The Kohn-Sham Model

We shall now show how the formalism examined above can be used to derive the standard Kohn-Sham scheme. We start by considering the Levy-Lieb energy functional (39), which is minimized under the normalization constraint 

The constant is here undetermined, due to the normalization constraint, J drp(r) = 0. If the functional LL [p] is Gateaux (Frechet) differentiable, then its Gateaux (Frechet) derivative will vanish at the minimum ([14], p. 460). 

In the Kohn-Sham model [2] we consider a system of noninteracting electrons, moving in a local potential vKS(r), 

Using the constrained search, the correspondence of the Levy-Lieb functional (38) is for the noninteracting system the minimum of the kinetic energy, 

The ground-state energy of the noninteracting system is obtained by minimizing this energy functional under the normalization constraint, 

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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. 

Podeszwa R, Szalewicz K (2005) Accurate interaction energies for argon, krypton, and benzene dimers from perturbation theory based on the Kohn-Sham model. Chem Phys Lett 412 488-493... 

In standard DFT an energy functional, E[p, is minimized under the constraint that the density p(r) is normalized to the number of electrons, which requires that the functional is differentiable with respect to the density at the minimum. In the Kohn-Sham model, an interacting system is simulated by a system of noninteracting electrons moving in a local potential, which requires that the derivative of the functionals involved can be represented by a local function. 

With the potential (79) the Kohn-Sham model (72) yields the same ground-state density as the original problem, and inserting this density into the HK functional (10) leads, in principle, to the ground-state energy. This is the basic Kohn-Sham... 

Related to the v-representability problem is the Kohn-Sham v-representability problem. That is, given a system of interest, can one always find an internal potential, w(r), such that the ground-state electron density of the Kohn-Sham model system is the same as that of the state of interest Again, the answer seems to be no [87], but if one allows fractional occupation numbers of the Kohn Sham orbitals, then no essential difficulties arise [3,88,89]. We note that in this case, the idempotency constraint, Eq. (71), is no longer appropriate, and the less stringent Eq. (70) should be used instead. 

It turns out that in the case analyzed (and so far only in this case), we can calculate the exact total energy E [Eq. (11.17)], wonder potential vq that in the Kohn-Sham model gives the exact density distribution p [Eq. (11.83)], exchange potential Vx and eorrelation potential Vc [Eqs. (11.70) and (11.71)]. Let us begin from the total energy. 

Energies for Argon, Krypton, and Benzene Dimers from Perturbation Theory Based on the Kohn-Sham Model. 

For this kind of confinement, the solution of the non-relativistic time independent Schrodinger equation has been tackled by different techniques. For confined many-electron atoms the density functional theory [6], using the Kohn-Sham model [7], has given some estimations of the non-classical effects [8-11], through the exchange-correlation functional. An elementary review of this subject can be found in Ref [12], where numerical techniques are discussed to solve the Kohn-Sham equations. Furthermore, in this reference, some chemical predictors are analyzed as a function of the confinement radii. 

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... 

The Kohn-Sham determinant is the single determinant which reproduces the electron density and minimises the kinetic energy [1,9].) They observed that for the Be atom, the Kohn-Sham orbitals were nearly indistinguishable from the HF orbitals, and on this evidence they claim that the problem of finding a physically meaningful wave function from an electron density is solved . Here, we merely note that there are a number of desirable features for our model ... 

A new and accurate quantum mechanical model for charge densities obtained from X-ray experiments has been proposed. This model yields an approximate experimental single determinant wave function. The orbitals for this wave function are best described as HF orbitals constrained to give the experimental density to a prescribed accuracy, and they are closely related to the Kohn-Sham orbitals of density functional theory. The model has been demonstrated with calculations on the beryllium crystal. 

Note that the Kohn-Sham Hamiltonian hKS [Eq. (4.1)] is a local operator, uniquely determined by electron density15. This is the main difference with respect to the Hartree-Fock equations which contain a nonlocal operator, namely the exchange part of the potential operator. In addition, the KS equations incorporate the correlation effects through Vxc whereas they are lacking in the Hartree-Fock SCF scheme. Nevertheless, though the latter model cannot be considered a special case of the KS equations, there are some similarities between the Hartree-Fock and the Kohn-Sham methods, as both lead to a set of one-electron equations allowing to describe an n-electron system. 

Modeling of biological systems frequently requires that the accuracy of calculated energy differences are at the kT level (which amounts to less than 1.0 kcal/mol in room temperature). In conventional ab initio methods, such accuracy has been achieved because of effective error cancelation, which is not always the case for the Kohn-Sham calculations. 

An alternative approach to conventional methods is the density functional theory (DFT). This theory is based on the fact that the ground state energy of a system can be expressed as a functional of the electron density of that system. This theory can be applied to chemical systems through the Kohn-Sham approximation, which is based, as the Hartree-Fock approximation, on an independent electron model. However, the electron correlation is included as a functional of the density. The exact form of this functional is not known, so that several functionals have been developed. 

Theoretical considerations leading to a density functional theory (DFT) formulation of the reaction field (RF) approach to solvent effects are discussed. The first model is based upon isolelectronic processes that take place at the nucleus of the host system. The energy variations are derived from the nuclear transition state (ZTS) model. The solvation energy is expressed in terms of the electrostatic potential at the nucleus of a pseudo atom having a fractional nuclear charge. This procedure avoids the introduction of arbitrary ionic radii in the calculation of insertion energy, since all integrations involved are performed over [O.ooJ The quality of the approximations made are discussed within the frame of the Kohn-Sham formulation of density functional theory. 

As compared with the Kohn-Sham functional for electronic systems, the nuclear Skyrme functional is less genuine. The main (Coulomb) interaction in the Kohn-Sham problem is well known and only exchange and corellations should be modeled. Instead, in the nuclear case, even the basic interaction is unknown and should be approximated, e.g. by the simple contact interaction in Skyrme forces. 

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