Lower Bounds for Nonrelativistic Atomic Energies

A recently developedlower bound theory for Coulombic problems(E. Pollak, R. Martinazzo, J. Chem. Theory Comput. 2021, 17, 1535) is further developedand applied to the highly accurate calculation of the ground-stateenergy of two- (He, Li+, and H-) andthree- (Li) electron atoms. The method has been implemented with explicitlycorrelated many-particle basis sets of Gaussian type, on the basisof the highly accurate (Ritz) upper bounds they can provide with relativelysmall numbers of functions. The use of explicitly correlated Gaussiansis developed further for computing the variances, and the necessarymodifications are here discussed. The computed lower bounds are ofsubmilli-Hartree (parts per million relative) precision and for Lirepresent the best lower bounds ever obtained. Although not yet asaccurate as the corresponding (Ritz) upper bounds, the computed boundsare orders of magnitude tighter than those obtained with other lowerbound methods, thereby demonstrating that the proposed method is viablefor lower bound calculations in quantum chemistry applications. Amongseveral aspects, the optimization of the wave function is shown toplay a key role for both the optimal solution of the lower bound problemand the internal check of the theory.