  The development of computational resources has made it possible to determine
upper bounds for atomic and molecular energies with high precision. Yet, error
bounds to the computed energies have been available only as estimates. In this
paper, the Pollak-Martinazzo lower-bound theory, in conjunction with correlated
Gaussian basis sets, is elaborated and implemented to provide
sub-parts-per-million convergence of the ground- and excited-state energies for
the He, Li, and Be atoms. The quality of the lower bounds is comparable to that
of the upper bounds obtained from the Ritz method. These results exemplify the
power of lower bounds to provide tight estimates of atomic energies.
