Variational calculation of continuum corrections to overlap

A method is derived for determining the accuracy of an approximate wavefunction by variationally calculating its overlap with the (unknown) true wavefunction. The variational flexibility serves to generalize a previous result due to Wang, and corresponds formally to the inclusion of continuum states in Weinberger‐like modifications of the ordinary Eckart criterion for overlap. We use this procedure to calculate improved overlaps for two helium ground‐state wavefunctions: (i) a simple screened‐hydrogenic approximation, and (ii) a nearly exact 135‐term Hylleraas‐type function, in each case significantly surpassing the Eckart and Weinberger limits. A novel feature of these calculations is the evaluation of integrals over ${\mathcal{H}}^3$ , the cube of the Hamiltonian operator.