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Optical rotation second-order effects
Chemical Physics 280 (2002) 103–109 www.elsevier.com/locate/chemphys
Optical rotation second-order effects M. Maestro a, F. Mavelli a, G. Paiano b,*, E. Polacco c b
a Dipartimento di Chimica dell’Universita di Bari, via Orabona 4, I-70125 Bari, Italy Dipartimento di Fisica dell’Universita di Bari and INFN Sezione di Bari, via Orabona 4, I-70125 Bari, Italy c Dipartimento di Fisica dell’Universita di Pisa, Piazza Torricelli 2, I-56100 Pisa, Italy
Received 5 November 2001; in final form 13 March 2002
Abstract The optical rotatory power (ORP) classical phenomenological theory is developed to a higher order with respect to the one usually dealt with. All the formulae needed for the passage from the molecular parameters to the macroscopic ones which are present in the correct phenomenological material medium equations are deduced. In particular a minor term which has already been considered by Condon has been corrected. The possibility that an experimental measurement on a racemate could allow a determination of the ORP of its components is discussed. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Optical rotatory power; Second-order effects
1. Introduction The problem of the asymmetric synthesis has worried the chemists for a long time and only rather recently it seems to have been solved (at least in some cases) [1,2]. But even the preliminary problem of the control of the optical purity cannot be considered as trivial [3]. Apart from this, all the phenomenology concerning the optical activity has kept its interest and its up-to-dateness, also for physicists who are interested in problems of basic invariance principles [4]. From a more general point of view, a large literature on non-linear interactions between EM fields and matter is at disposal at least since the late
*
Corresponding author.
sixties when the work mainly of Buckingham group (see [5,6]) laid a solid theoretical basis of the phenomenology. More recently, this subject revealed interesting technical aspects in some fields as, for instance, in characterising polymers [7]. At the same time, the rapid extension of the calculation facilities has driven a renewed interest for the theoretic ab initio calculations of more and more new tiny observables of higher order (see [8–11]). A great deal of new ‘‘effects’’, which added to the classic Kerr and Cotton Mouton effects, has been defined and identified, sometimes with not so transparent acronyms [12]. In some cases the problem of suggesting and setting up the correct and precise experimental procedures in order to verify the calculated results, deserves a particular care (see [13,14]). In this frame, we considered that the development of the classical phenomenological
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M. Maestro et al. / Chemical Physics 280 (2002) 103–109
theory of OPR up to a higher order for evidencing some possible new effects to be detected experimentally could be worth exploring. In particular the idea of bringing out some quadratic effect in the molecular parameter which characterises the OPR, in order to distinguish a racemate from a compound containing two active centres but inactive for internal compensation, can appear rather naive, but we think that it deserves some attention. In the landmark paper by Condon [15], a second-order term is quoted, the effect of which is a simple small shift of the refractive index. A coherent development of the theory brings to the emergence of some new terms, while the wanted quadratic term is obtained. 2. Dielectric properties of the active media Optical rotatory power (ORP) can be phenomenologically described [1] by the following material medium equations: _; D ¼ eE gH ð1Þ B ¼ lH þ gE_ : The parameter g is directly bound to the optical rotation r being k2 r; 4p2 c where c and k are the radiation speed and the wavelength, respectively. Eq. (1) must be dealt with together with the Maxwell equations system: g¼
div D ¼ 0; B_ rot E ¼ ; c
div B ¼ 0; rot H ¼
_ D c
ð2Þ
in order to deduce all the physics of the electromagnetic field in the active medium. For the sake of simplicity we will consider the case of an isotropic, homogeneous and not conducting medium (typically, but not exclusively a liquid or a solution of some active organic compound). Notwithstanding such restrictions, the results are generalisable. The well-known theory of ORP [15–17] introduces the auxiliary (macroscopic) vectors polarisation P, and magnetisation Q
D ¼ E þ 4pP; B ¼ H þ 4pQ;
ð3Þ
where P and Q are related to the electric and magnetic molecular moments R and M induced by the field in the medium by the simple relations: P ¼ NR and Q ¼ N M; where N is the number of active molecular centres (N ffi 1019 in solids). The Lorenz field approximation: 4pP ; Ei ¼ E þ 3 ð4Þ 4pQ Hi ¼ H 3 is then utilised in order to describe the field inside the medium. The induced molecular moments R and M can be developed as functions of the macroscopic field and polarisation vectors E, P, H and Q via the molecular parameters a; b=c; c and j (see [16]). These last quantities can in turn be defined with reference to their quantum definitions: _ _ m a b b a h jP j i h jP j i X ba 2 aa ¼ ; 3h b m2 m2 ba _ _ m a b b a h jM j i h jM j i X ba 2 ja ¼ ; 2 3h b m m2 n ba_ o ð5Þ _ I a b b a h jP j i h jM j i X ba 1 ¼ ; 3ph b c m2ba m2 n _ o _ 2 X mba R hajPjbihbjMjai ca ¼ 3h b m2ba m2 and, in principle, be evaluated on the basis of molecular eigenfunctions. _ _ In Eq. (5) P and M are the operators of the electric and magnetic moments. One has b _ R ¼ aEi H i þ cHi ; c ð6Þ b_ M ¼ jHi þ Ei þ cEi : c From Eqs. (4) and (6) and from the definitions of P and Q one obtains:
M. Maestro et al. / Chemical Physics 280 (2002) 103–109
_ þ 3c H 4p N M _ 4p N M 3aE 3 bc H 3 3
R¼ M¼
3 4pN a b _ 4p _ 3jH þ 3 c E þ 3 N R þ 3c E þ 4p NR 3 3 þ 4pN j
;
: ð7Þ
Eq. (7) are used for deducing the relations between the molecular quantities a; j; b and c and the corresponding macroscopic ones e; l and g present in the phenomenological Eq. (1). In order to bring the procedure to a higher order one must simply derive with respect to time Eq. (7) and operate the suitable substitutions. Limiting ourselves to the terms in b2 =c2 , the following equations are valid: € 3aE ; 3 4pN a € € ¼ 3jH : M 3 þ 4pN j
105
Eqs. (9a) and (9b) with the correlated definitions are the required generalisations of the well-known first-order formulae [18] and must replace the material medium equation (1). It is easy to see that to the same order b2 =c2 it results g ffi h and p ffi q while v 6¼ w. In deducing the Eqs. (9a) and (9b) we have supposed that c is of the same order as xðb=cÞ. This last assumption is not justified (as far as we know) on the basis of some known results, since in general the quantity c has been considered devoid of practical interest [15,17]; but on the other hand it seems to be the most rational on the basis of the definitions (5). And besides, as it will be seen, a correct higher order treatment cannot avoid taking into account this quantity.
€¼ R
ð8Þ 3. The propagation of waves in the active media
Eqs. (3), (7) and (8) result in the following final formulae for D and B: _ þ vE €; D ¼ eE þ pH gH
ð9aÞ
€; B ¼ lH þ qE þ hE_ wH
ð9bÞ
where e¼
ð3 þ 8pN aÞð3 þ 4pN jÞ 64p2 N 2 c2 ; ð3 4pN aÞð3 þ 4pN jÞ 16p2 N 2 c2
l¼
ð3 4pN aÞð3 þ 16pN jÞ þ 64p2 N 2 c2 ; ð3 4pN aÞð3 þ 4pN jÞ þ 16p2 N 2 c2
g¼
36pN bc ; ð3 4pN aÞð3 þ 4pN jÞ 16p2 N 2 c2
36pN bc ; ð3 4pN aÞð3 þ 4pN jÞ þ 16p2 N 2 c2 36pN c p¼ ; ð3 4pN aÞð3 þ 4pN jÞ 16p2 N 2 c2 36pN c q¼ ; ð3 4pN aÞð3 þ 4pN jÞ þ 16p2 N 2 c2
h¼
2
v¼
144p2 N 2 bc2 ; ð3 4pN aÞ½ð3 4pN aÞð3 þ 4pN jÞ 16p2 N 2 c2 2
144p2 N 2 bc2 w¼ : ð3 þ 4pN jÞ½ð3 4pN aÞð3 þ 4pN jÞ þ 16p2 N 2 c2 ð10Þ
The active medium is characterised by two different refractive indexes for two circular (l ¼ left and r ¼ right) polarised waves. For sake of simplicity we deal with a purely dispersive medium, i.e. our refractive indexes nl and nr are real quantities. Following the classical procedure [18] in order to deduce these two quantities one must study a wave solution of the Maxwell equations describing the active media by means of Eqs. (9a) and (9b). Electromagnetic vectors that propagate into the active medium are described by the following general formula: n h z io Aðz; tÞ ¼ R A0 exp i x t n : ð11Þ c For every wave solution the following equations are valid: € ¼ x2 E; E € ¼ x2 H: H _ the time As a result, by solving with respect to H derivative of Eq. (9b) and from the third Maxwell Eq. (2) one has _ ¼ H
1 ðc rot E qE_ þ gx2 EÞ ðl þ wx2 Þ
ð12Þ
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M. Maestro et al. / Chemical Physics 280 (2002) 103–109
and therefore
Z 1 _ H¼ c rot E dt qE gE : ðl þ wx2 Þ ð13Þ By inserting Eqs. (12) and (13) in Eq. (9a) one has 1 D ¼ e vx2 E þ ðl þ wx2 Þ Z
q c rot E dt qE 2 gð c rot E þ gx EÞ and finally 1 D¼ ðl þ wx2 Þ Z
n2E þ s2 x2 E c q rot Edt g rot E ; ð14Þ where the following notations have been introduced: n2E ¼ n20 q2 , n20 ¼ el and s2 ¼ ew lv g2 and a term of higher order has been neglected. Vector D can be expressed in terms of E and its derivatives by utilising the second Maxwell equation for rot. One has Z c D ¼ c rot H dt ¼ ðl þ wx2 Þ Z Z Z
c rot rot E dt dt q rot E dt Z g rot E_ dt : ð15Þ From Eqs. (14) and (15) and by means of the wellknown equation for rot rot (being div E ¼ 0), one has Z Z ðn2E þ s2 x2 ÞE þ cg rot E c2 r2 E dt dt Z þ g rot E_ dt ¼ 0: ð16Þ From the chosen form for the field vectors (11), one has: n2 x2 nx2 Ey ; r E ¼ 2 ; rot E_ x ¼ c c inxEy inxEx rot Ex ¼ ; rot Ey ¼ ; c c nx2 Ex : rot E_ y ¼ c 2
ð17Þ
Therefore Eq. (16) can be reduced to the following secular equation in n n2 þ s2 x2 n2 1 2ingx E 2 2 2 2 ¼ 0: 2 2ingx nE þ s x n ðl þ wx Þ ð18Þ Eq. (18) can be easily solved. The positive real solutions for n are the following: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ¼ xg þ n2E þ fx2 ffi nF xg; ð19Þ where f ¼ ðew lvÞ and nF ¼ nE þ ðfx2 =2nE Þ. From Eq. (19), to the same order g2 , one has n2 ¼ n2E 2nE xg þ x2 g2 þ fx2 :
ð20Þ
It is easy to see that in the usual first-order limit, when one assumes l ¼ 1 one has also s ¼ q ¼ w ¼ 0 and n2E ¼ n20 , so that one has n2 ¼ n20 jgjx and, since jgjx << n20 it will be n ffi n0 þ d, where d ¼ jgjx=2n0 , a well-known result [18]. Two enantiomers are distinguished by the sign of g which immediately is related to that of b, the corresponding molecular parameter. In the following treatment we prefer to introduce only jgj and therefore we will obtain different formulae for right- and left-handed media. To deduce the phenomenology of the light propagation in the active media one can resort to the classic procedure [19]; this implies the explicit evaluation of the eigenvectors of the E matrix already introduced which can be easily found. For nþ one has ðn2E s2 x2 n2þ ÞEx 2inþ gEy ¼ 0 and therefore Exþ ¼ i Eyþ and, analogously for n Ex ¼i Ey and then, the normalised eigenvectors are the following i i eþ ¼ ; e ¼ : ð21Þ 1 1 Eigenvectors in Eq. (21) are the same as those obtained in the case of the first-order approxima-
M. Maestro et al. / Chemical Physics 280 (2002) 103–109
tion. As a consequence, to the second order no new effect can be expected regarding the ellipticity of the light which crosses the active medium. At this point, following the method of Yariv and Yeh [19], it is easy to describe a plane-polarised wave in the x direction and to deduce its shape after it has propagated along the z direction covering a path Z in the time interval t. For that one must develop the original Cartesian vectors on the basis of (21) 1 1 1 1 x¼ ¼ þ ; 0 i i 2 1 0 i i y¼ ¼ þ : 1 1 1 2 Then one has for a right-handed medium
1 Z ER ðZ; tÞ ¼ CR exp ix t n þ x c i
1 Z þ exp ix n c i
Z ¼ 2C cos x t nF c Z Z
cos x2 jgj ~ I sen x2 jgj ~ J ; c c ð22Þ where C ¼ ðjEð0; 0ÞjÞ=2 and ~ I and ~ J are the original cartesian vectors. In Eq. (22) ER x ðZ; tÞ is the electric field component of the wave at time t and after a path length Z of a plane wave which at time t ¼ 0 was polarised in the plane xz. Analogously one has
Z R Ey ðZ; tÞ ¼ 2C cos x t nF c
Z Z~ 2 2 ~
senx jgj I þ cos x jgj J : ð23Þ c c The corresponding formulae for the left-handed forms are simply obtained by changing the signs before sinus functions. To deduce the result for a wave crossing a racemic mixture, one can suppose that light covers successively half path in each of two enantiomers and simply apply successively Eqs. (22) and (23) and their corresponding for the left-handed
107
I and ~ J. enatiomer putting ExL and EyL in place of ~ The results are:
Z ~ ERac ðZ; tÞ ¼ 2C cos x t n I ð24Þ F x c and
Z ~ ERac ðZ; tÞ ¼ 2C cos x t n J: F y c
ð25Þ
It is interesting to compare (24) and (25) with the corresponding formulae for a hypothetical corresponding compound which is inactive for internal compensation. To do this, we recall Eq. (19) nF ¼ nE þ ðfx2 =2nE Þ, where f ¼ ev lw and, in turn v and w are defined in Eq. (10) while n2E ¼ n20 q2 . In this compound one simply has g ¼ q ¼ f ¼ 0 and therefore
Z ~ E0x ¼ 2C cos x t n0 I; ð26Þ c where n20 ¼ el: The shift n20 q2 was already present in Condon paper [15]. The new term in x2 makes nF correct to the second order. In the following section we shortly discuss the possibility of an experimental discrimination between (24) and (26).
4. Discussion on a possible experimental test Eqs. (24) and (26) refer to two physical situations that are in principle different, even though they cannot be distinguished in a first-order treatment of optical rotation; experimentally to discriminate between them some further points must be discussed. The first one is the choice of a suitable chemical compound. The main problem originates from a possible different crystal structure of the active and the inactive forms which in principle imply some very different values of refraction indexes of racemic mixtures and of inactive forms, apart from the intrinsic difference emphasised by Eqs. (24) and (26). On the other hand resorting to solutions could not solve the problem since in this case one should take into account the problem of the weight
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M. Maestro et al. / Chemical Physics 280 (2002) 103–109
of different configurations. It is for these reasons that a well-known molecule such as tartaric acid must be discarded, whereas asymmetrically substituted spiro compounds and allenes can be considered as possibly suitable candidates. Here, we will not deal with this problem, but in principle it should not be impossible to select some rigid structured molecule that can prevent the difficulties just examined. Another problem seems to involve the level of homogeneity of the racemic mixture. But a simple calculation shows that this point is not critical. In fact, considering relatively small molecules (up to a 3 ) their molecular volume of some hundreds of A number in a sphere of radius of the same order as the light wavelength k should ensure the proper statistical weight of two enantiomers. This condition is met also for solutions, at least for high concentrations (ffi 1 M). Apart from these and some other points that can possibly be taken into account, the most relevant problem is the smallness of the difference between nF and n0 . At any rate it is only this point that will now be considered. Other observables show the same quadratic dependence on the optical rotation parameters: for instance the intensity of the scattered Raman radiation at right angle when the sample is crossed by a properly polarised coherent wave (see [6,20]) and it is nice that in experiences based on the violation of the principle of gravitational equivalence [4], a (mechanical) squared frequency turns out to be the discriminating element. Our choice of the refractive index variation is probably simpler to be made. To do this, we will consider an ideal molecule with physical data representative of many real cases. Table 1 shows the starting data and the resulting calculated parameters. From
Table 1 Typical data for a suitable chemical compound d (Density) M (Molecular weight) k (Light wavelength) r (Optical rotation at the given k) e (Relative dielectric constant) l (Relative magnetic suscept.)
2 g=cm3 200 Da 6:33 105 cm 100 deg/cm 4 1.0002
these data and from Avogadro number NA ¼ 6:02 1023 , one has N ¼ 6:02 1021 cm3 ; g ¼ 5:30887 1021 s; x ¼ 2:975757 1015 s1 : These data are not sufficient to determine all the microscopic quantities which appear in Eq. (10); at least a further assumption has to be done. Assuming c ¼ xðb=cÞ as already discussed, the values for q, a; b=c, j; v and w can be evaluated by Eq. (10). To the second order in b=c it results q p ¼ xg ¼ 1:758338 105 and then a ¼ 1:98287 1023 cm3 ; j ¼ 1:057997 1027 cm3 ; b=c ¼ 3:905511 1044 cm3 s; c ¼ 1:1621864 1028 cm3 ; v ¼ 1:163846 1041 s2 ; w ¼ 5:818821 1042 s2 ; f ¼ ew lv ¼ 1:1634498 1041 s2 : From the given input data one has n20 ¼ el ¼ 4:0008, therefore the quantities to be compared are n20 ; q2 ¼ 3:09175 1010 and fx2 ¼ 1:03025 1010 and finally one obtains Dn ¼ n0 nF ¼ 5:1 1011 : Is this effect experimentally detectable? We consider an experimental set-up formed by a Michelson interferometer (see Fig. 1) equipped with a He–Ne laser source (k ¼ 633 nm) with a power I0 ¼ 2 103 W, and test tubes length L=2 ¼ 50 cm. If / is the phase difference between the two optical patterns on the two arms of the interferometer, the light intensity which impinges on the photodiode is I ¼ I0 =2ð1 cos uÞ and therefore the maximum sensitivity is obtained when u ¼ p=2. If the two samples have an equal optical path length, for Dn ¼ 5:1 1011 one has Du ¼ 2pDnðL=kÞ ¼ 5:09 104 rad. The induced variation of light intensity on the photodiode is DI ¼ ðI0 =2Þ sin uDu ffi 5:09 107 W. If the photodiode efficiency is 0.4 A/W and the operational
M. Maestro et al. / Chemical Physics 280 (2002) 103–109
109
Acknowledgements The authors wish to thank Dr. C. Cardellicchio for useful discussions on the subject of the paper and Dr. S. Pascazio for clarifying some analytical points.
References
Fig. 1. Michelson interferometer scheme, L: laser source, m1 and m2 : mirrors, R: operational amplifier resistance, B: beamsplitter, S1 and S2 : samples holders.
amplifier resistance is R ¼ 104 X one obtains a tension variation DV ffi 2 103 V. This DV value is much higher than the shot and thermal noise limits for the photodiode, implying that, on experimental ground, such an effect can be measured. The most critical thing is then the choice of the samples for which the only optical difference should consist in the second-order effect of ORP.
5. Conclusions In this paper we tried to show that the classical theory of the ORP still has some predictive capabilities. As a consequence of a full second-order perturbative treatment, a squared frequency term comes out which, in principle, could discriminate two enantiomers in a racemic mixture. Just in this context, a possible detection experiment was also discussed.
[1] G.R. Stephenson, Advanced Asymmetric Synthesis, Chapman & Hall, London, 1996. [2] E.N. Jacobsen, A. Pfaltz, H. Yamamoto, Comprehensive Asymmetric Catalysis I–III, Springer, Berlin, 1999. [3] G. Helmchen, R.W. Hoffmann, J. Muller, E. Schaumann, Stereoselective Synthesis in Holden and Weyl – Methods of Organic Chemistry, forth ed., Thieme, Stuttgart, 1995. [4] A.M. Schwartz, Novel Equivalence Principle Test, preprint, August 2001. [5] A.D. Buckingham, Adv. Chem. Phys. 12 (1967) 107. [6] L.D. Barron, A.D. Buckingham, Mol. Phys. 20 (1971) 1111. [7] F. Ghebremichael, M.G. Kuzyk, H.S. Lackritz, Prog. Polym. Sci. 22 (1997) 1147. [8] S.M. Cybulski, D.M. Bishop, Chem. Phys. Lett. 250 (1996) 471. [9] S. Coriani, C. Haettig, A. Rizzo, J. Chem. Phys. 111 (1999) 7828. [10] A. Rizzo, S. Coriani, A. Halkier, C. Haettig, J. Chem. Phys. 113 (2000) 3007. [11] M. Jaszunski, A. Rizzo, P. Joergensen, Theor. Chim. Acta 116 (2001) 251. [12] D.M. Bishop, Adv. Quantum Chem. 25 (1994) 1. [13] M. Surma, Mol. Phys. 90 (1997) 993. [14] R. Zawodny, S. Wozniak, G. Wegniere, Mol. Phys. 91 (1997) 165. [15] F.U. Condon, Rev. Modern Phys. 9 (1937) 144. [16] C. Bruhat, Optique, sixth ed., Masson et C, Paris, 1956. [17] Rosenfeld, Z. Phys. 52 (1928) 161. [18] H. Eyring, J. Walter, G.E. Kimball, Quantum Chemistry, Wiley, New York, 1944. [19] A. Yariv, P. Yeh, Optical Waves in Crystal, Wiley, New York, 1984. [20] P.W. Atkins, L.D. Barron, Mol. Phys. 16 (1964) 453; L. Blum, H.L. Frisch, J. Chem. Phys. 9 (1970) 52; L. Hecht, L.A. Nafie, Mol. Phys. 72 (1990) 441.
Optical rotation second-order effects M. Maestro a, F. Mavelli a, G. Paiano b,*, E. Polacco c b
a Dipartimento di Chimica dell’Universita di Bari, via Orabona 4, I-70125 Bari, Italy Dipartimento di Fisica dell’Universita di Bari and INFN Sezione di Bari, via Orabona 4, I-70125 Bari, Italy c Dipartimento di Fisica dell’Universita di Pisa, Piazza Torricelli 2, I-56100 Pisa, Italy
Received 5 November 2001; in final form 13 March 2002
Abstract The optical rotatory power (ORP) classical phenomenological theory is developed to a higher order with respect to the one usually dealt with. All the formulae needed for the passage from the molecular parameters to the macroscopic ones which are present in the correct phenomenological material medium equations are deduced. In particular a minor term which has already been considered by Condon has been corrected. The possibility that an experimental measurement on a racemate could allow a determination of the ORP of its components is discussed. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Optical rotatory power; Second-order effects
1. Introduction The problem of the asymmetric synthesis has worried the chemists for a long time and only rather recently it seems to have been solved (at least in some cases) [1,2]. But even the preliminary problem of the control of the optical purity cannot be considered as trivial [3]. Apart from this, all the phenomenology concerning the optical activity has kept its interest and its up-to-dateness, also for physicists who are interested in problems of basic invariance principles [4]. From a more general point of view, a large literature on non-linear interactions between EM fields and matter is at disposal at least since the late
*
Corresponding author.
sixties when the work mainly of Buckingham group (see [5,6]) laid a solid theoretical basis of the phenomenology. More recently, this subject revealed interesting technical aspects in some fields as, for instance, in characterising polymers [7]. At the same time, the rapid extension of the calculation facilities has driven a renewed interest for the theoretic ab initio calculations of more and more new tiny observables of higher order (see [8–11]). A great deal of new ‘‘effects’’, which added to the classic Kerr and Cotton Mouton effects, has been defined and identified, sometimes with not so transparent acronyms [12]. In some cases the problem of suggesting and setting up the correct and precise experimental procedures in order to verify the calculated results, deserves a particular care (see [13,14]). In this frame, we considered that the development of the classical phenomenological
0301-0104/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 ( 0 2 ) 0 0 4 0 5 - 6
104
M. Maestro et al. / Chemical Physics 280 (2002) 103–109
theory of OPR up to a higher order for evidencing some possible new effects to be detected experimentally could be worth exploring. In particular the idea of bringing out some quadratic effect in the molecular parameter which characterises the OPR, in order to distinguish a racemate from a compound containing two active centres but inactive for internal compensation, can appear rather naive, but we think that it deserves some attention. In the landmark paper by Condon [15], a second-order term is quoted, the effect of which is a simple small shift of the refractive index. A coherent development of the theory brings to the emergence of some new terms, while the wanted quadratic term is obtained. 2. Dielectric properties of the active media Optical rotatory power (ORP) can be phenomenologically described [1] by the following material medium equations: _; D ¼ eE gH ð1Þ B ¼ lH þ gE_ : The parameter g is directly bound to the optical rotation r being k2 r; 4p2 c where c and k are the radiation speed and the wavelength, respectively. Eq. (1) must be dealt with together with the Maxwell equations system: g¼
div D ¼ 0; B_ rot E ¼ ; c
div B ¼ 0; rot H ¼
_ D c
ð2Þ
in order to deduce all the physics of the electromagnetic field in the active medium. For the sake of simplicity we will consider the case of an isotropic, homogeneous and not conducting medium (typically, but not exclusively a liquid or a solution of some active organic compound). Notwithstanding such restrictions, the results are generalisable. The well-known theory of ORP [15–17] introduces the auxiliary (macroscopic) vectors polarisation P, and magnetisation Q
D ¼ E þ 4pP; B ¼ H þ 4pQ;
ð3Þ
where P and Q are related to the electric and magnetic molecular moments R and M induced by the field in the medium by the simple relations: P ¼ NR and Q ¼ N M; where N is the number of active molecular centres (N ffi 1019 in solids). The Lorenz field approximation: 4pP ; Ei ¼ E þ 3 ð4Þ 4pQ Hi ¼ H 3 is then utilised in order to describe the field inside the medium. The induced molecular moments R and M can be developed as functions of the macroscopic field and polarisation vectors E, P, H and Q via the molecular parameters a; b=c; c and j (see [16]). These last quantities can in turn be defined with reference to their quantum definitions: _ _ m a b b a h jP j i h jP j i X ba 2 aa ¼ ; 3h b m2 m2 ba _ _ m a b b a h jM j i h jM j i X ba 2 ja ¼ ; 2 3h b m m2 n ba_ o ð5Þ _ I a b b a h jP j i h jM j i X ba 1 ¼ ; 3ph b c m2ba m2 n _ o _ 2 X mba R hajPjbihbjMjai ca ¼ 3h b m2ba m2 and, in principle, be evaluated on the basis of molecular eigenfunctions. _ _ In Eq. (5) P and M are the operators of the electric and magnetic moments. One has b _ R ¼ aEi H i þ cHi ; c ð6Þ b_ M ¼ jHi þ Ei þ cEi : c From Eqs. (4) and (6) and from the definitions of P and Q one obtains:
M. Maestro et al. / Chemical Physics 280 (2002) 103–109
_ þ 3c H 4p N M _ 4p N M 3aE 3 bc H 3 3
R¼ M¼
3 4pN a b _ 4p _ 3jH þ 3 c E þ 3 N R þ 3c E þ 4p NR 3 3 þ 4pN j
;
: ð7Þ
Eq. (7) are used for deducing the relations between the molecular quantities a; j; b and c and the corresponding macroscopic ones e; l and g present in the phenomenological Eq. (1). In order to bring the procedure to a higher order one must simply derive with respect to time Eq. (7) and operate the suitable substitutions. Limiting ourselves to the terms in b2 =c2 , the following equations are valid: € 3aE ; 3 4pN a € € ¼ 3jH : M 3 þ 4pN j
105
Eqs. (9a) and (9b) with the correlated definitions are the required generalisations of the well-known first-order formulae [18] and must replace the material medium equation (1). It is easy to see that to the same order b2 =c2 it results g ffi h and p ffi q while v 6¼ w. In deducing the Eqs. (9a) and (9b) we have supposed that c is of the same order as xðb=cÞ. This last assumption is not justified (as far as we know) on the basis of some known results, since in general the quantity c has been considered devoid of practical interest [15,17]; but on the other hand it seems to be the most rational on the basis of the definitions (5). And besides, as it will be seen, a correct higher order treatment cannot avoid taking into account this quantity.
€¼ R
ð8Þ 3. The propagation of waves in the active media
Eqs. (3), (7) and (8) result in the following final formulae for D and B: _ þ vE €; D ¼ eE þ pH gH
ð9aÞ
€; B ¼ lH þ qE þ hE_ wH
ð9bÞ
where e¼
ð3 þ 8pN aÞð3 þ 4pN jÞ 64p2 N 2 c2 ; ð3 4pN aÞð3 þ 4pN jÞ 16p2 N 2 c2
l¼
ð3 4pN aÞð3 þ 16pN jÞ þ 64p2 N 2 c2 ; ð3 4pN aÞð3 þ 4pN jÞ þ 16p2 N 2 c2
g¼
36pN bc ; ð3 4pN aÞð3 þ 4pN jÞ 16p2 N 2 c2
36pN bc ; ð3 4pN aÞð3 þ 4pN jÞ þ 16p2 N 2 c2 36pN c p¼ ; ð3 4pN aÞð3 þ 4pN jÞ 16p2 N 2 c2 36pN c q¼ ; ð3 4pN aÞð3 þ 4pN jÞ þ 16p2 N 2 c2
h¼
2
v¼
144p2 N 2 bc2 ; ð3 4pN aÞ½ð3 4pN aÞð3 þ 4pN jÞ 16p2 N 2 c2 2
144p2 N 2 bc2 w¼ : ð3 þ 4pN jÞ½ð3 4pN aÞð3 þ 4pN jÞ þ 16p2 N 2 c2 ð10Þ
The active medium is characterised by two different refractive indexes for two circular (l ¼ left and r ¼ right) polarised waves. For sake of simplicity we deal with a purely dispersive medium, i.e. our refractive indexes nl and nr are real quantities. Following the classical procedure [18] in order to deduce these two quantities one must study a wave solution of the Maxwell equations describing the active media by means of Eqs. (9a) and (9b). Electromagnetic vectors that propagate into the active medium are described by the following general formula: n h z io Aðz; tÞ ¼ R A0 exp i x t n : ð11Þ c For every wave solution the following equations are valid: € ¼ x2 E; E € ¼ x2 H: H _ the time As a result, by solving with respect to H derivative of Eq. (9b) and from the third Maxwell Eq. (2) one has _ ¼ H
1 ðc rot E qE_ þ gx2 EÞ ðl þ wx2 Þ
ð12Þ
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and therefore
Z 1 _ H¼ c rot E dt qE gE : ðl þ wx2 Þ ð13Þ By inserting Eqs. (12) and (13) in Eq. (9a) one has 1 D ¼ e vx2 E þ ðl þ wx2 Þ Z
q c rot E dt qE 2 gð c rot E þ gx EÞ and finally 1 D¼ ðl þ wx2 Þ Z
n2E þ s2 x2 E c q rot Edt g rot E ; ð14Þ where the following notations have been introduced: n2E ¼ n20 q2 , n20 ¼ el and s2 ¼ ew lv g2 and a term of higher order has been neglected. Vector D can be expressed in terms of E and its derivatives by utilising the second Maxwell equation for rot. One has Z c D ¼ c rot H dt ¼ ðl þ wx2 Þ Z Z Z
c rot rot E dt dt q rot E dt Z g rot E_ dt : ð15Þ From Eqs. (14) and (15) and by means of the wellknown equation for rot rot (being div E ¼ 0), one has Z Z ðn2E þ s2 x2 ÞE þ cg rot E c2 r2 E dt dt Z þ g rot E_ dt ¼ 0: ð16Þ From the chosen form for the field vectors (11), one has: n2 x2 nx2 Ey ; r E ¼ 2 ; rot E_ x ¼ c c inxEy inxEx rot Ex ¼ ; rot Ey ¼ ; c c nx2 Ex : rot E_ y ¼ c 2
ð17Þ
Therefore Eq. (16) can be reduced to the following secular equation in n n2 þ s2 x2 n2 1 2ingx E 2 2 2 2 ¼ 0: 2 2ingx nE þ s x n ðl þ wx Þ ð18Þ Eq. (18) can be easily solved. The positive real solutions for n are the following: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ¼ xg þ n2E þ fx2 ffi nF xg; ð19Þ where f ¼ ðew lvÞ and nF ¼ nE þ ðfx2 =2nE Þ. From Eq. (19), to the same order g2 , one has n2 ¼ n2E 2nE xg þ x2 g2 þ fx2 :
ð20Þ
It is easy to see that in the usual first-order limit, when one assumes l ¼ 1 one has also s ¼ q ¼ w ¼ 0 and n2E ¼ n20 , so that one has n2 ¼ n20 jgjx and, since jgjx << n20 it will be n ffi n0 þ d, where d ¼ jgjx=2n0 , a well-known result [18]. Two enantiomers are distinguished by the sign of g which immediately is related to that of b, the corresponding molecular parameter. In the following treatment we prefer to introduce only jgj and therefore we will obtain different formulae for right- and left-handed media. To deduce the phenomenology of the light propagation in the active media one can resort to the classic procedure [19]; this implies the explicit evaluation of the eigenvectors of the E matrix already introduced which can be easily found. For nþ one has ðn2E s2 x2 n2þ ÞEx 2inþ gEy ¼ 0 and therefore Exþ ¼ i Eyþ and, analogously for n Ex ¼i Ey and then, the normalised eigenvectors are the following i i eþ ¼ ; e ¼ : ð21Þ 1 1 Eigenvectors in Eq. (21) are the same as those obtained in the case of the first-order approxima-
M. Maestro et al. / Chemical Physics 280 (2002) 103–109
tion. As a consequence, to the second order no new effect can be expected regarding the ellipticity of the light which crosses the active medium. At this point, following the method of Yariv and Yeh [19], it is easy to describe a plane-polarised wave in the x direction and to deduce its shape after it has propagated along the z direction covering a path Z in the time interval t. For that one must develop the original Cartesian vectors on the basis of (21) 1 1 1 1 x¼ ¼ þ ; 0 i i 2 1 0 i i y¼ ¼ þ : 1 1 1 2 Then one has for a right-handed medium
1 Z ER ðZ; tÞ ¼ CR exp ix t n þ x c i
1 Z þ exp ix n c i
Z ¼ 2C cos x t nF c Z Z
cos x2 jgj ~ I sen x2 jgj ~ J ; c c ð22Þ where C ¼ ðjEð0; 0ÞjÞ=2 and ~ I and ~ J are the original cartesian vectors. In Eq. (22) ER x ðZ; tÞ is the electric field component of the wave at time t and after a path length Z of a plane wave which at time t ¼ 0 was polarised in the plane xz. Analogously one has
Z R Ey ðZ; tÞ ¼ 2C cos x t nF c
Z Z~ 2 2 ~
senx jgj I þ cos x jgj J : ð23Þ c c The corresponding formulae for the left-handed forms are simply obtained by changing the signs before sinus functions. To deduce the result for a wave crossing a racemic mixture, one can suppose that light covers successively half path in each of two enantiomers and simply apply successively Eqs. (22) and (23) and their corresponding for the left-handed
107
I and ~ J. enatiomer putting ExL and EyL in place of ~ The results are:
Z ~ ERac ðZ; tÞ ¼ 2C cos x t n I ð24Þ F x c and
Z ~ ERac ðZ; tÞ ¼ 2C cos x t n J: F y c
ð25Þ
It is interesting to compare (24) and (25) with the corresponding formulae for a hypothetical corresponding compound which is inactive for internal compensation. To do this, we recall Eq. (19) nF ¼ nE þ ðfx2 =2nE Þ, where f ¼ ev lw and, in turn v and w are defined in Eq. (10) while n2E ¼ n20 q2 . In this compound one simply has g ¼ q ¼ f ¼ 0 and therefore
Z ~ E0x ¼ 2C cos x t n0 I; ð26Þ c where n20 ¼ el: The shift n20 q2 was already present in Condon paper [15]. The new term in x2 makes nF correct to the second order. In the following section we shortly discuss the possibility of an experimental discrimination between (24) and (26).
4. Discussion on a possible experimental test Eqs. (24) and (26) refer to two physical situations that are in principle different, even though they cannot be distinguished in a first-order treatment of optical rotation; experimentally to discriminate between them some further points must be discussed. The first one is the choice of a suitable chemical compound. The main problem originates from a possible different crystal structure of the active and the inactive forms which in principle imply some very different values of refraction indexes of racemic mixtures and of inactive forms, apart from the intrinsic difference emphasised by Eqs. (24) and (26). On the other hand resorting to solutions could not solve the problem since in this case one should take into account the problem of the weight
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M. Maestro et al. / Chemical Physics 280 (2002) 103–109
of different configurations. It is for these reasons that a well-known molecule such as tartaric acid must be discarded, whereas asymmetrically substituted spiro compounds and allenes can be considered as possibly suitable candidates. Here, we will not deal with this problem, but in principle it should not be impossible to select some rigid structured molecule that can prevent the difficulties just examined. Another problem seems to involve the level of homogeneity of the racemic mixture. But a simple calculation shows that this point is not critical. In fact, considering relatively small molecules (up to a 3 ) their molecular volume of some hundreds of A number in a sphere of radius of the same order as the light wavelength k should ensure the proper statistical weight of two enantiomers. This condition is met also for solutions, at least for high concentrations (ffi 1 M). Apart from these and some other points that can possibly be taken into account, the most relevant problem is the smallness of the difference between nF and n0 . At any rate it is only this point that will now be considered. Other observables show the same quadratic dependence on the optical rotation parameters: for instance the intensity of the scattered Raman radiation at right angle when the sample is crossed by a properly polarised coherent wave (see [6,20]) and it is nice that in experiences based on the violation of the principle of gravitational equivalence [4], a (mechanical) squared frequency turns out to be the discriminating element. Our choice of the refractive index variation is probably simpler to be made. To do this, we will consider an ideal molecule with physical data representative of many real cases. Table 1 shows the starting data and the resulting calculated parameters. From
Table 1 Typical data for a suitable chemical compound d (Density) M (Molecular weight) k (Light wavelength) r (Optical rotation at the given k) e (Relative dielectric constant) l (Relative magnetic suscept.)
2 g=cm3 200 Da 6:33 105 cm 100 deg/cm 4 1.0002
these data and from Avogadro number NA ¼ 6:02 1023 , one has N ¼ 6:02 1021 cm3 ; g ¼ 5:30887 1021 s; x ¼ 2:975757 1015 s1 : These data are not sufficient to determine all the microscopic quantities which appear in Eq. (10); at least a further assumption has to be done. Assuming c ¼ xðb=cÞ as already discussed, the values for q, a; b=c, j; v and w can be evaluated by Eq. (10). To the second order in b=c it results q p ¼ xg ¼ 1:758338 105 and then a ¼ 1:98287 1023 cm3 ; j ¼ 1:057997 1027 cm3 ; b=c ¼ 3:905511 1044 cm3 s; c ¼ 1:1621864 1028 cm3 ; v ¼ 1:163846 1041 s2 ; w ¼ 5:818821 1042 s2 ; f ¼ ew lv ¼ 1:1634498 1041 s2 : From the given input data one has n20 ¼ el ¼ 4:0008, therefore the quantities to be compared are n20 ; q2 ¼ 3:09175 1010 and fx2 ¼ 1:03025 1010 and finally one obtains Dn ¼ n0 nF ¼ 5:1 1011 : Is this effect experimentally detectable? We consider an experimental set-up formed by a Michelson interferometer (see Fig. 1) equipped with a He–Ne laser source (k ¼ 633 nm) with a power I0 ¼ 2 103 W, and test tubes length L=2 ¼ 50 cm. If / is the phase difference between the two optical patterns on the two arms of the interferometer, the light intensity which impinges on the photodiode is I ¼ I0 =2ð1 cos uÞ and therefore the maximum sensitivity is obtained when u ¼ p=2. If the two samples have an equal optical path length, for Dn ¼ 5:1 1011 one has Du ¼ 2pDnðL=kÞ ¼ 5:09 104 rad. The induced variation of light intensity on the photodiode is DI ¼ ðI0 =2Þ sin uDu ffi 5:09 107 W. If the photodiode efficiency is 0.4 A/W and the operational
M. Maestro et al. / Chemical Physics 280 (2002) 103–109
109
Acknowledgements The authors wish to thank Dr. C. Cardellicchio for useful discussions on the subject of the paper and Dr. S. Pascazio for clarifying some analytical points.
References
Fig. 1. Michelson interferometer scheme, L: laser source, m1 and m2 : mirrors, R: operational amplifier resistance, B: beamsplitter, S1 and S2 : samples holders.
amplifier resistance is R ¼ 104 X one obtains a tension variation DV ffi 2 103 V. This DV value is much higher than the shot and thermal noise limits for the photodiode, implying that, on experimental ground, such an effect can be measured. The most critical thing is then the choice of the samples for which the only optical difference should consist in the second-order effect of ORP.
5. Conclusions In this paper we tried to show that the classical theory of the ORP still has some predictive capabilities. As a consequence of a full second-order perturbative treatment, a squared frequency term comes out which, in principle, could discriminate two enantiomers in a racemic mixture. Just in this context, a possible detection experiment was also discussed.
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