Kohn-Sham construction

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. 

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... 

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. 

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... 

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 ... 

Wang, Y. and R. Parr. Construction of exact Kohn-Sham orbitals from a given electron density. Phys. Rev. A 47, R1591. 

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... 

Similar relations can be obtained for the nonlinear/ functions. Kohn-Sham orbital formulations of these nonlinear responses can be constructed along the lines described previously [32] and will be presented elsewhere. 

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