Kohn-Sham system

On the other hand, asymptotic density of the corresponding Kohn-Sham system is determined completely by the highest occupied orbital and is given as... 

At this point one question must be answered Is the potential calculated in the manner above path independent [21] Equivalently, is the field given by Equation 7.33 curl-free For one-dimensional cases and within the central field approximation for atoms, it is. For other systems, there is a small solenoidal component [21,22] and we will see later that it arises from the difference in the kinetic energy of the true system and the corresponding Kohn-Sham system (in this case the HF system and its Kohn-Sham counterpart). For the time being, we explore whether the physics of calculating the potential in the manner prescribed above is correct in the cases where the curl of the field vanishes. 

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... 

The noninteracting Kohn-Sham system is defined by adiabatic connection,... 

We have applied a slight variation of this general idea to the exchange-correlation potential of the He atom [18], The virial theorem applied to the Kohn-Sham system yields [39] ... 

Here, we should mention that there exists an extensive discussion in the literature on the capabilities of spin-DFT regarding, for instance, the question whether the Kohn-Sham spin density has to be equal to the spin density of the fully interacting system of electrons (and in the case of open-shell singlet broken-symmetry (BS) determinants (see below) for binuclear transition-metal clusters this is certainly not the case see Ref. (33) for a more detailed discussion). But the situation is much more subtle and one may basically set up the variational procedure in a Kohn-Sham framework such that the spin density of the Kohn-Sham system of noninteracting fermions represents the true spin density. However, the frame of this review is not sufficient to present all details on this matter (34,35). 

In this section we shall derive [69] a formally exact representation of the linear density response ni(r,t ) of an interacting many-electron system in terms of the response function of the corresponding (non-interacting) Kohn-Sham system and a frequency-dependent xc kernel. 

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 ... 

The only result is that the electron full density determines the potential, but there is still the original many-body problem. One possibility is to replace the original interacting-particle problem with one that can be more easily solved, that is, the Kohn-Sham auxiliary system. This alternative is a non-interacting electron system assumed to have the same density as the interacting system. The Kohn-Sham system presumes to have the same density as the true interacting system. Thus,... 

Let us describe a couple of cases for which we know this conjecture to be true. Consider a system of two particles. Let the interacting system have density n. Then we can construct a noninteracting Kohn-Sham system with ground state wavefunction... 

Because Eq. (45) depends on the electron density and Eq. (46) depends on the Kohn-Sham orbitals, these two equations must be solved self-consistently. The procedure for solving the Kohn-Sham system, then, is to guess an electron density, construct vj[p r] + vxc[p r], and solve Eq. (45), subsequently obtaining a new electron density from Eq. (46). Unless the electron density from Eq. (46) equals the guess density, one proceeds to construct a (suitably improved) guess for the electron density and repeats the process until the input density and the output density are the same. 

This general form, in which the value of a property is computed expressed in terms of its value for the Kohn-Sham system plus a correction dependent on the exchange-correlation energy, recurs throughout Kohn-Sham density-functional theory. 

Linear-Scaling Methods for Solving the Kohn-Sham System... 

Often, the bottleneck in linear-scaling density-functional theory is the evaluation of the Coulomb potential the trade olf between the simple and direct method of integrating Eq. (94) and the more sophisticated linear-scaling approaches is evidenced by the fact that, for moderately large systems, linear-scaling density-functional techniques are often less efficient than direct solution to the Kohn-Sham system. As the size of the system increases beyond 10 to 20 A, however, linear-scaling techniques become essential. 

Up to this point we have discussed DFT in terms of the charge (or particle) density n(r) as fundamental variable. In order to reproduce the correct charge density of the interacting system in the noninteracting (Kohn-Sham) system, one must apply to the latter the effective KS potential vs = v + Vh + vxc, in which the last two terms simulate the effect of the electron-electron interaction on the charge density. This form of DFT, which is the one proposed originally [24], could also be called charge-only DFT. It is not the most widely used DFT in practical applications. Much more common is a formulation that employs one density for each spin, n-f(r) and rq(r),... 

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