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The non-interacting electrons occupy the single-particle eigenstates according to the Fermi—Dirac statistics. Eqs. (331)—(334) have to be solved self-consistently. To do this one needs approximate expressions for the exchangecorrelation potential. For local density approximations and calculations see [183—186] The adiabatic-connection expression for the exchange-correlation expression was given by Perdew . 2. Density functional theory for excited states The density functional theory was originally formalized for the ground-state .
From the genesis of the local density approximation, one would expect that this approximation is reasonably accurate, if the density varies slowly. The approximation gives, however, acceptable results for real electronic systems, where this condition is (strongly) violated. One reason for this is that the local density exchange-correlation-hole satisfies the exact sum rule. Nevertheless, the local density approximation leads to a number of deficiencies, the most serious being a rather imperfect cancellation of self-interaction effects, which leads to the incorrect asymptotic limit of the local density exchange-correlation potential.
4. A new way of treating excited states within the density functional theory recently proposed by Go¨rling  is outlined in the next subsection. 6. 7). "¹K # »K #»K (377) ? represents a continuous connection between a non-interacting system and the real system. Here not only the ground-state but also the kth eigenstate ? of the coupling constant Hamiltonian is I considered. The additional assumption here is that the energetic order of eigenstates ? of HK ?