Optical Activity of Anisotropic Solutions. I

Abstract

General theory of optical rotation is developed for transparent solutions in which the orientational distribution of solute molecules is given arbitrary. The theory of Stephen and Hameka is generalized to derive a formula giving the rotation angle in terms of the orientational distribution function and matrix elements of the solute molecule. The rotation angle is a sum of molecular rotations (contribution from individual molecules), which depend on the orientation of the molecules relative to the incident light. The molecular rotation is given as a series, which converges very rapidly when the characteristic length of the molecule is small compared with the wavelength of the incident light. If we approximate the series by the first two terms, which is valid for most cases, the optical rotation α is given by the equation, $$\mathrm{\alpha =A+}{\displaystyle \sum _{\text{i=1}}^5}{b}_i{B}_i.$$ Here, anisotropy parameters of the solution b_i are defined by $$b_i=\frac{1}{{\text{8π}}^2}{\displaystyle \int }\sin {\mathrm{\theta d\theta d\phi d\psi f(\theta ,\phi ,\psi )b}}_i\text{(θ,φ,ψ),\hspace{0.17em}\hspace{0.17em}(i=1,2,···,5),}$$ where f(θ, φ, ψ) is the orientational distribution function of optically active solute molecules and b_i (θ, φ, ψ) are certain functions of Eulerian angle. The value of the anisotropy parameters b_i and the expressions B_i are dependent on the choice of the body fixed coordinate system and behave as five‐dimensional irreducible tensors of Rank 3 for rotations of the system. When the solution is isotropic, the anisotropy parameters vanish and the formula reduces to α=A, which is identical with the formula derived by Rosenfeld. In cases of rotationally symmetric molecules the number of the anisotropy parameters reduces to one.