The Kohn-Sham scheme

Now we know that it is possible to use the density as the fundamental variable, but is there a good way to do this The answer is most certainly yes. The year after Hohenberg and Kohn published their article [14], Kohn and Sham published another article [28] in which they presented a computational scheme called the 

Kohn-Sham scheme. The main error in the earlier approaches to find a density functional theory was to approximate the kinetic energy as a local density functional. All of those approximations gave large errors, and it was clear that some new way had to be found to get around this problem. 

Let us have a look at the exact kinetic energy. In general, it is written as 

Since the Kohn-Sham system is a system of non-interacting electrons giving the same density as the real system, we can write for its orbitals  

Here the subscript s on the potential denotes that we are now solving single- electron equations, since we have a non-interacting system. But what is the potential in the equation above By isolating the non-interacting kinetic energy (Ts) and the Coulomb interaction (U) in our energy functional EVo(n) we get 

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Here (Oj is the excitation energy ErE0 and the sum runs over all excited states I of the system. From equation (5-37) we immediately see that the dynamic mean polarizability a(co) diverges for tOj=co, i. e has poles at the electronic excitation energies 0)j. The residues fj are the corresponding oscillator strengths. Translated into the Kohn-Sham scheme, the exact linear response can be expressed as the linear density response of a non-interacting... 

Up to this point, exactly the same formulae also apply in the Hartree-Fock case. The difference is only in the exchange-correlation part. In the Kohn-Sham scheme this is represented by the integral,... 

That is, we now aim to describe in a more appropriate way the interaction part of the kinetic energy that is introduced to the ex-change-correlation functional in the Kohn-Sham scheme. Including the kinetic energy corrections increases the computational requirements substantially, but the accuracy is also much improved compared with conventional gradient-corrected functionals. 

Before the progress with the relativistic gradient expansion of the kinetic energy took place, and due to a growing interest of applying the Kohn-Sham scheme of density functional theory [19] in the relativistic framework, an explicit functional for the exchange energy of a relativistic electron gas was found [20,21] ... 

The Kohn-Sham scheme then provides a mapping from the true interacting system to a Slater determinantal approximation. 

In Refs [10, If] we have shown that Eqn (30) is an expression for the first-order shell correction term in the EKS-DFT frame. As we pointed it out, the extended version [26,27] of the Kohn-Sham scheme [46] is appropriate because it allows fractional occupation numbers, thus permitting the... 

The methodology focuses, as many density-functional schemes do, on the key role of the electron density. The Schrodinger equation is then solved self-consistently in the Kohn-Sham scheme.86 Initial approaches dealt with a jellium-adatom system, which would at first sight seem rather unchemical, lacking microscopic detail. But there is much physics in such an effective medium theory, and with time the atomic details at the surface have come to be modeled with greater accuracy. 

In practice, one uses the Kohn-Sham scheme that consists in solving self-consistently the following one-electron equations [16] ... 

If we combine this result with previous theorems we obtain the following important consequence for the Kohn-Sham scheme ... 

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,... 

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