Molecular quadrupole moments. Quantum correction to the classical formula

Abstract

The classical formula relating the refractive index difference induced in a gas by an electric field-gradient F_xx=–F_yy, is of the form n_x–n_y=(AT^–1+B)F_xx at constant density, where A is proportional to the product of the quadrupole moment and the anisotropy in the polarizability of a molecule, and B to the effect of a field-gradient on the polarizability of a non-rotating molecule. A full quantum-mechanical expression for n_x–n_y is derived, and applied in detail to diatomic molecules. The effects of quantization of the rotational motion are found to be negligible for CO₂, but to lead to corrections of about 25% for H₂ at room temperature. The effects of centrifugal distortion of the molecule are also considered, but are generally less important. The (2J+ 1)-fold degeneracy of the rotational states is completely lifted by the field-gradient, and this first-order splitting, in association with distortion of the rotational states by the field-gradient, leads to the term in T^–1; all the rotational states contribute to A. At low temperatures, para-H₂(J= 0) should have a temperature-independent birefringence, whereas ortho-H₂(J= 1) should obey a T^–1 law, though not the classical one.