Table
6
The
di†erence
*D (%) \ M[D (observed)
ij ij
This shape results because V A V whereas Caminati et al.
4
2
[ D (calculated)]/D (observed)N ] 100 for ethylbenzene dissolved in
calculated the reverse relative magnitudes, which produces a
ij
ij
I35, ZLI 1132, I52 and EBBA
minimum at 90¡ and a maximum at 0¡. Fixing V \ 0 and
4
varying the other eight variables gave a higher, but acceptable,
*D (%)
ij
rms value, and V (t) now has essentially the same shape as
calculated by Caminati et al., but still with unreasonable
values for the angular distortion parameters.
When the Ðtting procedures used for 4-chloroethylbenzene
were repeated for the new data on ethylbenzene, the same
kind of unacceptable results were obtained. Quite clearly,
therefore, the NMR data are not consistent with the results
from the molecular orbital calculations. The most probable
reason for this disagreement is that the geometry calculated
by the molecular orbital method is for an isolated molecule,
ij
I35
ZLI 1132
I52
EBBA
9,10
9,11
9,12
9,13
9,14
9,16
10,11
10,12
10,14
10,16
11,14
11,16
14,15
14,16
16,17
[0.7
[0.9
[1.1
[2.7
0.8
0.8
0.0
0.9
[1.3
[1.7
[1.6
2.0
[0.4
[0.6
[1.1
[0.7
0.2
[0.1
[0.1
[0.3
0.7
0.5
[7.7
[0.4
[1.7
0.4
0.1
0.9
1.3
1.4
3.2
1.4
0.8
1.6
0.7
2.4
which is not undergoing vibrational motion, whereas the D
are averages over these modes, and are for a molecule which is
[5.5
[7.2
[4.8
[5.4
ij
1.3
1.1
1.7
2.0
[5.4
0.7
2.1
[2.4
[7.2
[1.7
2.1
[0.1
[5.2
[03
2.3
[5.4
0.4
in a liquid phase, and hence may be a†ected by intermolecular
forces. A method for averaging the D over vibrational modes
[0.6
ij
when all these are of small amplitude, and intermolecular
4.9
[5.5
e†ects are ignored, has been developed, but is not applicable
to a molecule like ethylbenzene which has bond rotational
motion. The magnitudes of the changes produced by vibra-
tional averaging on interproton couplings in more rigid,
simpler molecules have been found to be of the order of about
0È5%, and so we have explored the possibility of Ðnding a
structure for ethylbenzene which will Ðt the observed coup-
lings within this kind of error on the individual couplings. To
do this it was simpler to assume that the molecule rotates as a
rigid entity, that is with Ðxed values of the angles a and b. The
data on the solution in I35 were Ðtted Ðrst.
tropic phase, whilst U (t) is Ðnite in all phases and is the
int
rotational potential for the molecule in an isotropic environ-
ment. We will identify U (t) with V (t) for the rotation about
int
the ringÈC bond for a molecule in the isotropic phase of the
mesogenic solvent molecules.
In the AP model U (b,c,t) is written as
ext
U
(b,c,t) \ [ e (t)C (b) [ 2e (t)C (B,c) (12)
ext
2, 0
2, 0
2, 2
2, 2
The Ðrst step was to determine the relative positions of the
where the C (b,c) are modiÐed spherical harmonics. The
2, m
protons in the aromatic ring which Ðt the inter-ring D
conformation dependence of the interaction coefficients is
ij
exactly. This can be done by using eqn. (3) with the S
modelled by expressing them as sums of contributions, e ( j),
ab
replaced by local order parameters,3 SR . To do this it is
ab
2, n
from each of the j rigid subunits in the molecule:
necessary to Ðx one of the interproton distances, and the
e
(t) \ ; ; e (j)D2 (X )
2,m 2,n n,m
(13)
choice was made of r
dent inter-ring proton couplings were then used to obtain the
\ r
\ 2.48 A. The six indepen-
j
9, 10
12, 13
j
n
D2 (X ) is the Wigner function describing the orientation of
two order parameters, SR and SR [ SR , and the coordinates
n, m
j
zz
xx
yy
the jth fragment in reference axes Ðxed in some rigid fragment,
for example the benzene ring in ethylbenzene. The rigid frag-
ments in ethylbenzene are the aromatic ring and the alkyl
x , z , x , z , with x \ z \ 0. The bond lengths and
9
9
10 10
11
11
angles in the aromatic ring were then adjusted to give these
proton coordinates, and the resulting structure is shown in
Fig. 1.
group. The ring has C symmetry and requires eR and eR
.
2v
2, 0
2, 2
The alkyl group is sub-divided into a CÈC bond and Ðve
equivalent CÈH bonds. These bonds have axial symmetry and
require just eCC or eCH . It has been found in studies on many
The second step was to explore the e†ect of changing the
structure of the ethyl group so as to produce an acceptable
rms error by varying the values of the e ( j) and the coeffi-
2, 0
2, 0
2, n
similar compounds that setting eCH \ 0 does not a†ect the
cients deÐning the rotational potential V (t). This was done by
2, 0
precision with which observed and calculated dipolar coup-
determining the sensitivity of the Ðt to changes in the bond
lengths and angles. This led to an acceptable Ðt being
obtained by changing the CÈH bond lengths in the methyl
and methylene groups to 1.1 A, and increasing the angle HCH
in the methylene group from 109.5 to 113.5¡.
lings can be brought into agreement, and the same situation is
found to hold for ethylbenzene.
Note that in the calculations it is easier to use equivalent
Cartesian forms of these tensor components. Thus, eR is
2, 0
replaced by (2/3)1@2eR , eR
by 1(eR [ eR ), and eCC by
The potential describing rotation about the C ÈC bond
zz 2, 2
2 xx
yy
2, 0
1
7
(2/3)1@2e , where the CC represents the direction of the
was simpliÐed to
CC
C ÈC bond.
V (t) \ V cos 2t ] V cos 4t
The e†ect was explored too of raising the restriction that rota-
(14)
7
8
2
4
The data for 4-chloroethylbenzene was Ðtted to the
observed D by varying these three interaction coefficients,
tion of the methyl group about the C ÈC bond can be
ij
and some or all of the parameters deÐning the bond rotational
7
8
approximated as jumps between just three, equivalent posi-
tions. To do this the potential was described as
potential. Thus, it was found that Ðxing V , V , a , *a and
2
4
0
0
the bond lengths and angles at the values determined by
Caminati et al.5, and varying the three interaction coefficients,
did not give a good Ðt to the NMR data, that is, the rms error,
R \ ; M[D (observed) [ D (calculated)]2/FN1@2, is unaccept-
V (/) \ V cos 3/
(15)
3
with a value of [6 kJ mol~1 for V (/ \ 0¡ corresponds to
3
the C ÈH bond being in the xz plane as shown in Fig. 1).
ij
ij
ij
8
16
ably large. A good Ðt to 14 of the D could be obtained by
Doing this has only a small e†ect on the Ðt obtained. In fact,
ij
introducing a similar dependence of the angle b on t as that
in eqn. (9) for a, and varying all nine variables (the seven
above plus b and *b), but the values obtained for a and *a
are very di†erent from those calculated by Caminati et al., and
were judged to be unlikely. Moreover, the shape obtained for
V (t) is quite di†erent in character from that calculated,
having minima at 0¡ and 90¡ with a maximum at about 45¡.
with V \ [6 kJ mol~1 it was found that the full e†ects on
3
ij
the D of methyl rotation could be obtained by allowing the
protons to oscillate through ^30¡ about their minimum
0
0
0
energy positions.
This structural and dynamic model for ethylbenzene was
used to Ðt the D obtained for all four data sets and the
ij
results are summarized in Table 6, which gives the percentage
Phys. Chem. Chem. Phys., 2000, 2, 3405È3413
3411