Lower bounds to expectation values

A formula is presented giving a rigorous lower bound to the true quantum-mechanical expectation value < ψ ψ F ψ ψ > of a positive operator F >or= 0 in terms of some approximate wave function ϕ. This bound is compared with previous results of Bazley and Fox and of Jennings and Wilson as well as with a much more general lower-bound expression which is often applicable even when matrix elements of F 2 do not exist. It is shown that the non-linearities of the lower-bound expressions can be exploited by choosing `symmetric sum' operators to optimize the lower bound. As an illustrative application lower bounds are calculated for various powers of the nuclear-electronic and interelectronic distances r 1 , r 12 in the normal helium atom using a simple screened hydrogenic approximation.