Kohn-Sham electron density

The Kohn-Sham wave function, KS, is not expected to be a good approximation to the exact wave function indeed, it is a worse approximation to the exact wave function than the Hartree-Fock wave function. Flowever, unlike the electron density obtained from the Flartree-Fock equations, the Kohn-Sham method yields, in principle, the exact electron density. Thus we do not need to use the Kohn-Sham wave function to compute the properties of chemical systems. Rather, motivated by the first Hohenberg-Kohn theorem, we compute properties directly from the Kohn Sham electron density. How one does this, for any given system and for any property of interest, is an active topic of research. 

The field-induced reconstruction was studied in the ground state Kohn-Sham electron density functional simulations for fixed atomic structures of the surface layers [283,284]. Their energies are compared... 

Most practical electronic structure calculations using density functional theory [1] involve solving the Kohn-Sham equations [2], The only unknown quantity in a Kohn-Sham spin-density functional calculation is the exchange-correlation energy (and its functional derivative) [2]... 

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). 

For closed-shell and open-shell molecules, spin-restricted Kohn-Sham (RKS) and spin-unrestricted Kohn-Sham (UKS) density functional calculations were employed, respectively. Except for the calculations of excited states and the cases where pure states are sought, we have employed an approximation in which electron density is smeared among the closely spaced orbitals near the Fermi levels. In this procedure, fractional occupations are allowed for those frontier orbitals with energy difference within 0.01 hartree to avoid the violation of the Aufbau principle (46). 

In Kohn-Sham (KS) density functional theory (DFT), the occupied orbital functions of a model state are derived by minimizing the ground-state energy functionals of Hohenberg and Kohn. It has been assumed for some time that effective potentials in the orbital KS equations are always equivalent to local potential functions. When tested by accurate model calculations, this locality assumption is found to fail for more than two electrons. Here this failure is explored in detail. The sources of the locality hypothesis in current DFT thinking are examined, and it is shown how the theory can be extended to an orbital functional theory (OFT) that removes the inconsistencies and paradoxes. 

Key words Density functional theory - Kohn-Sham -Electron correlation - Molecular orbital theory... 

Hanson and Martin applied the same approach to investigate the rupture of isoprene and butadiene oligomers in order to investigate covalent bond rupture in rubber.Using density functional theory, they identified the point of rupture where the unrestricted solution to the Kohn-Sham electronic wave functions falls below the restricted solution. This implies that the rupture process should be heterolytic and so this concept can only be applied for instances in which radical species are formed in the initial rupture event. 

Kohn and Sham presented the concept of a system with non-interacting electrons, subject to some wondef external field VQ(r) (instead of that of the nuclei), such that the resulting density p remains identical to the exact ground-state density distribution pQ. This fictitious system of electrons plays a very important role in the DFT. Since the Kohn-Sham electrons do not interact, their wave function represents a single Slater determinant (called the Kohn-Sham determinant). 

For the electronic ground state, i.e., k = 0, Kohn-Sham (KS) density functional theory is commonly used. In this case, the energy is given by... 

By introducing this expression for the electron density and applying the appropriate variational condition the following one-electron Kohn-Sham equations result ... 

The total electron density is just the sum of the densities for the two types of electron. The exchange-correlation functional is typically different for the two cases, leading to a set of spin-polarised Kohn-Sham equations ... 

The premise behind DFT is that the energy of a molecule can be determined from the electron density instead of a wave function. This theory originated with a theorem by Hoenburg and Kohn that stated this was possible. The original theorem applied only to finding the ground-state electronic energy of a molecule. A practical application of this theory was developed by Kohn and Sham who formulated a method similar in structure to the Hartree-Fock method. 

In this formulation, the electron density is expressed as a linear combination of basis functions similar in mathematical form to HF orbitals. A determinant is then formed from these functions, called Kohn-Sham orbitals. It is the electron density from this determinant of orbitals that is used to compute the energy. This procedure is necessary because Fermion systems can only have electron densities that arise from an antisymmetric wave function. There has been some debate over the interpretation of Kohn-Sham orbitals. It is certain that they are not mathematically equivalent to either HF orbitals or natural orbitals from correlated calculations. However, Kohn-Sham orbitals do describe the behavior of electrons in a molecule, just as the other orbitals mentioned do. DFT orbital eigenvalues do not match the energies obtained from photoelectron spectroscopy experiments as well as HF orbital energies do. The questions still being debated are how to assign similarities and how to physically interpret the differences. 

Kohn-Sham orbitals functions for describing the electron density in density functional theory calculations... 

Implementation of the Kohn-Sham-LCAO procedure is quite simple we replace the standard exchange term in the HF-LCAO expression by an appropriate Vxc that will depend on the local electron density and perhaps also its gradient. The new integrals involved contain fractional powers of the electron density and cannot be evaluated analytically. There are various ways forward, all of which... 

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