The Kohn-Sham construction

A particular mapping - is determined by the Kohn-Sham construction (KSC) minimize the kinetic energy orbital functional T = JT(j t i) for specified spin-indexed electron density p. This applies Hohenberg-Kohn logic to a 

If an exact local exchange potential does not exist, there is no reason for UHF, OEP, and KSC results to be the same in the UHF model. That the OEP density is not exactly equal to that of the UHF ground state is indicated by an analysis of OEP results [412], and by recent test calculations [69], This would imply that the density constraint in KSC is a true variational constraint for more than two electrons, so that Eksc E0ep- The calculations considered here may not have sufficient numerical accuracy to establish this evidently small energy difference. 

---

Hohenberg-Kohn theorems, but use the Kohn-Sham construction and local approximations to such non-local potentials and often lump together the exchange and the correlation energies into an exchange-correlation energy Exc[n], This yields a local exchange-correlation potential vxc(t) in the Kohn-Sham equations that determine the Kohn-Sham spin orbitals j, i.e. 

The Kohn-Sham construction is a pragmatic one, justified by computational utility. Of special computational utility is the fact that each Kohn-Sham orbital experiences the same potential and that this potential, in turn, is a functional of the electron density alone. This allows us to rewrite the Kohn-Sham energy in terms of the first-order density matrix,... 

Even if problems with negative-energy states were eliminated by the construction of the energy functional, Eq. (3), they are re-introduced at the one-electron level, Eq. (10), due to the Kohn-Sham construction based on the Dirac form of the kinetic energy, Eq. (5) [41]. 

The Kohn-Sham construction of an auxiliary system rests upon two assumptions ... 

Fh p) = Ec p) + Fxc p) + FM + FEAPhFT p) (3.15) (with subscripts C, XC, eN, Ext, and T denoting Coulomb, exchange-correlation, electron-nuclear attraction, external, and kinetic energies respectively). It is CTucial to remark that (3,15) is not the Kohn-Sham decomposition familiar in conventional presentations of DFT. There is no reference, model, nor auxiliary system involved in (3.15). From the construction presented above it is clear that in order to maintain consistency and to define functional derivatives properly all... 

It is a truism that in the past decade density functional theory has made its way from a peripheral position in quantum chemistry to center stage. Of course the often excellent accuracy of the DFT based methods has provided the primary driving force of this development. When one adds to this the computational economy of the calculations, the choice for DFT appears natural and practical. So DFT has conquered the rational minds of the quantum chemists and computational chemists, but has it also won their hearts To many, the success of DFT appeared somewhat miraculous, and maybe even unjust and unjustified. Unjust in view of the easy achievement of accuracy that was so hard to come by in the wave function based methods. And unjustified it appeared to those who doubted the soundness of the theoretical foundations. There has been misunderstanding concerning the status of the one-determinantal approach of Kohn and Sham, which superficially appeared to preclude the incorporation of correlation effects. There has been uneasiness about the molecular orbitals of the Kohn-Sham model, which chemists used qualitatively as they always have used orbitals but which in the physics literature were sometimes denoted as mathematical constructs devoid of physical (let alone chemical) meaning. 

We now need to discuss how these contributions that are required to construct the Kohn-Sham matrix are determined. The fust two terms in the parenthesis of equation (7-12) describe the electronic kinetic energy and the electron-nuclear interaction, both of which depend on the coordinate of only one electron. They are often combined into a single integral, i. e ... 

Here the vector zs(r) is constructed from the kinetic energy tensor obtained by employing the solutions of the Kohn-Sham equation in Equation 7.42. Thus it is, in general, different from the vector z(r). A comparison of Equations 7.41 and 7.46 gives... 

Now we discuss the differential virial theorem for HF theory and the corresponding Kohn-Sham system. The Kohn-Sham system in this case is constructed [41] to... 

Evidently, the LSD and GGA approximations are working, but not in the way the standard spin-density functional theory would lead us to expect. In Ref [36], a nearly-exact alternative theory, to which LSD and GGA are also approximations, is constructed, which yields an alternative physical interpretation in the absence of a strong external magnetic field. In this theory, Hf(r) and rti(r) are not the physical spin densities, but are only intermediate objects (like the Kohn-Sham orbitals or Fermi surface) used to construct two physical predictions the total electron density n(r) from... 

The basic concepts of the one-electron Kohn-Sham theory have been presented and the structure, properties and approximations of the Kohn-Sham exchange-correlation potential have been overviewed. The discussion has been focused on the most recent developments in the theory, such as the construction of from the correlated densities, the methods to obtain total energy and energy differences from the potential, and the orbital dependent approximations to v. The recent achievements in analysis of the atomic shell and molecular bond midpoint structure of have been... 

In comparing Eq. (13) to the Kohn-Sham equations Eq. (3) one concludes that E(.r, x E), since it is derived from exact many-electron theory [22], is the exact Coulomb (direct) plus exchange-correlation potential. It is non-local and also energy-dependent. In view of this it is hard to see how the various forms of constructed local exchange correlation potentials that are in use today can ever capture the full details of the correlation problem. 

---