Kohn-Sham scheme

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... 

Of course, this self-correction error is not limited to one electron systems, where it can be identified most easily, but applies to all systems. Perdew and Zunger, 1981, suggested a self-interaction corrected (SIC) form of approximate functionals in which they explicitly enforced equation (6-34) by substracting out the unphysical self-interaction terms. Without going into any detail, we just note that the resulting one-electron equations for the SIC orbitals are problematic. Unlike the regular Kohn-Sham scheme, the SIC-KS equations do not share the same potential for all orbitals. Rather, the potential is orbital dependent which introduces a lot of practical complications. As a consequence, there are hardly any implementations of the Perdew-Zunger scheme for self-interaction correction. 

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] ... 

Many density functional applications to atoms, molecules, clusters, and solid state systems have been made, based on the spin-polarized Kohn-Sham scheme... 

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

STRUTINSKY S SHELL-CORRECTION METHOD IN THE EXTENDED KOHN-SHAM SCHEME APPLICATION TO THE IONIZATION POTENTIAL, ELECTRON AFFINITY, ELECTRONEGATIVITY AND CHEMICAL HARDNESS OF ATOMS... 

Abstract. Calculations of the first-order shell corrections of the ionization potential, 6il, electron affinity, 5 A, electronegativity, ix, and chemical hardness. Sir] are performed for elements from B to Ca, using the previously described Strutinsky averaging procedure in the frame of the extended Kohn-Sham scheme. A good agreement with the experimental results is obtained, and the discrepancies appearing are discussed in terms of the approximations made. 

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... 

We also discuss the generalization of density-functional theory to n-partical states, nDFT, and the possible extension of the local density approximation , nLDA. We will see there that the difficulty of describing the state of a system properly in terms of n-particle states presents no formal difficultie since DFT is directed only at the determination of the particle density rather than individual-particle wave functions. The extent to which practical applications of nDFT within a generalized Kohn-Sham scheme will provide a viable procedure is commented upon below. 

Given the formal similarity between the Hamiltonians defined in Eqs.(2) and (5), it follows that the ground-state energy, E, is given in terms of a universal functional of the pair (or n-particle) density, n(x), which attains its minimum value for the exact pair density. Furthermore, within a Kohn-Sham scheme, the form of this functional is identical to the functional of ordinary DFT but is given in terms of the correlated pair density. The details of this derivation... 

Gorling A, Levy M (1994) Exact Kohn-Sham scheme based on perturbation theory, Phys Rev A, 50 196-204... 

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