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We extend the Eckart theorem, from the ground state to excited statew, whichintroduces an energy augmentation to the variation criterion for excitedstates. It is shown that the energy of a very good excited state trial functioncan be slightly lower than the exact eigenvalue. Further, the energy calculatedby the trial excited state wave function, which is the closest to the exacteigenstate through Gram-Schmidt orthonormalization to a ground stateapproximant, is lower than the exact eigenvalue as well. In order to avoid thevariation restrictions inherent in the upper bound variation theory based onHylleraas, Undheim, and McDonald [HUM] and Eckart Theorem, we have proposed anew variation functional Omega-n and proved that it has a local minimum at theeigenstates, which allows approaching the eigenstate unlimitedly by variationof the trial wave function. As an example, we calculated the energy and theradial expectation values of Triplet-S(even) Helium atom by the new variationfunctional, and by HUM and Eckart theorem, respectively, for comparison. Ourpreliminary numerical results reveal that the energy of the calculated excitedstates 3rd Triplet-S(even) and 4th Triplet-S(even) may be slightly lower thanthe exact eigenvalue (inaccessible by HUM theory) according to the GeneralEckart Theorem proved here, while the approximate wave function is better thanHUM.

Extended Eckart Theorem and New Variation Method for Excited States of Atoms