Theory of Molecular Light Scattering Spectra Using the Linear-Dipole Approximation

Abstract

We extend existing classical molecular theories for the total light scattered by a random array of molecules, deriving by quantum mechanics the spectrum of light scattered by arbitrarily dense, randomly moving, and rotating molecules which polarize linearly in response to the instantaneous fluctuating local optical fields which they see. Using this “linear‐dipole approximation,” we obtain an expression for the spectrum of light scattered from a polarization state $a$ to state $b$ in terms of a Fourier transform $f_{\mathrm{ba}}\mathrm{(\omega )}$ of a two‐time correlation function $F_{\mathrm{ba}}{\mathrm{(t)\; \propto \; 〈C}}_{\mathrm{ba}}{\mathrm{†(t)C}}_{\mathrm{ba}}\text{(0)〉}$ involving a “susceptibility fluctuation operator” $C_{\mathrm{ba}}\mathrm{(t)}$ which is a complicated but completely defined function of the position, orientation, and vibration coordinates of all the scattering molecules. This spectral expression gives qualitatively all features observed in scattered spectra, and it is felt that difficult but feasible numerical evaluations of this expression will give a quantitative account of the depolarized and Rayleigh wing spectra recently observed from the noble gases, superfluid helium, and liquid CCl₄. A crude approximation of our expression leads to existing phenomenological expressions of the wing spectrum of more complicated molecules in terms of the two‐time correlations of their polarizability tensors. A hydrodynamic approximation for the correlations leads to standard theories of the polarized central Rayleigh and Brillouin components which are presently better understood than the depolarized scattering. The correlation function $F_{\mathrm{ba}}\mathrm{(t)}$ can be calculated classically if that is appropriate. The quantum expression for $f_{\mathrm{ba}}\mathrm{(\omega )}$ can be re‐expressed in the form of a scattering Golden Rule in which the interaction Hamiltonian is proportional to $C_{\mathrm{ba}}\text{(0)}$ . From this we derive the correct symmetry between intensities scattered at frequencies lower (Stokes) and higher (anti‐Stokes) than the incident beam frequency, a symmetry that cannot be understood classically. It is not clear when the widely used linear‐dipole approximation (LDA) by which we have treated the electromagnetic interactions of the molecules is a valid approximation for treating light scattering. Therefore, rather than embark first on a detailed spectral calculation, we derive general consequences of the LDA that permit experimental tests of its validity that are independent of the nature of the intermolecular forces and of whether the scattering medium is a mixture of molecular species. One consequence is that, in the classical regime, the total depolarized scattering equals the ac Kerr constant times the temperature, the refractive index, and other factors. This relation accounts more precisely for local field fluctuations at high molecular densities than other similar relations proposed previously. The nonlinear refractive index is also related to the polarized scattering under slightly more restricted conditions. We examine in detail scores of published accounts of Kerr and Rayleigh scattering data and find that the Kerr constant is predicted with our LDA‐based relations from Rayleigh scattering coefficients (or vice versa) to within at least 20% for liquid CS₂, and within at least 60% for benzene, CCl₄, cyclohexane, hexane, and octane (the only liquids for which existing data permits comparisons). Discrepancies among the published observations preclude determining at present whether the LDA is even more accurate. The above relations must be modified if electronic non‐linearities contribute significantly to the Kerr effect. That this might be occurring in the latter four liquids is suggested by an examination of some recent, though inconclusive, data.