[en] The focus of this work is the bonded contribution to the intrinsic Helmholtz free energy of molecules. A weighted density approximation (WDA) for this contribution is presented within the interaction site model (ISM) for ring and cage polymers. The resulting density functional theory (ISM/WDA) for these systems is no more complex than theories for a pure simple fluid, and much less complex than density functional approaches that treat the bonding functional exactly. The ISM/WDA bonding functional is much more accurate than either the ISM/HNC or ISM/PY bonding functionals, which are related to the reference interaction-site model (RISM)/HNC and RISM/PY integral equations respectively, for ideal ring polymers. This means that the ISM/WDA functional should generally be more accurate for most 'real' ring or cage polymer systems when any reasonable approximation for the 'excess' contribution to the intrinsic Helmholtz free energy is employed