Single-Particle Eigenvalues and Excited-State Energies

Single-Particle Eigenvalues and Excited-State Energies 

Equation (64) shows that Eo is not simply the sum of all e,-. If Eo can be written approximately as (where the are not the same as the Kohn-Sham eigenvalues e, ), the system can be described in terms of N weakly interacting quasiparticles, each with energy Fermi-liquid theory in metals and effective-mass theory in semiconductors are examples of this type of approach. The however, are not the same as the Kohn-Sham eigenvalues e,-, nor are they necessarily approximated by them. 

In fact, it should be clear from our derivation of (62) that the 6 are introduced as completely artificial objects they are the eigenvalues of an auxiliary single-body equation whose eigenfunctions (orbitals) yield the correct density. It is only this density that has strict physical meaning in the Kohn-Sham scheme. The Kohn-Sham eigenvalues, on the other hand, in general bear only a semiquantita-tive resemblance to the true energy spectrum, but are not to be trusted quantitatively. 

The early suggestion of Gunnarsson and Lundqvist to use a symmetry-dependent exchange-correlation functional to calculate the lowest-energy excited state of each symmetry class has been implemented approximately by von Barth, but suffers from lack of knowledge of the symmetry dependence of the functional. More recent work on this dependence is provided in Ref [87]. 

Excitation energies can also be calculated from excited-state densities. A mapping between excited-state densities and the external potential, and a Kohn-Sham-like scheme for calculating the density and energy of a specific excited state of interest have been constructed, but the definition of a proper Kohn-Sham procedure is not trivial.  

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