Calculation of Energies of Excited States

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Abstract
Using arguments previously employed for derivation of lower‐bound principles, we derive a new method for obtaining an upper bound on the nth eigenvalue of a Schrödinger‐type equation. The method requires a knowledge only of the single eigenvalue immediately below that being sought—the corresponding wavefunction is not needed, as trial functions need not be orthogonalized to lower states. We illustrate the method on the 2Σg+ states of H2+, using a two‐parameter trial function. For instance, we obtain −0.3265 a.u. as an upper limit on the energy of the first excited level (true value=−0.36086 a.u.) and −0.07 a.u. for the second (true value = −0.24 a.u.).

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