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On the possibility of measuring quadrupole transition moments in achiral systems by circular polarization spectroscopy
CHEhfICAL PHYSICS LETTERS
Volume 49. number 2
ON THE POSSIBILITY OF MEASURING QUADRUPOLE IN ACHIRAL SYSTEMS BY CIRCULAR POLARIZATION B. EHRENBERG
15 July 1977
TRANSITION MOMENTS SPECTROSCOPY
and I.Z. STEINBERG
Department of Chemical Physics, Weizmann Institute of Science, Rehovot. Israel Received 1 February 1977
An electronic transition that has both dipoiar and quadrupolar character will have a different probability of excitation when induced by circularly polarized light of opposite sense, depending on the orientation of every individual chromophore being excited relative to the light beam. if brownian motion is frozen during the lifetime of the excited state, the excited population of molecules photoselected by circularly polarized li_ght of a given sense will emit partially circularly polarized tight, the extent of which depends on the molecular spect;oscopic parameters.
1. Introduction Electric quadrupole transition moments contribute only a minute amount to the total intensity of allowed transitions, characterized by a strong electric transition dipole moment. Their measurement may be of interest. however, since their dependence on the wavefunctions involved in the electronic transition is different than the dependence of the electric transition dipole moment, the quadrupole moment depending more sensitively on the variation with space of the wavefunctions. While the evaluation of quadrupole contributions to the intensity of strong bands seems to be a hopeless task, some information may be obtained about these contributions from suitable experiments with circularly polarized absorption and emission, as will be explored in this communication. The probability, p, of absorption (or emission) of light of an arbitrary polarization expressed by the vector potentialA is proportional to [l] : p = KIP-2
+ i(E/2Ac)k.Q*A
+M(k=z)12,
(1)
where P, Q and M are the electric dipole, electric quadrupole and magnetic dipole transition moments. respectively, h = 27rtZis Plan&s constant, and k is a unit vector in the direction of propagation of the light wave. taken to be the z direction in the following and K is a proportionality factor. For right handed and left handed circularly polarized light, A = 2-1/2A(i + ti), and 2-l12A(I’ - ij), respectively, where A is the amplitude and i andj are unit vectors in the x and y directions, respectively. For achiral systems, M either vanishes or is perpendicular to P. We shall drop M in the following considerations and comment later on the case in which M is finite and perpendicular to P. Inspection of eq. (1) reveals that generally for an arbitrary orientation of the molecule p will be different for right handed and left handed circularly polarized light, i.e., Q will contribute to the circuiar dichroism. CD, or the circular polarization of the luminescence, CPL. However, as was already shown by Condon et al. [I] this difference will average out to zero for a collection of molecules which is isotropically oriented in space. For non-isotropic collections of molecules a contribution of Q to the circular dichroism is expected to persist [2] _ In previous studies [3-O] it was shown that effects of non-isotropy in chiral molecules can be obtained in circularly polarized spectroscopy (i.e., CPL and fluorescence detected CD, FDCD) by photoselection, i.e., excitation of the sample molecules by light of specified polarization in a frozen medium in which brownian rotation is neg295
15 July 1977
CHEMICAL PHYSICS LEnERS
Volume 49, number 2
legible during the lifetime of the excited state_ Thus, excitation by non-polarized light results in circularly polarized luminescence that is different from that obtained from isotropic systems and contains contributions from the component ofM that is perpendicular to P, as well as contributions from Q. Similar results are obtained for FDCD when the fluorescence light is collected through a linear polarizer or without a polarizer at all. The CPL or FDCD of achiral molecules vanishes, however, under these conditions even if rotatory brownian motion is frozen and the systems are therefore non-isotropic. In contrast to the above, if a frozen solution of achiral molecules is excited by circularly polarized light and the luminescence is studied, quadrupole contributions to the absorption and emission processes may cause the emitted light to be partly circularly polarized. As will be shown below, a study of the circular polarization of the emitted light under such circumstances may yield information about the electric quadrupole transition moments involved.
2. Theoretical treatment It is not difficult to see the physical reason why electric quadrupole transition moments may cause the emitted light to be partly circularly polarized in frozen media, if the excitation is performed with circularly polarized light. It is true that the overall absorption of the sample is the same for right handed and left handed circularly polarized light, if the molecules are achiral. However, every individual molecule will have a different probability of absorption of right and left handed circularly polarized light, as follows from eq. (l), depending on P and Q and on the molecular orientation. It is the average CD for the whole collection of molecules that cancels out to zero. Thus, excitation by circularly polarized light will preferentially excite molecules of such orientations that their P
and Q moments absorb and emit preferentially
light of circular polarization
of a given sense in a given direction.
The quantitative aspects will be presented below and are based on approaches described previously [3-6]_ For the special application to achiral systems we shall assume the magnetic dipole transition moment to vanish and express the optical tensor explicitly in terms of its quadrupole components. The intensity, 4 that is detected from the luminescent sample is proportional to the probability of excitation of the system multiplied by the probability of emission. Using eq. (l), one obtains: I= KIPabs -&,s
+ i(~~bs/c)kabs.Qabs.~~bs121Pern
-zcrn + i(“~,,/~)~e,
-Q,,-&,i*,
(2)
where the subscripts abs and em refer to quantities involved in the absorption and emission processes, respectively, and K is a proportionality factor. Since the quadrupole contributions are much smaller than the dipolar contributions, we shall in the developments of eq. (2) discard terms which contain Q in powers higher than 2. Furthermore, we shall illustrate the use of eq. (2) for the case in which fabs and P,, coincide (i.e., excitation is accomplished by light absorption at the long wavelength band of absorption under normal circumstances), PZbs = P,, = P, and the fluorescence is detected at 0’ to the excitation beam, i.e., kabs = k,, = k. We thus obtain, I=K{IP-~I&,IP-~i~m-
- (2X&
/c)lPAIfb,
(2m;,,/c)Im[A-P(k-Q-~)] Im[A -P(k-Q-zj]
abslP-Zlzm
em + (m;,,lc)21k.Q.~I~b,IP.~l~m
+ (nu,mlc)21P~~l~b,[k~Q*~l~m + (47r%absvem/c2) Im[A*P(k~Q*~)]
l
absIm[A P(k*Q*x)] em)_
(3)
The sample is excited by circularly polarized light; therefore Aabs = 2-l12A(i + ij) if it is right handed. The emitted light is analyzed by a right handed and a left handed circular polarizer, i.e. A,, = 2-1/2A(i f ij) the up-
per sign designating a right handed circular analyzer. It is pertinent to note that eq. (3) applies to emission of light by a molecule, or a set of molecules, of a defined orientation in space. To obtain the intensity of a collection of molecules one has to average over all orientations. The procedures for doing this have been described elsewhere [3--61 and we shall only present the results. Without loss in generality we shall designate the direction ofP as the
molecular z-axis. 296
Volume 49, number 2
CHEMICAL PHYSICS LETTERS
15
July 1977
(4)
where N is the difference in intensity upon detection of the luminescence through a left banded and a right handed analyzer_ I is the total emitted intensity regardless of polarization_ Terms containing the quadrupole transition moments were omitted from 1 because of their relatively small magnitude. At/1 is thus the extent of circular polarization of the emitted light. In eq. (4) and subsequent equations the upper and lower signs relate to excitation by right handed and left handed circularly polarized light, respectively. It should be noted that the terms which contain a product off’ and Q in the first power will vanish for achiral molecules, which possess a plane or center of symmetry, since for a finite P the Q components associated with the corresponding P components in eq. (4) will be zero and vice versa. One is thus left with the following expression for &
It is evident that N= 0 if the excitation beam is linearly polarized or unpolarized, and can thus be represented, respectively, as a coherent or incoherent combination of two circularly polarized beams of opposite sense of rotation, and no CPL will then be detected. Using symmetry considerations similar to those used by Eyring et al. [7] it can be shown that in centro-symmetric molecules there are identically no quadrupole contributions to the intensity of an allowed transition. In contrast, if the symmetry of the molecule is planar but devoid of centro-symmetry, there may be quadrupole contributions and two cases may be distinguished_ If the electric dipole transition moment (which is also the molecular z-axis) is in the plane of symmetry, taken to be the x-z plane, the following expression holds:
(6) (We use the quantity u/(1/2) in order to conform to the definition of the g factor used in chiral systems [8] )_ If the electric transition dipole moment is perpendicular to the plane of symmetry, the following expression applies:
Eqs. (6) and (7) have been derived for cases in which the molecular symmetry in the ground and excited states are upon excitation, one should go back to eq. (5) in every specific case and examine which terms in this equation should be retained. On the other hand, if the molecules studied are rigid and do not change appreciably upon electronic excitation, and if the excitation is performed at the same electronic transition as that involving the emission process, Qabs may be approximately equal to Qem> with concomitant simplification of eqs. (6) and (7).
the same. If there is a change in symmetry
3. Discussion The considerations presented above show that achiral molecules may emit circularly polarized light, due to contributions of electric quadrupole transition moments to the intensities of absorption and emission. To obtain this effect rotatory brownian motion should be frozen during the lifetime of the electronically excited state. Naturally, molecular symmetry may make the existence of an electric transition dipole moment and some components of the transition quadrupole moment mutually exclusive_ Thus, center symmetry precludes the presence of quadrupole transition moments in transitions that involve electric transition dipole moments. However, for plane symmetric molecules such mutual exclusion is less extensive and circular polarized luminescence may indeed be emitted by achiral molecules. As a matter of fact, the only quadrupole component that does not contribute to the 297
Volume 49, numbcr 2
CHEMICAL PHYSICS LETTERS
15 July 1977
CPL is 833, although it is not forbidden to accompany an allowed electric dipole transition moment [see eqs. (S)-(7)] _The physical reason for this is not difficult to see: since QS3 responds to the same component of the radiative electric field as B it cannot sense the phase difference of the orthogonal components of the electric field, which makes the light circularly polarized.
The extent of circular polarization expected for the quadrupole contributions in achiral systems is of the order of Qg/lI’12 [see eqs. (5)-(7)], where Qji are components of the quadrupole transition moment. In chiral systems quadrupole contributions to CD may be of the same order as those of magnetic transition dipoles; we thus expect Q@‘12 to be of the order ofg2, g being the asymmetry factor and typically falling in the range of 10m2 to low4 for c&al systems. Thus, the extenr of circular polarization in the luminescence of achiral systems excited by circularly polarized light is expected to be IO4 or less. In chiral molecules the effects described by eqs. (.5)-(7) will probably be masked by the inherent optical activity of the molecules; but in achiral systems the circular polarization of the luminescence due to quadrupole contributions should be detectable in many cases. Present day instruments [VJO] readily measure degrees of circular polarization that are significantly lower than 10m4 and can probably be markedly improved. The case of the existence of a magnetic dipole transition moment which is perpendicular to P needs some comment_ This is a legitimate occurrence in achiral systems, and is known for example in symmetric carbonyl compounds. As was shown previously [3], photoselection may lead to contributions to CPL by components ofM that
are orthogonal to P. In the case that the electric dipole transition moments in absorption and emission are parallel to one another [which also applies to eqs. (3)-(7)] the contributions ofM tog are of the order ofM2/Pz (see ref. [3] ), which may be of similar magnitude of Qf/P*_ If the molecules possess a center of symmetry it can be shown that Al vanishes ifP is finite. Similarly, for mo ecules possessing a plane of symmetry, P and M cannot coexist in the plane of symmetry. For all other cases of molecules which possess a plane of symmetry, eqs. (6) and (7) can be generalized to include contributions of a magnetic transition moment by replacing Q13 by (IMI + Ql 3), de- fining the molecu!ary-axis so that it points in the direction ofM (taken to be the same in absorption and emission). The existence of such an M component may be readily diagnosed by exictation at a wavelength whose P is not parallel to the P associated with the emission; under such circumstances the contribution of M to N/(1/2) is expected to be of the order ofgc3), which can be experimentally easily verified. It may be noted that octupole transition moments will not contribute to circular polarization effects since the octupole contributions are in phase with those ofP, and were not, therefore, considered in the present study.
Whiie eqs. (6) and (7) were obtained assuming that the emitted light is collected at 0” to the excitation light, calculations were performed also for the case of light detection at 90” to the excitation light. The results obtained were identical to those expressed in eqs. (6) and (7). Obviously, by the same techniques the above equations can be extended to cases in which Pabs does not coincide with Pem_
References [ 11 E-U. Condon, W. Altar and H. Eyring, J. Chem. Phys. 5 (1937) 753. [2] I. Tinoco and W.G. Hammcrlc, J. Phys. Chcm. 60 (1956) 1619.
[3] 1-Z. Steinberg and B. Ehrenberg, J. Chem. Phys. 61 (1974) 3382. [4] B. Ehrenberg and 1-Z. Steinberg, J. Am. Chcm. Sot. 98 (1976) 1293. [S 1 J.P. Riehl and F.S. Richardson, J. Chem. Phys. 65 (1976) 1011. [6 J I. Tinoco, B. Ehrenberg and I.Z. Steinberg, J. Chem. Phys. 66 (1977) 916. [7] H. Cyring, J. Walter and G.E. Kimball, Quantum chemistry (Wiley, New York, 1965) p. 346. [S] I.Z. Steinberg, in: Concepts in biochemical fluorescence, Vol. 1, eds. R.F. Chen dnd H. Edelhoch (Dekker, New York, 1976) p. 79. [VI I.Z. Steinberg and A. Gafni, Rev. Sci. Instr. 43 (1972) 409. [lo] C.K. Luk and I-S. Richardson, J. Am. Chem. Sot. 96 (1974) 2006.
298
Volume 49. number 2
ON THE POSSIBILITY OF MEASURING QUADRUPOLE IN ACHIRAL SYSTEMS BY CIRCULAR POLARIZATION B. EHRENBERG
15 July 1977
TRANSITION MOMENTS SPECTROSCOPY
and I.Z. STEINBERG
Department of Chemical Physics, Weizmann Institute of Science, Rehovot. Israel Received 1 February 1977
An electronic transition that has both dipoiar and quadrupolar character will have a different probability of excitation when induced by circularly polarized light of opposite sense, depending on the orientation of every individual chromophore being excited relative to the light beam. if brownian motion is frozen during the lifetime of the excited state, the excited population of molecules photoselected by circularly polarized li_ght of a given sense will emit partially circularly polarized tight, the extent of which depends on the molecular spect;oscopic parameters.
1. Introduction Electric quadrupole transition moments contribute only a minute amount to the total intensity of allowed transitions, characterized by a strong electric transition dipole moment. Their measurement may be of interest. however, since their dependence on the wavefunctions involved in the electronic transition is different than the dependence of the electric transition dipole moment, the quadrupole moment depending more sensitively on the variation with space of the wavefunctions. While the evaluation of quadrupole contributions to the intensity of strong bands seems to be a hopeless task, some information may be obtained about these contributions from suitable experiments with circularly polarized absorption and emission, as will be explored in this communication. The probability, p, of absorption (or emission) of light of an arbitrary polarization expressed by the vector potentialA is proportional to [l] : p = KIP-2
+ i(E/2Ac)k.Q*A
+M(k=z)12,
(1)
where P, Q and M are the electric dipole, electric quadrupole and magnetic dipole transition moments. respectively, h = 27rtZis Plan&s constant, and k is a unit vector in the direction of propagation of the light wave. taken to be the z direction in the following and K is a proportionality factor. For right handed and left handed circularly polarized light, A = 2-1/2A(i + ti), and 2-l12A(I’ - ij), respectively, where A is the amplitude and i andj are unit vectors in the x and y directions, respectively. For achiral systems, M either vanishes or is perpendicular to P. We shall drop M in the following considerations and comment later on the case in which M is finite and perpendicular to P. Inspection of eq. (1) reveals that generally for an arbitrary orientation of the molecule p will be different for right handed and left handed circularly polarized light, i.e., Q will contribute to the circuiar dichroism. CD, or the circular polarization of the luminescence, CPL. However, as was already shown by Condon et al. [I] this difference will average out to zero for a collection of molecules which is isotropically oriented in space. For non-isotropic collections of molecules a contribution of Q to the circular dichroism is expected to persist [2] _ In previous studies [3-O] it was shown that effects of non-isotropy in chiral molecules can be obtained in circularly polarized spectroscopy (i.e., CPL and fluorescence detected CD, FDCD) by photoselection, i.e., excitation of the sample molecules by light of specified polarization in a frozen medium in which brownian rotation is neg295
15 July 1977
CHEMICAL PHYSICS LEnERS
Volume 49, number 2
legible during the lifetime of the excited state_ Thus, excitation by non-polarized light results in circularly polarized luminescence that is different from that obtained from isotropic systems and contains contributions from the component ofM that is perpendicular to P, as well as contributions from Q. Similar results are obtained for FDCD when the fluorescence light is collected through a linear polarizer or without a polarizer at all. The CPL or FDCD of achiral molecules vanishes, however, under these conditions even if rotatory brownian motion is frozen and the systems are therefore non-isotropic. In contrast to the above, if a frozen solution of achiral molecules is excited by circularly polarized light and the luminescence is studied, quadrupole contributions to the absorption and emission processes may cause the emitted light to be partly circularly polarized. As will be shown below, a study of the circular polarization of the emitted light under such circumstances may yield information about the electric quadrupole transition moments involved.
2. Theoretical treatment It is not difficult to see the physical reason why electric quadrupole transition moments may cause the emitted light to be partly circularly polarized in frozen media, if the excitation is performed with circularly polarized light. It is true that the overall absorption of the sample is the same for right handed and left handed circularly polarized light, if the molecules are achiral. However, every individual molecule will have a different probability of absorption of right and left handed circularly polarized light, as follows from eq. (l), depending on P and Q and on the molecular orientation. It is the average CD for the whole collection of molecules that cancels out to zero. Thus, excitation by circularly polarized light will preferentially excite molecules of such orientations that their P
and Q moments absorb and emit preferentially
light of circular polarization
of a given sense in a given direction.
The quantitative aspects will be presented below and are based on approaches described previously [3-6]_ For the special application to achiral systems we shall assume the magnetic dipole transition moment to vanish and express the optical tensor explicitly in terms of its quadrupole components. The intensity, 4 that is detected from the luminescent sample is proportional to the probability of excitation of the system multiplied by the probability of emission. Using eq. (l), one obtains: I= KIPabs -&,s
+ i(~~bs/c)kabs.Qabs.~~bs121Pern
-zcrn + i(“~,,/~)~e,
-Q,,-&,i*,
(2)
where the subscripts abs and em refer to quantities involved in the absorption and emission processes, respectively, and K is a proportionality factor. Since the quadrupole contributions are much smaller than the dipolar contributions, we shall in the developments of eq. (2) discard terms which contain Q in powers higher than 2. Furthermore, we shall illustrate the use of eq. (2) for the case in which fabs and P,, coincide (i.e., excitation is accomplished by light absorption at the long wavelength band of absorption under normal circumstances), PZbs = P,, = P, and the fluorescence is detected at 0’ to the excitation beam, i.e., kabs = k,, = k. We thus obtain, I=K{IP-~I&,IP-~i~m-
- (2X&
/c)lPAIfb,
(2m;,,/c)Im[A-P(k-Q-~)] Im[A -P(k-Q-zj]
abslP-Zlzm
em + (m;,,lc)21k.Q.~I~b,IP.~l~m
+ (nu,mlc)21P~~l~b,[k~Q*~l~m + (47r%absvem/c2) Im[A*P(k~Q*~)]
l
absIm[A P(k*Q*x)] em)_
(3)
The sample is excited by circularly polarized light; therefore Aabs = 2-l12A(i + ij) if it is right handed. The emitted light is analyzed by a right handed and a left handed circular polarizer, i.e. A,, = 2-1/2A(i f ij) the up-
per sign designating a right handed circular analyzer. It is pertinent to note that eq. (3) applies to emission of light by a molecule, or a set of molecules, of a defined orientation in space. To obtain the intensity of a collection of molecules one has to average over all orientations. The procedures for doing this have been described elsewhere [3--61 and we shall only present the results. Without loss in generality we shall designate the direction ofP as the
molecular z-axis. 296
Volume 49, number 2
CHEMICAL PHYSICS LETTERS
15
July 1977
(4)
where N is the difference in intensity upon detection of the luminescence through a left banded and a right handed analyzer_ I is the total emitted intensity regardless of polarization_ Terms containing the quadrupole transition moments were omitted from 1 because of their relatively small magnitude. At/1 is thus the extent of circular polarization of the emitted light. In eq. (4) and subsequent equations the upper and lower signs relate to excitation by right handed and left handed circularly polarized light, respectively. It should be noted that the terms which contain a product off’ and Q in the first power will vanish for achiral molecules, which possess a plane or center of symmetry, since for a finite P the Q components associated with the corresponding P components in eq. (4) will be zero and vice versa. One is thus left with the following expression for &
It is evident that N= 0 if the excitation beam is linearly polarized or unpolarized, and can thus be represented, respectively, as a coherent or incoherent combination of two circularly polarized beams of opposite sense of rotation, and no CPL will then be detected. Using symmetry considerations similar to those used by Eyring et al. [7] it can be shown that in centro-symmetric molecules there are identically no quadrupole contributions to the intensity of an allowed transition. In contrast, if the symmetry of the molecule is planar but devoid of centro-symmetry, there may be quadrupole contributions and two cases may be distinguished_ If the electric dipole transition moment (which is also the molecular z-axis) is in the plane of symmetry, taken to be the x-z plane, the following expression holds:
(6) (We use the quantity u/(1/2) in order to conform to the definition of the g factor used in chiral systems [8] )_ If the electric transition dipole moment is perpendicular to the plane of symmetry, the following expression applies:
Eqs. (6) and (7) have been derived for cases in which the molecular symmetry in the ground and excited states are upon excitation, one should go back to eq. (5) in every specific case and examine which terms in this equation should be retained. On the other hand, if the molecules studied are rigid and do not change appreciably upon electronic excitation, and if the excitation is performed at the same electronic transition as that involving the emission process, Qabs may be approximately equal to Qem> with concomitant simplification of eqs. (6) and (7).
the same. If there is a change in symmetry
3. Discussion The considerations presented above show that achiral molecules may emit circularly polarized light, due to contributions of electric quadrupole transition moments to the intensities of absorption and emission. To obtain this effect rotatory brownian motion should be frozen during the lifetime of the electronically excited state. Naturally, molecular symmetry may make the existence of an electric transition dipole moment and some components of the transition quadrupole moment mutually exclusive_ Thus, center symmetry precludes the presence of quadrupole transition moments in transitions that involve electric transition dipole moments. However, for plane symmetric molecules such mutual exclusion is less extensive and circular polarized luminescence may indeed be emitted by achiral molecules. As a matter of fact, the only quadrupole component that does not contribute to the 297
Volume 49, numbcr 2
CHEMICAL PHYSICS LETTERS
15 July 1977
CPL is 833, although it is not forbidden to accompany an allowed electric dipole transition moment [see eqs. (S)-(7)] _The physical reason for this is not difficult to see: since QS3 responds to the same component of the radiative electric field as B it cannot sense the phase difference of the orthogonal components of the electric field, which makes the light circularly polarized.
The extent of circular polarization expected for the quadrupole contributions in achiral systems is of the order of Qg/lI’12 [see eqs. (5)-(7)], where Qji are components of the quadrupole transition moment. In chiral systems quadrupole contributions to CD may be of the same order as those of magnetic transition dipoles; we thus expect Q@‘12 to be of the order ofg2, g being the asymmetry factor and typically falling in the range of 10m2 to low4 for c&al systems. Thus, the extenr of circular polarization in the luminescence of achiral systems excited by circularly polarized light is expected to be IO4 or less. In chiral molecules the effects described by eqs. (.5)-(7) will probably be masked by the inherent optical activity of the molecules; but in achiral systems the circular polarization of the luminescence due to quadrupole contributions should be detectable in many cases. Present day instruments [VJO] readily measure degrees of circular polarization that are significantly lower than 10m4 and can probably be markedly improved. The case of the existence of a magnetic dipole transition moment which is perpendicular to P needs some comment_ This is a legitimate occurrence in achiral systems, and is known for example in symmetric carbonyl compounds. As was shown previously [3], photoselection may lead to contributions to CPL by components ofM that
are orthogonal to P. In the case that the electric dipole transition moments in absorption and emission are parallel to one another [which also applies to eqs. (3)-(7)] the contributions ofM tog are of the order ofM2/Pz (see ref. [3] ), which may be of similar magnitude of Qf/P*_ If the molecules possess a center of symmetry it can be shown that Al vanishes ifP is finite. Similarly, for mo ecules possessing a plane of symmetry, P and M cannot coexist in the plane of symmetry. For all other cases of molecules which possess a plane of symmetry, eqs. (6) and (7) can be generalized to include contributions of a magnetic transition moment by replacing Q13 by (IMI + Ql 3), de- fining the molecu!ary-axis so that it points in the direction ofM (taken to be the same in absorption and emission). The existence of such an M component may be readily diagnosed by exictation at a wavelength whose P is not parallel to the P associated with the emission; under such circumstances the contribution of M to N/(1/2) is expected to be of the order ofgc3), which can be experimentally easily verified. It may be noted that octupole transition moments will not contribute to circular polarization effects since the octupole contributions are in phase with those ofP, and were not, therefore, considered in the present study.
Whiie eqs. (6) and (7) were obtained assuming that the emitted light is collected at 0” to the excitation light, calculations were performed also for the case of light detection at 90” to the excitation light. The results obtained were identical to those expressed in eqs. (6) and (7). Obviously, by the same techniques the above equations can be extended to cases in which Pabs does not coincide with Pem_
References [ 11 E-U. Condon, W. Altar and H. Eyring, J. Chem. Phys. 5 (1937) 753. [2] I. Tinoco and W.G. Hammcrlc, J. Phys. Chcm. 60 (1956) 1619.
[3] 1-Z. Steinberg and B. Ehrenberg, J. Chem. Phys. 61 (1974) 3382. [4] B. Ehrenberg and 1-Z. Steinberg, J. Am. Chcm. Sot. 98 (1976) 1293. [S 1 J.P. Riehl and F.S. Richardson, J. Chem. Phys. 65 (1976) 1011. [6 J I. Tinoco, B. Ehrenberg and I.Z. Steinberg, J. Chem. Phys. 66 (1977) 916. [7] H. Cyring, J. Walter and G.E. Kimball, Quantum chemistry (Wiley, New York, 1965) p. 346. [S] I.Z. Steinberg, in: Concepts in biochemical fluorescence, Vol. 1, eds. R.F. Chen dnd H. Edelhoch (Dekker, New York, 1976) p. 79. [VI I.Z. Steinberg and A. Gafni, Rev. Sci. Instr. 43 (1972) 409. [lo] C.K. Luk and I-S. Richardson, J. Am. Chem. Sot. 96 (1974) 2006.
298