11546
J. Phys. Chem. 1996, 100, 11546-11558
Linear and Circular Dichroism Spectroscopic Study of β,β′-Dimethylmesobilirubin-XIIIr Oriented in a Nematic Liquid Crystal Danuta Bauman,† Claudia Killet,‡ Stefan E. Boiadjiev,§ David A. Lightner,§ Alfred Scho1 nhofer,⊥ and Hans-Georg Kuball*,‡ Institute of Physics, Poznan´ UniVersity of Technology, Piotrowo 3, 60-965 Poznan´ , Poland; Fachbereich Chemie, UniVersita¨ t Kaiserslautern, D-67653 Kaiserslautern, Germany; Department of Chemistry, UniVersity of NeVada, Reno, NeVada 89577-0020; and Institut fu¨ r Theoretische Physik, Technische UniVersita¨ t Berlin, D-10623 Berlin, Germany ReceiVed: February 20, 1996; In Final Form: April 29, 1996X
(βR,β′R)-dimethylmesobilirubin-XIIIR, an optically active synthetic analogue of bilirubin, the yellow pigment of jaundice, and xanthobilirubinat, a dipyrrinone analogue for one-half of a rubin pigment, dissolved in the nematic liquid crystal ZLI 1695, have been studied by means of UV-vis polarized absorption and circular dichroism (CD) spectroscopy. The order parameters have been evaluated from the temperature dependence of the degree of anisotropy. The method of “vanishing spectral features” has also been taken into consideration. The reduced absorption spectra have been obtained. It has been found that the orientational properties as well as the polarization of the absorption bands of the two compounds are quite different. Moreover, the UV-vis absorption and CD spectra of β,β′-dimethylmesobilirubin-XIIIR have been interpreted on the basis of the exciton coupling model, assuming the point symmetry group C2. The orientation of the principal axes of the orientational distribution tensor (order tensor) with respect to the molecular frame has been determined. That the anisotropic CD (ACD) spectra are not very different from the CD spectra can be understood from the spectroscopic analysis taking into account the orientational order.
1. Introduction Bilirubin-IXR is the yellow-orange lipophilic cytotoxic pigment of jaundice, which plays an important role in life processes. In normal metabolism, an adult human generates and eliminates about 300 mg per day of this pigment, which is produced by catabolism of hemoglobin, mainly from red blood cells.1,2 Bilirubin is produced in the spleen and transported to the liver, where it is transformed and secreted into the gall ducts as a component of bile. The main components of bile are water, phospholipids, cholesterol, and bile salts, which form a lyotropic liquid crystalline system.3 Thus, in living organisms bilirubin resides in an anisotropic medium. Therefore, the study of bilirubin in any oriented matrix may provide information about the properties of this pigment in a natural environment. Bilirubin-IXR and its analogues consist of two interacting dipyrrinone fragments covalently conjoined to the central methylene group at C10.2 Both dipyrrinone groups are able to rotate independently about this central CH2 (C10) group. These rotations allow the possibility of a large number of conformations. However, computations for bilirubin-IXR and its symmetric analogue in which the vinyl groups are replaced by ethyl (mesobilirubin-XIIIR) showed that the folded shape with the two dipyrrinone planes oriented to form a dihedral angle of about 100°4-6 is energetically the most favorable conformation. Such a conformation of bilirubin was also found in the solid by X-ray crystallography.7-9 It is stabilized through intramolecular hydrogen bonds and permits bilirubin to exist as a pair of enantiomeric conformers. When dissolved in achiral solvents, the pigment forms a racemic mixture of rapidly interconverting * Corresponding author. † Poznan ´ University of Technology, Poznan´. ‡ Universita ¨ t Kaiserslautern. § University of Nevada, Reno. ⊥ Technische Universita ¨ t Berlin. X Abstract published in AdVance ACS Abstracts, June 15, 1996.
S0022-3654(96)00508-4 CCC: $12.00
conformational enantiomers at room temperature. This equilibrium of enantiomers, however, can be shifted toward one or the other conformer either by addition of chiral agents,10-12 complexing with protein,13-15 or through methyl substitution in the propionic acid side chains,5,6,16 which makes evident the optical activity of the pigment. In this paper we study some properties of (βR,β′R)-dimethylmesobilirubin-XIIIR (1) dissolved in a nematic liquid crystal by means of UV-vis absorption and circular dichroism (CD) spectroscopy. The polarized absorption spectra of a solute in oriented anisotropic matrices provide information about alignment of molecules, order parameters, and polarization of electronic bands,17 whereas the measurement of the CD of oriented molecules (ACD ) CD of anisotropic samples) can be of further help for a quantitative analysis of molecular spectroscopic properties.18,19 2. Theoretical Background 2.1. Linear Dichroism. The molecular decadic absorption coefficients for light polarized parallel (1) and perpendicular (2) to the optical axis of a uniaxial sample are given by18,20 + 1 ) ∑ gij33ij ) ∑ aij2 g* jj33ii
(1a)
+ 2 ) ∑ gij11ij ) ∑ aij2 g* jj11ii
(1b)
i,j
i,j
i,j
i,j
ii+ are the absorption coefficients for light beams polarized linearly parallel to the x+ i axes in a completely oriented system. axes are the principal axes of the absorption tensor The x+ i ij.20-22 aij are the elements of the orthogonal matrix which + transforms the x* i coordinates into the xi coordinates. The coordinates x*i refer to the common principal axes of the order tensors gijkk (k ) 1, 2, 3) and the following convention © 1996 American Chemical Society
β,β′-Dimethylmesobilirubin-XIIIR
J. Phys. Chem., Vol. 100, No. 28, 1996 11547
22 with for numbering the axes is used: g* 3333 g g* 2233 g g* 1133 x*3 as the “orientation axis”. For a uniaxial system the orientational distribution coefficients gijkl are defined as follows:
gijkl )
1 ∫ f(β,γ) aik(R,β,γ) ajl(R,β,γ) sin β dR dβ dγ (2) 8π2
R, β, γ are the Eulerian angles. f(β,γ) is the R-independent orientational distribution function of the uniaxial system with the optical axis parallel to the space-filled x′3 axis. Here aij(R,β,γ) are the elements of the orthogonal transformation matrix from the space-fixed x′i to the molecule fixed xi coordinate system. The ii+ are related to the absorption coefficient of the isotropic solution by
) 1/3(1 + 22) ) 1/3 ∑ ii+
(3)
i
If there is no temperature dependence of the orientation of the x* i axes relative to the molecular frame, an appropriate description of the anisotropic absorption, the linear dichroism 1 - 2, is given by18
x3 2 - ) S* + 1 - 2 ) (* (* - * 11)D* 3 33 2 22
D* )
(5)
x3 x3 (2g* (g* - g* 2233 + g* 3333 - 1) ) 1133) (6) 2 2 2233
where S* characterizes the orientational order of the x* 3 axis with respect to the optical axis of the uniaxial phase and is the mean value of a function of the angle β and the parameter D* is a measure of the deviation from a rotationally symmetric distribution of the molecules about their x*3 axis and is the mean value of a function of β and γ. Dividing eq 4 by 3 yields the following relation between the degree of anisotropy (R) and the order parameters, S* and D*:
R)
∆ ) L - R ) 1/3 ∑ ∆ii
(4)
because in this case the * ii are temperature independent. S* and D* are Saupe’s order parameters,23 which are sufficient to characterize the long-range orientational order of a uniaxial phase with respect to the absorption process if the phase is characterized by a molecular biaxial orientational distribution. They are defined by
S* ) 1/2(3g* 3333 - 1)
spectra 1(νj) and 2(νj). This was obtained from the temperature dependence of the degree of anisotropy.21,22,24,25 In the present paper, the above-mentioned method is extended to molecules of low symmetry and is applied to a case where only one x* i principal axis is fixed by symmetry. It can also be applied to the case of two bands of mixed polarization with all x* i axes being symmetry-fixed. The “vanishing of the spectral features”, conditions similar to those introduced by Michl and Thulstrup,17 provide us with an additional information; i.e., that, as a spectroscopic rule, the condition * ii g 0 must be fulfilled. Using computer simulation with a proper fitting procedure, we were able to obtain various q* ii values for the chosen absorption bands, in addition to the order parameters S* and D*, and to decompose the absorption spectra into absorption tensor coordinates with respect to the x* i axes (* 11, * 22, * 33 reduced spectra17). For the best fit, the sum of squares of the deviations of the experimental R values from those calculated should be minimal. 2.2. ACD Spectroscopy. From the theoretical point of view, the difference between the molar extinction coefficients of lefthanded (L) and right-handed (R) circularly polarized light is the sum of three quantities:
1 - 2 1 x3 ) (3q* (q* - q* 33 - 1) S* + 11) D* (7) 3 2 2 22
where
q*ii ) * ii/3; 0 e q* ii e 1, ∑ q* ii ) 1 i
In order to obtain information about the order parameters and the directions of the transition moments responsible for the given absorption band from the linear dichroism, we used the method described in detail previously.21,22,24,25 This method was elaborated for the case where one purely polarized band (at νj1) and another band (at νj2) with mixed polarization exist in the absorption spectrum, for molecules with x+ i coordinate i and x* systems coinciding by symmetry. Thus, it is not directly applicable to molecules with, e.g., the point symmetry group C2. In the method presented previously, only one further piece of information was needed in addition to the polarized absorption
(8)
i
where ∆ii are the diagonal elements of the circular dichroism tensor ∆ij.18 For ∆ii the index i indicates the direction of the light propagation whereas for the absorption tensor elements ii+ in eq 3 it gives the direction of the light polarization. The values of each ∆ii are a new possible source of information about molecular properties of the substance investigated.26 Moreover, ∆ii can be separated into two parts: an electric dipole-magnetic dipole (∆µm ii ) and an electric di22,27,28 This depole-electric quadrupole (∆µQ ii ) contribution. composition depends, except for special cases, on the choice of the origin of the xi coordinate system. With a line-shape spectral function Gon(νj) of the transition |o〉 f |n〉 we obtain28 µQ ∆11 ) ∆µm 11 + ∆11
)
Bνj 2
(9)
∑n Im{〈µ2〉on〈m2〉no + 〈µ3〉on〈m3〉no}Gon(νj) + Bνj
ωno[-〈µ2〉on〈Q31〉no + 〈µ3〉on〈Q21〉no]Gon(νj) ∑ 2c n
where
B)
32π3NA 103hc ln 10
NA is Avogadro’s number, h is Planck’s constant, c is the speed of light in vacuo, and νj is the wavenumber. ∆22 and ∆33 are derived by cyclic permutation of the indices. 〈µi〉on, 〈mi〉no, and 〈Qij〉no are coordinates of the electric dipole, magnetic dipole, and electric quadrupole transition moments, respectively. ωon ) (En - Eo)/p, where En and Eo are the energies of the states |n〉 and |o〉. In an isotropic solution, the second term in eq 9 averages to zero, and the contribution ∆iiµQ cannot be deterµQ mined. Both contributions, ∆µm ii and ∆ii can be obtained 17,18,26,28 separately only from ACD spectroscopy. For the ACD of low-symmetry molecules where the orientation of the principal axes of the order tensor gij33 relative to the
11548 J. Phys. Chem., Vol. 100, No. 28, 1996
Bauman et al.
molecular frame is temperature-dependent, one may use the general formula
∆A ) ∑gij33∆ij ) ∑aii2g* jj33∆° ii i,j
(10)
i,j
∆A means here the circular dichroism of the anisotropic sample. In eq 10 the light beam is assumed to propagate parallel to the optical axis (x′3). The eigenvalues ∆°ii describe the molecular property ∆ij in its system of principal axes (x°i). For these quantities eq 8 also holds. aij are here the elements of the matrix which transforms the x*i coordinates into the x°i coordinates. If there is no or only a negligible temperature dependence of the orientation of the principal axes (x* i) relative to the molecular frame, it is of advantage to use instead of eq 10 the relation18
be interpreted on the basis of the exciton coupling model.29,30 The applicability depends on the coupling strength and the relative orientation of the chromophoric systems.29 Let us consider a molecule consisting of two identical chromophores m and n with strongly allowed ππ* transitions. The interaction between the chromophores splits the energy level of the excited state into two levels with higher and lower energy relative to the undisturbed excited state. The excitation to these levels generates the bisignate CD spectrum with oppositely signed Cotton effects (“couplet”). If the interaction between both chromophores is small, perturbation theory30 yields the following energy values of the two excited molecular states, R and β, arising from the exciton splitting of the undisturbed localized transition |N〉 f |K〉 in both chromophores: R
1 (∆* ∆ - ∆ ) (∆* 33 - ∆)S* + 22 - ∆* 11)D* x3 A
(11)
because in this case the ∆* ii are temperature independent. In order to eliminate the frequency dependence, for the electronic transition |N〉 f |K〉 one may calculate the anisotropic rotational strength RA related to ∆A by18,26
R ) A
3 B
∫band
∆A(νj) νj
dνj ) 3 ∑gij33RijNK
(12)
i,j
D )
12
∆ij(νj) 1 ) ∫band dνj B νj
B
∫band
A(νj)
(13)
dνj ) 3 ∑ gij33DijNK
νj
(14)
i,j
where
DijNK )
ij(νj) 4 dνj ∫ band B νj
(15)
is the transition moment tensor. For an isotropic solution, gij33 ) 1/3 δij, and RA becomes the rotational strength
RNK ) ∑RiiNK ) i
3
∫ B band
∆(νj) νj
dνj
(16)
dνj
(17)
Similarly,
DNK ) ∑DiiNK ) i
12
∫ B band
(νj) νj
The decomposition of RiiNK corresponding to eq 9 yields
RNK 11
)
Rµm 11
+
RµQ 11
mn is the excitation energy from the ground (N) to the where ENK mn is the excited (K) state of the chromophores and ∆ENK dipole-dipole interaction energy between the two transition mn is the exciton (Davydov) splitting energy. moments. 2∆ENK mn ∆ENK can be calculated in the point dipole approximation according to ref 30 by
mn m n ) [〈µ b〉NK 〈µ b〉NK ∆ENK -2
is the tensor of rotation. Analogously to eq 12 the anisotropic dipole strength of the electronic transition |N〉 f |K〉 is given by A
(19)
m n 3|R Bmn| (〈µ b〉NK B Rmn)(〈µ b〉NK B Rmn)]|R Bmn|-3 (20)
where 3/B ) 22.96 × 10-40 in cgs units and
RijNK
mn mn Eβ ) ENK - ∆ENK
m n and 〈µ b〉NK are the electric dipole transition where 〈µ b〉NK moments of the chromophores m and n, respectively, and B Rmn is the interchromophoric distance vector connecting the centers of the groups m and n. In general, in absorption spectra in the case of exciton coupling only one maximum is observed as a result of the excitation to the R and β states. The band is broadened in comparison to the bands of the single fragments and the absorption coefficient is approximately twice as large as that of a single fragment, so that the dipole strength is approximately doubled. The spectrum of the chromophores in the undisturbed state is not always known; it can be estimated from the spectrum of a compound similar to the interacting component chromophores. On the basis of absorption measurements it is difficult to state whether there is exciton coupling or not. A possible aid may be the polarized absorption spectrum because the polarization for transitions to R and β states is different. In combination with the CD spectrum the existence of exciton coupling can be determined with more certainty because a bisignate CD curve should not very often appear accidentally. The appearance of the “couplet” shows the excited state interaction in weakly coupled electronic systems. Furthermore, it has been demonstrated5,6,10-16,29,30 that the sign of the first (long-wavelength) and second (short-wavelength) Cotton effect may often be used to determine the absolute configuration of a chiral arrangement of two chromophores causing exciton coupling (exciton chirality method30).
3. Experimental Section
(18)
1 ) (Im {〈µ2〉NK〈m2〉KN + 〈µ3〉NK〈m3〉KN} - 〈µ2〉NK〈Q31〉KN + 2 〈µ3〉NK〈Q21〉KN) NK RNK 22 and R33 are derived by cyclic permutation of the indices. 2.3. Exciton Coupling. The UV-vis absorption and CD spectra of molecules with two identical or similar groups can
Figure 1 shows the molecular structures of (βR,β′R)-dimethylmesobilirubin-XIIIR (1) in linear and folded conformation5,6 as well as of dipyrrinone 2 and 3. We investigated both enantiomers of β,β′-dimethylmesobilirubin-XIIIR: (βR, β′R) and (βS, β′S). For measuring the UV-vis absorption and ACD spectra the compounds were dissolved in the liquid crystal ZLI 1695 (Merck, Darmstadt), which is a mixture of four 4-n-alkyl-4′cyanobicyclohexanes and has a nematic phase in the temperature
β,β′-Dimethylmesobilirubin-XIIIR
J. Phys. Chem., Vol. 100, No. 28, 1996 11549 direction of the light beam into six positions differing by 30°. If there is no or only a small difference between the ∆A values for these six positions, the mean value can be used to evaluate further data. In order to check for possible decomposition of the substance during the long experimental time period, comparisons were made, between ∆ at T ) 80 °C (isotropic phase) measured before and after measuring ∆A (six times for each temperature). For both the polarized absorption and ACD measurements, the baseline for pure ZLI 1695 at the same positions of the cell were recorded. To control the temperature of the samples, the temperature units were calibrated with a Pt100 resistor. The temperature was measured with an accuracy of (0.5°. 4. Results
Figure 1. Molecular structures of (βR,β′R)-dimethylmesobilirubinXIIIR (1) in linear (a) and folded (b) conformation and of dipyrrinone (2, 3) (c).
range from 13 to 72 °C. The concentration of 1 in ZLI 1695 was about 1.5 × 10-3 mol/L and that of 2 and 3 was about 2.4 × 10-3 mol/L. The polarized UV-vis absorption spectra were recorded using a Varian Cary 2200 spectrophotometer equipped with Glan polarizers. The measurements were carried out in a “sandwich” cell of 0.01 cm thickness. The planar homogeneous orientation of the liquid crystal and solute molecules was achieved by applying an ac field of about E ) 1.5 × 106 V/m parallel to the surfaces of the plates of the cell. The absorption of the pigments dissolved in ZLI 1695 was recorded at four orientations of the cell: the optical axis of the liquid crystal sample was oriented at the angles 0°, 90°, 180°, and 270° with respect to the polarization plane and perpendicular to the propagation direction of the light beam. 1 and 2 were obtained as the average values from the measurements at the angles 0°, 180° and 90°, 270°, respectively. For ACD measurements, the (βR,β′R)-dimethylmesobilirubinXIIIR-liquid crystal mixture was filled into a special cuvette,31 where the chiral induction caused by solving chiral molecules in nematic phases is compensated by an electric ac field of about E ) 5 × 106 V/m. The sample was adjusted at the CD spectrophotometer Jobin Yvon Mark IV. The optical axis of the uniaxial nematic liquid crystal phase was chosen to be parallel to the propagation direction of the light beam. In order to ensure that the results are not falsified by linear birefringence and dichroism artifacts, various ∆A measurements were performed with the sample being rotated about the propagation
4.1. Spectral Properties. Table 1 includes the CD and UVvis experimental spectral data of (βR,β′R)-dimethylmesobilirubin-XIIIR (1) in the isotropic phase of ZLI 1695 at T ) 80 °C and the data for this pigment in some other solvents, taken from refs 5, 32, and 33. It is seen that the amplitudes ∆(νjmax) ∆(νjmin) of the Cotton effects in the CD spectrum of 1 dissolved in the liquid crystal are larger than those in the case of other solvents even at elevated temperatures. Measurements of the polarized absorption and ACD were accomplished at various temperatures in the range from 28 to 65 °C. We have found that the addition of (βR,β′R)-dimethylmesobilirubin-XIIIR (1) or dipyrrinones 2 and 3 to ZLI 1695 at a concentration used in our experiment does not change the nematic-isotropic phase transition temperature of the liquid crystal. Thus, the reduced temperature TR ) T/TNI (T is the temperature of the measurement and TNI is the clearing point temperature in K) is the same for both ZLI 1695 and ZLI 1695 doped with the substance investigated. Therefore, in the following we can use the temperature in °C instead of the reduced temperature values. Figures 2 and 3 show the absorption spectra of β,β′dimethylmesobilirubin-XIIIR (1) and dipyrrinone (3), respectively, dissolved in ZLI 1695. The spectra of 2 are approximately equal to those of 3. In Figures 2a and 3a the polarized absorption spectra (1 and 2) at T ) 28 and 65 °C are presented, whereas Figures 2b and 3b give the absorption spectra in the isotropic phase (T ) 80 °C). From the latter spectra the halfbandwidth of the long-wavelength absorption band of 1 was determined to be ∆νj1/2 ) 4125 cm-1, whereas that of 3 is ∆νj1/2 ) 3425 cm-1. The dipole strengths D are 85.9 × 10-36 and 49.8 × 10-36 cgs, respectively. Additionally, in Figures 2b and 3b the degree of anisotropy R at three different temperatures is shown as a function of the wavenumber. A distinct difference between the degree of anisotropy of 1 and that of 3 is observed which indicates qualitatively a change in the spectroscopic properties as well as different kinds of orientation of these two molecules. Figures 4 and 5 present the results obtained from ACD measurements of 1 in ZLI 1695 at three different temperatures.
TABLE 1: Circular Dichroism and Ultraviolet-Visible Spectral Data for (βR,β′R)-Dimethylmesobilirubin-XIIIr (1) in Various Solvents
solvent
dielectric const
∆(νjmax)/ L mol-1 cm-1 (νjmax/103 cm-1)
νj0/103 cm-1 (∆ ) 0)
∆(νjmin)/ L mol-1 cm-1 (νjmin/103 cm-1)
(νjmax)/ 103 L mol-1 cm-1 (νjmax/103 cm-1)
ZLI 1695a THFb ethanolb acetonitrileb
5.5c 7.3d 24.3d 36.2d
+318 (22.62) +256 (23.10) +206 (23.15) +232 (23.36)
24.16 24.57 24.63 24.81
-184 (25.33) -145 (25.72) -133 (25.77) -137 (26.04)
52.49 (23.15) 56.00 (23.20) 56.20 (23.47) 53.70 (23.64)
At T ) 80 °C, c ≈ 1.5 × 10-3 mol/L. b At T ) 22 °C, c ≈ 1.3 × 10-3 mol/L. c Merck catalogue. d Gordon, A. J.; Ford, R. A. The Chemist’s Companion; Wiley: New York, 1972. a
11550 J. Phys. Chem., Vol. 100, No. 28, 1996
Bauman et al.
Figure 4. ∆A of (βR,β′R)-dimethylmesobilirubin-XIIIR in ZLI 1695 at T ) 28 °C (2), T ) 43 °C (b), and T ) 65 °C (9).
Figure 2. Absorption spectra of (βR,β′R)-dimethylmesobilirubin-XIIIR (1) in ZLI 1695: (a, top) polarized absorption spectrum 1 at T ) 28 °C (2) and 65 °C (9) and 2 at T ) 28 °C, (4) and 65 °C (0). (b, bottom) Absorption spectrum of (T ) 80 °C) ([) and degree of anisotropy R at T ) 28 °C (2), T ) 43 °C (b), and T ) 65 °C (9).
Figure 5. ∆A - ∆ of (βR,β′R)-dimethylmesobilirubin-XIIIR in ZLI 1695 at T ) 28 °C (2), T ) 43 °C (b), and T ) 65 °C (9).
TABLE 2: Relative Absorption Coefficients q* ii for the Three Absorption Bands of (βR,β′R)-Dimethylmesobilirubin-XIIIr (1) and of Dipyrrinone (3) in ZLI 1695 substance
νj/103 cm-1
q*11
q* 22
q* 33
(βR,β′R)-dimethylmesobilirubin-XIIIR (1)
νj1 ) 41.07 νj1′ ) 34.52 νj2 ) 25.91 νj1 ) 24.51 νj1′ ) 34.52 νj2 ) 42.74
0.21 0.15 0.27 0.14 -0.11 ≈ 0a 0.00
0.00 0.11 0.57 0.16 0.51 0.62
0.79 0.74 0.16 0.70 0.60 0.38
dipyrrinone (3)
a
Figure 3. Absorption spectra of dipyrrinone (3) in ZLI 1695: (a, top) polarized absorption spectrum 1 at T ) 28 °C (2) and 65 °C (9) and 2 at T ) 28 °C (4) and 65 °C (0). (b, bottom) Absorption spectrum (T ) 80 °C) ([) and degree of anisotropy R at T ) 28 °C (2), T ) 43 °C (b), and T ) 65 °C (9).
In Figure 4 the values of ∆A are shown whereas Figure 5 illustrates the difference between the circular dichroism in the
From fitting procedure q* 11 ) -0.11; error see ref 34.
nematic (∆A) and isotropic (∆) phases of the liquid crystal. It is seen that this difference is relatively small and diminishes with increasing temperature. 4.2. Qualitative Estimation of the Polarization of the Absorption Bands. In order to determine the diagonal elements of the absorption tensor (relative absorption coefficients q* ii) with respect to the x*i axes and next the order parameters S* and D*, two values of the degree of anisotropy at the wavenumbers νj1 or νj1′ and νj2 are selected; i.e., the degrees of anisotropy for the wavenumbers νj1 and νj1′ are used alternatively for the evaluation of the order parameters and the reduced spectra. In the case of molecular symmetry the number of necessary parameters q*ii(νj) is reduced as shown earlier. For molecules without symmetry a qualitative estimation of the transition moment directions with respect to the x* i axes may provide an information which is helpful for the quantitative analysis. The values of νj1, νj1′, and νj2 for two substances investigated in ZLI 1695 are listed in Table 2. Then, the dependence of R(νj1) and R(νj2) on temperature is analyzed for the assignment of the transition moment directions with respect to the x*i axes as shown in Figure 6 for 1. The three criteria are of importance here:21,22 (1) the position of points in the R(νj1),
β,β′-Dimethylmesobilirubin-XIIIR
J. Phys. Chem., Vol. 100, No. 28, 1996 11551
Figure 6. R(νj2 ) 25.91 × 103 cm-1) vs R(νj1 ) 41.07 × 103 cm-1) (3) as a function of temperature for (βR,β′R)-dimethylmesobilirubinXIIIR (1).
R(νj2) plane, (2) the slope of the curve R(νj2,T) ) Φ(R(νj1,T)), and (3) its curvature. The relation R(νj2,T) ) Φ(R(νj1,T)) can be derived from eq 7 assuming D* to be a function of S* as calculated, e.g., in the mean field theory.35 Starting with the assumption of a point symmetry group C2 for 1, the transitions are polarized either parallel to the C2 rotation axis or in a plane perpendicular to it. Using additionally the criteria (1) to (3), we have concluded that the transition moment connected with the absorption band at νj1 ) 41.07 × 103 cm-1 and also νj1′ ) 34.52 × 103 cm-1 of 1 lies in the x*1, x* 3 plane, mainly in the x* 3 direction (q* 33 > j2 ) 25.91 × 103 cm-1 q*11 * 0; q* 22 ) 0), whereas the band at ν is polarized in the x* 2 direction (parallel to the C2 axis) but j2) (i ) 1, 2, 3) because of overlapping bands at νj2 all three q* ii(ν are different from zero. The dipyrrinone fragment (3) is approximately planar,36 and thus a point symmetry group Cs may be assumed. The transition moment responsible for the long-wavelength absorption band at νj1 ) 24.51 × 103 cm-1 and also at νj1′ is mainly directed j1) > q* j1)) whereas for the band at νj2 along the x* 3 axis (q* 33(ν 22(ν ) 42.74 × 103 cm-1 q* (ν j ) > q* j1). The molecular plane is 22 1 33(ν j1), q* j1′), and q* j2) are the x*2, x* 3 plane. Furthermore, q* 11(ν 11(ν 11(ν equal to zero within the experimental error. 4.3. Quantitative Determination of the Reduced Absorption Spectra by Computer Simulation. The values of the q*ii, S*, and δ in D* ) f(S*,δ) can be estimated using computer simulation. δ is the ratio of two potential parameters from the mean field theory. Assuming this ratio to be temperature independent relates the order parameters S* and D* by the function given in ref 35. The relation between the degrees of anisotropy (eq 7) measured at two different wavenumbers21,22,24,25
R(νj2, T) ) Φ(R(νj1, T))
(21)
is calculated varying the order parameter S* within 0 e S* e 1 or for a smaller interval S* 1 e S* e S* 2 with an arbitrarily chosen set of parameters q*ii(νj1), q*ii(νj2) for i ) 1 or 2, or 3, and also with an arbitrary δ value. The sum of the squares of the vertical distances of the experimental points from this calculated curve for all S* values is evaluated and taken as a measure for the quality of the approximation by the chosen parameter set. j1), q* j2), and δ are varied Subsequently, all parameters q* ii(ν ii(ν within their limits. Those sets for which the sum of squares is within the experimental error are possible solutions of the fitting j1) and q* j2) which yield the smallest procedure. With the q* ii(ν ii(ν sum of quadratic deviations, the order parameters S* and D* are calculated by eq 7 from the experimental R(νj1, T), R(νj2, T) values for each temperature. Then, by a multiple regression for R(S*,D*,qii(νjk)), k ) 1, 2 with the variables S*, D* the jj) are calculated for all wavenumbers νjj of the spectrum. q* ii(ν jj), S*, D* which These resulting sets of parameters q* ii(ν
Figure 7. Reduced absorption spectra * 11 (2), * 22 (1), * 33 (b), and *11 + *33 (9) of (βR,β′R)-dimethylmesobilirubin-XIIIR (1) in ZLI 1695.
TABLE 3: Order Parameters S* and D* for (βR,β′R)-Dimethylmesobilirubin-XIIIr (1) and Dipyrrinone (3) in ZLI 1695 at Different Temperatures (βR,β′R)-dimethylmesobilirubin-XIIIR (1)
dipyrrinone (3)
S* D* S* D* T/°C (∆S* ) (0.01) (∆D* ) (0.02) (∆S* ) (0.01) (∆D* ) (0.02) 28 33 38 43 48 58 65
0.51 0.48 0.47 0.42 0.41 0.32 0.26
0.07 0.07 0.08 0.04 0.07 0.05 0.08
0.65 0.62 0.60 0.55 0.47 0.41
0.08 0.10 0.07 0.11 0.11 0.10
describe the experimental curve within the experimental error can be further differentiated by using the “vanishing of spectral features” conditions * ii g 0 in the whole spectral region. That j1), q*ii it makes sense to use all spectroscopic parameters q* ii(ν (νj2) and also δ follows from the fact that R(νj2,T) ) Φ(R(νj1;T)) is a sufficiently complicated function, in general. j1 and Table 2 gathers the relative absorption coefficients q* ii at ν νj2 received from the procedure given above for both compounds investigated. jj) allows us to obtain the reduced The knowledge of the q* ii(ν absorption spectra * ii which are shown for 1 in ZLI 1695 in NK (NK ) R, β) of Figure 7. The values of the coordinates D* ii the transition moment tensor for the long-wavelength absorption band of the pigment have been calculated by eqs 15 and 17: D*11R ) 24.3 × 10-36 cgs; D*22β ) 43.4 × 10-36 cgs; D*33R ) 17.5 × 10-36 cgs. The dipole strength obtained from the reduced spectra is thus DNK ) 85.2 × 10-36 cgs which is in good agreement with the value calculated from the UV-vis absorption spectrum with eq 17 for T ) 80 °C (Figure 2b). The maximum of the IR band at νjR ) 22.83 × 103 cm-1 and the Iβ band at νjβ ) 24.38 × 103 cm-1 leads to an exciton splitting of mn ) ∆νjE ) 1.55 × 103 cm-1. The center of both 2(hc)-1∆ENK bands is at νjc ) 23.39 × 103 cm-1 and is shifted by 1.15 × 103 cm-1 with respect to the maximum of the dipyrrinone band. In Table 3 the values of the order parameters S* and D* calculated from eq 7 for two compounds oriented in ZLI 1695 are listed. Three independent measuring series have been carried out and the results presented in Table 3 are average values. 5. Discussion 5.1. Linear Dichroism Spectra. 5.1.1. Phenomenological Comparison of the Polarized UV Spectra of 1 and 3. From the results presented in Table 3 it is seen that the order parameter S* for the two compounds oriented in ZLI 1695 depends strongly on temperature whereas the biaxiality parameter D* is nearly temperature independent (taking into account the uncer-
11552 J. Phys. Chem., Vol. 100, No. 28, 1996 tainty in the calculations). No drastic difference between the S* and D* values for 1 and those for the dipyrrinone fragment 3 is observed. However, the degrees of anisotropy R differ. This leads to the conclusion that the directions of the orientation axes (x*3) or, in general, of all x* i axes, are very different with the two compounds. This is not an unexpected result taking into account the molecular structure and keeping in mind the folded conformation of 1. It has to be mentioned here that the order parameters of 3 are mean values for a mixture of various conformers resulting mostly from the region of the flexible chain. This seems not to be important for the dipyrrinone as the molecules 2 and 3 have nearly equal S*, D* values in spite of their different chain lengths. From the values of the degree of anisotropy (Figures 2b and 3b) and of the q*ii (Table 2) there follows that the polarization of the absorption bands of the pigment 1 is quite different from that of the dipyrrinone fragment in spite of the fact that the absorption bands of 1 are built up from those of the fragment 3. This shows again the effect of the exciton coupling and the fact that the orientation axes (x* 3) or, in general, the x* i axes are oriented quite differently in the two molecules. With 1 both short-wavelength bands (at νj ) 34.52 × 103 cm-1 and νj ) 42.07 × 103 cm-1) are polarized nearly parallel to each other. They are polarized in the x*1, x*3 plane with a significant dominance of the x*3 axis polarization. In the case of 3 the band at νj ) 34.52 × 103 cm-1 has a polarization (q* 11 ) 0, q*22 ) 0.51, q*33 ) 0.60 within the experimental error) quite different from that of the band at νj ) 42.74 × 103 cm-1 which may be of mixed polarization with a strongly prevalent contribution along the x*2 axis. The different direction of the transition moments of these two bands is an unexpected result when compared to the polarized absorption of 1 and can only be understood taking into account the folded structure of 1. A quantitative interpretation will be given in the following section. The long-wavelength absorption band of 3 at νj ) 24.51 × 103 cm-1 is polarized nearly parallel to the band at νj ) 34.52 × 103 cm-1 and the transition moment is directed approximately along the x* 3 axis. For 1, two bands with various polarizations can be distinctly recognized in the long-wavelength region on the basis of the reduced spectra (Figure 7): the first band (at longer wavelengths) is polarized in the x* 1, x* 3 plane whereas the second one (at shorter wavelengths) results as nearly purely polarized in the x*2 direction. This is a consequence of the point symmetry group C2 assumed for 1 which dictates that one electronic transition must be polarized along the symmetry axis (A f A transition) and the second one perpendicular to this axis (A f B transition). The angles of the transition moment directions against the principal axes of the order tensor are known from the q* ii only up to their sign, i.e., the direction of the first (νj ) 25.91 × 103 cm-1) transition moment is (48°, that of the second one (νj ) 34.52 × 103 cm-1) is (22° to (26°, and that of the third (νj ) 41.07 × 103 cm-1) transition moment is (26° to (28° with respect to the x*3 axis in the x* 1, x* 3 plane. The second band is polarized parallel to the x* 2 direction. 5.1.2. Calculation of the Polarized Spectra of 1 by Means of the Exciton Theory. The two dipyrrinone residues m and n of 1 are identical units in case of the assumed C2 symmetry. If they produce only a small mutual perturbation, we may treat 1 by means of the exciton coupling model.29,30 The absolute values of the electric dipole transition moments of the two fragments are assumed to be equal and the absorption intensity of 1 should be twice that of dipyrrinone. Because the dipole strengths of 1 for the long-wavelength absorption as well as for the bands at about 34 × 103 and 41 × 103 cm-1 are twice
Bauman et al.
m Figure 8. Orientation of the dipole transition moments 〈µ b〉NK and n 〈µ b〉NK with respect to the central carbon atom C10 in the moleculefixed xi coordinate system. The x1 axis is perpendicular to the plane spanned by the C10-C9 and the C10-C11 bond directions. The x2 axis lies within this plane and is the bisector of the angle C9-C10C11. H-C-H defines the x1, x2 plane. For C2 symmetry: η′ ) 360° - η, φ ) φ′, and γ ) 360° - γ°. xˆ , xˆ ′ are unit vectors parallel to m n 〈µ〉NK , 〈µ〉NK , respectively.
as large as the dipole strengths of 3 within about 20% (DNK(1) ≈ 1.73DNK(3)), this presupposition is approximately fulfilled for 1. The theory of exciton coupling30 yields in a simple approximation for the electric dipole transition moments of the exciton transitions to the states R and β the difference and the sum of the electric dipole transition moments of the fragments m and n, respectively:
〈µ b〉βR )
1 m n - 〈µ b〉NK ) (〈µ b〉NK x2
(22)
The - and the + sign in eq 22 correspond to the R and β transition, respectively. With the two identical fragments m and n in 1, these transition moments of the exciton bands are then given in the coordinate system of Figure 8 by
〈µ b〉R )
x
〈µ b〉β )
x
DNK {-2 sin φ sin γ, 0, 2 cos η cos φ sin γ + 2 2 sin η cos γ} (23) DNK {0, -2 sin η cos φ sin γ + 2 cos η cos γ, 0} 2 (24)
(where {...} indicates a column vector) respectively, where DNK m 2 n 2 ) DNK,m ) |〈µ b〉NK | ) DNK,n ) |〈µ b〉NK | . η is the angle between the C10-C11 bond direction and the molecule-fixed x2 axis (C2 symmetry axis) shown in Figure 8. A vector ˆt parallel to the C10-C11 bond direction together with the m in Figure 8 span the plane of transition moment vector 〈µ b〉NK m and the dipyrrinone fragment m. γ is the angle between 〈µ b〉NK the C10-C11 bond direction and φ the angle of rotation of the molecular plane of the dipyrrinone fragment m about the C10C11 bond direction (torsion angle). The angles η′, γ′, and φ′ for the fragment n are defined analogously: η′ ) 360° - η is the angle between the C10-C9 bond direction (tˆ′) and the x2 axis, and γ′ ) 360° - γ is the angle between the dipole n and the C10-C9 bond direction. transition moment 〈µ b〉NK Furthermore, for a molecule of symmetry group C2 as assumed for 1, φ′ ) φ. φ ) φ′ ) 0 is the situation with the carbonyl group of the dipyrrinone fragments being oriented towards the x2 axis. The angle Θ between the planes of the two dipyrrinone fragments of 1 (Figure 1b) is then given by
cos Θ ) 1 - 2 sin2 η sin2 φ
(25)
β,β′-Dimethylmesobilirubin-XIIIR
J. Phys. Chem., Vol. 100, No. 28, 1996 11553
m and and the angle ϑ between the transition moments 〈µ b〉NK n 〈µ b〉NK of the fragments m and n by
cos ϑ ) -1 + 2(cos γ cos η - sin γ sin η cos φ)2
DR11 + DR33 ) (〈µ1〉R)2 + (〈µ3〉R)2
(27)
Dβ22 ) (〈µ2〉β)2
(28)
For the relative absorption coefficients in the coordinate system of Figure 8 there follows R
R qβ ij
R
ij
)
)
∑k
R Dβ kk
R
〈µi〉β〈µj〉β
DNK(1 - cos ϑ)
(29)
In the coordinate system chosen, there are nondiagonal elements qijR different from zero. The experimentally determinable reduced spectra and thus the dipole strengths are always given with respect to the principal axes of the order tensor (x* i). Therefore, the values calculated by eqs 22-24 and eqs 27-29 have to be transformed to the x*i coordinate system. Because of symmetry the x* 2 and the x2 axis are identical; the orientation axis x* 3 lies in the x1, x3 plane. The transformation is given by
〈µ*1〉R ) 〈µ1〉R cos ψ + 〈µ3〉R sin ψ R
R
R
compd
transition
R 36 D* 11 × 10 R (* (ν j ) × max 11 10-3)
1
IR Iβ IIR IIIR I II III
24.3 (16.8) 0 (0) 1.2 (1.1) 4.8 (4.5) 7.5 (6.2) ∼0 (0) ∼0 (0)
(26)
Furthermore, the dipole strengths of both exciton bands (R, β) resulting from one transition in each fragment are given in the coordinate system of Figure 8 by
Dβ
TABLE 4: Dipole Strengths of the Exciton Bands Ir, Iβ, IIr, and IIIr of 1 and of the Bands I, II, and III of 3
(30)
〈µ* 3〉 ) -〈µ1〉 sin ψ + 〈µ3〉 cos ψ
(31)
β β 〈µ* 2〉 ) 〈µ2〉
(32)
where ψ is the angle enclosed by the orientation axis (x* 3) and the x3 axis. The dipole strengths of the molecule with two identical chromophores can be written in the new coordinate system as R NK D*11R + D* 33 ) D (1 - cos ϑ)
(33)
β NK D* 22 ) D (1 + cos ϑ)
(34)
The polarization direction of the exciton transition to the β state has to be parallel to the x2 axis (eqs 24 and 28) which coincides with the assumed C2 symmetry axis of the molecule. From eqs 23 and 27 there follows a polarization of the R band β β R in the x1, x3, i.e., x* 1, x* 3 plane. Thus, D22 ) D* 22 and D11 + R R R R R R D33 ) D*11 + D* 33 and, in general D11 * D* 11 and D33 * D*33R. 5.1.3. Determination of the Molecular Parameters. There are three transitions I, II, and III in 2 or 3 which can be used for the analysis of the spectra of 1. From these three transitions in each of the dipyrrinone fragments, six transitions IR, Iβ, IIR, IIβ, IIIR, and IIIβ appear in 1. In spite of the fact that the q*ii of the fragments 2 and 3 are known, the transition moment directions with respect to their molecular frame are unknown for I to III because of the unknown orientation of the orientation axes (x*3) in 2 and 3. For the determination of γi ) γ′i (i ) IR, IIR, IIIR) and the transition moment directions of IR, IIR, and IIIR in 1 and the γDi (i ) I, II, III) in 2 or 3, the orientation of the orientation axes with respect to the transition moment
3
R 36 D* 22 10 R (* (ν j )× max 22 10-3)
D*33R × 1036 R (* jmax) × 33 (ν 10-3)
0 (24.3) 43.4 (0) 0 (0) 0 (0) 6.0 (6.4) 3.4 (2.7) 6.4 (5.9)
17.5 (13.2) 0 (0) 7.2 (4.5) 20.0 (16.2) 35.2 (25.6) 3.4 (2.4) 4.3 (3.5)
directions for 1, 2, and 3, the reduced spectra of 1 (Figure 5) and 2 or 3,34 and results from the exciton theory can be taken into account. For the calculation of the properties of the exciton transitions of 1 the angles Θ ) 96.2° (Figure 1b) and η ) 57.2° determined from the X-ray data of bilirubin7 have been used. φ has been calculated by eq 25 to be φ ) 62.4° for 1 and is known for bilirubin and its derivates from molecular mechanics calculations5,32,33 to be φ ) 60°. The angle ϑ between the m n and 〈µ b〉NK of the dipyrrinone fragtransition moments 〈µ b〉NK ments (eq 26) can be determined by fitting the absorption bands IR and Iβ of 1 in the isotropic phase of ZLI 1695. Here the spectral function (νj) of dipyrrinone 3 is used taking into account the band shifts ∆νjs of the center of the exciton transitions IR and Iβ and the change of intensity (by a factor F) of the bilirubin absorption in comparison to the monomer (3) (Table 4):
(νj) ) F{R(νj) + β(νj) + (R(νj) - β(νj))cos ϑ} (35) The wavenumber dependence of the IR and Iβ band follows from the dipyrrinone band by substituting νj by νj - ∆νjE - ∆νjS in the case of IR and by νj + ∆νjE - ∆νjS in the case of Iβ mn is the splitting of the exciton states in where hc∆νjE ) 2∆ENK cm-1. Integration of eq 35 yields approximately R β R R NK ≈ (* jmax) + F ) (D* 11 + D* 22 + D* 33 )/D 11 (ν β R * jmax) + * jmax))/m(νjmax) (36) 22 (ν 33 (ν
m(νj) ) n(νj) is the molar decadic absorption coefficient of the fragments m and n. Furthermore, from eqs 33 and 34 there follows
V)
D*22β R D* 11 + D* 33
) R
β * jmax) 22 (ν 1 + cos ϑ ≈ R 1 - cos ϑ * (νj ) + * R(νj 11
max
33 β
)
max)
(νjmax) R(νjmax)
(37)
R R β 1 β with R ) 1/3(* 11 + * 33 ) and ) /3* 22 . The fitting of the UV spectrum (Figure 9a) yields cos ϑ ) -0.28 (ϑ ) 106°) and F ) 0.87 from which V ) 0.56 follows. The exciton splitting ∆νjE ) 1550 cm-1 has been taken from the band splitting of the reduced spectra (Figure 7). Here the shortwavelength maximum of *22β has been used. The shoulder must be considered as artificial. ∆νjS is obtained from the shift of the center of the splitting with respect to the position of the transition I in 2 and 3. For the CD spectra of IR and Iβ, i.e., for ∆R(νj) and ∆β(νj) the same wavenumber dependence as for the allowed UV band of 3 can be assumed. The spectrum ∆(νj) of the couplet is
11554 J. Phys. Chem., Vol. 100, No. 28, 1996
Bauman et al. strengths presented in Table 4 gives a hint that the contribution of the transition Iβ may be overestimated. This conjecture is based on the appearance of the small shoulder at the longwavelength side of the band which seems to be a consequence of an incomplete decomposition of the IR and Iβ band. From R(νjmax) ) 29.8 × 103 L mol-1 cm-1 and β(νjmax) ) 24.3 × 103 L mol-1 cm-1 (Figure 7, Table 4) there follows V ) 0.81. These values are of higher accuracy because of a more reliable band resolution in the region of the maximum of the absorption band. With ϑ ) 106° and 93° from the UV and the CD band of the isotropic solution, respectively, and with η ) 57.2 and φ ) 60° from molecular mechanics calculation,32,33 the transition moment direction of the band IR, i.e., γ, can be calculated by eq 26 to be γI(1) ) 113°, γI(2) ) 171°, and γI(3) ) 142°, respectively. With ϑ ) 93° from the CD band also no solution for γ exists with eq 39 because the resulting V ) 0.91 is too large. With the dipole strengths of the transitions of the fragment 2 or 3, the transition moment directions and the dipole strengths of the exciton transitions R(A f B) and β(A f A) can be calculated by eqs 23, 24, and 27, 28 as functions of γ. Thus, the ratio V can be evaluated for these transitions of 1 as a function of γ:
V(γ) )
Figure 9. (a, top) Fitting of the UV absorption of the IR and Iβ transition of 1 by use of the spectral function of the transition I of 3; experimental spectrum of 1 (‚‚‚), calculated spectrum (s); νjR ) 23.13 × 103 cm-1, νjβ ) 24.68 × 103 cm-1, ∆νj ) 1550 cm-1. The decrease of the mean intensity F of (νj) with respect to the absorption of dipyrrinone (3) yields F ) 0.87. The angle between the two transition m n and 〈µ b〉NK results to be ϑ ) 106°. (b, bottom) Fitting moments 〈µ b〉NK of the CD couplet of the exciton transitions IR and Iβ of 1 by use of the spectral function of the transition I of 3; experimental CD spectrum of 1 (‚‚‚), calculated CD spectrum (s); νjR ) 23.13 × 103 cm-1, νjβ ) 24.68 × 103 cm-1, ∆νj ) 1550 cm-1, the intensity 4RRf /DNK of the f calculated spectrum ∆(νj) with respect to the amplitude of couplet of m 1 is 0.0123 and the angle between the two transition moments 〈µ b〉NK n R and 〈µ b〉NK results to be ϑ ) 93°. The rotational strength is Rf ) - Rβf ) 13.3 × 10-38 cgs.
then given by
∆(νj) )
4RR (∆R(νj) - ∆β(νj) - (∆R(νj) + ∆β(νj)) cos ϑ) DNK (38)
RR is the rotational strength of the band R (eq 43; see also section 5.2). The best fit with respect to the amplitude of the couplet, as shown in Figure 9b, is obtained for 4RfR/DfNK ) 1.23 × 10-2 and cos ϑ ) -0.06 (ϑ ) 93°) from which V ) 0.91 follows. With the dipole strength of the dipyrrinone fragment in 1, there results DfNK ) FDNK ) 43 × 10-36 cgs, and the rotational strength RfR ) 13.3 × 10-38 cgs. The CD band as well as the UV band cannot be fitted very well at the long-wavelength side. This is a consequence of the band structure of the transition I of the dipyrrinones 2 and 3 (Figure 3) which has a smaller slope than the band IR/Iβ of 1. For this latter fact no interpretation can be given here. With the dipole strengths from the reduced spectra (Table 4) eq 37 yields the ratio V ) 1.04 ( 0.1 for the long-wavelength transitions IR(A f B) and Iβ(A f A). Because of the large bandwidth of the transition Iβ (Figure 7), the ratio of the dipole
Dβ22(γ) DR11(γ) + DR33(γ)
(39)
By comparing the calculated ratio V(γ) (eq 39) with the experimentally obtained ratio (eq 37) for IR and Iβ, IIR and IIβ, IIIR and IIIβ possible values γ ) γ(k) i (i ) I, II, III; k ) number for a special solution of eqs 26 and 39) can be found. With the dipole strengths of the transitions IR and Iβ, i.e., for V ) 1.04 ( 0.2 for the long-wavelength transitions IR(A f B) and Iβ(A f A), η ) 57.2° and φ ) 60° no solution for γ exists. This is a further indication for the overestimation of the intensity of the band IR from the integrated reduced spectra of 1. For the value V ) 0.81 from the absorption maxima of the bands IR and Iβ there are two solutions of eq 39: γI(4) ) 129° and γI(5) ) 155° which lie in the interval derived from the UV and CD spectra given above. From the corresponding data of the bands IIR and IIβ and IIIR and IIIβ there follows Dβ22 , DR11 + DR33 and thus the angles γ for both transitions lie within 54° ( 15°. Because V(γ) is nearly constant in the neighborhood of γ ) 54°, one has to assume an error of about (15° for both bands. From UV, CD, and the polarized spectroscopy, γ has been calculated to be 113° or 171°, 142°, and 129° or 155°, respectively. The measured splitting of the IR and Iβ band 2∆νjE ) 1550 cm-1 also allows the calculation of γ by eq 20 if the Rmn has been interchromophoric distance vector B Rmn is known. B chosen to be the vector between the centers of the straight lines drawn between the atoms C2-C8 and C12-C18 of the dipyrrinone fragments. From X-ray data of bilirubin7 the Bmn| ) 6.03 Å is absolute value is |R Bmn| ) 6.49 Å whereas |R obtained for η ) 57.2° and φ ) 60°, and the value from X-ray Bmn| ) 6 Å. With data Θ ) 96.2. Lightner et al.10 have used |R mn ) hc∆νjE from eq 20 the latter value there follows with ∆ENK the angle γ ) 116°. For solutions from the UV experiment and V(γ), i.e., for γI(2) ) 171° and γI(5) ) 155° the energy splitting is too large. This is a first indication to omit these mn is positive and values. In accordance with literature37 ∆ENK thus the R excited state of 1 corresponds to B symmetry (A f B transition) and β to A symmetry (A f A transition) as assumed for the evaluation of the reduced spectra. It is difficult to estimate the error of the five γ values determined from UV, CD, polarized spectroscopy and energy
β,β′-Dimethylmesobilirubin-XIIIR
J. Phys. Chem., Vol. 100, No. 28, 1996 11555 moment directions of Iβ, IIβ, and IIIβ are fixed by symmetry (x2 axis ) x*2 axis). Additional experimental information, the angles β* between the orientation axis (x*3) and the transition moment directions can also be calculated up to the sign from
x
tan β* ) (
R D* 11 R D* 33
≈(
x
R * 11 R * 33
(41)
Only for the long-wavelength transition IR(AfB) the ratio R R R R D* 11 /D* 33 ) 1.39 or the ratio * 11 /* 33 ) 1.27 of the maxima of the x*1 and x*3 polarized contribution of the IR band (Figure 7) are sufficiently accurate to allow the determination of the orientation of the orientation axis with respect to the xi R coordinate system (Figure 10a). From the ratio * 11 /* 33 for IR one gets β* ) 48°. For IIR, β* is approximately 22°-26° and for IIIR approximately 26°-28°. With β* ) 48° there follows
ψ ) β(γI) - β* )
Figure 10. (a, top) Transition moment directions of the bands IR, IIR, and IIIR in the x1, x3 plane, and the orientation of the orientation axis (x*3) of 1 with respect to the right-handed orthogonal xi coordinate system. The dipyrrinone fragments are placed above the plane of the paper. (b, bottom) Transition moment directions of the band I, II, and III and the orientation axis (x*3) in the molecular plane x1, x2 of 3. The shaded areas given in Figure 10a,b represent approximately the uncertainty of the estimated transition moment directions for IIR and IIIR resulting from the experimental findings that no couplet has been found experimentally for both bands.
splitting of the exciton bands IR/Iβ. Thus, for the following calculations a mean value γI ) 121° ( 20° is taken. The value from the CD measurement is also omitted here because the CD spectrum cannot be described with sufficient accuracy by the exciton coupling (see section 5.2). This value, taken into account, would increase the mean value of γ only by 9° which is within the experimental error. Furthermore, the values 155° and 171° can be excluded. [For γ ) 155° and 171° there follows from eq 40 β ) 29° and β ) 10°. These β values can be ruled out because the orientations (eq 41) of the corresponding orientation axes lead to contradictions.] With γI ) 121° ( 20° for the transition IR and γ ) 54° ( 15° for the transitions IIR and IIIR of 1 and the intensity ratio XR(AfB)/Xβ(AfA) of the exciton bands with X ) I, II, and III, the transition moment directions of the bands IR, IIR, and IIIR of 1 in the x1, x3 plane (Figure 8) and the ratio of the dipole strengths DR11(γ)/DR33(γ) can be calculated by the formula
tan β(γ) )
〈µ1〉R R
〈µ3〉
)-
sin φ sin γ ) cos η cos φ sin γ + sin η cos γ (
x
DR11(γ) DR33(γ)
(40)
Here, β(γi) (i ) I, II, III) is the angle between the transition moment direction of the transition IR, IIR, or IIIR, and the x3 axis, respectively (Figure 10a). With η ) 57.2° and φ ) 60° eq 40 leads to β(γI ) 121°) ) 75° for band I and to β(γII ) γIII ) 54°) ) -44° ( 10° for the bands II and III. Thus, the β(γi) values, i.e., the transition moment directions are known for IR, IIR, and IIIR; the transition
{
27° 123°
(42)
Further analysis and a check of the results can be obtained from the interesting fact that the degrees of anisotropy of both short-wavelength bands II and III of 2 and 3 at νjII ) 34.52 × 103 and νjIII ) 42.74 × 103 cm-1 are different (Figure 3b) whereas the corresponding bands of 1 (Figure 2b) are polarized almost parallel to each other. The origin of this effect must result from the facts that at first in 1 the short-wavelength transitions also yield exciton bands and, second, that the orientations of the orientation axes in 1 and 2 or 3 are very different. However, in CD no exciton couplets are found in the spectral region of the bands IIR,β and IIIR,β. This is R consistent with Rβ (γ) (eq 42) calculated with η ) 57.2°, φ ) 60°, and γ ) 54°. For γ ) γII ) γIII ≈ 54° a change of sign of a CD couplet occurs and thus the amplitude of couplets in these UV band with 50° e γ e 60° is too small to be measured. If we assume the transition moment directions of all transitions in the fragments m, n of 1 to be equal to those of dipyrrinone, then for the transitions I, II, and III in 2 and 3 D ) 54° ( 15°. γD ) 0° there follows γID ) 121° and γIID ) γIII means the direction of the CH3-C1 bond. From the ratios R *22R/* 33 of the transitions I to III of 3, the angles between the dipole transition moments and the orientation axis (x* 3) of 3 can be calculated up to the sign to be γ*I ) (27°, γ*II ) (47, and γ*III ) (52°. For the orientation axis there follows ψD ) 121° ( γ*I ) 94° or 148°. Because the bands II and III are polarized within the shaded area of Figure 10b (54° ( 15°), only γ ) 94° is an acceptable solution. The results for 3 in the discussion above are summarized in Figure 10b. In the same way, the transition moment directions of the bands II and III are obtained as well as the orientation axis (x*3) which are all in the x* 2,x* 3 plane given in Figure 10, a and b. According to these results, the apparently strange appearance of the degree of anisotropy of the bands IIR/β and IIIR/β in 1 and 3 (equal in 1 and different in 3) can be quantitatively explained by the exciton coupling model and the different principal axes of the order tensors in both molecules 1 and 3. 5.2. Circular Dichroism Spectra. The oppositely signed rotational strengths RR and Rβ of a chiral molecular exciton are given by30 R
(π2)E
Rβ ) (
)(
(π2)E
mn mn m R ‚(〈µ b〉NK NKB
mn mn R ‚(〈µ b〉R NKB
n × 〈µ b〉NK )
× 〈µ b〉β)
(43a)
(43b)
11556 J. Phys. Chem., Vol. 100, No. 28, 1996
Bauman et al.
TABLE 5: Comparison of Spectroscopic and Geometric Data of 1 and 3 of This Paper with Data from the Literature compd 3
properties I(νjmax), L mol-1 cm-1 II(νjmax) L mol-1 cm-1 m(νjmax) L mol-1 cm-1 γID, deg γIID, deg D γIII , deg β*D(I), deg β*D(II), deg β*D(III), deg ΨD, deg DINK, cgs DIINK, cgs NK DIII , cgs NK D11 (I), cgs DNK 22 (I), cgs DNK 33 (I), cgs * jmax)(I), L mol-1 cm-1 11(ν *22(νjmax)(I), L mol-1 cm-1 *33(νjmax)(I), L mol-1 cm-1
1
I(νjmax), L mol-1 cm-1 II(νjmax), L mol-1 cm-1 III(νjmax), L mol-1 cm-1 γI, deg
γII, deg γIII, deg β(γI), deg β(γII), deg β(γIII), deg β*(I), deg β*(II), deg β*(III), deg Ψ, deg DINK, cgs DIINK, cgs NK DIII , cgs D*11(I), cgs D*22β(I), cgs D*33R(I), cgs * jmax)(I), L mol-1 cm-1 11(ν *22(νjmax)(I), L mol-1 cm-1 *33(νjmax)(I), L mol-1 cm-1 ∆(νjmax)(I), L mol-1 cm-1 ∆(νjmin)(I), L mol-1 cm-1 A(I), L mol-1 cm-1 Af(I), L mol-1 cm-1 |Rmn|, Å Θ, deg η, deg φ, deg ϑ, deg
our resultsa 36.5 × 103 (νjmax ) 24.54× 103 cm-1) 3.76 × 103 (νjmax ) 33.56 × 103 cm-1) 9.09 × 103 (νjmax ) 42.64 × 103 cm-1) 121 43 38 27 47 52 94 49.8 × 10-36 5.5 × 10-36 12.0 × 10-36 7.5 × 10-36 6.0 × 10-36 35.2 × 10-36 6.2 × 103 (νjmax ) 25.13 × 103 cm-1) 6.4 × 103 (νjmax ) 24.30 × 103 cm-1) 25.6 × 103 (νjmax ) 24.42 × 103 cm-1) 52.5 × 103 (νjmax ) 23.15 × 103 cm-1) 6.16 × 103 (νjmax ) 34.52 × 103 cm-1) 20.8 × 103 (νjmax ) 42.37 × 103 cm-1) 121
54 ( 15 54 ( 15 75 -29 -31 48 26 28 123 85.9 × 10-36 12.1 × 10-36 30.3 × 10-36 24.3 × 10-36 43.4 × 10-36 17.5 × 10-36 16.8 × 103 (νjmax ) 22.99 × 103 cm-1) 24.3 × 103 (νjmax ) 24.27 × 103 cm-1) 13.2 × 103 (νjmax ) 22.86 × 103 cm-1) 318 (νjmax ) 22.62 × 103 cm-1) -184 (νjmin ) 25.35 × 103 cm-1) 502 (νj∆)0 ) 24.16 × 103 cm-1) 475 6.03 (φ ) 60°) 6.15 (φ ) 62.4°), γ0 ) 113.8 calc: 93 (φ ) 60°) 96 (φ ) 62.4°) [eq 25] used: 57.2 calc: 62.4 (Θ ) 96.2°) [eq 25] used: 60 calc: 101 (φ ) 60°) 110 (φ ) 62.4°) [eq 26] fit: 93-106 (UV, CD)
Lightner et al.
other literature
37.0 × 103 (νjmax ) 24.39 × 103 cm-1) in CHCl36
44.0 × 10-36 in CHCl36,10
55.5 × 103 (νjmax ) 23.20 × 103 cm-1) in CHCl36
parallel to the long axis of the dipyrrinone fragments6,10,32,33
X-ray: ∠(C8-C2,C9-C10): 113.8 ∠(C12-C18,C10-C11): 113.8 ∠(C30-C8,C9-C10): 101.8 ∠(C35-C12,C10-C11): 103.17
337 (νjmax ) 23.04 × 103 cm-1) -186 (νjmin ) 25.50 × 103 cm-1) in CHCl332 523 (νj∆)0 ) 24.70 × 103 cm-1) in CHCl36 610
X-ray: 6.497
1006,10,32 calc: 856 not used
X-ray: 96.27 104.08 X-ray: ∠(C9, C10, C11): 114.33 half: 57.1657 X-ray: 59.8, 63.732
calc: 59.1, 58.9 used: 60 ϑ≈Θ
X-ray: ∠(C2-C8,C12-C18): 1097
β,β′-Dimethylmesobilirubin-XIIIR
J. Phys. Chem., Vol. 100, No. 28, 1996 11557
TABLE 5 (Continued) compd
properties θ, θ′, deg mn ∆ENK , cm-1 RRβ(I), cgs RfR,β, cgs νj0, cm-1 R(νjmax)(I), L mol-1 cm-1 β(νjmax)(I), L mol-1 cm-1 ∆R(νjmax)(I), L mol-1 cm-1 ∆β(νjmin)(I), L mol-1 cm-1
a
our resultsa calc: 128, 52 (φ ) 60°) exp: 775 calc: 950 (φ ) 60°) calc: (5.8 × 10-38 fit: (13.3 × 10-38 RfR ) 7.7 × 10-38 Rfβ ) -4.5 × 10-38 νj0 ) 23.9 × 103 calc: 40.4 × 103 (νjmax ) 23.10 × 103 cm-1) 23.6 × 103 (νjmax ) 24.68 × 103 cm-1) calc: 297 (νjmax ) 22.65 × 103 cm-1) calc: -179 (νjmin ) 25.10 × 103 cm-1)
Lightner et al.
other literature
130, 5010 1074 (φ1, φ2 ) 60°)32 calc: ≈(10 × 10-38
νj0 ) 23.5 × 103 32 calc: 40.0 × 103 (νjmax ) 22.42 × 103 cm-1) 36.0 × 103 (νjmax ) 24.57 × 103 cm-1)32 calc: 324 (νjmax ) 23.04 × 103 cm-1) calc: -183 (νjmin ) 25.50 × 103 cm-1)6
In ZLI 1695.
where the upper and lower sign apply to the transition to the mn is the excitation excited states R and β, respectively, and ENK energy of the transition I taken here as 23.9 × 103 cm-1 from the center of the long-wavelength exciton bands of 1 (Figure 7) and |R Bmn| ) 6.0 Å.10 The rotational strength RR ) -Rβ calculated with eq 43 then is 7.3 × 10-38 cgs. For the (βR,β′R) enantiomer of 1 there follows from eq 43 RR > 0 and Rβ < 0. The fitting of the CD band in isotropic phase of ZLI 1695 yields RfR ) -Rfβ ) 13.3 × 10-38 cgs. From the CD -38 cgs for the positive experiment one gets RNKR exp ) 5.1 × 10 NKβ -38 cgs for the negative band band and Rexp ) -4.7 × 10 -38 cgs and the - RNKβ which yields RNKR exp exp ) 9.8 × 10 amplitude Aiso ) 502 L mol-1 cm-1. The corresponding values from the curve fitting (Figure 9b) are RfNKR ) 7.7 × 10-38, RfNKβ ) -4.5 × 10-38, and RfNKR - RfNKβ ) 12.2 × 10-38 cgs leading to an amplitude Af ) 475 L mol-1 cm-1. RfNKR is too large because of the insufficent fit of the UV and CD band at the long-wavelength side of IR and Iβ which can be seen also from the difference of the intensities of the two lobes of the NKβ couplet RfNKR - RfNKβ > RNKR exp - Rexp . R In comparison to the value Rf ) - Rfβ ) 13.3 × 10-38 cgs resulting from the curve fitting (Figure 9b) of the CD of 1 in the isotropic solution the value from the exciton theory (eq 43) is too small (55 % of the experimental result). Because R(γ) with γ ) 121° is near its maximum, a false value of γ cannot be the reason for this discrepancy found. A variation of R as a function of φ has only a small influence on the calculated R
Rβ . Therefore, a false value of φ cannot be the reason for a rotational strength too small. Furthermore, a variation of φ by more than 2-4 deg seems not to be realistic in view of the results found in literature.32 A more critical point and thus a possible origin of a failure of the exciton theory may be the choice of the interchromophoric distance vector B Rmn. The mn direction of the vector B R is fixed if the origins of the multipole expansions are chosen symmetrically in the two fragments m, n. But its length |R Bmn| then still depends on the position of these points which should be chosen in such a way that the dipole-dipole term outweighs the higher multipole terms. This remark concerns the exciton splitting energy as well as the factor m n b〉NK × 〈µ b〉NK in the expression given in eq 43a for the B Rmn‚〈µ R
rotational strength Rβ . A shift of these points will change the distance |R Bmn|. With γ ≈ 121° the experimental value of the rotational strength demands |R Bmn| > 12 Å which seems too large to be realistic. It should be mentioned here that the extension of the fragments is large compared to their distance which actually is a severe restriction for the applicability of the present approximation of the exciton theory. Furthermore, the change
of the band structure gives an indication for a change of the potential curves for the excited fragment state involved in the exciton states. In general, it should be possible by ACD spectroscopy to decompose the contributions of both exciton bands experimentally if the x* 3 orientation axis has a suitable orientation with respect to the transition moment directions of both bands IR and Iβ. For 1 the difference between ∆A and ∆ (Figures 4 and 5) is very small and thus qualitatively consistent with the theory of exciton coupling for the case where the orientation axis is nearly perpendicular to both exciton transitions and, therefore, the ratio of the ∆A values of both bands is independent of the order parameters:18
∆A(AfA) ∆A(AfB)
) -1
(44)
The amplitude A of a positive couplet can be calculated, neglecting the electric dipole-electric quadrupole term, from the expression18
A ) [∆A(νjmax;AfB) - ∆A(νjmax;AfA)] + [∆A(νjmin;AfB) - ∆A(νjmin;AfA)] ≈ Aiso(1 + S*/2) (45) νjmax and νjmin are the wavenumbers of the positions of the maximum and the minimum of the couplet, respectively. Aiso is the amplitude of the CD measurement in the isotropic phase. Taking S* ) 0.51 for (βR,β′R)-dimethylmesobilirubin-XIIIR in ZLI 1695 at T ) 28 °C, there results from eq 45 the value A ) 630 L mol-1 cm-1, which is in very good agreement with the value A ) 599 L mol-1 cm-1 obtained from ACD measurement (Figures 4 and 5). The electric dipole-electric quadrupole contribution cannot be estimated because of the smallness of the ACD effect. 6. Conclusion In Figure 10a,b the orientations of the transition moment directions of the transitions of symmetry A f B (C2) of 1 and A f A (Cs) of 3 in the x1, x3 and in the molecular (x2, x3) plane, respectively, are placed together. The angles of the orientation axis of 3 against the CH3-C1 direction and the transition moment vectors of the transitions I, II, and III are γD D ) 94°, γID ) 121°, γIID ) 43°, and γIII ) 38°. The error of the absolute values of these angles is difficult to estimate. It is assumed to be about (15°. Because the orientation axis divides the angle enclosed by the transition moment directions of the transitions I and II, the degrees of anisotropy of I and II are
11558 J. Phys. Chem., Vol. 100, No. 28, 1996 comparable (that of II is about 15% smaller than that of I) whereas the R value of III is much smaller. This situation seems to be in contrast to the hierarchy of the R values of 1. Here, R is about zero or even negative in the spectral region of the transition IR whereas R > 0 and approximately equal for IIR and IIIR (βIIR ≈ βIIIR ≈ 44°). The latter fact is a consequence of the exciton coupling with the transitions II and III which creates four exciton transitions in 1. Two of them (IIR and IIIR) are polarized in the x1, x3 plane and the other two (IIβ and IIIβ) in the x2 direction. From the direction of the transition moments of II and III in 3 there follows that IIR and IIIR gain almost all the intensity whereas IIβ and IIIβ are very weak and thus cannot be seen in the experimental spectra. The angle enclosed by the transition moment vectors of the m n and 〈µ b〉NK , of 1 has been determined to be ϑ transition I, 〈µ b〉NK ) 106° from which γ ) 121° follows. From the fact that the exciton coupling does not lead to a measurable CD couplet, there follows γII ) γIII ) 54° ( 15° and for the transition moment directions of IIR and IIIR the value β ) -44° ( 10°. The orientation axis of 1 is oriented approximately along the line connecting C2 and C8 (113.8°) at ψ ) 123°. All data are summarized in Table 5. Acknowledgment. This research was supported by the Deutsche Forschungsgemeinschaft and the Fonds der chemischen Industrie, Germany, and in part by the Poznan´ University of Technology Research Project. D.B. also thanks the Alexander von Humboldt Stiftung for financial support during the stays in Kaiserslautern. We thank Dr. H. Schulze for developing the programs of the fitting procedure for the evaluation of the spectroscopic constants and the order parameters. S.E.B. and D.A.L. thank the U.S. National Institutes of Health for support. References and Notes (1) McDonagh, A. F.; Lightner, D. A. Pediatrics 1985, 75, 443. (2) Ostrow, J. D., Ed. Bile Pigments and Jaundice; Marcel-Dekker: New York, 1986. (3) Brown, G. H.; Wolken, J. J. In Liquid Crystals and Biological Structures; Academic Press: New York, 1979. (4) Falk, H. In The Chemistry of Linear Oligopyrroles and Bile Pigments; Springer Verlag: New York, 1989. (5) Person, R. V.; Boiadjiev, S. E.; Peterson, B. R.; Puzicha, G.; Lightner, D. A. Lect. Posters 4th Int. Conf. CD, Bochum, Germany; 1991; p 55. (6) Lightner, D. A.; Boiadjiev, S. E.; Person, R. V.; Puzicha, G.; Knobler, C.; Maverick, E.; Trueblood, K. N. J. Am. Chem. Soc. 1992, 114, 10123. (7) Bonnett, R.; Davies, J. E.; Hursthouse, M. B.; Sheldrick, G. M. Proc. R. Soc. London, Ser. B 1978, 202, 249. (8) Becker, W.; Sheldrick, W. S. Acta Crystallogr. 1978, B34, 1298.
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