Equations for calculating accurate 4-point and 5-point bridge diagrams in terms of h-bonds have been presented and solved for various phase points of the Lennard-Jones fluid. A method of finding a self-consistent solution for the bridge function and the radial distribution function is demonstrated. The significance of this result over bridge diagrams expressed as f-bonds, in terms of its applicability to charged and dipolar models is discussed. Two very simple phenomenological bridge diagram forms for the bridge function for this model are examined and found to give results almost as accurate and in some cases more accurate than previous forms in the literature. This work represents the first use of directly calculated 5-point bridge diagrams in terms of h-bonds, and the many extra orders of f-bond diagrams which they include, in an integral equation result.