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

We extend the Eckart theorem, from the ground state to excited statew, which introduces an energy augmentation to the variation criterion for excited states. It is shown that the energy of a very good excited state trial function can be slightly lower than the exact eigenvalue. Further, the energy calculated by the trial excited state wave function, which is the closest to the exact eigenstate through Gram-Schmidt orthonormalization to a ground state approximant, is lower than the exact eigenvalue as well. In order to avoid the variation restrictions inherent in the upper bound variation theory based on Hylleraas, Undheim, and McDonald [HUM] and Eckart Theorem, we have proposed a new variation functional Omega-n and proved that it has a local minimum at the eigenstates, which allows approaching the eigenstate unlimitedly by variation of the trial wave function. As an example, we calculated the energy and the radial expectation values of Triplet-S(even) Helium atom by the new variation functional, and by HUM and Eckart theorem, respectively, for comparison. Our preliminary numerical results reveal that the energy of the calculated excited states 3rd Triplet-S(even) and 4th Triplet-S(even) may be slightly lower than the exact eigenvalue (inaccessible by HUM theory) according to the General Eckart Theorem proved here, while the approximate wave function is better than HUM.