Chemical
Physics 12 (1976) 113 @ NorthHolland Publishing Company
OPTICAL ACTIVITY OF ORIENTED MOLECULES II. Theoretical description of the optical activity * HansGeorg
KUBALL,
Ties KARSTENS
I:achberciclr Cfiemie der Uttiversihit Kaiserslauterrt.
Kniserrlm~tern,
Germany
and Alfred SCHijNHOFER
Rcccivcd 8 July 1975
The theory of the optical activity of oriented molecules is developed starting from an expression given by GE. In order to describe the experiments, a tcnsioral rotational strength RT=gk,33RzK is introduced. The oticntational distribution coefficientsg~~33 determine the influence of the type and magnitude of the orientation Lo RT. Xhc tensor of rotation RF is a function of magnetic dipole, electric dipole and electric quadrupole transition moments. Not all the coordinates of the tensor ciln be determined cxpcrimcntally at prcscnt. It is shown howcvcr, that for molecules of a sufficiently hish symmetry the tensor will have a vcty simple form. For tuo unsaturated keto steroids the magnitude of the coore.g. Dzsymr+ x and RyjK IS csrjmatcd from cspcrimcntal results. dinntcs Ryt, RzZh
1. introduction The first attempt to measure the anisotropy of the optical activity of small molecules, oriented in an electric field, was not successful, because the effect was too small [I]. This anisotropy was found years later by Tinoco [2] with polymer solutions in an electric field or in a streaming solvent. He developed the fit theoretical description of the phenomena starting from the theory of optical activity due to Rosenfeld and Kirkwood [3]. In the following years the effect has been used in the research field of polymer solutions, especially biopolymer solutions 141. Orientation was mostly produced by an electric field or in a streaming medium. The systems were uniaxial. Measurements perpendicular to the optical axis were not very successfull, because the effect is very small [S]. Furthermore, the optical activity is obscured here by the linear dichroism 2nd birefringence. The separation of the different effects is not withqut problems [6]. In spite of the fact that the measurement of the anisotropy for small molecules was not very successful (for a more recent attempt see ref. [lo]), the theory has been treated in some detail by Co [7], Buckingham and Dunn [8] and Chiu [9]. The difficulties in measuring the effect can be seen from the. investigation of the coriditions under which the anisotropy is measurable [ 1 I].
* Part 1 An:ew,Chcm.
87 (1975) 200; Intern. Ed. 75 (1975)
176.
Contrary to it mmmonbut unfortunate habit, WC mean by co~~por~e~~ of s tensor y= ways Ien~or~ composing 7, e.g. the qUaIItitieS Tqk_.,
[email protected] [email protected]@... 'Ihe COt?ffiCicflrS Tj$ ~telative to the basis dclined by the vectors Ui.
. . . alXi,j,k.._Tijk_..ui~uj8uk CalI coordinates of tie tensor
we
H.G. Kuball et aLfOprico1
2. 2. The problem
acriviry of oriented
molecules.
II
.
.By Using a.new technique we were able to measure the optical activity (circular dichroisin) of oriented steroid ketones in an oriented liquid crystal solvent [12]. For the evaluation of the experimental data, a suitable theoretical description of the effect is necessary, but none is given in the literature which fulfills all conditions we need. The theory of Buckingham and Dunn (81 starts from the phenomenological description and for the explicit description of the dispersion
the WignerWeisskopf
sults, is only applicable
for transitions
formalism between
is used. But this formalism,
from which a Lore&z
function
re
two single isolated
states. If there are overlapping rotational, vibrational and electronic states, the WignerWeisskopf formalism is not correct as was shown by us in the theory of electrochromism and Kerr effect’ [13]*. GE’s theory [7], which is based on the result by a paper of Stephen (151, @KS the same result as the theory of Buckingham and Dunn [8], except for the dispersion. The circular dichroism MS. described by G6 as a KramersKronig transform of the r&ation. In this paper we want to develop the theory of G6 in order to derive equations for the evaluation of our experimental results. Furthermore, we want to discuss the theoretical results with respect to the molecular structure.
3. The starting The complex &=*+ie=
equation optical
rotation
Nd 9 8n2 V
& derived by Ga [7] for an oriented
f(% P, ~)H;l(e,
P, Y) sin P dadfldy
ensemble
of molecules
,
is given by** (1)
N is the number of molecules in the volume V;d is the optical path length; C, represents the sum over all electrons of a molecule; m and e are the mass and charge of the electron; V= wavenumber [cm‘] ; w = 25rcr, c is the vejocity of light ina vacuum; k = o/c; q is a positive infiitesirnal;p~ 7 operator of linear momentum; wno = e‘(E,, E&E,, E,, energyof the states In>, IO); primed symbols as xi, xi, xi etc. refer to the space fmed coordinate system, unprimed letters are related to the coordinate system fmed to the molecule; latin indices denote vector and tensor coordinates. Q is the optical rotation (ORD) a;td 0 represents the ellipticity, tg 0 = b/a, where a and t, are the major and the minor axis of the ellipse, respectively. The equation is valid for small rotation and ellipticity. This condition is not fulfilled in general. Especially, one has to be careful with measurements perpendicular to the o#ical axis. We want to discuss this problem at a later time. For the experimental results presented hete, the condition is sufficiently fulfilled because we want to discuss measurements parallel to the optical axis. i?“] is an eletient of the interaction matrix l!$ of a photon with a space fured molecule. The incident light is prodistribution function pagating in the jz; direction and is polarized in the xix; plane.f(a, 0,~) is th e orientational and Q, 0,~ are the eulerian angles as defined in appendix 1. Furthermore, there holds the equation
[email protected],r)siw’3dadW= 1. .’ l
The r&Its
differ by a factor of about tluee in Ihe amplitude.
‘* lhi signis s+nged with .espectto the pnpcrsof ‘,
Gu in.order to have the us4
convention for the sign of the rotation and ellipl
. i&y.
:_ .’ : :., : : ..
_
,.
,,
:
:
._
,_::
_
H.G. Kuball
4. The transition moment tensor Dy
et ol./Opricd
arriviry of cwie~rredmolecuk
3
II
and the tensor of rotaCon R$’
Using the relation Pie *GjOn(p;ei%jnO where the wavefunctions e*t%,
Ii) are chosen
= 1 k ax;ll
eq. (1) can be transformed
The following D;?
(3)
real, and the expansion
,
(4) into
abbreviations
are used here:
= (o~;ln) (nlP;lo’ = $U;:)&L&o
p;. = 
,
= @;ei&)
[email protected];ei%),o
,
(6a)
Cd N’ Y
R;y3 = 
(6b) QQ)~
The third rank tensor R$! R;?; = R;;;
).
(’
&
.
(6c)
no shows the follcwing
behaviour
,
(7a) (7b) (7c)
The third rank tensor, antisymmetric rank two (see appendix 2):
in the first pair of indices,
can be replaced
by a symmetric
pseudotensor
of
of ekli are equal to zero. If there is a El23 = E312 = E231 = 1; El32 = E~13 = E321 =  1; all other components repeated tensor suffm, it means a summation over this index taking the values 1,2 and 3. From this it follows R’O” =R’b 123 33
.
With eq. (9) and the transformation can be written as
where 6ii is the Kronecker gklij
= L&a,
symbol
P, y)~~i~,j
of D;>@’and R$
to a body fixed coordinate
sy’stem (see appendix
l), eq. (5)
and sin P da dp dy 
The new tensor Rg which we want to c& the tensor netic dipole and an electric quadtupole contribution:
of rotation
(Rdtationstensor)
can be separated
into a mag
‘. _‘. :
. .
.. ..
‘.:..’
H.G.Kuball et al&pica1
4
activity of oriented molecules. II
(124
(12b) (12c) (126) (nlijh 0 and (Qii)no represent the magnetic dipole transition moment ment respectivkly.gklii are the orientational distribution coefficients.
5. The frequency
dependence
and the electric
quadrupole
transition
me
of $ (dispersion)
In order to examine the frequency dependence of the complex function 6 in eq. (10). one has to introduce librational states as Moffitt and Moscow&z have done in the theory of the optical activity of isotropic solutions [ 161. Therefore we introduce the following model: A solute molecule is surrounded by a large number (N’) of solvent molecules interacting by intermolecular forces. This weakly bound system has 3(N’ + 1)  6 normal coordinates of vibration which can be identified with the librational coordinates. The force constants of these librations are small which means that the energy gap between two adjacent librational states is small, as compared with the energy differences of the vibrational and electronic states*. Therefore we may consider the librational states as 2. quasicontinuum. So we represent our states by
,
IO) = !NnK+.+‘)~
IN = wm~&d”)~
.
(13)
The energy gap between these states is fiw,ro = h(w”  w’).N and K are the quantum numbers ground state and an excited electronic state respectively. n and k are the quantum numbers of \X,.,,,,(W’)~ and \XKk(,: J”)) describe the librational states inNn and Kk. If and K respectively**. ground and the excited states is given by gNn(w’) andgA.(w”), the sum over n in eq. (IO) has sum over II, Kk and an integration over w’ and a”. Therefore, from eq. (10) follows
w
X (a”
0’)’
 J
{(a” 
w’)D$(u’, w”)SilSj2 
ZiwR$‘(w’,
o”)6i36i3)do’
for the electronic the vibrations inN the density of the to be replaced by a
dw”;
 2iqw
~~,,(a’) gives the probability that tie molecule is in the electronic ground We assume .that only one state N is populated_ As Aown in appendix 3, eq. (14) can be transformed to the equation
stateN
and ir? the vibrational
(14) state n.
‘.&I alterna!ivc model we have dcsdribed in an earlier paper [ 131. It is not necessary to consider the natural line width of the molecules [ 161. ** For siniplicity we speak here already in terms if the BornOppenheimer approximation. However, the la& is not yet assumed in ou_‘fqu&ons; we might interpret Wn), IKk)asvibronic states as well. Explicit use of the BornOppenhe#ner approximation i!_ made in eq. (19).
,, .’ .:. .” ..
. ...,,
._
.
H.G. Krrball er al./Opricnl
For this equation cfiK’(w)
acriviry of onknred molecules. II
5
we have to detine: = 1
[E&J’, w’ + w) +g(&,
o’  w)] dw’
[@.a’,
w’
,
(174
0 FN”Kk(u)
= r
w’
+ w)

g(o’,

w)]
dw’ ,
(17b) (174
f symbolizes Cauchy’s principal value of the integral. $rK’ and RpAx are given by eqs. (6) and (12) if 10) and In) are substituted by wn> and [MC). The terms GLD and @‘cD are responsible for the rotation of the plane of polarization and BLD and 0,~ for the (a,, ellipticity. In the approximation used the linear birefringence (aLD, 0 LD) and the circular birefringence tI,,) are additive. Furthermore, we have !to pronounce that eq. (15) only describes the anisotropy as an effect of the anisotropic distribution of rhe molecules. The effect of the orienting forces on the molecule is not included (e.g. effects like the electrochromism, if the orienting force is an external electric field). flnm(w) and @‘IKk (w) are the spectral functions of the rotation and the ellipticity. OLDand qCD are the GarnersKronig transforms of @‘LD and 8,, (see appendix 3).
6. Discussion
of the dispersion
phenomena
The complex optical activity 6 in eq. (15) is determined by four terms. Two of them are concerned with the linear birefringence (*‘LD, 19~~) and the other two with the circular birefringcnce (@CD, 13~~). In the LD the rctation is due to the absorption effect, whereas in the CD the ellipticity is due to the absorption_ In the same way, *CD tid BLD arise from the elastic effects (effects of the refractive index)_ This different behaviour of the rotation and the ellipticity in LD and CD can also be seen from the KmmersKronig transforms which appeared in eq. (15). In both cases (LD and CD), the elastic effect is given in the equation as the transform of the dissipative effect. This means that ineq. (i.5) BL~ is the transform of +‘LD and @‘cD is the transform of 0,,. With respect to the measurable quantity (rotation,
[email protected]) the elastic and the dissipative effect change their roles. This is shown once more in table 1. Lf a sample exhibits LD as well as CD, its interaction with light has to be described as an elliptical birefringence [17]. That means the sample splits the light beam into two orthogonally elliptically polarized light beams. The whole effect is then given as a superposition of two elliptically polarized light waves of different phases and amplitudes. Within the approximation used here (small + and 0), the effect can be represented as a superposition of linear and circular birefringence”, as one can see from eq. (16). From that additivity one can derive the dispersion of these phenomena: * For large Q and 0 this is not true.
miation ellipticity
0 e
LD
CD
eu Q
“L“R
ma“1
‘LER
3) rtu, nI and ~1, EL are the refraclive index and the absorption coefficient of light polarized linearly parallel (0) and perpendicular (I) to the optical axis. “L, “Rand EL, ER are the refractive index and &Sorption coefficient of left (L) or right (R) circularly polarized light .
:
.,
..:
H.G. Kuball et ab/Opticol acrivify of onenred
6
molecules. If
Fig. 90
In the superposition, one has to keep in mind that the LD effect is about three orders of magnitude larger than the CD except when the LD is zero for syrnmetj reasons (e.g. parallel to the optical axis). One can see that the elastic effects (OLD, @CD) are @en by sigmoid curves, While the dissipative effects (+LD, 8~0) are represented by curves similar to those for an absorption process.
7. The tensorial
rotational
strength
RT
In the theory and the description of experimental results, the concept of the partial rotation was very successful in studying isotropic solutions. This concept can also be used for anisotropic & can be divided into contributions of the varions electronic transitions K *N.
and ellipticity [ 161 systems because
(18) The components in eq. (18) represent the contributions to LD and CD of the different absorption bands K +N. Very often it is sufficient to know the whole contribution of an absorption band and not its detailed wavelength dependence. In order to get this information we have to integrate over the contribution of the band. For the ellipticity we then get the expression
For this we have used the approximation R$fi
= 83
I(nNlkKl12
and the relations
+ instead of the
[email protected] and the concentration mostly use the circular dichroism’ Ae = E Ea (NL = Avogadro’s instead ofN/Y. With the relations fi,, = z ln 10 AeAC’d and N/lr = .103N, we can transform eq. (19) into
Experimentalists in C [mol/liter]
number),
 . AeA(U) dS__ 22.9 X lOA hc IO3 h~ lo s 3 32&JL ‘. in
RT ;;Wj3R&=
 &A(P) s
0
dv
[cgs] .
”
deA is the circtilar dichcoism of an anisotropic solution measured with light propagat& alqng &e “3” direction. We want to call RT the tensorial rotational strength (tensbrielie Rotationssttike). For an isotropic $lution this expression reduces to me usual rotatio&;trength RT.=Rik [eq. (21)l.
.
_. :
_‘... : .:‘_:y
.. .’ :.
.
.
,...:
H.G. Kubdl ci dfOpt%d
8. Special orientational
distributions
distribution function In order to calculate RfiK, the orientational ferent integrals gklj3(k, I = 1,2,3); only five of them are independent, g/&33
7
activity of oriented molecules. II
,(a, 0, y) has to be known. because the relation
There
are 6 dif
= 1 ,
holds. Furthermore, gk133 does not depend on LYand therefore the effect is invariant against a common rotation of the molecules about the xi axis (direction of the propagation of light). This is plausible because the effect [ 131 is due to circularly polarized light which has no favoured direction perpendicular to the axis of propagation. In the following part of this section, we want to show how elements of RgK can be calculated from measumments if special orientational distribution functions are chosen. (a) Isotropic
solutions
(21) (b) Completely parallel alignment In this case we can write
of the molecules
(fixed molecules)
(22)
~‘(*,P,~)=(~IT~Is~;~P)~(~~~)SCOP~)~(~Y~). It follows
for R’
R= = Rz
+ sir+
 sin 27u Ryf]
[RF 
 RE
+
sir?
yO
sin 2&, [cos 7u Rff
(Rff 
 Rrf)
sin 70 RT$
] .
(23)
AU coordinates of the tensor can be measured if we orient the molecules in 6 different ways with respect to the direction of the propagating light beam and if the circular and linear effects can be separated experirnentahy. This is shown in table 2. and the system of principal axes is thus obIf all coordinates are known, the tensor RF! can be diagonalized tained from experiment. (c)f(a, &7) with rotational The distribution function
[email protected],
A 7) = (Wsin
symmetry about the fixed xg axis is given by:
0) 6 (a  QO) 6 (P PO)
,
RT = R!$$ +
$ sin2
& [Ryf
With the choice of two different elements cannot be measured.
NK  2R + Rz2 $fl
(25)
angles flu. the constants
shown in table 3 can be measured.
The nondiagonal
(d) Fraser model In context with the anisotropy of infrared spectra in foils, Fraser [ 181 has represented an anisotropic distribution by a fraction (1 p) of isotropically distributed molecules and a fraction p of molecules which are distributed as described in (c). Theorientational distribution function is then given by f(o, 8,$
Tik, tensorial
= 1 P + P(4+nlj) rotational
strength
6(o  ou)W
 Bu) .
W)
will then be (27)
with two speciai~vahres of&, (0, :n), Ryf + RNK and RTf can be calculated if p is known. One can calculate the NJ.2 from a measurement with &, = 0 and the measurement of an isosame coordinates of the tensor of rotation Rkl tropic solution. Measurements with more than two values of flu do not, however, give information about 0. From .eqs. (21) and (27), one gets
Ran=(RT  Ri,)fp +Ri, R~
+R~~=3Ri,
9. The tensor
m3)
)
Ran,
(29)
of rotation R$K for molecules with D, symmetry
The second rank pseudotensor is symmetric and there are 6 unknown independent coordinates because the system cf principal axes in general is unknown. If all coordinates are measured,.the tensor can be diagonaltied and the systemof principal axes be determined. The.problem can be simplified using symmetry conditions. One of the most suitable cases is that of molecules possessing~D2 symmetry. If we consider transitions between pure symmetry states, only the diagonal elements are different from zero. The principal axes are given here by symmetry (the three C, axes). For the four different types of transitions from a ground state k, we have put the results together in table 4. A measurement paraUel to the optical axis and one of an isotropic solution are sufficient to give alI the desired infomiatioit.
.
10. Experimeritil
. ;‘~$eme&ured . . . ,,:
reidts
[ I.21 m:.an oriented liquid cry&l
. :
matrix as a.solvent the CD fur t&o unsaturated ketosteroids
Table 4 =I
a) All other coordinates
of RflK are equal to zero.
Table 5 ‘) I
RT0q) ~*
= RT  Rizo
Rko
0%)
II
0.33
1.21
PC) @.q) Rise (H) AEmns (HI Cma~ (HI
0.3 1 (29.9)b) 3.88 X 104’ 1.44 (29.6jJ) 39.4 (29.7)b)
45.3
(except RT
3.99x
1040
1.69 (ZS.SO)b) 1.30 (29SO)b) 33.2 (29.61b)
(28.33)b)
Af,Aax @.q) A’mns (Q.q) emax (QJi)
a) Solvent: (P.q) = liquid crystal, (H) = whcptanc b) Wavenumber [cm‘],
1.77 x 1040 2.22 x IO40
0.24 X IO+’ 3.76 x 1040 3.52 x lo=
(n.q)
(29.85,b) (29.9)b)
0.2 I (29.6)b) 1040 1.40 (29.3)b) 31.5 (29.7P) 4.32~
and Rise all IneaSUIClllentS
at T = 20°C).
=) P = (El, el)/(eu* EL). I
II
Fig. 1.
[I, II (fig. l), table 51 with light propagating parallel to the optical axis (T Z=36OC) and in the isotropic state (T= 80°C). in organic solvents the temperature dependence of the CD is small in this tempeiature region and therefore we believe that we can neglect to a good approximation $e influence of the temperature and combine these two measurements. The degree of polarization is also determined [12] (table 5). The data for the molecules dissolved in the liquid crystal are very similar to those in nheptane.(table 5). Therefore, the specific interactions of .the liquid crystal matrix with the solved optically active molecules cannot be very large. Because ART =RT Ri, ha the meaning of an anisotropy parameter (eq. 30)
[email protected] to that used in the description of.the Kerr effect, for I and 11 these numbers are given in table 5
10
H.G. Kuball
et al. fOpticaioctivity
of orienred molecules.
II
As we see from table 5, there are drastic effects especially for 1 where we have a change in sign between RT and Rim and furthermore MT is larger than RT by a factor of 16. In the case of II, RT and AR’ have different signs and.Ri, is larger than RT by a factor of 2. In spite of these effects both compounds have  except for the sign a rather similar behaviour in the CD in isotropic solution [19]. From this result one may hope that the measurements of the anisotropy of the optical activity will give more Information about the molecularstrucfurc than that obtainable from an Isotropic solution.
11.
Estimation of the coordinates of RF
is not well known. With a very rough model which The orientational distribution function and themforegk133 Fraser has used for describing the anisotropy of 1R spectra (section 8, model d), we can estimate the.magnitude of the tensor coordinates RF . In order to determine p with a method Yogev et al. [20] have used for the interpretation of the anisotropy of W spectra, we need two further assumptions: (a)The transition moments of the na* transition in I and I1 are of the same magnitude but have different directions (nearly 240” between each other). (b) The fractionpof oriented molecules is the same for I and II in the liquid crystal matrix (this is rather a crude assumption but in polyethylene sheets it works very well 1201). From the degree of polarization (table 5), the direction of the transition moment can be estimated for I at about 25’ and for II at about 215’ with respect to the long axis of the molecule (x3 axis, fig. 1). p can be estimated* to 0.3. The tensor coordinates calcuiated by eqs. (28) and (29) are given in table 6. In the theory of optical activity, the concept of the optical activity of the chromophore in a molecule is often used with some success. The chromophores in I and II
x
HL ‘c’
c
'C'
I
II 0 are of the szrne type. From the optical activity of the isotropic solution is has been concluded [ 191 that the chromophores (of the most populated conformation) are the Same in both compounds except that they are enantiomeric. If this were true th: different tensorial rotational strengths Rr would be a consequence of the different orientations of the chromophores with respect to the skeleton. We can investigate this furthx if we assume that the
chromophoric groups~in I and II are perpendicular to each other. The correct angle is 120”, but with this value the matrixelements cannot be estimated, because at the moment there is no knowledge of the nondiagonal elements. The tensor coordinates of ITare transformated to a new coordinate system & : x1 = yl , x2 = y3, x3 = yz) in order to have equal coordinates with respect,to the chromophores in I and II. We get the expressions (31)
with,the.experjmenG information (table 6) for . ~~~he~th&.ti&l description of the absorption prkcesi and the_explidt calculation will be given in part III of this &es .:.:.
.” ./ .’ . : ..
iZlj.
.: .. :, ._‘:._.’ “.
:. : .
..
.: . ..
H.G. Kuball et al./Optical
activity
of oriented
molecules.
11
iI
Table 6=) I 1040 R$$
9
104’ (Ry,K+RNK 22
"R~lK,R$!!and R$ x,.x2
and13
II
1
3.4
20
15
arethecoordinates of the tensor of rotation which one would measure with light propagating along the
axis.
(32) The tensor coordinates ‘R1, are now comparable to those of ‘Rrr. We can calculate the individual coordinates by eq. (32) and the results of table 6. But one has to keep in mind that the chromophore in I1 is the enantiomerie form of that in I. Therefore we have to change the sign of the coordinates of II in table 6 for the calculation. According to the calculation the tensor coordinates of the chromophoric system in I are ‘Ryf = 24 X 10H40; r# = 3 X 1(r4$ IRE = 9 X 1O4o (cgs). The results are not very plausible. It is not easily understood why rR is so large because the magnetic and electric moments should be mainly in the 2,3 plane. Whether the quadrupole transition moment can give a contribution in this order of magnitude is not known at the moment. On the other hand one should ask whether the assumption that the chromophoric system is the same in I and 11 is realistic. From other o, #Iunsaturated ketones we have investigated [21], one has also to conclude that the chromophoric system does not seem to be the same in different Q, fl unsaturated ketones as one would believe from the results of the investigations of isotropic solutions.
Acknowledgement Financial support from the “Deutsche Forschungsgemeinschaft”
is gratefully acknowledged.
Appendix 1: Coordinate systems and transformations xi K = nodal line A i = aii A,! ,
(AlI Aij= aikairAie .
/ 1
A
\ \ \ \ Xl
cosacos/3cosysinasiny aij =
cosacosj9sin~ssinrrcos~
cosasinp
sinacosj3cos7+cosasin7 ~sinrrcospsiny+coslYcosy
sinasinp
sinflcos7 sinpsin7 cosP
62) )
12
H.G. Kuball er cil./Opticai
Appendix.2:
Reduction
acriviry of ohrued
nmlenrles.
II
of the third rank tensor
The symmetric second rank tensor
Di7induces the represedtation (A31
of O(3) while the third rank tensor Riik = Riik induces (A4) r%; it has no comG is only given by the coordinates Dlt and R;,,. Diz belongs to the irreducible representation roU + rzU and has no component in lTlu as one can see, for inponent in rag. Riz3 belongs to the representation stance from the first paper of GG. A symmetric pseudotensor of the second rank Rb = Rii induces the same’representation rou + rzU of o(3) as the special coordinate R;,,. Therefore this coordinate can be expressed linearly by the coordinates Rij of such a pseudotensor. We define Rii = t(EX,iR~,j+
and get R;,,
Appendix
645)
=~,iR;rli) ,
=R&_
3: Derivation of eq. (15) (frequency dependence
One has to transform
at first the frequency
I
1
(a”  W’)2  6J2  2iqw = 2(0”  0’) 9
1
=
2(r_/  w’)
(
2(&J” w’) (&‘_ &)2  ,2
of 6)
factor of eq. (14);
(
1
W”  a’
w + *ina+
1 u” _w’ + w + 2iqw 1
+ ix {b(cd”  w’  cd)  15(u”  Cd’ + W))
)
,
(W
where the relation I/(x
Civ) =9(1/x)
7 ins(x)
has been used. Insertion of (A6) into (14) together with the substitution w” = w’ + D yields a new equation in which the librational states have to be separated from the electronic and vibrational states. The approximation used is shown in eq.(13). Because&w’G) = 0 ifw’
2 f 0
‘LD=;;
dw
zj2_,2
IA71
’
.’ 2 *CD=,
(A%

are also valid for a system .case of electrochromism.
of oriented
molecules.
Eq. (A7) has been given in an earlier paper
...
_: _ ,‘..
_.
[ 131 for the special
.
...
:
..
. .
.::
._:,
: ..;
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