NEW APPROACH TO THE THEORY OF CIRCULAR DICHROISM

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1 Theory of Circular Dichroism 1187 NEW APPROACH TO THE THEORY OF CIRCULAR DICHROISM Jaroslav ZAMASTIL 1, Lubomir SKALA 2, Petr PANCOSKA 3 ad Oldrich BILEK 4 Faculty of Mathematics ad Physics, Charles Uiversity, Prague 2, Czech Republic; 1 zamastil@quatum.karlov.mff.cui.cz, 2 skala@quatum.karlov.mff.cui.cz, 3 pacoska@tigger.uic.edu, 4 bilek@quatum.karlov.mff.cui.cz Received April 16, 1998 Accepted Jue 12, 1998 Dedicated to Professor Rudolf Zahradik o the occasio of his 70th birthday. Usig the semiclassical approach for the descriptio of the propagatio of the electromagetic waves i optically active isotropic media we derive a ew formula for the circular dichroism parameter. The theory is based o the idea of the time damped electromagetic wave iteractig with the molecules of the sample. I this theory, the Lambert Beer law eed ot be take as a empirical law, however, it follows aturally from the requiremet that the electromagetic wave obeys the Maxwell equatios. Key words: Circular dichroism; Optically active media; CD spectroscopy; Quatum chemistry. I this paper, we are iterested i the theory of the circular dichroism i isotropic media. It is well-kow that the propagatio of the left or right haded circularly polarized electromagetic waves (LHCP or RHCP) i optically active media is differet. If the molecule absorbs at the frequecy of the electromagetic wave, the absorptio of the LHCP ad RHCP waves is ot equal. This effect is called the circular dichroism (CD) ad is characterized by the differece of the correspodig absorptio coefficiets. This differece is usually deoted as the circular dichroism parameter (see e.g. refs 1,2 ) b = b L b R, (1) where b L ad b R are defied by the Lambert Beer law I L (z) I L (0) = exp ( b Lz), (2) I R (z) I R (0) = exp ( b Rz). (3)

2 1188 Zamastil, Skala, Pacoska, Bilek: Here, I(z) is the itesity of the wave propagatig alog the z axis. The stadard theories of the circular dichroism assume the electromagetic field i the form of the moochromatic plae wave 1,2. I case of absorptio, such approach is ot theoretically justified sice the plae wave does ot obey the Maxwell equatios. The Lambert Beer law is take as a empirical law 1. The aim of this paper is to preset a more cosistet theory i which we assume a damped electromagetic wave istead of the plae wave. This approach has two advatages. First, this wave satisfies the Maxwell equatios. The secod advatage of our approach is that the Lambert Beer law follows aturally from the damped wave assumptio. I the ext sectio, we summarize the semiclassical approach ad itroduce the basic physical quatities. We briefly discuss difficulties of the usual approaches ad itroduce the idea of a damped electromagetic wave, ad the we solve the Maxwell equatios ad derive the formula for the circular dichroism parameter. I the last sectio, discussio of the results is preseted. The averagig procedure of molecular quatities is described i Appedix A. I Appedix B, a order estimate of the dampig costat is performed. SEMICLASSICAL APPROACH We assume that the electromagetic waves are solutios of the Maxwell equatios. Assumig further that there are o free charges ad currets i the sample the Maxwell equatios read E = 1 ε 0 P, 1 µ 0 B ε 0 t E = M + t P, B = 0, E + t B = 0. To solve these equatios we have to kow the material relatios P = P(E,B), M = M(E,B). (4) I this paper, we assume liear material relatios. I this case we get from the first Maxwell equatio

3 Theory of Circular Dichroism 1189 E = 0. Therefore, we require the gauge coditios A mac = 0, ϕ = 0, (5) where A mac ad ϕ deote the macroscopic vector ad scalar potetials. I this way, we get from the Maxwell equatios the wave equatio 1 µ 0 2 A mac + ε 0 2 t 2A mac = t P + M. (6) To derive the material relatios (4), we proceed as follows. We assume that the optical activity is produced by the iteractio of the electromagetic field with the idividual molecules of the sample. We are iterested i the electroic states ad assume that the iteractio amog the molecules is weak ad the wave fuctios of the idividual molecules are kow. For the vibratioal states, these assumptios are ot justified. First, we calculate d ad m, the quatum mechaical average of the electric ad magetic dipole momets of the molecule iteractig with the electromagetic field. These quatities are defied by the equatio Q = ψ Q^ ψ, (7) where Q deotes d or m. To calculate (7), we have to kow the state vector ψ. It is the solutio of the Schrödiger equatio ih t ψ(t) = (H^ 0 + W^ (t)) ψ(t), (8) where H^ 0 is the Hamiltoia of the free molecule ad W^ describes the iteractio betwee the molecule ad the local electromagetic field i the first order of A loc W^ (t) = e m i A^ loc (r i ) p^i. (9)

4 1190 Zamastil, Skala, Pacoska, Bilek: Here, e = e ad m deote the charge ad mass of the electro, respectively, i rus over all electros i the molecule. Keepig liear terms i A loc oly, we solve Eq. (8) i the first order of the perturbatio theory. To solve Eq. (8), we expad ψ(t) ito the eigestates ψ m (t) of the free hamiltoia H^ 0 with the eergies E m = h ω m where ψ(t) = ψ 0 (t) + m c m (t) ψ m (t), (10) ψ m (t) = exp (iω m t) m. (11) Here, ψ 0 deotes the groud state of the molecule. The sum over statioary states icludes the groud state, too. By isertig expasio (10) ito Eq. (8) we get i the first order ih ċ = ψ (t) W^ (t) ψ 0 (t), (12) where the orthoormality relatios ψ m ψ = δ m (13) are used. Itegratio of Eq. (12) yields c = 1 t ih dt ψ (t) W^ (t) ψ 0 (t), (14) t 0 where t 0 is the iitial time of the iteractio betwee the field ad the molecule. By isertig expasio (10) ito the defiitio (7) we obtai i the first order of A loc Here, Re deotes the real part, ψ Q^ ψ = Q Re c Q 0 exp ( iω 0 t). (15) Q 0 = 0 Q^ (16)

5 Theory of Circular Dichroism 1191 ad ω 0 = ω ω 0. (17) I the secod step, we covert the local electric ad magetic momets d ad m ito the macroscopic polarizatio P ad magetizatio M. We do it by calculatig average over all orietatios of the molecules ad by multiplyig the averaged quatities by the umber of the molecules i the uit volume P = N d, M = N m. (18) Fially, we have to trasfer from the local field A loc to the macroscopic field A mac. I this paper, we use the usual assumptio 1 A mac = S A loc, (19) where S is a pheomeological costat. DAMPED WAVE APPROACH The usual approach to the solutio of the wave equatio (6) is based o the plae wave assumptio 1,2. It has two disadvatages. First, this wave does ot obey Eq. (6). The secod disadvatage is that the Lambert Beer law does ot follow from the theory, however, it must be take as a empirical law. I this paper, we replace the plae wave assumptio by a more accurate damped wave approximatio. To perform the itegratio i Eq. (14), we have to assume the time depedece of the vector potetial A loc = A loc (r,t). I the usual approach 1,2, the followig time depedece is assumed A loc (r,t) = A c (r) cos ωt + A s (r) si ωt. (20) The, we ca write W^ (t) = W^ + exp (iωt) + W^ exp ( iωt), (21)

6 1192 Zamastil, Skala, Pacoska, Bilek: where W^ ± ca be obtaied from compariso of Eqs (21), (20) ad (9). With this otatio, we evaluate the itegral i Eq. (14) ad get 10 where c = 1 ih W exp [i(ω + ω + 0)t] 0 + W exp [i( ω + ω 0)t] i(ω + ω 0 ) 0 i( ω + ω 0 ), (22) ± ± W 0 = W^ 0. (23) It is see that i case of absorptio (ω = ω 0 ) the assumptio (20) leads to c = 1 ih W 0 (t t 0 ) + W + exp (2iωt) 0 2iω. (24) Because of the term proportioal to t t 0, this result is i cotradictio with the assumptio (20). Therefore, this approach caot be used i case of absorptio. This problem ca be solved i two ways: either usig the Kramers Kroig relatios 1,3 or takig ito accout the spotaeous emissio 2,4,5. However, these approaches are ot satisfactory sice it is ecessary to make rather crude approximatios 6. I this paper, we fid the time depedece of the solutio from the heuristic cosideratios. I case of resoace, molecules are excited to higher eergy levels by the absorptio of idividual photos. After very short times, molecules emit the electromagetic waves ad deexcitate. The electromagetic waves propagate i all directios. Therefore, the amplitude of the wave propagatig i oe directio is damped. First we summarize the semiclassical approach. Whe solvig the Schrödiger equatio (8) we have to assume some time depedece of A which must obey the wave equatio (6). Therefore, we have to guess the time depedece of the electromagetic wave i advace. Thus, the wave appearig o the right had side of Eq. (8) is ot the exteral wave, but the iteral wave iteractig with idividual molecules. We believe that the picture described above ca be obtaied by takig ito accout the oliear terms i A. Therefore, we cosider the wave A loc (r,t) = (A c (r) cos ωt + A s (r) si ωt) exp ( κt), (25) where κ is a pheomeological parameter. If we were able to solve the problem of the propagatig electromagetic waves i the matter exactly, we would get the depedece (25) i the limit case of the weak field. I this case, we would get κ as a fuctio of

7 Theory of Circular Dichroism 1193 parameters describig the iteractio betwee the wave ad matter. The value of κ is estimated i Appedix B. We assume that dampig of the moochromatic electromagetic wave is give by a sigle dampig factor. We ote first that A loc (r,t) is multiplied i Eq. (12) by the factor exp (iω 0 t). Further, this fuctio of t is itegrated i Eq. (14) ad divided by exp (iω 0 t) (see Eq. (15)). Fially, S A loc (r,t) is iserted ito A mac, P ad M i Eq. (6) ad the derivatives i this equatio are performed. It follows from the resultig equatio that the time depedece (25) is the oly depedece which is the solutio of the wave equatio (6). CALCULATION OF THE DAMPED WAVE I case of the time depedece (25) we ca write W^ (t) = exp ( κt)[w^ + exp (iωt) + W^ exp ( iωt)]. (26) Itegratio of Eq. (14) with this time depedece yields c = 1 ih W exp [i(ω + ω iκ)t] 0 + W exp [i( ω + ω 0 + iκ)t] i(ω + ω 0 + iκ) 0 i( ω + ω 0 + iκ) By isertig this formula ito Eq. (15) we obtai Q = Q 00 2 exp ( κt) h. (27) Re Q + W 0 exp (iωt)(ω 0 + ω iκ) 0 (ω 0 + ω) 2 + κ 2 + W 0 exp ( iωt)(ω 0 ω iκ) (ω 0 ω) 2 + κ 2.(28) To describe the circular dichroism it is sufficiet to cosider just two terms of the Taylor expasio of the space depedece of A loc i the molecule A loc (r,t) = A loc (0,t) + r A loc (r,t) r = 0. (29) Here poit r = 0 correspods to the ceter of the molecule. Further, we put d = e r i (30) i

8 1194 Zamastil, Skala, Pacoska, Bilek: ad m = e 2m r i p i (31) i ito Q, covert these quatities ito P ad M (see Appedix A), covert the local field ito the macroscopic oe usig Eq. (19) ad use the well-kow relatio p 0 = imω 0 r 0. (32) Takig the real part of the resultig expressio for Q we obtai 11 P = exp ( κt) [(α 1 + ωα 2 )(A s cos ωt A c si ωt) + (β 1 + ωβ 2 )(B s cos ωt B c si ωt) + + κα 2 (A c cos ωt + A s si ωt) + κβ 2 (B c cos ωt + B s si ωt)] (33) ad M = exp ( κt)[(β 3 + ωβ 1 )(A c cos ωt + A s si ωt) κβ 1 (A s cos ωt A c si ωt)], (34) where B c = A c, B s = A s, (35) α 1 = N 3Sh ω2 0 d (ω 0 + ω) 2 + κ 2 1 (ω 0 ω) 2 + κ 2, (36) α 2 = N 3Sh ω 0 d (ω 0 + ω) 2 + κ (ω 0 ω) 2 + κ 2, (37) β 1 = N 3Sh ω 0 Im { m 0 d 0 } 1 (ω 0 + ω) 2 + κ 2 1 (ω 0 ω) 2 + κ 2,(38) β 2 = N 3Sh Im { m 0 d 0 } 1 (ω 0 + ω) 2 + κ (ω 0 ω) 2 + κ 2, (39)

9 Theory of Circular Dichroism 1195 β 3 = N 3Sh 2 ω 0 Im { m 0 d 0 } 1 (ω 0 + ω) 2 + κ (ω 0 ω) 2 + κ 2. (40) Here, we eglected the terms proportioal to m 0 m 0 which are small. I the limit κ 0 whe the absorptio disappears we get α 1 + ωα 2 = ωα, (41) β 1 + ωβ 2 = + ωβ, (42) β 2 + ωβ 1 = ω 2 β, (43) where α is the polarizability 1 α = 2N 3Sh ad β is the optical rotatioal parameter 1 β = 2N 3Sh ω 0 d ω 0 ω 2 (44) Im { m 0 d 0 } 2 ω 2. (45) Usig Eq. (33) for P, Eq. (34) for M ad the time depedece (25) for A i the wave equatio (6) we obtai ω 0 1 µ 2 (A c cos ωt + A s si ωt) + ε 0 [(κ 2 ω 2 )(A c cos ωt + A s si ωt) 0 2κω (A s cos ωt A c si ωt)] = κα 1 (A s cos ωt A c si ωt) [α 2 (κ 2 + ω 2 ) + α 1 ω](a c cos ωt + A s si ωt) 2κβ 1 (B s cos ωt B c si ωt) [β 2 (κ 2 + ω 2 ) β 3 ](B c cos ωt + B s si ωt). (46)

10 1196 Zamastil, Skala, Pacoska, Bilek: Comparig terms with the same cos ωt ad si ωt depedece o both sides of Eq. (46) we obtai two equatios. By puttig A s = ±ia c we ca reduce these two equatios ito oe equatio 1 µ 0 2 A c + ε 0 (κ 2 ω 2 )A c 2iεκωA c = iκα 1 A c [α 2 (κ 2 + ω 2 ) + α 1 ω]a c 2iκβ 1 B c [β 2 (κ 2 + ω 2 ) β 3 ]B c. (47) Now we search for the solutio i the form A c = A 0 exp (ik r), (48) where k = ω c. (49) Here, is a refractio idex, ω is a circular frequecy, c is the velocity of the propagatio of the electromagetic waves ad A 0 is the complex amplitude of the vector potetial. From the gauge coditio (5) we obtai k A 0 = 0. (50) Therefore, we ca put the z axis alog the directio of the k vector. The we ca write A 0 = (A 1, A 2, 0). (51) From Eq. (47) we get a system of two liear algebraic homogeous equatios (ε a Re + ia Im )A 1 = i(b Re + ib Im )A 2, (52) where (ε a Re + ia Im )A 2 = i(b Re + ib Im )A 1, (53)

11 Theory of Circular Dichroism 1197 a Re = ε 0 (κ 2 ω 2 ) + α 2 (ω 2 + κ 2 ) + α 1 ω ω 2, (54) a Im = κα 1 2ε 0 κω ω 2, (55) b Re = β 2 (ω2 + κ 2 ) β 3 ωc (56) ad b Im = 2κβ 1 ωc. (57) To solve Eqs (52) ad (53), the determiat of this system of equatios for A 1 ad A 2 must equal zero. The solutio of this coditio with respect to is 1,2 = b Re + ib Im ± 4ε 0 (a Re + ia Im ), (58) 2ε 0 3,4 = b Re ib Im ± 4ε 0 (a Re + ia Im ). (59) 2ε 0 Here, we eglected the term (b Re + ib Im ) 2 i the discrimiat which is quadratic i m 0. The solutios with the mius sig i frot of the discrimiat i Eqs (58) ad (59) are uphysical sice the amplitude of the correspodig wave would icrease. As a result, we get the left haded polarized wave A 2 = i A 1 (60) correspodig to 1 ad the right haded polarized wave A 2 = i A 1 (61) correspodig to 3.

12 1198 Zamastil, Skala, Pacoska, Bilek: Calculatig I L = A L 2 ad I R = A R 2 we get the Lambert Beer law, Eqs (2) ad (3). Rather complex expressios for b L ad b R will ot be give explicitly here. The ratio of the itesities I L ad I R equals I L = A L 2 = exp ( bz), (62) I R A R 2 where b = 2ωb Im ε 0 c = 4κµ 0 N 3Sh ω 0 Im {m 0 d 0 } 1 (ω 0 ω) 2 + κ 2 1 (ω 0 + ω) 2 + κ 2 (63) is the fial formula for the circular dichroism parameter. I cotrast to ref. 2, Eq. (63) takes ito accout the summatio over all states. We ote also that the secod term i the brackets is ot cosidered i ref. 2. However, for ω closed to ω 0 this term ca be eglected. We emphasize that, i cotrast to ref. 2, κ does ot describe the atural lie width but the effective lie width resultig from the absorptio of the electromagetic wave by the medium. I the usual approximate approach to the theory of circular dichroism 1, the expressio for b reads b = 4µ 0N 3Sh κ ωim { m k0 d 0k } (64) for ω = ω k0 ad b = 0 otherwise. We see that this formula ca be obtaied from our Eq. (63) as a approximatio for ω = ω k0 if the secod term i the brackets i Eq. (63) is eglected ad the summatio i this equatio is reduced to just oe k-th term. It is obvious that the approximate formula (64) ca be used oly if the eergy levels of the molecule are sufficietly distat. I cotrast to Eq. (64) where the spectral lies have δ-like character our equatio (63) yields more realistic Loretzia shape of spectral lies. It is obvious that for achiral molecules with the ceter of symmetry the scalar product Im { m k0 d 0k } is equal to zero (compare with ref. 2 ). Therefore, the circular dichroism parameter b equals zero for such molecules.

13 Theory of Circular Dichroism 1199 CONCLUSIONS I this paper, we have used the semiclassical approach to derive a ew formula (63) for the circular dichroism parameter b. The stadard expressios for the circular dichroism parameter follow from our more geeral formula as a special case. I our approach, we do ot have to take the Lambert Beer law as a empirical law, however, this law follows aturally from the discussio of the possible forms of the solutios of the Maxwell equatios. APPENDIX A I this Appedix, we describe the averagig procedure i more detail. We follow the procedure itroduced i ref. 7. We assume that the matter is isotropic. It meas that the molecules have all possible spatial orietatios. We limit our attetio to the vector character of Eq. (28) Q i Q i + A j p j Q i + j r j A l p l Q i, (65) where the summatio covetio is used. A arbitrary rotatio of the vector α i is give by α λ = R λi α i, (66) where α λ is the rotated vector, R λi is the matrix of rotatio i the three dimesioal space expressed i terms of the Euler agels. The explicit form of the rotatio matrix is ot ecessary for our purposes. Rotatig the expressio i Eq. (65) we get Q λ Q λ + A µ p µ Q λ + µ r µ A ν p ν Q λ. (67) Here, we do ot rotate the field quatities. The electromagetic wave propagates through the matter, where it iteracts with differet rotatig electric ad magetic momets of the molecules. Usig Eq. (66) we obtai Q λ R λi Q i + A µ R µj R λi p j Q i + µ A ν R λi R νl R µj r j p l Q i. (68) Averagig over all spatial orietatios meas the itegratio of the terms cotaiig the matrices R λi over the Euler agels. Here, we ca use the idetities 3,7,8

14 1200 Zamastil, Skala, Pacoska, Bilek: R λi = 0, (69) R λi R µj = 1 3 δ λµ δ ij (70) ad R λi R µj R νl = 1 6 ε λµν ε ijl. (71) Usig these equatios we get from Eq. (68) Q λ A 1 µ 3 δ λµ δ ij p j Q i + µ A 1 ν 6 ε λµν ε ijl Q i r j p l. (72) This formula is used i the trasitio from Eq. (28) to Eqs (33) ad (34). APPENDIX B I this Appedix, we make a order estimate of κ. By usig the estimates N m 3, µ N C 2 s 2, ω c λ 1015 s 1, Im m k0 d 0k C 2 m 3 s 1, h Js i Eq. (64) we get b 1011 Sκ m 1. I the case of a box of the legth z 10 2 m we have bz 109 Sκ.

15 Theory of Circular Dichroism 1201 A typical experimetal value of bz for the cosidered molecular cocetratio ad legth of the box 9 is bz 10 5 l 10. Compariso of the last two estimates yields the order estimate of the product Sκ Sκ s 1. This work was supported by the Grat Agecy of the Czech Republic ad the Grat Agecy of Charles Uiversity. REFERENCES AND NOTES 1. Caldwell D. J., Eyrig H.: The Theory of Optical Activity. Wiley, New York Barro L. D.: Molecular Light Scatterig ad Optical Activity. Cambridge Uiversity Press, Cambridge Healy W.: J. Phys. B: At., Mol. Opt. Phys. 1974, 7, Davydov A. S.: Quatum Mechaics. Gosizdat, Moscow Heitler W.: The Quatum Theory of Radiatio. Claredo Press, Oxford Zamastil J.: M.S. Thesis. Charles Uiversity, Praha Power E. A., Thiruamachadra T.: J. Chem. Phys. 1974, 60, Weyl H.: The Classical Groups. Priceto Uiversity Press, Priceto Easma G. D., Ed.: Circular Dichroism ad the Coformatioal Aalysis of Biomolecules. Pleum Press, New York The molecules i the sample are excited at differet times t 0. The macroscopic polarizatio ad magetizatio are give by a average over a large umber of the molecules. Due to the oscillatig form of c, we ca assume that the cotributios of the lower bouds i Eq. (14) cacel so that the lower bouds eed ot be cosidered. 11. Sice the Hamiltoia H^ 0 describig the molecule is real, both the real ad imagiary parts of the wave fuctio have to obey the statioary Schrodiger equatio. Hece, we may assume that the wave fuctio is real. It follows from this assumptio that we ca take real d 0 ad purely imagiary m 0.