Lower bounds on par with upper bounds for few-electron atomic energies

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 subparts-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.

Automated summary

The development of computational resources allows us to determine the upper bounds of atomic and molecular energies accurately. However, the error bounds for computed energies are only available as estimates. In this paper, the Pollak-Martinazzo lower-bound theory combined with correlated Gaussian basis sets is used to achieve subparts-per-million convergence of ground and excited-state energies for He, Li, and Be atoms. The quality of these lower bounds is comparable to the upper bounds obtained from the Ritz method. These results demonstrate the power of lower bounds in providing precise estimates of atomic energies.