In this paper we explore the construction of arbitrarily tight αBB relaxations of C2 general non-linear non-convex functions. We illustrate the theoretical challenges of building such relaxations by deriving strict conditions under which it is possible for an αBB underestimator to provide exact bounds. We subsequently propose a methodology to build αBB underestimators which may provide arbitrarily tight bounds in “sub-optimal” do- mains, assuming exact eigenvalue calculations. For this purpose, we propose the transforma- tion of the original function into a μ-subenergy function and the derivation of αBB under- estimators for the new function. We prove that this transformation results in a number of de- sirable bounding properties in certain sub-optimal domains. These theoretical results are val- idated in computational test cases where the tightest possible μ-subenergy underestimators, derived using sampling, are compared to the tightest possible classical αBB underestimators. Our tests show that μ-subenergy underestimators produce much tighter bounds, and succeed in fathoming nodes which, are impossible to fathom using classical αBB.