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Linear polarization Raman optical activity: a new form of natural optical activity
Volume 174, number 6
CHEMICAL PHYSICS LETTERS
23 November 1990
Linear polarization Raman optical activity: a new form of natural optical activity Lutz Hecht and Laurence A. Nafie Department
oj Chemistry,Syracuse University, Syracuse, NY 13244-4100, USA
Received 22 May 1990; in final form 12 September 1990
A new form of natural optical activity is identified on the basis of a general theory of natural Raman optical activity (ROA). This novel ROA phenomenon is predicted to occur as the difference in Raman-scattered intensity for pairs of linearly polarized measurements and depends on the real part of the product of the transition-polarizability and optical-activity tensors. This is the tint form of ROA to be identified that does not directly involve the use or measurement of circularly polarized radiation.
1. Introduction Natural Raman optical activity (ROA) has previously been described in the most general terms as arising from the differential Raman scattering intensity of right versus left circularly polarized radiation [ l-41. ROA was originally defined [ 5 ] as the difference in Raman scattered intensity for right minus left circularly polarized incident radiation with linearly polarized or unpolarized [ 3 ] scattered radiation. The result of such a measurement is termed a circular-polarization intensity difference (CID) and is defined as Z,”-Z,“, and the normalized CID is formed by dividing by the corresponding circularpolarization intensity sum (CIS), ZE+ Zk, where the superscripts refer to the polarization state of the incident radiation and the subscripts refer to the polarization state of the scattered radiation. More recently, the definition of ROA was extended to include the measurement of circular-polarization components of the scattered radiation, with linearly polarized incident radiation [3,6]. In this case the measured quantity was termed degree of circular polarization and was defined as the ratio of the Stokes vector component for the circularity of the scattered radiation to the Stokes parameter for the total intensity. Subsequently, a particular method of measuring the degree of circular polarization was proposed and demonstrated [ 71. It was simply the direct
analogue of the original ROA measurement technique, namely the CID, Z$ -If. This particular approach to ROA was termed scattered circular-polarization (SCP) ROA, and analogously, the original form of ROA was called incident circular-polarization (ICP) ROA. Following this conceptual direction, a new form of circular-polarization ROA was discovered [ 81 and named dual circular-polarization (DCP) ROA, where the measured CIDs are Zi - Zi and Zk -If and where the former is referred to as DCPiROA and the latter as DCPuROA. A comprehensive theoretical description of the various forms of circular-polarization ROA has recently been completed [ 91. In this paper, we extend the known forms of ROA beyond those involving CIDs. We define a new class of ROA measurements involving linear-polarization intensity differences (LIDS) by analogy to the four CIDs described above. Hence, as in the case of circular-polarization (CP) ROA, three forms of measurement exist for linear-polarization (LP) ROA: incident (ILP), scattered (SLP), and dual linear-polarization (DLP) ROA. In the case of dual polarization ROA it is possible to realize experiments that combine CP and LP ROA into a single measurement and to measure circular-linear-polarization intensity differences, CLIDs, or the reverse, LCIDs, depending on whether the circular-polarization modulation is performed for the incident or scattered radiation,
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respectively. The identification of LP ROA arose from considerations of the relation of CP ROA invariants, contained in the general theory, to possible ROA experiments that could be realized to isolate the maximum number of theoretical invariant combinations [ 9]_ An intriguing aspect of the theoretical description of LP ROA is its complete dependence on the real parts of the usual product of the complex transitionpolarizability and optical-activity tensors, as opposed to the imaginary parts of the same tensors that appear in the CP ROA theory. A consequence of this property is the vanishing of LP ROA in the far-fromresonance condition of Raman scattering. Thus, LP ROA is a phenomenon that is a direct probe of the presence of resonance Raman intensity enhancement. The purpose of this paper is to define LP ROA in general, conceptual terms and present a precise theoretical description for a particular case of 90” scattering, using only linearly polarized radiation. This experimental configuration is likely to be a convenient, practical approach to LP ROA measurements. A comprehensive treatment of the theory of LP ROA, including a wide range of experimental configurations will be provided in a subsequent publication [ lo]. The theoretical expressions given below have been derived using a new polarization-propagation vector theory of ROA and employing a generalized Stokes-Mueller matrix theory, both of which were developed for our general theoretical treatments of CP and LP ROA [9,10]. Finally, we note that the basic LP ROA expressions provided by this paper were contained (to within a few minor differences) within expressions for the Stokes vector description of Raman scattered light in a previously published paper [ 61. The LP ROA contributions, however, were not isolated or identified as a form of ROA. Nevertheless, these prior expressions were a useful source of theoretical reference for the present work.
2. Definitions The tensor that underlies the scattering of light by the Raman effect is the complex Cartesian polarizability tensor for the n~rn vibronic transition, given by [3,6,1.1-131 576
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(1) where fia is the electric dipole operator; lj) represents a complete manifold of intermediate excited states; Wj, is the angular transition frequency between molecular states j and n; w is the angular frequency of the incident laser radiation; 5 is the bandwidth of the jth excited state and becomes important as the resonance condition Wjn- w is approached or realized; and greek subscripts are used to refer to Cartesian directions in the molecular frame. A tilde over a symbol indicates a complex quantity and the circumflex indicates a quantum-mechanical operator. The corresponding ROA transition tensors can be defined [ 6,8,14] by appropriate substitution of A with Hi,, the magnetic dipole operator, or with &, the electric quadrupole operator, by means of the conversion relations (&,),,+
(G&Jl,
(&&?ln-‘(
%&?l”
(%p)mn+
bL,ByL%n
(%7)
Qa)
by&+&, by /L-ad bYFr+&y bYCi,-ha,,
mn --) (~,asLn
>
(2b) mc)
3 Pp&
(2d)
Using these tensors, one can define the following standard tensor products [ 9 ] : (W
ao= (&a)mn(qJmn, aI = (&Jdnwz(%p)m
1
(3b)
&2=
9
(3c)
co = (&&I,($),”
3
(3d)
CI = (&shnn(G,),n
3
Ve)
G = (&,)rn,(c’+,)mn
2
(w
(&,)mn(&Y)m,
A = ~iW~,y6(~L,8)mn(Aj,dS)mn,
(3%)
22 = Wa~(4&d*,~~L~
(3h)
23 = fiW~,~(~,B),,(~,,~)mn.
(3i)
The symbols used above for these expressions have been used previously [ 8 ] except that tildes were not included, a0 and Go were written as (Y~and GS, and
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zX was omitted. The corresponding script standard tensor products g,, and .& can be defined in relation to the Roman standard tensor products by substitution of the corresponding script ROA tensor, eqs. (2b) and (2d), for the Roman one,, eqs. (2a) and (2c). Alternatively, the isotropic and anisotropic invariants for Raman and LP ROA can be defined as
(da) (4b) (4c)
(4d)
-d PM=
le
-i -e
2
(6h)
I ,
where ni and nd are the incident and scattered propagation vectors, and 2’ and dd are the corresponding complex polarization vectors. Finally, we define the following set of primitive standard tensor-product combinations that emerge directly from the isotropic averaging process [ 91 and span all known forms of ROA: s, =2G,,
(7a)
S,=GcrGc,-Al,,
(7b)
s,=-GJ+2G, -&A;
+ kr~(Ala,ti)*l”~>>
(W
where the symmetric and antisymmetric forms of tensors defined above are given by
(T,B)~,=f[(TaB)rnn+(TBn)rnnl 9
(sa)
(~~~)~“=f[(~~~,m,-(~~)mnl.
(5b)
The prime on the five LP ROA invariants indicates the real part, in contrast to the corresponding unprimed quantities that are widely used for the imaginary part of these same tensor combinations 191. The corresponding script ROA invariants are defined with the usual script for Roman tensor substitutions. Next, we define the polarization-propagation vector products (6a) jj22=Bi.;d@*,
~~‘9=nd.~iX&*,
~33=jd.giXgi*lli_gd*,
~,o=gi.pdXgdand.p,
(6b) (6~)
+A;,
(7c)
S.=-G)tG,tA;,
(7d)
&=-2G),
(7e)
&=C, -&-&,
(7f)
ST=-2G2,
(7g)
3. General polarization ROA expressions Using the polarization-propagation vector products in eqs. (6) and the primitive standard tensorproduct combinations in eqs. (7), we can write the following genera1 expression, involving two sets of polarization states labelled ( 1) and (2), for an ROA measurement of an isotropically averaged sample of chiral molecules,
= Fkil
Im[ (pii)-P’p))(
-iQ]
,
(8)
where K= $ ( co’p&(‘) / 4xR ) 2, defined previously [ 6,8,9 I. Eq. ( 8) can be regarded as a master polarization-propagation vector equation for all natural ROA measurements. In the case of CP ROA mea577
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surements, the two polarization factors in eq. (8) are complex conjugates of one another and, via Pi’) = &)+ =pk, eq. (8) becomes Alor.= Fkii
Irn(l’k) Im(&) .
(9)
In the case of pure LP ROA measurements (no use of circular-polarization states), the polarization products are real and eq. (8) reduces to NLp= Fk$,
(Pi’) -Pi2)) Re( -$J
.
(10)
For real polarization vectors the polarization-propagation vector products in eqs. (6a)-(6~) vanish and, further, the identities P4 = P5 =P6 = P7 and P,, =P,z=P,3=P,4 hold. By combining the primitive standard tensor-product combinations for these two sets of equal polarization-propagation vector products, we have &e=
c6K[(Pp-Pp)
xRe(3f?9-2c1
+3G,-&+A;)
+(Pl.;)-Pp) x Re(3@0-2~,-3@Z+&+&).
(11)
A general expression for the total Raman intensity may be written as [ 91 Z=Z”‘+I”‘=6K[ +P,5(-cYo--(Y*
( --cy0+4cy, -_(y2)
+4a,)+P,,(4a,-a,
average. The convention [ 15,161 that we use here for the Stokes vectors is that so is associated with the total intensity, s, with a projection of the polarization ellipse on the +45” or -45” axis, s2 with the degree and sense of circularity, and s3 with a projection of the polarization ellipse on the 0” or 90” axis. All four components of a Stokes vector will be written below in the form (so, sl, s,, s3).
4. LP ROA expressions for right-angle scattering As a typical set of examples of LP ROA, we consider the cases of right-angle scattering for ILP, SLP, DCPr and DCPrI measurements where only linearpolarization states are employed throughout. Other specific cases will be considered elsewhere [ lo]. We begin by considering the ILP experiment, depicted in fig. la, in which the incident polarization state is modulated between a polarization angle, @,of +45” and -45 ’ with respect to the laboratory x-axis, and the scattered light is vertically polarized along this same laboratory axis. The polarization and propagation vectors needed for eqs. ( 11) and ( 12) are e:= (u,+rr,)/Jz, A&U,, @l= (%-Ily)/J2, e& = uX , n& = u,, where u, is a unit vector in the (Ydirection. With these vectors we have Pi’) = l/2, Pi’) = - l/2, and PI5 =P16 = l/2 which yields AlILp=I+-I6K = cRe(3G,,-2G,+3eZ-&+&),
-az)]. (12)
An alternative approach to the expression of ROA intensities for various types of experiments is the Stokes-Mueller matrix theory [ 15,16 1. The master equation governing ROA, including total Raman intensity, for a sample of isotropically averaged chiral molecules is given by
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(Ida)
and IILp=I++I-=33K(~O$6~I+~2),
(14b)
where the + and - superscripts refer to positive and negative incident polarization angles, using the angular convention for @, shown in fig. la. Alternatively, via si= ( 1, -t 1, 0,O ) and sd = ( 1, 0, 0, l), eq. (13) becomes r’=~K[(Moo)+(M,,)~((M,,)+(M,,))l, (15a)
where sf and sf are the Stokes vector components associated with the intensity and polarization state of the incident and scattered radiation, respectively, and (MI_,> is the 4 x4 Raman Mueller matrix [ 111 where the angular brackets indicate isotropic spatial 578
where (M,o)=~[~~2t13B,(~)2+15~~(~)2],
(15b)
(M30~=~[451y2+8s(~)2-5B,(~~zl,
(15c)
4
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which yields
ILP ROA
hl ILP
=
[45((uG)’
?
+8:(G)‘-sB(G)2-B:(A’)2+P:(A’)Z], (15e)
I,,,=ZK[45a2+7B,(~)2+58,(6)2].
(15f)
When expressed in terms of Raman and ROA transition tensors, eqs. (14) and ( 15) give identical results. The experimental configuration for the right-angle SLP ROA experiment is given in fig. 1b. Here, the polarization and propagation vectors are: et,* =u,, i n1.2 =a,, d n ,.2 =
e;‘=(u,+u,)l$Z
e!=(u,--u,)/$,
rr, . The nonzero polarization-propagation vecproducts are PI1 ( 1) = k/2, P, , (2) = - I /2,
tor PIs=P16= l/2, the SLP ROA intensity is
W
M sI_p=z+ -I_
SLP ROA
= FRe(330-2@,
+3%+&
+Jj),
(16)
and the total Raman intensity is the same as in the ILP measurement above. The Stokes vectors for this example are si=(l,O,O, 1) and sd=(l, &l,O,O) which leads to r,=~K[(M,)+(Mo,)f((M,,)+(M,,))l> (17a)
(Ml3~=W30)
(17b)
9
(M,,}={M,~)=~I45(cu’ri)’ +j?:( 8)2-5/%( BFtg:(~)‘+8:(&2], Fig. I. (a) Scattering diagram for right-angle ILP ROA showing polarization and propagation vectors and incident linear-polarization angle; (b) scattering diagram for right-angle SLP ROA showing polarization and propagation vectors and scattering linear-polarization angle.
(17c)
and thus GLP
=
F
[45(0!9)’
+B:(~)2_58:(~)2+8l(~)a+B:(~)2]
) (17d)
(MOI > = = &
[45(aG)’
+BJ(G)2-5S:(G)2-8:(A1)2+8:(a)2]
)
(15d)
and ZsLpis the same as JILp.We have chosen the convention of the direction of the positive polarization angle, 6, to specify the first of the two Raman intensity measurements in eq. ( 16) and hence to define the sense of positive SLP ROA intensity. This 579
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is analogous to CP ROA where right circularly polarized light is used to define the sense of positive ROA intensity. The experimental diagrams for DLPl and DLPu right-angle ROA measurements are given in figs. 2a and 2b, respectively. The polarization vectors for the
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DLPI measurement are the modulated polarization vectors of the ILP and SLP measurements above. The resulting nonzero polarization-propagation vector products are P!j’)=1/4, Pa’)=-l/4, PIi)=1/4, PI:’ = - l/4, and PI5=P16= l/4. The resulting DLP, and total Raman intensities are AIDLPl=Z$ -Jr
= FRe[3(eo+
go)
DLPI ROA -2(G,+$)+3(G~++*)-&+Al,+&+J&], (18a) I DLPI=I: +I: =;K( -Ly, t 14a, -(Yz) I
(18b)
The Stokes vectors for the DLPI experiment are ?=(l, + 1, 0, 0) and sd=( 1, * 1, 0, 0) where the upper signs correspond to the first measurement and the lower to the second. These vectors lead to
AIDLn= :{45[(aG)‘+(o9)‘] •t MW+IM@)‘l
-5M@2-V:(
- [8:(~12-P:(~‘)21+
b)
DLPn ROA
IDL,Q=K[&dt
[Bb(W+/U
13fis(&)2t
15fi,(di)*]
%‘I
R21L (19b) ,
(19c)
where (M,, ) equals zero. The corresponding expressions for the DLPII ROA measurements are the same for the total Raman intensity expressions and differ from the DLP, ROA intensity expressions by a change in the signs of all terms involving the script tensor invariants. The signs of Pif'and Ply) change and xd= ( 1, T 1, 0,0)_These changes give AxDLPu =I?
-I:
=
FRe[3(G-
go)-2(G,
- 3,)
+3(~~--~)-AlztA;-~,--~],
(2Oa)
=~{45[(c~G)‘-(cuY)‘]t[j?;(8)~-~:(~)~]
Fig. 2. (a) Scattering diagrams for right-angle DLP, ROA and (b) DLP,, ROA showing all polarization and propagation vectors and polarization angles.
580
-51B:~~~2-Ph~~)21-~8:~~)2~P:~~)21 + K(J)2-P:(Ja211.
(20b)
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5. Discussion
Excluding higher-order, non-linear processes, natural optical activity involving electromagnetic radiation consists of three principal classes of phenomena, two of which are closely related. The classical forms of optical activity are optical rotation and circular dichroism; the complete spectra of each are Kramer+Kronig transforms of one another [ 3,17 1. Light-scattering optical activity, embodied by Rayleigh and Raman optical activity, is recognized as a completely new form of optical activity [ 3,5 1. As a coherent two-photon event, CP ROA is neither circular dichroism nor optical rotation: however, since it involves a differential intensity with respect to opposite circular-polarization states, it is more closely related to circular dichroism than optical rotation. Optical rotation is traditionally determined, for small angles of rotation, by a direct measurement of the rotation of the plane of linearly polarized light that has passed through a sample. This can be accomplished by passing light through a polarizer before the sample and rotating a second polarizer placed after the sample until a maximum transmitted intensity is achieved, or in practice, 90” from this position where an optical null is achieved. An alternative approach is to measure the difference in transmitted intensity for settings of the second polarizer at + 45 ’ relative to the first polarizer [ 18 1. In this way, optical rotation can be measured as a linear-polarization differential intensity, in direct analogy to circular dichroism, which is measured as a circular-polarization differential intensity. On the basis qf these considerations, we suggest that LP ROA is the light-scattering analogue of one-photon optical rotation and ihat CP ROA is the light-scattering analogue of one-photon circular dichroism. In
the case of LP ROA, each vibrational Raman transition rotates the plane of polarization of the incident or scattered radiation by a small amount and in a direction determined by the theoretical expressions given above. ILP ROA measures a Raman rotation associated with the incident radiation, SLP ROA measures the same associated with the scattered radiation, and DLP ROA measures both simultaneously, either together or in opposition. A more fundamental set of analogies are those with two-photon optical activity. Two-photon circular di-
23 November 1990
chroism has been described theoretically [ 19,201; it is directly analogous to CP ROA and corresponds to various types of circular differential two-photon absorption measurements. On the other hand, no theoretical description of two-photon optical rotation has been advanced, although that phenomenon should exist and would be directly analogous to the LP ROA scattering described in this paper. Analysis of the resonance-frequency dependence of the principal contributions to CP ROA and LP ROA on the incident laser radiation reveals [ LO]that CP ROA possesses an absorption bandshape dependence as the laser frequency is tuned through the absorption maximum of the resonant excited state, whereas LP ROA exhibits a dispersion bandshape dependence, passing through zero intensity and changing sign as the laser frequency is tuned though resonance. This frequency behaviour of CP and LP ROA further strengthens the broad analogy of these two forms of ROA to circular dichroism and optical rotatory dispersion, respectively. Although the sense of ILP and SLP ROA measurements was defined in the same way with respect to positive polarization angles of the incident and scattered radiation, respectively, inspection of the relative signs of the corresponding Roman and script tensors reveals that they have the opposite relative signs in ILP and SLP ROA compared to their relative signs in ICP and SCP ROA. As a result, measured ILP and SLP ROAs may tend to have opposite signs, and DLPr ROA may tend to be smaller than DLP,, ROA. Because the experimental configurations for LP ROA allow for effects of ordinary one-photon optical rotation to be observed, a slowly varying background relative to the widths of vibrational Raman bands is expected to be present in measurements of ILP ROA. Similarly, circular dichroism backgrounds can be avoided in CP ROA by carrying out ROA measurements outside regions of electronic absorption bands. This approach cannot be employed for LP ROA because optical rotation, in contrast to circular dichroism, persists well beyond the bounds of absorbing regions of the spectrum. As demonstrated here, LP ROA depends on the real parts of the traditional products of complex optical-activity and polarizability tensors. Since the product of the operators and coefftcients of these 581
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tensors is pure imaginary, and additional imaginary contribution must arise from the wavefunctions or frequency denominators in eq. (2) in order for the real parts of the tensor products in eqs. (3d)-(3i) to be non-zero. In general, this will not occur far from regions of electronic absorption bands where the effects of resonance Raman scattering are not obsetied. Hence, LP ROA appears to be a purely resonance Raman-enhanced optical-activity phenomenon. Other ROA phenomena in this general class are Stokes-anti-Stokes asymmetry [ 61, differences in corresponding ICP and SCP ROA measurements [ 71, and the observation of DCPn ROA [ 81. The fact that LP ROA depends on the real parts of optical-activity tensor products and ordinary optical rotation depends on the imaginary part of an opticalactivity tensor is an additional point of distinction between these two analogous phenomena. Finally, we note that the phenomena of magnetic LP RAO should exist by analogy to magnetic CP ROA [ 3,2 1 ] and the Faraday effect, which is magnetic-field-induced optical rotation [ 3,221, and since a uniform static external magnetic field oriented parallel to the propagation direction of a light beam (of arbitrary polarization) constitutes a truly chiral physical influence [ 23 1.
Acknowledgement
We thank the Deutsche Forschungsgemeinschaft (DFG) for a Postdoctoral Scholarship (III 02-He 158811-I) to LH and the National Institutes of Health for a Research Grant (GM-23567) to LAN.
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References [ I] L.A. Nafie and C.G. Zimba, in: Biological applications of Raman spectroscopy, Vol. 1, ed. T.G. Spiro (Wiley, New York, 1987) p. 307. [2] L.D. Barron and L. Hecht, Collection Czech. Chem, Commun., in press. [ 3I L.D. Barron, Molecular light scattering and optical activity (Cambridge Univ. Press, Cambridge, 1982). [ 41W. Hug, in: Raman spectroscopy, eds. .I. Lascombe and P.V. Huong (Wiley, New York, 1982) p. 3. [5] L.D. Barron and A.D. Buckingham, Mol. Phys. 20 (1971) 1111. [ 61 L.D. Barron and J.R. Escribano, Chem. Phys. 98 (1985) 437. [ 7I K.M. Spencer, T.B. Freedman and L.A. Nalie, Chem. Phys. Letters 149 (1988) 367. [S] L.A. Nafie and T.B. Freedman, Chem. Phys. Letters 154 (1989) 260. [9] L. Hecht and L.A. Nalie, Mol. Phys., in press. [ 101 L. Hecht and L.A. Nafie, Mol. Phys., to be submitted for publication. [ II] A.C. Albrecht, J. Chem. Phys. 34 (I 961) 1476. [ 121J.A. Koningstein, Introduction of the theory of the Raman effect (Reidel, Dordrecht, I97 I ) p. 146: [ 131B.B. Johnson and W.L. Peticolas, Ann. Rev. Phys. Chem. 27 (1976) 465, and references therein. [ 141A.D. Buckingham and R.E. Raab, Proc. Roy. Sot. A 345 (1975) 365. [ 151L. Hecht, B. Jordanov and B. Schrader, Appl. Spectry. 41 (1987) 295. [16] L. Hecht andL.D. Barron, Appl. Spectry. 44 (1990) 483. [ 171 P. Salvadori and F. Ciardelli, in: Optical rotatory dispersion and circular dichroism, eds. F. Ciardelli and P. Salvadori (Heyden, London, 1973) p. 3. [ 181E.-H. Korte and B. Schrader, in: Advances in infrared and Raman spectroscopy, Vol. 8, eds. R.J.H. Clark and R.E. Hester (Heyden, London, 1981) p. 226. 1191I. Tinoco Jr., J. Chem. Phys. 62 (1975) 1006. [ 201E.A. Power, J. Chem. Phys. 63 (1975) 1348. [ 211L.D. Barron and A.D. Buckingham, Mol. Phys. 23 (1972) 145. [ 221A.D. Buckingham and P.J. Stephens, Ann. Rev. Phys. Chem. 17 ( 1966) 399. [23] L.D. Barron, Chem. Sot. Rev. 15 (1986) 189.
CHEMICAL PHYSICS LETTERS
23 November 1990
Linear polarization Raman optical activity: a new form of natural optical activity Lutz Hecht and Laurence A. Nafie Department
oj Chemistry,Syracuse University, Syracuse, NY 13244-4100, USA
Received 22 May 1990; in final form 12 September 1990
A new form of natural optical activity is identified on the basis of a general theory of natural Raman optical activity (ROA). This novel ROA phenomenon is predicted to occur as the difference in Raman-scattered intensity for pairs of linearly polarized measurements and depends on the real part of the product of the transition-polarizability and optical-activity tensors. This is the tint form of ROA to be identified that does not directly involve the use or measurement of circularly polarized radiation.
1. Introduction Natural Raman optical activity (ROA) has previously been described in the most general terms as arising from the differential Raman scattering intensity of right versus left circularly polarized radiation [ l-41. ROA was originally defined [ 5 ] as the difference in Raman scattered intensity for right minus left circularly polarized incident radiation with linearly polarized or unpolarized [ 3 ] scattered radiation. The result of such a measurement is termed a circular-polarization intensity difference (CID) and is defined as Z,”-Z,“, and the normalized CID is formed by dividing by the corresponding circularpolarization intensity sum (CIS), ZE+ Zk, where the superscripts refer to the polarization state of the incident radiation and the subscripts refer to the polarization state of the scattered radiation. More recently, the definition of ROA was extended to include the measurement of circular-polarization components of the scattered radiation, with linearly polarized incident radiation [3,6]. In this case the measured quantity was termed degree of circular polarization and was defined as the ratio of the Stokes vector component for the circularity of the scattered radiation to the Stokes parameter for the total intensity. Subsequently, a particular method of measuring the degree of circular polarization was proposed and demonstrated [ 71. It was simply the direct
analogue of the original ROA measurement technique, namely the CID, Z$ -If. This particular approach to ROA was termed scattered circular-polarization (SCP) ROA, and analogously, the original form of ROA was called incident circular-polarization (ICP) ROA. Following this conceptual direction, a new form of circular-polarization ROA was discovered [ 81 and named dual circular-polarization (DCP) ROA, where the measured CIDs are Zi - Zi and Zk -If and where the former is referred to as DCPiROA and the latter as DCPuROA. A comprehensive theoretical description of the various forms of circular-polarization ROA has recently been completed [ 91. In this paper, we extend the known forms of ROA beyond those involving CIDs. We define a new class of ROA measurements involving linear-polarization intensity differences (LIDS) by analogy to the four CIDs described above. Hence, as in the case of circular-polarization (CP) ROA, three forms of measurement exist for linear-polarization (LP) ROA: incident (ILP), scattered (SLP), and dual linear-polarization (DLP) ROA. In the case of dual polarization ROA it is possible to realize experiments that combine CP and LP ROA into a single measurement and to measure circular-linear-polarization intensity differences, CLIDs, or the reverse, LCIDs, depending on whether the circular-polarization modulation is performed for the incident or scattered radiation,
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respectively. The identification of LP ROA arose from considerations of the relation of CP ROA invariants, contained in the general theory, to possible ROA experiments that could be realized to isolate the maximum number of theoretical invariant combinations [ 9]_ An intriguing aspect of the theoretical description of LP ROA is its complete dependence on the real parts of the usual product of the complex transitionpolarizability and optical-activity tensors, as opposed to the imaginary parts of the same tensors that appear in the CP ROA theory. A consequence of this property is the vanishing of LP ROA in the far-fromresonance condition of Raman scattering. Thus, LP ROA is a phenomenon that is a direct probe of the presence of resonance Raman intensity enhancement. The purpose of this paper is to define LP ROA in general, conceptual terms and present a precise theoretical description for a particular case of 90” scattering, using only linearly polarized radiation. This experimental configuration is likely to be a convenient, practical approach to LP ROA measurements. A comprehensive treatment of the theory of LP ROA, including a wide range of experimental configurations will be provided in a subsequent publication [ lo]. The theoretical expressions given below have been derived using a new polarization-propagation vector theory of ROA and employing a generalized Stokes-Mueller matrix theory, both of which were developed for our general theoretical treatments of CP and LP ROA [9,10]. Finally, we note that the basic LP ROA expressions provided by this paper were contained (to within a few minor differences) within expressions for the Stokes vector description of Raman scattered light in a previously published paper [ 61. The LP ROA contributions, however, were not isolated or identified as a form of ROA. Nevertheless, these prior expressions were a useful source of theoretical reference for the present work.
2. Definitions The tensor that underlies the scattering of light by the Raman effect is the complex Cartesian polarizability tensor for the n~rn vibronic transition, given by [3,6,1.1-131 576
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(1) where fia is the electric dipole operator; lj) represents a complete manifold of intermediate excited states; Wj, is the angular transition frequency between molecular states j and n; w is the angular frequency of the incident laser radiation; 5 is the bandwidth of the jth excited state and becomes important as the resonance condition Wjn- w is approached or realized; and greek subscripts are used to refer to Cartesian directions in the molecular frame. A tilde over a symbol indicates a complex quantity and the circumflex indicates a quantum-mechanical operator. The corresponding ROA transition tensors can be defined [ 6,8,14] by appropriate substitution of A with Hi,, the magnetic dipole operator, or with &, the electric quadrupole operator, by means of the conversion relations (&,),,+
(G&Jl,
(&&?ln-‘(
%&?l”
(%p)mn+
bL,ByL%n
(%7)
Qa)
by&+&, by /L-ad bYFr+&y bYCi,-ha,,
mn --) (~,asLn
>
(2b) mc)
3 Pp&
(2d)
Using these tensors, one can define the following standard tensor products [ 9 ] : (W
ao= (&a)mn(qJmn, aI = (&Jdnwz(%p)m
1
(3b)
&2=
9
(3c)
co = (&&I,($),”
3
(3d)
CI = (&shnn(G,),n
3
Ve)
G = (&,)rn,(c’+,)mn
2
(w
(&,)mn(&Y)m,
A = ~iW~,y6(~L,8)mn(Aj,dS)mn,
(3%)
22 = Wa~(4&d*,~~L~
(3h)
23 = fiW~,~(~,B),,(~,,~)mn.
(3i)
The symbols used above for these expressions have been used previously [ 8 ] except that tildes were not included, a0 and Go were written as (Y~and GS, and
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zX was omitted. The corresponding script standard tensor products g,, and .& can be defined in relation to the Roman standard tensor products by substitution of the corresponding script ROA tensor, eqs. (2b) and (2d), for the Roman one,, eqs. (2a) and (2c). Alternatively, the isotropic and anisotropic invariants for Raman and LP ROA can be defined as
(da) (4b) (4c)
(4d)
-d PM=
le
-i -e
2
(6h)
I ,
where ni and nd are the incident and scattered propagation vectors, and 2’ and dd are the corresponding complex polarization vectors. Finally, we define the following set of primitive standard tensor-product combinations that emerge directly from the isotropic averaging process [ 91 and span all known forms of ROA: s, =2G,,
(7a)
S,=GcrGc,-Al,,
(7b)
s,=-GJ+2G, -&A;
+ kr~(Ala,ti)*l”~>>
(W
where the symmetric and antisymmetric forms of tensors defined above are given by
(T,B)~,=f[(TaB)rnn+(TBn)rnnl 9
(sa)
(~~~)~“=f[(~~~,m,-(~~)mnl.
(5b)
The prime on the five LP ROA invariants indicates the real part, in contrast to the corresponding unprimed quantities that are widely used for the imaginary part of these same tensor combinations 191. The corresponding script ROA invariants are defined with the usual script for Roman tensor substitutions. Next, we define the polarization-propagation vector products (6a) jj22=Bi.;d@*,
~~‘9=nd.~iX&*,
~33=jd.giXgi*lli_gd*,
~,o=gi.pdXgdand.p,
(6b) (6~)
+A;,
(7c)
S.=-G)tG,tA;,
(7d)
&=-2G),
(7e)
&=C, -&-&,
(7f)
ST=-2G2,
(7g)
3. General polarization ROA expressions Using the polarization-propagation vector products in eqs. (6) and the primitive standard tensorproduct combinations in eqs. (7), we can write the following genera1 expression, involving two sets of polarization states labelled ( 1) and (2), for an ROA measurement of an isotropically averaged sample of chiral molecules,
= Fkil
Im[ (pii)-P’p))(
-iQ]
,
(8)
where K= $ ( co’p&(‘) / 4xR ) 2, defined previously [ 6,8,9 I. Eq. ( 8) can be regarded as a master polarization-propagation vector equation for all natural ROA measurements. In the case of CP ROA mea577
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surements, the two polarization factors in eq. (8) are complex conjugates of one another and, via Pi’) = &)+ =pk, eq. (8) becomes Alor.= Fkii
Irn(l’k) Im(&) .
(9)
In the case of pure LP ROA measurements (no use of circular-polarization states), the polarization products are real and eq. (8) reduces to NLp= Fk$,
(Pi’) -Pi2)) Re( -$J
.
(10)
For real polarization vectors the polarization-propagation vector products in eqs. (6a)-(6~) vanish and, further, the identities P4 = P5 =P6 = P7 and P,, =P,z=P,3=P,4 hold. By combining the primitive standard tensor-product combinations for these two sets of equal polarization-propagation vector products, we have &e=
c6K[(Pp-Pp)
xRe(3f?9-2c1
+3G,-&+A;)
+(Pl.;)-Pp) x Re(3@0-2~,-3@Z+&+&).
(11)
A general expression for the total Raman intensity may be written as [ 91 Z=Z”‘+I”‘=6K[ +P,5(-cYo--(Y*
( --cy0+4cy, -_(y2)
+4a,)+P,,(4a,-a,
average. The convention [ 15,161 that we use here for the Stokes vectors is that so is associated with the total intensity, s, with a projection of the polarization ellipse on the +45” or -45” axis, s2 with the degree and sense of circularity, and s3 with a projection of the polarization ellipse on the 0” or 90” axis. All four components of a Stokes vector will be written below in the form (so, sl, s,, s3).
4. LP ROA expressions for right-angle scattering As a typical set of examples of LP ROA, we consider the cases of right-angle scattering for ILP, SLP, DCPr and DCPrI measurements where only linearpolarization states are employed throughout. Other specific cases will be considered elsewhere [ lo]. We begin by considering the ILP experiment, depicted in fig. la, in which the incident polarization state is modulated between a polarization angle, @,of +45” and -45 ’ with respect to the laboratory x-axis, and the scattered light is vertically polarized along this same laboratory axis. The polarization and propagation vectors needed for eqs. ( 11) and ( 12) are e:= (u,+rr,)/Jz, A&U,, @l= (%-Ily)/J2, e& = uX , n& = u,, where u, is a unit vector in the (Ydirection. With these vectors we have Pi’) = l/2, Pi’) = - l/2, and PI5 =P16 = l/2 which yields AlILp=I+-I6K = cRe(3G,,-2G,+3eZ-&+&),
-az)]. (12)
An alternative approach to the expression of ROA intensities for various types of experiments is the Stokes-Mueller matrix theory [ 15,16 1. The master equation governing ROA, including total Raman intensity, for a sample of isotropically averaged chiral molecules is given by
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(Ida)
and IILp=I++I-=33K(~O$6~I+~2),
(14b)
where the + and - superscripts refer to positive and negative incident polarization angles, using the angular convention for @, shown in fig. la. Alternatively, via si= ( 1, -t 1, 0,O ) and sd = ( 1, 0, 0, l), eq. (13) becomes r’=~K[(Moo)+(M,,)~((M,,)+(M,,))l, (15a)
where sf and sf are the Stokes vector components associated with the intensity and polarization state of the incident and scattered radiation, respectively, and (MI_,> is the 4 x4 Raman Mueller matrix [ 111 where the angular brackets indicate isotropic spatial 578
where (M,o)=~[~~2t13B,(~)2+15~~(~)2],
(15b)
(M30~=~[451y2+8s(~)2-5B,(~~zl,
(15c)
4
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which yields
ILP ROA
hl ILP
=
[45((uG)’
?
+8:(G)‘-sB(G)2-B:(A’)2+P:(A’)Z], (15e)
I,,,=ZK[45a2+7B,(~)2+58,(6)2].
(15f)
When expressed in terms of Raman and ROA transition tensors, eqs. (14) and ( 15) give identical results. The experimental configuration for the right-angle SLP ROA experiment is given in fig. 1b. Here, the polarization and propagation vectors are: et,* =u,, i n1.2 =a,, d n ,.2 =
e;‘=(u,+u,)l$Z
e!=(u,--u,)/$,
rr, . The nonzero polarization-propagation vecproducts are PI1 ( 1) = k/2, P, , (2) = - I /2,
tor PIs=P16= l/2, the SLP ROA intensity is
W
M sI_p=z+ -I_
SLP ROA
= FRe(330-2@,
+3%+&
+Jj),
(16)
and the total Raman intensity is the same as in the ILP measurement above. The Stokes vectors for this example are si=(l,O,O, 1) and sd=(l, &l,O,O) which leads to r,=~K[(M,)+(Mo,)f((M,,)+(M,,))l> (17a)
(Ml3~=W30)
(17b)
9
(M,,}={M,~)=~I45(cu’ri)’ +j?:( 8)2-5/%( BFtg:(~)‘+8:(&2], Fig. I. (a) Scattering diagram for right-angle ILP ROA showing polarization and propagation vectors and incident linear-polarization angle; (b) scattering diagram for right-angle SLP ROA showing polarization and propagation vectors and scattering linear-polarization angle.
(17c)
and thus GLP
=
F
[45(0!9)’
+B:(~)2_58:(~)2+8l(~)a+B:(~)2]
) (17d)
(MOI > =
[45(aG)’
+BJ(G)2-5S:(G)2-8:(A1)2+8:(a)2]
)
(15d)
and ZsLpis the same as JILp.We have chosen the convention of the direction of the positive polarization angle, 6, to specify the first of the two Raman intensity measurements in eq. ( 16) and hence to define the sense of positive SLP ROA intensity. This 579
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is analogous to CP ROA where right circularly polarized light is used to define the sense of positive ROA intensity. The experimental diagrams for DLPl and DLPu right-angle ROA measurements are given in figs. 2a and 2b, respectively. The polarization vectors for the
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DLPI measurement are the modulated polarization vectors of the ILP and SLP measurements above. The resulting nonzero polarization-propagation vector products are P!j’)=1/4, Pa’)=-l/4, PIi)=1/4, PI:’ = - l/4, and PI5=P16= l/4. The resulting DLP, and total Raman intensities are AIDLPl=Z$ -Jr
= FRe[3(eo+
go)
DLPI ROA -2(G,+$)+3(G~++*)-&+Al,+&+J&], (18a) I DLPI=I: +I: =;K( -Ly, t 14a, -(Yz) I
(18b)
The Stokes vectors for the DLPI experiment are ?=(l, + 1, 0, 0) and sd=( 1, * 1, 0, 0) where the upper signs correspond to the first measurement and the lower to the second. These vectors lead to
AIDLn= :{45[(aG)‘+(o9)‘] •t MW+IM@)‘l
-5M@2-V:(
- [8:(~12-P:(~‘)21+
b)
DLPn ROA
IDL,Q=K[&dt
[Bb(W+/U
13fis(&)2t
15fi,(di)*]
%‘I
R21L (19b) ,
(19c)
where (M,, ) equals zero. The corresponding expressions for the DLPII ROA measurements are the same for the total Raman intensity expressions and differ from the DLP, ROA intensity expressions by a change in the signs of all terms involving the script tensor invariants. The signs of Pif'and Ply) change and xd= ( 1, T 1, 0,0)_These changes give AxDLPu =I?
-I:
=
FRe[3(G-
go)-2(G,
- 3,)
+3(~~--~)-AlztA;-~,--~],
(2Oa)
=~{45[(c~G)‘-(cuY)‘]t[j?;(8)~-~:(~)~]
Fig. 2. (a) Scattering diagrams for right-angle DLP, ROA and (b) DLP,, ROA showing all polarization and propagation vectors and polarization angles.
580
-51B:~~~2-Ph~~)21-~8:~~)2~P:~~)21 + K(J)2-P:(Ja211.
(20b)
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5. Discussion
Excluding higher-order, non-linear processes, natural optical activity involving electromagnetic radiation consists of three principal classes of phenomena, two of which are closely related. The classical forms of optical activity are optical rotation and circular dichroism; the complete spectra of each are Kramer+Kronig transforms of one another [ 3,17 1. Light-scattering optical activity, embodied by Rayleigh and Raman optical activity, is recognized as a completely new form of optical activity [ 3,5 1. As a coherent two-photon event, CP ROA is neither circular dichroism nor optical rotation: however, since it involves a differential intensity with respect to opposite circular-polarization states, it is more closely related to circular dichroism than optical rotation. Optical rotation is traditionally determined, for small angles of rotation, by a direct measurement of the rotation of the plane of linearly polarized light that has passed through a sample. This can be accomplished by passing light through a polarizer before the sample and rotating a second polarizer placed after the sample until a maximum transmitted intensity is achieved, or in practice, 90” from this position where an optical null is achieved. An alternative approach is to measure the difference in transmitted intensity for settings of the second polarizer at + 45 ’ relative to the first polarizer [ 18 1. In this way, optical rotation can be measured as a linear-polarization differential intensity, in direct analogy to circular dichroism, which is measured as a circular-polarization differential intensity. On the basis qf these considerations, we suggest that LP ROA is the light-scattering analogue of one-photon optical rotation and ihat CP ROA is the light-scattering analogue of one-photon circular dichroism. In
the case of LP ROA, each vibrational Raman transition rotates the plane of polarization of the incident or scattered radiation by a small amount and in a direction determined by the theoretical expressions given above. ILP ROA measures a Raman rotation associated with the incident radiation, SLP ROA measures the same associated with the scattered radiation, and DLP ROA measures both simultaneously, either together or in opposition. A more fundamental set of analogies are those with two-photon optical activity. Two-photon circular di-
23 November 1990
chroism has been described theoretically [ 19,201; it is directly analogous to CP ROA and corresponds to various types of circular differential two-photon absorption measurements. On the other hand, no theoretical description of two-photon optical rotation has been advanced, although that phenomenon should exist and would be directly analogous to the LP ROA scattering described in this paper. Analysis of the resonance-frequency dependence of the principal contributions to CP ROA and LP ROA on the incident laser radiation reveals [ LO]that CP ROA possesses an absorption bandshape dependence as the laser frequency is tuned through the absorption maximum of the resonant excited state, whereas LP ROA exhibits a dispersion bandshape dependence, passing through zero intensity and changing sign as the laser frequency is tuned though resonance. This frequency behaviour of CP and LP ROA further strengthens the broad analogy of these two forms of ROA to circular dichroism and optical rotatory dispersion, respectively. Although the sense of ILP and SLP ROA measurements was defined in the same way with respect to positive polarization angles of the incident and scattered radiation, respectively, inspection of the relative signs of the corresponding Roman and script tensors reveals that they have the opposite relative signs in ILP and SLP ROA compared to their relative signs in ICP and SCP ROA. As a result, measured ILP and SLP ROAs may tend to have opposite signs, and DLPr ROA may tend to be smaller than DLP,, ROA. Because the experimental configurations for LP ROA allow for effects of ordinary one-photon optical rotation to be observed, a slowly varying background relative to the widths of vibrational Raman bands is expected to be present in measurements of ILP ROA. Similarly, circular dichroism backgrounds can be avoided in CP ROA by carrying out ROA measurements outside regions of electronic absorption bands. This approach cannot be employed for LP ROA because optical rotation, in contrast to circular dichroism, persists well beyond the bounds of absorbing regions of the spectrum. As demonstrated here, LP ROA depends on the real parts of the traditional products of complex optical-activity and polarizability tensors. Since the product of the operators and coefftcients of these 581
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tensors is pure imaginary, and additional imaginary contribution must arise from the wavefunctions or frequency denominators in eq. (2) in order for the real parts of the tensor products in eqs. (3d)-(3i) to be non-zero. In general, this will not occur far from regions of electronic absorption bands where the effects of resonance Raman scattering are not obsetied. Hence, LP ROA appears to be a purely resonance Raman-enhanced optical-activity phenomenon. Other ROA phenomena in this general class are Stokes-anti-Stokes asymmetry [ 61, differences in corresponding ICP and SCP ROA measurements [ 71, and the observation of DCPn ROA [ 81. The fact that LP ROA depends on the real parts of optical-activity tensor products and ordinary optical rotation depends on the imaginary part of an opticalactivity tensor is an additional point of distinction between these two analogous phenomena. Finally, we note that the phenomena of magnetic LP RAO should exist by analogy to magnetic CP ROA [ 3,2 1 ] and the Faraday effect, which is magnetic-field-induced optical rotation [ 3,221, and since a uniform static external magnetic field oriented parallel to the propagation direction of a light beam (of arbitrary polarization) constitutes a truly chiral physical influence [ 23 1.
Acknowledgement
We thank the Deutsche Forschungsgemeinschaft (DFG) for a Postdoctoral Scholarship (III 02-He 158811-I) to LH and the National Institutes of Health for a Research Grant (GM-23567) to LAN.
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References [ I] L.A. Nafie and C.G. Zimba, in: Biological applications of Raman spectroscopy, Vol. 1, ed. T.G. Spiro (Wiley, New York, 1987) p. 307. [2] L.D. Barron and L. Hecht, Collection Czech. Chem, Commun., in press. [ 3I L.D. Barron, Molecular light scattering and optical activity (Cambridge Univ. Press, Cambridge, 1982). [ 41W. Hug, in: Raman spectroscopy, eds. .I. Lascombe and P.V. Huong (Wiley, New York, 1982) p. 3. [5] L.D. Barron and A.D. Buckingham, Mol. Phys. 20 (1971) 1111. [ 61 L.D. Barron and J.R. Escribano, Chem. Phys. 98 (1985) 437. [ 7I K.M. Spencer, T.B. Freedman and L.A. Nalie, Chem. Phys. Letters 149 (1988) 367. [S] L.A. Nafie and T.B. Freedman, Chem. Phys. Letters 154 (1989) 260. [9] L. Hecht and L.A. Nalie, Mol. Phys., in press. [ 101 L. Hecht and L.A. Nafie, Mol. Phys., to be submitted for publication. [ II] A.C. Albrecht, J. Chem. Phys. 34 (I 961) 1476. [ 121J.A. Koningstein, Introduction of the theory of the Raman effect (Reidel, Dordrecht, I97 I ) p. 146: [ 131B.B. Johnson and W.L. Peticolas, Ann. Rev. Phys. Chem. 27 (1976) 465, and references therein. [ 141A.D. Buckingham and R.E. Raab, Proc. Roy. Sot. A 345 (1975) 365. [ 151L. Hecht, B. Jordanov and B. Schrader, Appl. Spectry. 41 (1987) 295. [16] L. Hecht andL.D. Barron, Appl. Spectry. 44 (1990) 483. [ 171 P. Salvadori and F. Ciardelli, in: Optical rotatory dispersion and circular dichroism, eds. F. Ciardelli and P. Salvadori (Heyden, London, 1973) p. 3. [ 181E.-H. Korte and B. Schrader, in: Advances in infrared and Raman spectroscopy, Vol. 8, eds. R.J.H. Clark and R.E. Hester (Heyden, London, 1981) p. 226. 1191I. Tinoco Jr., J. Chem. Phys. 62 (1975) 1006. [ 201E.A. Power, J. Chem. Phys. 63 (1975) 1348. [ 211L.D. Barron and A.D. Buckingham, Mol. Phys. 23 (1972) 145. [ 221A.D. Buckingham and P.J. Stephens, Ann. Rev. Phys. Chem. 17 ( 1966) 399. [23] L.D. Barron, Chem. Sot. Rev. 15 (1986) 189.