Surpassing the Temple Lower Bound

A non-traditional derivation of the Temple lower bound formula shows how the Temple bound can be improved using expectation values of H³. Improved upper bounds result as a byproduct as well. Since expectation values of H² are already quite difficult in analytical calculations we also introduce an approximate lower bound which avoids the need for H³ but which is rigorous only in a limiting sense. Examples of the two bounds are given using the lithium and perturbed hydrogen atoms.