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Optical Properties of Molecular Aggregates. II. Classical Theory of the Refraction, Absorption, and Optical Activity of Solutions and Crystals
1.H. DeVoe, J. Chem. Phys. 41, 393 (1964).
2.Throughout this paper we use a complex notation to express the time dependence of those quantities which oscillate at angular frequency ω. Thus a complex quantity Q stands for the real time‐dependent function . Note that
3. is correctly given by where R and are the reflectivity and incident intensity, respectively.
4.R. W. Ditchburn, Light (Interscience Publishers, Inc., New York, 1953), Chap. 15.
5.In practice the properties usually measured are the rotation of linearly polarized light, and the circular dichroism (the difference in absorption of right and left circularly polarized light).
6.E. U. Condon, Rev. Mod. Phys. 9, 432 (1937).
7.For Eq. (10) to be valid, the sample must be effectively homogeneous down to a scale much smaller than the wavelength Thus the monomer spacing must be much smaller than the wavelength, and we ignore the statistical concentration fluctuations in liquids which are responsible for Rayleigh scattering.
8.It would be more accurate to use oscillator polarizabilities which are appropriate to the monomer in the time‐average (static) field of its environment. The polarizabilities then might change on going from a vacuum or solvent environment to an aggregate. With this modification of the classical theory, small environmental spectral shifts would be predicted which in quantum theory are explained by a perturbation of the transition energy when the static monomer electric dipole is different in the ground and excited states [E. G. McRae, J. Phys. Chem. 61, 562 (1957)].
9.The phase change from the retardation between oscillators i and j is neglected; it is much smaller than the phase change caused by the wavelength [G. N. Ramachandran, Proc. Indian Acad. Sci. 33A, 217 (1951)].
10.Inherent in the way we use is the assumption that if electronic oscillators i and j belong to the same monomer or solvent molecule. This is equivalent to assuming that the electronic oscillators of a given monomer belong to different normal modes, or in quantum‐mechanical terms that the excited state wavefunctions are mutually orthogonal.
11.N. Q. Chako, J. Chem. Phys. 2, 644 (1934).
12.O. E. Weigang, Jr., J. Chem. Phys. 41, 1435 (1964).
13.A. Moscowitz, Advan. Chem. Phys. 4, 67 (1962).
14.There is a certain error in making this assumption when the aggregate is nonspherical. A better procedure in such cases is to form a spherical aggregate by including some solvent molecules as part of the aggregate.
15.Programs have been written for a Honeywell‐800 computer to perform these steps. The results for specific cases are being prepared for publication elsewhere.
16.For references see Ref. 6.
17.H. C. Bolton and J. J. Weiss, Nature 195, 666 (1962).
18.H. DeVoe, Nature 197, 1295 (1963).
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23.R. K. Nesbet, Mol. Phys. 7, 211 (1964).
24.Two minor points of disagreement are (1) Ref. 22 derives the rotational strength of a long helical polymer by ignoring phase differences, whereas we have found the phase differences must be taken into account for optical activity; (2) Ref. 23 derives the absorption of a dimer by assuming ε is proportional to (M is the dimer transition moment) while correctly ε is proportional to thus the results given in Ref. 23 are valid only for parallel transition moments.
25.A. S. Davydov, Theory of Molecular Excitons, translated by M. Kasha and M. Oppenheimer, Jr. (McGraw‐Hill Book Company, Inc., New York, 1962).
26.D. S. McClure, Solid State Phys. 8, 1 (1959).
27.D. Fox and R. M. Hexter, J. Chem. Phys. 41, 1125 (1964).
28.I. Tinoco, Jr., R. W. Woody, and D. F. Bradley, J. Chem. Phys. 38, 1317 (1963).
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31.W. Moffitt, D. D. Fitts, and J. G. Kirkwood, Proc. Natl. Acad. Sci. U.S. 43, 723 (1957).
32.Paper I (Ref. 1) derives the refraction and absorption of an aggregate, neglecting solvent, by finding normal modes of aggregate polarization. This procedure involves an expansion of the coefficients of the present paper in the eigenvalues of the matrix, making it possible to compare the classical theory with the exciton theory which is an eigenvalue problem. Normal modes have not been introduced in the present paper, because finding them takes extra computation time except in the case of identical oscillators for which the eigenvectors are the same at all frequencies.
33.I. Tinoco, Jr., J. Am. Chem. Soc. 82, 4785 (1960);
33.I. Tinoco, Jr., 83, 5047 (1961)., J. Am. Chem. Soc.
34.I. Tinoco, Jr., J. Chem. Phys. 33, 1332 (1960);
34.I. Tinoco, Jr., 34, 1067 (1961)., J. Chem. Phys.
35.I. Tinoco, Jr., Advan. Chem. Phys. 4, 113 (1962).
36.W. Rhodes, J. Am. Chem. Soc. 83, 3609 (1961).
37.H. DeVoe, J. Chem. Phys. 37, 1534 (1962).
38.J. Thiéry, J. Chem. Phys. 43, 553 (1965).
39.W. Kauzmann, Quantum Chemistry (Academic Press Inc., New York, 1957), p. 581.
40.The derivation of Eq. (80) is briefly as follows. Each coefficient as defined by Eq. (45), is a quotient of two polynomials in the oscillator polarizabilities of the form . We evaluate the integrals , along a closed path in the complex ω plane: along the real axis from to and then along a semicircle of radius in the upper half‐plane. Since is analytic in the upper half‐plane, both integrals are zero provided the denominator of has no zeros. Thiéry (Ref. 38) shows there are no zeros in the case of a dimer if the interaction energy of the two transition moments is less than the mean transition energy, and it is reasonable to assume the same criterion holds for larger aggregates. The first term in the expansion of about is proportional to so that the part of both integrals along the semicircular path goes to zero as ρ goes to infinity. Both ω Im and ω Im are even functions of ω along the real ω axis. Therefore the part of both integrals along the real axis from 0 to is zero, which gives Eq. (80).
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