  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.
