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Elliptical dichroism of oriented helical polymers
Volume 79, number
CHEhlICAL
1
ELLIPTICAL
~~C~~~~S~~
OF ORIENTED
David A RABENOLD Itzrlrrlre of dlolec~rlar Broplr\ sacs. TIE Florida State TallnIlarsee
Rorrda
Rcce~\ed 8 December
32306,
PHYSICS
HELKAL
i Aprii 1981
LETTERS
POLYMERS
Dtrr~ erntl,,
US4
f9SO
h hnear response theor? of elhptlcai drchrolsm IS obtamed for hght propagatrng perpendlcuku to the heh\ axes of a spstern of orwnted hehcsl uotvmers by solvmg the equntlon ot motion for the radI;ltlon field’s average vector potentml A nonl.mea.r mature of cucuhr and hear dvzluolsm and buerrmgcnce IS obtaned
There has been a contmumg interest in circul3.r dlchro~sm (CD) of tsotroprc as well as anisotroplc systems Formulatlons of the CD of anIsotropIc systems [l-9] are usually based on n quantum mechamcai scattermg theory [lo] and have used the mtultlve concept of a rotational strength tensor to describe the CD for various orlentatlons of the system to the dIrectIon of h&t propagation Other formulations have used lmear response theory [ll ] or clawnl polarirnbihty theory [lZ!j to express the CD for each of the three orthogonal ortentations Some awareness eusts of the dl~t~cuity of interpreting ltleasurements for hght propagatmg Perpendi~~llar to the optical ayts, say of a cylmdrlca!ly symmetric system (solution of altgned helical polymers) [4,1 I ] Moreover, It IS wetl hrlown that qundrupole contrlbutlons to the CD vamsb on the average over orientations while for mdwdual onentations they contribute In this note we show for a solution of ahgned hehcal polymers that the rotatIona strength tensor concept IS meanmgful only for mdlcatmg part of the strength of the transverse current linearly mduced by the radiation field Furrhermose, we show that the CD IS not defined for all orlentatlons because of the non&near mixture of finear and circular dIchrotsm and buefrmgence that IS obtamed for hght propagation perpendlcuiar to the hehx axis Instead of using a scattering theory our work rests on the solution to the equation of motion for the average vector potcnttal for the radmtlon field for I&t pawn g through the sample [13-151 That IS, our obJect Is to solve (w’fc’
- $1
A&, w} = - (-I+)
Cj(q. w)>,
where A(q. w) IS a Fourier component of the average vector potent131 with wavevector q and frequency w tj(q, w)> ISa Founer component of the transverse current Imearly mduced by the radtatlon field We assume perfect alignment, all hehcal axes parallel to one another Prewous work [14] provides tractable expresstons for wherem a multlpole expansion gives the current m terms of field quantltles and susceptlb&tles Tlus IS simplified by expandmg about 4 = 0 and keepmg terms up to ones hnear m Q_ At tius pomt quadrupole contrrbuttons to the current east for IndI~dual orlentattons of q to the hehx anus These vamsh for the isotroptc case. Let us first consider q = i,q_ _ wth the I aus bemg the hehx axts The hneariy Induced current IS [I 5 1 (4ir/c) (j(q,
w>> =
cG(W) (0
0 .l(,,K:)+(:=(”
;=+&p,,
-3
(Ii;)>
(2)
where, n~~ectlng the solvent, C@(U) and fi=(w) are stmply related to the complex susceptiblfitles responstble for ordmary absorption and CD (ako optlcal rotatory dxsperslon) respectiveiy. The symbol CY~(W)m&icates that the 86
Volume
79, number
1
CHEMICAL
PHYSICS
LETTERS
1 Aprd 1981
susceptrbihty contams transrtrons polarrzed perpendicular to the hehx axrs. The symbol p=(o) mdrcates that rt contains the approprrate susceptrbrhtres for z axrs propagation. Thrs has been presented earher [15] m detarl except for the fact that /3’(w) here contains quadrupole contnbutions. They were not mcluded m the prevrous work [15] which was for rsotroprc medra. The presence of quadrupole contributions to /3’(w), whrch wdl be presented r.n detail elsewhere, does not alter the form of (2). Eq. (1) wrth (2) substrtuted mto it may then be wrrtten as [q; - c&7-
- ocL(o) + #(w)]
A’ = 0,
where A’ = 2-lj2
(4 _~+ rA,).A+(A-)
corresponds
+ { [p’(w)]*/4
4; (w) = + p=(o)/‘! Defining
a complex
N,(w)
Oll(o)
to left (nght)
crrcularly
polanzed
lrght
Thrs gives
f aL(W)p2
(4)
mdex of refractron
w/c = [+J)
where 12’ mdrcates
+ cJ/c?-
(3)
+ lazy]
w/c = 4: (w),
(5)
the real part and U” the imaginary
= [12:(o)
-H:(W)]
and, for monochromatrc
0/2c
= -i
part, grves the elhptrcrty
0 (0)
per unit length
as
Im o’(w),
(6)
plane wave propagation,
A(=, f) = {Ai exp[r(w/c)
+A6
N+=]
exp[l(o/c)
N-z])
e-‘W’.
(7)
For thrs case, a hnear relatronshrp IS obtamed between O”(o) and p=(oj and, thus, the rotatronal strength mdrcates the strength of the CD For lrght propagating perpendrcular, say along X, to the hehx axis the induced current IS [ 151
directly
(8) The symbol o!(w) mdrcates the susceptrbrhty contams transrtrons polarrzed parallel to the hebx axrs B(w) contams the appropnate susceptrbrhtres for hght propagatron perpendicular to the hehx axis [ 151 p(w) contarns rotational strengths whrch hnearly drctate part of the strength of the mduced current but, as shown below, not the CD for thrs case Substrtutron of (8) mto (1) gives
q;fJ/c2(
d(cd)
‘;;
--‘4x
Eq
lq~~(w)
q; - cd*/2 - a”(W)
NW)
(9), symbohzed
as [
] A = 0,s
A, = 0.
)(AZ)
solved by transfonnmg
(9)
wrth
(10) asR[
] R-IRA
=Owhere
tan 2 x = [BY” The components =A-cosx-A
nents
[q; - Cd*/2
There are a number
= o/c
129, p(o)_
(11)
of RA are the left and nght elliptrcally polarrzed components AL = A- sn x + AC cos x and AE + SUIx respectively. As x goes to zero a”(o) = c&(w) and th e usual cucularly polarrzed compo-
are obtarned.
q; = [&c’
- c&(o)]
Eq. (9) becomes - $ [d(w)
+ d(w)]
of approxrmatrons +$ [a”(W) + Cd(o)]
+ (c/4w)[(Y”(O)
+ d-(w)]
T $ { [cY”(CtJ) - oc(w)] 2 + 4q,2~(o)p2]
that rmght be made + f {[d(w)
F (c/4w)
We employ
the sunplest
- CYl(w)]* + 4(0*/c*)
{
)l’2
A6 = 0
(12)
to obtain
p*(w)}‘1z]
l/* (13) 87
Volume
73, number
1
CHEMICAL
PHYSICS LETTERS
1 Apnl1981
and
d’(b)
=:
[m[q:(o)
-q;(o)]
* -(c/G>
lm {[a”(w)
- a’(w)]’
+4(&/c”)
~P(w))~“.
(14)
O’(o) IS the elhptlcal dlchrolsm (not the CD) and IS non-hnearly related to the hnear and circular bxefrmgence a?d dlchrotsm The rotational strengths m p(w) do not dictate the strength of 8’(w) It is clear then that the isotropic CD is not Zj 0” t$ f?‘, whereas the Induced current For an isotropic system 1s Cj> = $j” +$ iL ~mploy~ient of thx(j) m (1) ywlds the CD for an isotropic system as shown earher [13--151. Recently there has been some work on the CD of molecules of arbitrary size [16,17] The fiidmgs here suggest for rhx a more correct approach which IS to fLrst obtam the transverse current hnearly Induced by the field [l 1, 13 14.1 wlule avoldmg a multipole expansion i he wavevector dependent current-current susceptlblhty that accounts for the induced current does contam the mechamsm for rotatmg the ptane of polarlzatlon of the hght [i I, 131 An agerngmg procedure would be needed to mahe the result applicable to tsotroplc media. However, for both lsotroplc and amsotroptc cases the mdured current mtist be substituted Into (1) for solutron In order to obtam refractlve tndlces The model system used earher [17.18], an electron constrained to move on a hehx should be useful here The author IS grateful for support by Professor Wllham Rhodes through Contract between the Dtvrsron of Blomedwal and En~ror~~nental Research of the Department Unrvzrslt~
No DE-ASOS-78EV05784 of Energy and Florida State
References [l ] H -G [2j H-G 131 H -C [4j H-G
KubaU, J Akschuh 2nd 4 Schonhotfer. Chem Phss 43 (1979) 67 Kiub&, hl Acuwsctnd J Aitschuh, J L\m Chem Sot lOI(1979) 20 Kub.U, 3 \ltschuh, R Kulbsch and A Schonhoffer, Helv Chun Acta 61 (1978) 571 KubaU,T Karstersnnd A Schonhoffer, Chem Ph)s 12 (1976) 1 [ 51 I Tmoco Jr , 4dvan Chem Ph) s 4 (1962) 113 [6] J Snir and J Schellman, J Phys Chem 77 (1973) 1653 [7f FM Louom, J Chem Phys 51 (1969)4899 Phls Rev Bl (1971)) 858,Intern J Quantum IS] CW Deutsch, J Chem Phys 52 (1970) 3703 [Qj hl R Phdport J Chem Phys 56 (1972) 683 [lOI hl J Stephen, Proc CambrIdge PM Sot 54 (1958) 81 [ll] W Rhodes, J Chem Phys 53 (1970) 3650. W Rhodesand S X1 Redmann Jr , Chem Phys 22 (1977) 215 1121 A I. Levmand 1 Tmoco Jr, J Chem Phys 66 (1977) 349 tl3) R L Fulton,J Chem Phys 55 (197I.1 1386 [14) D A RabenoId, J Chem Phys 62 (1975) 376 [ 151 D A Rabenold, J Chem Phys. 74 (Jan 15,1981), to be pubhshed {161 1 Tobl3s.T R Brockland N L Balazs, J Chem. Phbs 62 (1975) 4181 [17] I Tmoco Jr, intern J QuantumChem 16 (1979) 111 [18] I Tmoco Jr and R \V Woody, J Chem Phys 40 (1964) 160
88
Chem
3S(f969)
147
CHEhlICAL
1
ELLIPTICAL
~~C~~~~S~~
OF ORIENTED
David A RABENOLD Itzrlrrlre of dlolec~rlar Broplr\ sacs. TIE Florida State TallnIlarsee
Rorrda
Rcce~\ed 8 December
32306,
PHYSICS
HELKAL
i Aprii 1981
LETTERS
POLYMERS
Dtrr~ erntl,,
US4
f9SO
h hnear response theor? of elhptlcai drchrolsm IS obtamed for hght propagatrng perpendlcuku to the heh\ axes of a spstern of orwnted hehcsl uotvmers by solvmg the equntlon ot motion for the radI;ltlon field’s average vector potentml A nonl.mea.r mature of cucuhr and hear dvzluolsm and buerrmgcnce IS obtaned
There has been a contmumg interest in circul3.r dlchro~sm (CD) of tsotroprc as well as anisotroplc systems Formulatlons of the CD of anIsotropIc systems [l-9] are usually based on n quantum mechamcai scattermg theory [lo] and have used the mtultlve concept of a rotational strength tensor to describe the CD for various orlentatlons of the system to the dIrectIon of h&t propagation Other formulations have used lmear response theory [ll ] or clawnl polarirnbihty theory [lZ!j to express the CD for each of the three orthogonal ortentations Some awareness eusts of the dl~t~cuity of interpreting ltleasurements for hght propagatmg Perpendi~~llar to the optical ayts, say of a cylmdrlca!ly symmetric system (solution of altgned helical polymers) [4,1 I ] Moreover, It IS wetl hrlown that qundrupole contrlbutlons to the CD vamsb on the average over orientations while for mdwdual onentations they contribute In this note we show for a solution of ahgned hehcal polymers that the rotatIona strength tensor concept IS meanmgful only for mdlcatmg part of the strength of the transverse current linearly mduced by the radiation field Furrhermose, we show that the CD IS not defined for all orlentatlons because of the non&near mixture of finear and circular dIchrotsm and buefrmgence that IS obtamed for hght propagation perpendlcuiar to the hehx axis Instead of using a scattering theory our work rests on the solution to the equation of motion for the average vector potcnttal for the radmtlon field for I&t pawn g through the sample [13-151 That IS, our obJect Is to solve (w’fc’
- $1
A&, w} = - (-I+)
Cj(q. w)>,
where A(q. w) IS a Fourier component of the average vector potent131 with wavevector q and frequency w tj(q, w)> ISa Founer component of the transverse current Imearly mduced by the radtatlon field We assume perfect alignment, all hehcal axes parallel to one another Prewous work [14] provides tractable expresstons for
w>> =
cG(W) (0
0 .l(,,K:)+(:=(”
;=+&p,,
-3
(Ii;)>
(2)
where, n~~ectlng the solvent, C@(U) and fi=(w) are stmply related to the complex susceptiblfitles responstble for ordmary absorption and CD (ako optlcal rotatory dxsperslon) respectiveiy. The symbol CY~(W)m&icates that the 86
Volume
79, number
1
CHEMICAL
PHYSICS
LETTERS
1 Aprd 1981
susceptrbihty contams transrtrons polarrzed perpendicular to the hehx axrs. The symbol p=(o) mdrcates that rt contains the approprrate susceptrbrhtres for z axrs propagation. Thrs has been presented earher [15] m detarl except for the fact that /3’(w) here contains quadrupole contnbutions. They were not mcluded m the prevrous work [15] which was for rsotroprc medra. The presence of quadrupole contributions to /3’(w), whrch wdl be presented r.n detail elsewhere, does not alter the form of (2). Eq. (1) wrth (2) substrtuted mto it may then be wrrtten as [q; - c&7-
- ocL(o) + #(w)]
A’ = 0,
where A’ = 2-lj2
(4 _~+ rA,).A+(A-)
corresponds
+ { [p’(w)]*/4
4; (w) = + p=(o)/‘! Defining
a complex
N,(w)
Oll(o)
to left (nght)
crrcularly
polanzed
lrght
Thrs gives
f aL(W)p2
(4)
mdex of refractron
w/c = [+J)
where 12’ mdrcates
+ cJ/c?-
(3)
+ lazy]
w/c = 4: (w),
(5)
the real part and U” the imaginary
= [12:(o)
-H:(W)]
and, for monochromatrc
0/2c
= -i
part, grves the elhptrcrty
0 (0)
per unit length
as
Im o’(w),
(6)
plane wave propagation,
A(=, f) = {Ai exp[r(w/c)
+A6
N+=]
exp[l(o/c)
N-z])
e-‘W’.
(7)
For thrs case, a hnear relatronshrp IS obtamed between O”(o) and p=(oj and, thus, the rotatronal strength mdrcates the strength of the CD For lrght propagating perpendrcular, say along X, to the hehx axis the induced current IS [ 151
directly
(8) The symbol o!(w) mdrcates the susceptrbrhty contams transrtrons polarrzed parallel to the hebx axrs B(w) contams the appropnate susceptrbrhtres for hght propagatron perpendicular to the hehx axis [ 151 p(w) contarns rotational strengths whrch hnearly drctate part of the strength of the mduced current but, as shown below, not the CD for thrs case Substrtutron of (8) mto (1) gives
q;fJ/c2(
d(cd)
‘;;
--‘4x
Eq
lq~~(w)
q; - cd*/2 - a”(W)
NW)
(9), symbohzed
as [
] A = 0,s
A, = 0.
)(AZ)
solved by transfonnmg
(9)
wrth
(10) asR[
] R-IRA
=Owhere
tan 2 x = [BY” The components =A-cosx-A
nents
[q; - Cd*/2
There are a number
= o/c
129, p(o)_
(11)
of RA are the left and nght elliptrcally polarrzed components AL = A- sn x + AC cos x and AE + SUIx respectively. As x goes to zero a”(o) = c&(w) and th e usual cucularly polarrzed compo-
are obtarned.
q; = [&c’
- c&(o)]
Eq. (9) becomes - $ [d(w)
+ d(w)]
of approxrmatrons +$ [a”(W) + Cd(o)]
+ (c/4w)[(Y”(O)
+ d-(w)]
T $ { [cY”(CtJ) - oc(w)] 2 + 4q,2~(o)p2]
that rmght be made + f {[d(w)
F (c/4w)
We employ
the sunplest
- CYl(w)]* + 4(0*/c*)
{
)l’2
A6 = 0
(12)
to obtain
p*(w)}‘1z]
l/* (13) 87
Volume
73, number
1
CHEMICAL
PHYSICS LETTERS
1 Apnl1981
and
d’(b)
=:
[m[q:(o)
-q;(o)]
* -(c/G>
lm {[a”(w)
- a’(w)]’
+4(&/c”)
~P(w))~“.
(14)
O’(o) IS the elhptlcal dlchrolsm (not the CD) and IS non-hnearly related to the hnear and circular bxefrmgence a?d dlchrotsm The rotational strengths m p(w) do not dictate the strength of 8’(w) It is clear then that the isotropic CD is not Zj 0” t$ f?‘, whereas the Induced current For an isotropic system 1s Cj> = $j” +$ iL ~mploy~ient of thx(j) m (1) ywlds the CD for an isotropic system as shown earher [13--151. Recently there has been some work on the CD of molecules of arbitrary size [16,17] The fiidmgs here suggest for rhx a more correct approach which IS to fLrst obtam the transverse current hnearly Induced by the field [l 1, 13 14.1 wlule avoldmg a multipole expansion i he wavevector dependent current-current susceptlblhty that accounts for the induced current does contam the mechamsm for rotatmg the ptane of polarlzatlon of the hght [i I, 131 An agerngmg procedure would be needed to mahe the result applicable to tsotroplc media. However, for both lsotroplc and amsotroptc cases the mdured current mtist be substituted Into (1) for solutron In order to obtam refractlve tndlces The model system used earher [17.18], an electron constrained to move on a hehx should be useful here The author IS grateful for support by Professor Wllham Rhodes through Contract between the Dtvrsron of Blomedwal and En~ror~~nental Research of the Department Unrvzrslt~
No DE-ASOS-78EV05784 of Energy and Florida State
References [l ] H -G [2j H-G 131 H -C [4j H-G
KubaU, J Akschuh 2nd 4 Schonhotfer. Chem Phss 43 (1979) 67 Kiub&, hl Acuwsctnd J Aitschuh, J L\m Chem Sot lOI(1979) 20 Kub.U, 3 \ltschuh, R Kulbsch and A Schonhoffer, Helv Chun Acta 61 (1978) 571 KubaU,T Karstersnnd A Schonhoffer, Chem Ph)s 12 (1976) 1 [ 51 I Tmoco Jr , 4dvan Chem Ph) s 4 (1962) 113 [6] J Snir and J Schellman, J Phys Chem 77 (1973) 1653 [7f FM Louom, J Chem Phys 51 (1969)4899 Phls Rev Bl (1971)) 858,Intern J Quantum IS] CW Deutsch, J Chem Phys 52 (1970) 3703 [Qj hl R Phdport J Chem Phys 56 (1972) 683 [lOI hl J Stephen, Proc CambrIdge PM Sot 54 (1958) 81 [ll] W Rhodes, J Chem Phys 53 (1970) 3650. W Rhodesand S X1 Redmann Jr , Chem Phys 22 (1977) 215 1121 A I. Levmand 1 Tmoco Jr, J Chem Phys 66 (1977) 349 tl3) R L Fulton,J Chem Phys 55 (197I.1 1386 [14) D A RabenoId, J Chem Phys 62 (1975) 376 [ 151 D A Rabenold, J Chem Phys. 74 (Jan 15,1981), to be pubhshed {161 1 Tobl3s.T R Brockland N L Balazs, J Chem. Phbs 62 (1975) 4181 [17] I Tmoco Jr, intern J QuantumChem 16 (1979) 111 [18] I Tmoco Jr and R \V Woody, J Chem Phys 40 (1964) 160
88
Chem
3S(f969)
147