxϪ␧yy , ␶0ϩ, hϩ , hϪ , and the angle ␣ gives a close fit
to the data and the parameters shown in Table V, column A.
In order to reduce the CPU requirements while keeping an
high resolution (1°) in exploring the conformational surface,
the calculations have been carried out by implementing a
‘‘smart,’’ properly normalizing algorithm which catches only
the essence of the distribution, without redundancies. It var-
ies step by step ␶ from 0° to 90° and ␶ from Ϫ␶ to ϩ␶1 .
*
␶1ϭ␶2ϭ56.5° by the B3LYP/6-31G method using the
GAUSSIAN 98 package.21 This is then used to calculate vibra-
tion corrections to the Dij , assuming that the order param-
eters are independent of the vibrational motion. The effect of
the additional averaging over the large amplitude bond rota-
tional motion is then done using the probabilistic method in
conjunction with the AP method for calculating how the or-
dering changes with this motion.
1
2
1
The obtained Dij are reported in Table II column A and the
probability PLC(␶ϩ ,␶Ϫ) obtained is shown in Fig. 8͑a1͒. It
has a maximum at ␶1ϭ␶2ϭ56.5°Ϯ0.5°, compared with
*
57.0° ͑Ref. 6͒ obtained by the B3LYP/6-31G calculation,
and a rather flat surface, as seen by the large values of hϩ
ϭ29.5° and hϪϭ28.7°. The optimized value of ␣ of
The ring geometry was first optimized with the inclusion
121.8°Ϯ0.5° is significantly larger than that of 114.6° deter-
6
of vibrational averaging. This produced
r11,21ϭ1.074
*
mined by the B3LYP/6-31G method.
Ϯ0.002 Å, and r9,10ϭ1.403Ϯ0.001 Å, which are changed
by Ͻ0.6%. Then, using all the vibrational modes and vary-
In Ref. 6, a strong correlation between ␣ and ␶ and ␶
1
2
was found, and the calculations with this theoretical relation-
ship were repeated to give the data of column B in Table V.
Note that the value of R is only slightly increased and the
position of the maximum probability changes by only 2.8°
(␶1ϭ␶2ϭ59.3°Ϯ0.5°). The resulting distribution, shown in
Fig. 8͑a2͒, has quite different cross sections, the widths hϩ
and hϪ differing noticeably (hϩϭ20.5°; hϪϭ47.4°) from
the previous ones.
R
R
R
ing ␧zz , ␧xxϪ␧yy , hϩ , hϪ , ␶0ϩ, and ␣ produced the results
shown in Table V column C. The error, R, is considerably
0
ϩ
worse than without vibrational averaging. The value of ␶ is
not changed significantly, corresponding to ␶1ϭ␶2ϭ58.8°
Ϯ0.5°.
The larger value of R is probably because the effect of
an oscillation involving ␶ and ␶ is counted twice: once as a
1
2
TABLE V. The interaction coefficients, ␧
and the parameters obtained by fitting observed dipolar couplings
2,m,
with the probabilistic approach without vibrational corrections: optimizing the ␣ angle ͑column A͒ and impos-
*
ing to the ␣ angle the behavior found from the B3LYP/6-31G calculation ͑column B͒; and with vibrational
contributions: using all the frequencies ͑column C͒ and without torsional frequencies ͑column D͒.
A
B
C
D
␧
␧
h
h
␶
zz /kJ molϪ1
0.881Ϯ0.001
1.292Ϯ0.001
20.5Ϯ0.7
47.4Ϯ0.4
118.6Ϯ1.0
0.882Ϯ0.001
1.277Ϯ0.001
28.0Ϯ0.5
25Ϯ19
120.0Ϯ1.0
120.7Ϯ0.5
6.5
0.915Ϯ0.001
1.274Ϯ0.001
34.7Ϯ0.1
21.3Ϯ3.7
110.2Ϯ1.0
121.3Ϯ0.5
1.0
0.926Ϯ0.001
1.286Ϯ0.001
29.5Ϯ0.1
28.7Ϯ1.5
113.0Ϯ1.0
121.8Ϯ0.5
0.4
xx-␧yy /kJ molϪ1
ϩ /°
Ϫ /°
0ϩ/°
␣/°
R/Hz
1.0
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6426
J. Chem. Phys., Vol. 118, No. 14, 8 April 2003
Celebre et al.
vibrational mode, and second by allowing the widths of the
gaussian functions to vary. Note, too, the very large error on
hϪ in column C of Table V, which is another indication that
this model is flawed. Excluding the vibrational modes
strongly dependent on ␶ and ␶ from the calculation of the
vibrational average leads to Rϭ1.0 Hz and the results in col-
umn D of Table V. A value of ␶1ϭ␶2ϭ55.1Ϯ0.5° is now
obtained. There is a large change in the values of hϩ and
hϪ , which is reflected in a large change in the shape of the
distribution PLC(␶1 ,␶2) as shown in Fig. 8͑b͒.
it suggests that this simple method of representing
P
LC(␶1 ,␶2) in terms of Gaussian functions is a good ap-
proximation.
1
2
ACKNOWLEDGMENTS
This work has been supported by MIUR PRIN ex 40%.
The authors gratefully acknowledge Dr. Rosa Saladino for
her help in the spectral analysis and Dr. J. Grunenberg for his
‘‘remote’’ support and kindness during the progress of the
work. J.W.E. thanks the Leverhulme Trust for the award of
an Emeritus Fellowship.
B. Using a jump model for PLC„␶1 ,␶2…
It is often assumed that only conformations correspond-
ing to the positions of energy minima need to be considered
when averaging dipolar or quadrupolar couplings. In the
probabilistic model this corresponds to collapsing the Gaus-
sians into delta functions at the positions ␶0ϩ. This approxi-
mation was explored, including averaging over all the vibra-
tional modes, the torsional ones too in order to simulate the
torsion as a libration about the minimum. In this way a mini-
mum in R of 19 Hz was found when ␶1ϭ␶2ϭ70°. Allowing
for a change of the HCH angle the value of R is reduced
to 1 Hz, but we obtain for that angle an unacceptable value
of 93°.
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VII. CONCLUSION
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interaction with the liquid crystal molecules, gives a distri-
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be taken to be a proof of the validity of the model used, but
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