On the dielectric theory of fluids

In this, the third paper of a series, we develop a rigorous theory of the static dielectric constant, ε, of non-polar fluids within the Quadrupole Approximation, whereby the effects of induced molecular quadrupoles are incorporated into the theory. The molecular quadrupoles are induced in response to the gradients of both the externally applied field and the fields due to the molecular dipoles and quadrupoles. Two hitherto distinct approaches to the problem of the electrostatic constitutive relationship are considered. The first is a grand canonical fluctuation formalism, formulated in terms of the local field and field gradient introduced in paper I of the series. The second, or direct approach to the problem is based on treating the externally applied field (field gradient) as the driving field (field gradient) in the microscopic linear response equations. By starting from non-local integral equations relating the macroscopic dipole (quadrupole) polarization, P(Q), to the external field and field gradient, and proceeding by a detailed graph theoretical analysis, we are led to local relations between P and Q and the local field and field gradient. Subject to certain enunciated constraints the dielectric constant is found to exist, and expressions for ε are obtained in both a virial type series and a renormalized expansion formulated in terms of physically meaningful renormalized polarizabilities. The renormalized theory is derived by means of a topological reduction of the virial series solution to the integral equations for the macroscopic multipole densities, and is applicable to a discussion of the dielectric constant at liquid densities. All quadrupole interactions are consistently incorporated into the theory. In addition we prove the arbitrary order equivalence of the virial series yielded by the fluctuation formalism and the corresponding series obtained from the direct approach to the problem.