Variational Theorem Bounds on Excited States

Just as the lowest eigenvalue has been shown to be an upper bound to the exact ground-state energy, more generally, any eigenvalue E_i can be shown to be an upper bound to the corresponding exact excited state energy ${\cal E}_i$ [14]. In fact, one can also show that as other N-electron basis functions are added to the CI procedure, the eigenvalues obey the MacDonald-Hylleraas-Undheim relations [14,15]

$$E_{i-1}^{(m)} \leq E_{i}^{(m+1)} \leq E_{i}^{(m)}$$ (tex2html_deferredtex2html_deferred3.tex2html_deferred22E_i-1^(m) E_i^(m+1) E_i^(m) )

where m is the number of N-electron basis functions.