Variational Extensions of Lower Bounds to Expectation Values

Generalized variational expressions are presented for calculating rigorous lower bounds to the true expectation value $〈\Psi \mathrm{}\left|F\right|\mathrm{}\Psi 〉$ of a positive semidefinite operator $F$. These extended formulas, which involve an additional variational function, always lead to an improvement over the results previously obtained. The improvement is illustrated numerically with an application to various properties of the helium-atom ground state.