Testing excited-state energy density functional and potential with the ionization potential theorem

The modified local spin density functional and the related local potential for excited states is tested by employing the ionization potential theorem. The functional is constructed by splitting $k$-space. Since its functional derivative cannot be obtained easily, the corresponding potential is given by analogy to its ground-state counterpart. Further to calculate the highest occupied orbital energy $\epsilon_{max}$ accurately, the potential is corrected for its asymptotic behavior by employing the van Leeuwen and Baerends correction to it. $\epsilon_{max}$ so obtained is then compared with the $\Delta$SCF ionization energy calculated using the MLSD functional. It is shown that the two match quite accurately.