Theory of polar fluids. III

Abstract

The equilibrium theory of fluids of polar, polarizable molecules is extended by systematic development of a graph-theoretical treatment introduced earlier. The process of graph condensation named renormalization is reconsidered and carried out explicitly in first and second order, corresponding to renormalization of the permanent dipole moment m and polarizability α. The complete second renormalization differs from an earlier prescription; it requires the introduction of two renormalized polarizabilities α′ and α″. The renormalized quantities m′, α′, and α″ are expressed in terms of m, α, and derivatives of the configurational chemical potential Δμ with respect to m and α. Hierarchies of sums of all graphs with appropriate connectors and any number n of labelled points are defined for three successive levels of renormalization. The relation between graph sums with n=1 at two successive levels of renormalization acts as a generator from which the relations for higher n can be deduced by functional differentiation. The generators of the first and second renormalizations are constructed explicitly; they are equivalent to graphical expressions for Δμ at different levels of renormalization. Similar expressions are found for the configurational Helmholtz free energy ΔA; they closely resemble the results for Δμ.