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Hardly anyone in the quantum chemistry community computes lower bounds to atomic and molecular energies, for good reason. All previous attempts at applying lower bound theory to Coulombic systems led to lower bounds whose quality was inferior to the Ritz upper bounds so that their added value was minimal. One of the central challenges is that the Coulomb potential leads to divergences when attempting to compute third and higher moments of the Hamiltonian, thus preventing the construction of Lanczos basis sets which are need for accurate lower bound computations. Professor R. Martinazzo and I have formulated a new lower bound theory which when applied to the H and He atoms shows much promise, providing lower bound estimates of the same quality as the upper bound Ritz estimates. The new theory overcomes the need for a Lanczos basis set, it needs as input only the Ritz upper bound energies and their standard deviations. This is much less information than needed for other lower bound theories, where typically one needs to compute the full Hamiltonian squared matrix, not just diagonal elements. In this talk I will present an overview of lower bound theories as well as the new Pollak-Martinazzo lower bound theory. Numerical examples will include the Hydrogen and He atoms. The prospects of turning this new method into a staple of variational based quantum chemistry programs will be discussed.
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