Kohn-Sham scheme/ orbitals

Of course, this self-correction error is not limited to one electron systems, where it can be identified most easily, but applies to all systems. Perdew and Zunger, 1981, suggested a self-interaction corrected (SIC) form of approximate functionals in which they explicitly enforced equation (6-34) by substracting out the unphysical self-interaction terms. Without going into any detail, we just note that the resulting one-electron equations for the SIC orbitals are problematic. Unlike the regular Kohn-Sham scheme, the SIC-KS equations do not share the same potential for all orbitals. Rather, the potential is orbital dependent which introduces a lot of practical complications. As a consequence, there are hardly any implementations of the Perdew-Zunger scheme for self-interaction correction. 

Since hybrid functionals, Meta-GGAs, SIC, the Fock term and all other orbital functionals depend on the density only implicitly, via the orbitals i[n, it is not possible to directly calculate the functional derivative vxc = 5Exc/5n. Instead one must use indirect approaches to minimize E[n and obtain vxc. In the case of the kinetic-energy functional Ts[ 0j[rr] ] this indirect approach is simply the Kohn-Sham scheme, described in Sec. 4. In the case of orbital expressions for Exc the corresponding indirect scheme is known as the optimized effective potential (OEP) [120] or, equivalently, the optimized-potential model (OPM) [121]. The minimization of the orbital functional with respect to the density is achieved by repeated application of the chain rule for functional derivatives,... 

If Cxc also depends on i or t, then exphcit differentiation with respect to p is no longer possible and the OEP method [47-55] should be used. In practice, the determination of the OEP is often avoided by departing slightly from the true Kohn-Sham scheme and minimizing the energy functional with respect to Kohn-Sham orbitals [271], that is, by assuming... 

In the general case of orbital-dependent functionals, minimization with respect to orbitals is only an approximation to the true Kohn-Sham scheme [281-285] (see also Ref. [58] concerning the gauge invariance problem with conventional r-dependent functionals). 

The relativistic Kohn-Sham scheme starts, in complete analogy to the non-relativistic case, with a representation of the four-current and of the non-interacting kinetic energy in terms of auxiliary spinor orbitals [8]. If one calculates the four-current of a system of fermions in an external potential (as indicated above), one obtains... 

In contrast to the Kohn-Sham scheme, no auxiliary orbitals are involved. Unfortunately, the presently available approximations to Tsb ] are only adequate for general estimates (rather than for results of chemical accuracy). The functional in question is derived from the exact kinetic energy... 

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. 

Just as in the unrestricted Hartree-Fock variant, the Slater determinant constructed from the KS orbitals originating from a spin unrestricted exchange-correlation functional is not a spin eigenfunction. Frequently, the resulting (S2) expectation value is used as a probe for the quality of the UKS scheme, similar to what is usually done within UHF. However, we must be careful not to overstress the apparent parallelism between unrestricted Kohn-Sham and Hartree-Fock in the latter, the Slater determinant is in fact the approximate wave function used. The stronger its spin contamination, the more questionable it certainly gets. In... 

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