We have developed expressions for the residual Helmholtz free energy and the residual chemical potential in terms of the correlation functions and bridge functions in the context of the interaction-site formalism for the Chandler-Silbey-Ladanyi equations. Unlike the corresponding expressions for the extended-RISM equation, these expressions are formally exact for systems described by interaction-site models. The new expressions are similar in form to those for multicomponent simple fluid mixtures and are found to reduce to them in the extended-atom limit, where the bond lengths approach infinity. We have also found that the residual Helmholtz free energy satisfies a variational principle for a certain class of closure relations. This finding could facilitate the development of more efficient methods for solving the Chandler-Silbey-Ladanyi equations. We have also derived explicit expressions for the residual Helmholtz free energy, residual chemical potential, residual pressure, and residual internal energy in the hypernetted-chain approximation of the Chandler-Silbey-Ladanyi equations. It is noteworthy that the derived expressions depend solely on the correlation functions of the system at full coupling, thus making the computation of the various fluid properties simpler and more efficient by eliminating the need to perform a numerical integration over a coupling constant. We have also found that the residual Helmholtz free energy associated with the hypernetted-chain approximation of the Chandler-Silbey-Ladanyi equations satisfies a variational principle. Furthermore, in the extended-atom limit, all the derived expressions associated with the hypernetted-chain approximation of the Chandler-Silbey-Ladanyi equations reduce to those corresponding to the multicomponent simple fluid mixtures.